2.1. Weyl symbol calculus on spacetime

2.1.1. Basic notation

We denote the time variable as $t$, space coordinates as ${\boldsymbol {x}} \equiv (x^1, x^2, \ldots, x^n)$, spacetime coordinates as ${\boldsymbol {\mathsf {x}}} \equiv ({\mathsf {x}}^0, {\mathsf {x}}^1, \ldots, {\mathsf {x}}^n)$, where ${\mathsf {x}}^0 \doteq t$ and ${\mathsf {x}}^i \doteq x^i$. The symbol $\doteq$ denotes definitions, and Latin indices from the middle of the alphabet ($i, j, \ldots$) range from 1 to $n$ unless specified otherwise. We assume the spacetime-coordinate domain to be $\smash {\mathbb {R}^{{\mathsf {n}}}}$.Footnote ³ Functions on ${\boldsymbol {\mathsf {x}}}$ form a Hilbert space $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$ with an inner product that we define as

(2.1)\begin{equation} {\left\langle {\xi |\psi}\right\rangle} \doteq \int \mathrm{d}{\boldsymbol {\mathsf{x}}}\, \xi^*({\boldsymbol {\mathsf{x}}}) \psi({\boldsymbol {\mathsf{x}}}). \end{equation}

The symbol $^*$ denotes complex conjugate,

(2.2)\begin{equation} \mathrm{d}{\boldsymbol {\mathsf{x}}} \doteq \mathrm{d}{\mathsf{x}}^0\, \mathrm{d}{\mathsf{x}}^1 \ldots \mathrm{d} {\mathsf{x}}^n = \mathrm{d} t\, \mathrm{d} x^1 \ldots \mathrm{d} x^n \end{equation}

(a similar convention is assumed also for other multi-dimensional variables used below), and integrals in this paper are taken over $(-\infty, \infty )$ unless specified otherwise. Operators on $\mathscr {H}_{{{\mathsf {x}}}}$ will be denoted with carets, and we will use indexes $_\text {H}$ and $_\text {A}$ to denote their Hermitian and anti-Hermitian parts. For a given operator $\smash {\widehat {{\mathsf {A}}}}$, one has $\smash {\widehat {{\mathsf {A}}}=\widehat {{\mathsf {A}}}_\text {H} + \mathrm {i} \widehat {{\mathsf {A}}}_\text {A}}$,

(2.3)\begin{equation} \widehat{{\mathsf{A}}}_\text{H} = \widehat{{\mathsf{A}}}_\text{H}^{{\dagger}} \doteq \frac{1}{2}\,(\widehat{{\mathsf{A}}} + \widehat{{\mathsf{A}}}^{{\dagger}}), \qquad \widehat{{\mathsf{A}}}_\text{A} = \widehat{{\mathsf{A}}}_\text{A}^{{\dagger}} \doteq \frac{1}{2\mathrm{i}}\,(\widehat{{\mathsf{A}}} - \widehat{{\mathsf{A}}}^{{\dagger}}), \end{equation}

where $^{{\dagger}}$ denotes the Hermitian adjoint with respect to the inner product (2.1). The case of a more general inner product is detailed in Dodin et al. (Reference Dodin, Ruiz, Yanagihara, Zhou and Kubo2019, supplementary material).

2.1.2. Vector fields

For multi-component fields $\smash {{\boldsymbol {\psi }} \equiv (\psi ^1, \psi ^2, \ldots, \psi ^M)}$${}^{\intercal }$ (our $\smash {^\intercal }$ denotes the matrix trans pose), or ‘row vectors’ (actually, tuples), we define the dual ‘column vectors’ $\smash {{\boldsymbol {\psi }}^{{\dagger}} \equiv (\psi _1^*, \psi _2^*, \ldots, \psi _M^*)^\intercal }$ via $\smash {{\boldsymbol {\psi }}^{{\dagger}} \doteq \boldsymbol{\mathsf {g}}{\boldsymbol {\psi }}^*}$. The matrix $\boldsymbol{\mathsf {g}}$ is assumed to be real, diagonal, invertible and constant; other than that, it can be chosen as suits a specific problem. (For example, a unit matrix may suffice.) This induces the standard rule of index manipulation

(2.4)\begin{equation} \psi_i = {\mathsf{g}}_{ij}\psi^j, \qquad \psi^i = {\mathsf{g}}^{ij}\psi_j, \qquad i, j = 1, 2, \ldots, M, \end{equation}

where $\smash {{\mathsf {g}}_{ij}}$ are elements of $\boldsymbol{\mathsf {g}}$ and $\smash {{\mathsf {g}}^{ij}}$ are elements of $\boldsymbol{\mathsf {g}}^{-1}$. Summation over repeating indices is assumed. The rules of matrix multiplication apply to row and column vectors as usual. Then, for $\smash {{\boldsymbol {\psi }} \equiv (\psi ^1, \psi ^2, \ldots, \psi ^M)}$${}^{\intercal }$ and $\smash {{\boldsymbol {\xi }} \equiv (\xi ^1, \xi ^2, \ldots, \xi ^M)^{\intercal }}$, the quantity $\smash {{\boldsymbol {\psi }}{\boldsymbol {\xi }}}$ is a matrix with elements $\smash {\psi ^i\xi ^j}$, $\smash {{\boldsymbol {\psi }}{\boldsymbol {\xi }}^{{\dagger}} }$ is a matrix with elements $\smash {\psi ^i\xi _j^*}$ and $\smash {{\boldsymbol {\xi }}^{{\dagger}} {\boldsymbol {\psi }}}$ is its scalar trace:Footnote ⁴

(2.5)\begin{equation} {\boldsymbol{\xi}}^{{\dagger}}{\boldsymbol{\psi}} = \operatorname{tr}({\boldsymbol{\psi}}{\boldsymbol{\xi}}^{{\dagger}}) = \xi_i^* \psi^i = {\mathsf{g}}_{ij} \xi^{i*} \psi^j, \qquad i, j = 1, 2, \ldots, M. \end{equation}

(Similarly, for $\smash {{\boldsymbol {\chi }} \equiv (\chi _1, \chi _2, \ldots, \chi _M)}$ and $\smash {{\boldsymbol {\eta }} \equiv (\eta _1, \eta _2, \ldots, \eta _M)}$, $\smash {{\boldsymbol {\eta }}{\boldsymbol {\chi }}}$ is a matrix with elements $\smash {\eta _i \chi _j}$.) We use (2.5) to define a Hilbert space $\smash {\mathscr {H}_{{{\mathsf {x}}}}^M}$ of $\smash {M}$-dimensional vector fields on ${\boldsymbol {\mathsf {x}}}$, specifically, by adopting the inner product

(2.6)\begin{equation} \left\langle{\boldsymbol{\xi}|\boldsymbol{\psi}}\right\rangle \doteq \int \mathrm{d}{\boldsymbol {\mathsf{x}}}\, {\boldsymbol{\xi}}^{{\dagger}}({\boldsymbol {\mathsf{x}}}) {\boldsymbol{\psi}}({\boldsymbol {\mathsf{x}}}). \end{equation}

Below, the distinction between $\mathscr {H}_{{{\mathsf {x}}}}^M$ and $\mathscr {H}_{{{\mathsf {x}}}}$ will be assumed but not emphasized. Also note that (2.5) yields

(2.7)\begin{equation} {\boldsymbol{\xi}}^{{\dagger}}({\boldsymbol{\psi}}{\boldsymbol{\psi}}^{{\dagger}}){\boldsymbol{\xi}} = ({\boldsymbol{\xi}}^{{\dagger}} {\boldsymbol{\psi}})({\boldsymbol{\psi}}^{{\dagger}} {\boldsymbol{\xi}}) = |{\boldsymbol{\xi}}^{{\dagger}} {\boldsymbol{\psi}}|^2 \geqslant 0, \end{equation}

for any $\smash {{\boldsymbol {\xi }}}$ and $\smash {{\boldsymbol {\psi }}}$. Thus, dyadic matrices of the form $\smash {{\boldsymbol {\psi }}{\boldsymbol {\psi }}^{{\dagger}} }$ are positive-semidefinite, even though $\smash {{\boldsymbol {\psi }}^{{\dagger}} {\boldsymbol {\psi }}}$ may be negative (when $\boldsymbol{\mathsf {g}}$ is not positive-definite).

For general matrices, the indices can be raised and lowered using $\boldsymbol{\mathsf {g}}$ and $\boldsymbol{\mathsf {g}}^{-1}$ as usual. The Hermitian adjoint $\smash {{{\boldsymbol {\mathsf {A}}}}^{{\dagger}} }$ for a given matrix $\smash {{\boldsymbol {\mathsf {A}}}}$ is defined such that $\smash {({{\boldsymbol {\mathsf {A}}}}^{{\dagger}} {\boldsymbol {\xi }})^{{\dagger}} {\boldsymbol {\psi }} = {\boldsymbol {\xi }}^{{\dagger}} ({{\boldsymbol {\mathsf {A}}}}{\boldsymbol {\psi }})}$ for any $\smash {{\boldsymbol {\psi }}}$ and $\smash {{\boldsymbol {\xi }}}$, which means

(2.8)\begin{equation} ({{\boldsymbol {\mathsf{A}}}}^{{\dagger}})_j{}^i = ({{\boldsymbol {\mathsf{A}}}})^{i*}{}_j \equiv {\mathsf{A}}^{i*}{}_j, \qquad i, j = 1, 2, \ldots, M. \end{equation}

The Hermitian and anti-Hermitian parts are defined as

(2.9)\begin{equation} {{\boldsymbol {\mathsf{A}}}}_\text{H} = {{\boldsymbol {\mathsf{A}}}}_\text{H}^{{\dagger}} \doteq \frac{1}{2}\,({{\boldsymbol {\mathsf{A}}}} + {{\boldsymbol {\mathsf{A}}}}^{{\dagger}}), \qquad {{\boldsymbol {\mathsf{A}}}}_\text{A} = {{\boldsymbol {\mathsf{A}}}}_\text{A}^{{\dagger}} \doteq \frac{1}{2\mathrm{i}}\,({{\boldsymbol {\mathsf{A}}}} - {{\boldsymbol {\mathsf{A}}}}^{{\dagger}}), \end{equation}

so $\smash {{{\boldsymbol {\mathsf {A}}}} = {{\boldsymbol {\mathsf {A}}}}_\text {H} + \mathrm {i} {{\boldsymbol {\mathsf {A}}}}_\text {A}}$. For one-dimensional matrices (scalars), one has ${{\boldsymbol {\mathsf {A}}}} = {\mathsf {A}}$,

(2.10)\begin{equation} {\mathsf{A}}_\text{H} = {\mathsf{A}}_\text{H}^* = \operatorname{re} {\mathsf{A}}, \qquad {\mathsf{A}}_\text{A} = {\mathsf{A}}_\text{A}^* = \operatorname{im} {\mathsf{A}}, \end{equation}

where $\smash {\operatorname {re}}$ and $\smash {\operatorname {im}}$ denote the real part and the imaginary part, respectively. We also define matrix operators $\smash {\widehat {\boldsymbol {{\boldsymbol {\mathsf {A}}}}}}$ as matrices of the corresponding operators $\smash {\widehat {{\mathsf {A}}}^{\,\,i}_j}$. Because $\boldsymbol{\mathsf {g}}$ is constant, index manipulation applies as usual. Also as usual, one has

(2.11)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}_\text{H} = \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}_\text{H}^{{\dagger}} \doteq \frac{1}{2}\,(\widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}} + \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}^{{\dagger}}), \qquad \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}_\text{A} = \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}_\text{A}^{{\dagger}} \doteq \frac{1}{2\mathrm{i}}\,(\widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}} - \widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}^{{\dagger}}), \end{equation}

and $\,\smash {\widehat {\boldsymbol {{\boldsymbol {\mathsf {A}}}}} = \widehat {\boldsymbol {{\boldsymbol {\mathsf {A}}}}}_\text {H} + \mathrm {i} \widehat {\boldsymbol {{\boldsymbol {\mathsf {A}}}}}_\text {A}}$, where $^{{\dagger}}$ is the Hermitian adjoint with respect to the inner product (2.6).

2.1.3. Bra–ket notation

Let us define the following operators that are Hermitian under the inner product (2.1):

(2.12)\begin{equation} \widehat{{\mathsf{x}\,}}^0 \equiv \widehat{t} \doteq t, \quad \widehat{{\mathsf{x}\,}}^i \equiv \widehat{x\,}^i \doteq x^i, \quad \widehat{{\mathsf{k}}}_0 \equiv{-}\widehat{\omega} \doteq{-}\mathrm{i}\partial_t, \quad \widehat{k}_i \doteq{-}\mathrm{i}\partial_i, \end{equation}

where $\partial _0 \equiv \partial _t \doteq \partial /\partial x^0$ and $\partial _i \doteq \partial /\partial x^i$. Accordingly,

(2.13)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{x}}}}} \equiv (\,\widehat{{\mathsf{x}}}^{\kern1.5pt0}, \widehat{{\mathsf{x}}}^{\kern1.5pt1}, \ldots, \widehat{{\mathsf{x}}}^{\kern1.5pt n}) = (\widehat{\,t}, \widehat{\boldsymbol{x}}), \qquad \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} \equiv (\widehat{{\mathsf{k}}}_0, \widehat{{\mathsf{k}}}_1, \ldots, \widehat{{\mathsf{k}}}_{n}) = (-\widehat{\omega}, \widehat{\boldsymbol{k}}) \end{equation}

are understood as the spacetime-position operator and the corresponding wavevector operator, which will also be expressed as follows:

(2.14)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{x}}}}} = {\boldsymbol {\mathsf{x}}}, \qquad \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} ={-} \mathrm{i}\partial_{\boldsymbol {\mathsf{x}}}. \end{equation}

Also note the commutation property, where $\smash {\delta _i^j}$ is the Kronecker symbol:Footnote ⁵

(2.15)\begin{equation} [\,\widehat{{\mathsf{x}}}^{\kern1.5pt i}, \widehat{{\mathsf{k}}}_j] = \mathrm{i} \delta^i_j, \qquad i, j = 0, 1, \ldots, n. \end{equation}

The eigenvectors of the operators (2.14) will be denoted as ‘kets’ $\smash {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}$ and $\smash {\left. {\boldsymbol {\mathsf {|k}}} \right\rangle}$:Footnote ⁶

(2.16)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{x}}}}} {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle} = {\boldsymbol {\mathsf{x}}} {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}, \qquad \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} {\left. {\boldsymbol {\mathsf {|k}}} \right\rangle} = {\boldsymbol {\mathsf{k}}} {\left. {\boldsymbol {\mathsf {|k}}} \right\rangle}, \end{equation}

and we assume the usual normalization

(2.17)\begin{equation} {\left\langle{\boldsymbol {\mathsf{x}}}_1|{{\boldsymbol {\mathsf{x}}}_2}\right\rangle} = \delta({\boldsymbol {\mathsf{x}}}_1 - {\boldsymbol {\mathsf{x}}}_2), \qquad {\left\langle{\boldsymbol {\mathsf{k}}}_1|{\boldsymbol {\mathsf{k}}}_2 \right\rangle} = \delta({\boldsymbol {\mathsf{k}}}_1 - {\boldsymbol {\mathsf{k}}}_2), \end{equation}

where $\delta$ is the Dirac delta function. Both sets $\lbrace {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}, {\boldsymbol {\mathsf {x}}} \in \mathbb {R}^{{\mathsf {n}}} \rbrace$ and $\lbrace {\left. {\boldsymbol {\mathsf {|k}}} \right\rangle}, {\boldsymbol {\mathsf {k}}} \in \mathbb {R}^{{\mathsf {n}}} \rbrace$, where ${{\mathsf {n}}} \doteq n + 1$, form a complete basis on $\mathscr {H}_{{{\mathsf {x}}}}$, and the eigenvalues of these operators form an extended real phase space $({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})$, where

(2.18)\begin{equation} {\boldsymbol {\mathsf{x}}} \equiv (t, {\boldsymbol{x}}), \qquad {\boldsymbol {\mathsf{k}}} \equiv (-\omega, {\boldsymbol{k}}). \end{equation}

The notation $\smash {{\boldsymbol {\mathsf {k}}} \cdot {\boldsymbol {\mathsf {s}}} \doteq -\omega \tau + {\boldsymbol {k}} \cdot {\boldsymbol {s}}}$ will be assumed for any ${\boldsymbol {\mathsf {s}}} \equiv (\tau, {\boldsymbol {s}})$ and $\smash {{\boldsymbol {k}} \cdot {\boldsymbol {s}} \doteq k_i s^i}$. In particular, for any $\psi$ and constant ${\boldsymbol {\mathsf {s}}}$, one has

(2.19)\begin{equation} \exp(\mathrm{i} \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} \cdot {\boldsymbol {\mathsf{s}}}) \psi({\boldsymbol {\mathsf{x}}}) = \exp({\boldsymbol {\mathsf{s}}} \cdot \partial_{\boldsymbol {\mathsf{x}}}) \psi({\boldsymbol {\mathsf{x}}}) = \psi({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}), \end{equation}

as seen from comparing the Taylor expansions in ${\boldsymbol {\mathsf{s}}}$ of the latter two expressions. (A generalization of this formula is discussed in § 4.1.) Also,

(2.20)\begin{equation} \left\langle{{\boldsymbol {\mathsf{x}}}|{\boldsymbol {\mathsf{k}}}}\right\rangle = \left\langle{{\boldsymbol {\mathsf{k}}}|{\boldsymbol {\mathsf{x}}}}\right\rangle^* = (2{\rm \pi})^{-{{\mathsf{n}}}/2} \exp(\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{x}}}), \end{equation}

and

(2.21)\begin{equation} \int \mathrm{d}{\boldsymbol {\mathsf{x}}}\,{\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {{{\boldsymbol {\mathsf{x|}}}}}}}} \right.} = \widehat{{\mathsf{1}}}, \qquad \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,{\left. {\boldsymbol {\mathsf {|k}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{k|}}}}}}} \right.} = \widehat{{\mathsf{1}}}. \end{equation}

Here, ‘bra’ ${\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{x|}}}}}}} \right.}$ is the one-form dual to ${\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}$, ${\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{k|}}}}}}} \right.}$ is the one-form dual to ${\left. {\boldsymbol {\mathsf {|k}}} \right\rangle}$, and $\widehat {{\mathsf {1}}}$ is the unit operator. Any field $\psi$ on ${\boldsymbol {\mathsf {x}}}$ can be viewed as the ${\boldsymbol {\mathsf {x}}}$ representation (‘coordinate representation’) of ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}$, i.e. the projection of an abstract ket vector ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle} \in \mathscr {H}_{{{\mathsf {x}}}}$ on $\smash {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}$:

(2.22)\begin{equation} \psi({\boldsymbol {\mathsf{x}}}) = \left\langle{{\boldsymbol {\mathsf{x}}}|\psi}\right\rangle. \end{equation}

Similarly, $\left\langle {{\boldsymbol {\mathsf {k}}}|\psi }\right\rangle$ is the ${\boldsymbol {\mathsf {k}}}$ representation (‘spectral representation’) of ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}$, or the Fourier image of $\psi$:

(2.23)\begin{equation} \mathring{\psi}({\boldsymbol {\mathsf{k}}}) \doteq \left\langle{{\boldsymbol {\mathsf{k}}}|\psi}\right\rangle = \frac{1}{(2{\rm \pi})^{{{\mathsf{n}}}/2}}\int \mathrm{d}{\boldsymbol {\mathsf{x}}}\, \mathrm{e}^{-\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{x}}}}\psi({\boldsymbol {\mathsf{x}}}). \end{equation}

2.1.4. Wigner–Weyl transform

For a given operator $\widehat {{\mathsf {A}}}$ and a given field $\psi$, $\widehat {{\mathsf {A}}}\psi$ can be expressed in the integral form

(2.24)\begin{equation} \widehat{{\mathsf{A}}}\psi({\boldsymbol {\mathsf{x}}}) = \int \mathrm{d} {\boldsymbol {\mathsf{x}}}' \left\langle{{\boldsymbol {\mathsf{x}}}|\widehat{{\mathsf{A}}}|{\boldsymbol {\mathsf{x}}}'}\right\rangle\psi({\boldsymbol {\mathsf{x}}}'), \end{equation}

where ${\left\langle {\boldsymbol {\mathsf {x}}}|\widehat {{\mathsf {A}}}|{\boldsymbol {\mathsf {x}}}'\right\rangle}$ is a function of $({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {x}}}')$. This is called the ${\boldsymbol {\mathsf {x}}}$ representation (‘coordinate representation’) of $\smash {\widehat {{\mathsf {A}}}}$. Equivalently, $\widehat {{\mathsf {A}}}$ can be given a phase-space, or Weyl, representation, i.e. expressed through a function of the phase-space coordinates, ${\mathsf {A}}({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})$Footnote ⁷:

(2.25)\begin{equation} \widehat{{\mathsf{A}}} = \frac{1}{(2{\rm \pi})^{{{\mathsf{n}}}}} \int \mathrm{d} {\boldsymbol {\mathsf{x}}} \,\mathrm{d}{\boldsymbol {\mathsf{k}}}\,\mathrm{d} {\boldsymbol {\mathsf{s}}}\, {{\boldsymbol {\mathsf{|x}}} + {\boldsymbol {\mathsf{s}}}/2} {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}} {\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{k|}}}}}}} \right.} {{\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{x|}}}}}}} \right.}-{\boldsymbol {\mathsf{s}}}/2|} \equiv \text{oper}_{{\mathsf{x}}} {\mathsf{A}}. \end{equation}

The function ${\mathsf {A}}({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})$, called the Weyl symbol (or just ‘symbol’) of $\widehat {{\mathsf {A}}}$, is given by

(2.26)\begin{equation} {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \doteq \int \mathrm{d}{\boldsymbol {\mathsf{s}}} \left\langle{{\boldsymbol {\mathsf{x}}}+{\boldsymbol {\mathsf{s}}}/2 | \widehat{{\mathsf{A}}} | {\boldsymbol {\mathsf{x}}}-{\boldsymbol {\mathsf{s}}}/2}\right\rangle \mathrm{e}^{-\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}} \equiv \text{symb}_{{\mathsf{x}}} \widehat{{\mathsf{A}}}. \end{equation}

The ${\boldsymbol {\mathsf {x}}}$ and phase-space representations are connected by the Fourier transform:

(2.27)\begin{equation} \left\langle{{\boldsymbol {\mathsf{x}}}| \widehat{{\mathsf{A}}} | {\boldsymbol {\mathsf{x}}}'}\right\rangle = \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}}\int\mathrm{d} {\boldsymbol {\mathsf{k}}}\, \mathrm{e}^{\mathrm{i}{\boldsymbol {\mathsf{k}}} \cdot({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{x}}}')} \, {\mathsf{A}}\left(\frac{{\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{x}}}'}{2}, {\boldsymbol {\mathsf{k}}}\right). \end{equation}

This also leads to the following notable properties of Weyl symbols:

(2.28)\begin{equation} \left\langle{{\boldsymbol {\mathsf{x}}}| \widehat{{\mathsf{A}}} | {\boldsymbol {\mathsf{x}}}}\right\rangle = \frac{1}{(2{\rm \pi})^{{{\mathsf{n}}}}}\int \mathrm{d} {\boldsymbol {\mathsf{k}}}\,{\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}), \qquad \left\langle{{\boldsymbol {\mathsf{k}}}| \widehat{{\mathsf{A}}} | {\boldsymbol {\mathsf{k}}}}\right\rangle = \frac{1}{(2{\rm \pi})^{{{\mathsf{n}}}}}\int \mathrm{d} {\boldsymbol {\mathsf{x}}}\,{\mathsf{A}}({\boldsymbol {\mathsf{x}}},{\boldsymbol {\mathsf{k}}}). \end{equation}

An operator unambiguously determines its symbol, and vice versa. We denote this isomorphism as $\smash {\widehat {{\mathsf {A}}} \leftrightarrow {\mathsf {A}}}$. The mapping $\smash {\widehat {{\mathsf {A}}} \mapsto {\mathsf {A}}}$ is called the Wigner transform, and $\smash {{\mathsf {A}} \mapsto \widehat {{\mathsf {A}}}}$ is called the Weyl transform. For uniformity, we call them the direct and inverse Wigner–Weyl transform. The isomorphism $\smash {\leftrightarrow }$ is natural in that it has the following properties:

(2.29)\begin{equation} \widehat{{\mathsf{1}}} \leftrightarrow 1, \quad \widehat{\boldsymbol{{\boldsymbol {\mathsf{x}}}}} \leftrightarrow {\boldsymbol {\mathsf{x}}}, \quad \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} \leftrightarrow {\boldsymbol {\mathsf{k}}}, \quad h(\,\widehat{\boldsymbol{{\boldsymbol {\mathsf{x}}}}}) \leftrightarrow h({\boldsymbol {\mathsf{x}}}), \quad h(\widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}}) \leftrightarrow h({\boldsymbol {\mathsf{k}}}), \quad \widehat{{\mathsf{A}}}^{{\dagger}} \leftrightarrow {\mathsf{A}}^*, \end{equation}

where $h$ is any function and $\smash {\widehat {{\mathsf {A}}}}$ is any operator. The product of two operators maps to the so-called Moyal product, or star product, of their symbols (Moyal Reference Moyal1949):

(2.30)\begin{equation} \widehat{{\mathsf{A}}}\,\widehat{{\mathsf{B}}}\,\leftrightarrow \, {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \star {\mathsf{B}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \doteq {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}})\mathrm{e}^{\mathrm{i}\widehat{\mathcal{L}}_{{\mathsf{x}}}/2}{\mathsf{B}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}), \end{equation}

which is associative:

(2.31)\begin{equation} \widehat{{\mathsf{A}}}\,\widehat{{\mathsf{B}}}\,\widehat{{\mathsf{C}}}\, \leftrightarrow \, ({\mathsf{A}} \star {\mathsf{B}}) \star {\mathsf{C}} = {\mathsf{A}} \star ({\mathsf{B}} \star {\mathsf{C}}) \equiv {\mathsf{A}} \star {\mathsf{B}} \star {\mathsf{C}}. \end{equation}

Here, $\smash {\widehat {\mathcal {L}}_{{\mathsf {x}}} \doteq \overset {{\scriptscriptstyle \leftarrow }}{\partial }_{\boldsymbol {\mathsf {x}}} \cdot \overset {{\scriptscriptstyle \rightarrow }}{\partial }_{\boldsymbol {\mathsf {k}}} - \overset {{\scriptscriptstyle \leftarrow }}{\partial }_{\boldsymbol {\mathsf {k}}} \cdot \overset {{\scriptscriptstyle \rightarrow }}{\partial }_{\boldsymbol {\mathsf {x}}}}$, and the arrows indicate the directions in which the derivatives act. For example, $\smash {{\mathsf {A}}\widehat {\mathcal {L}}_{{\mathsf {x}}}{\mathsf {B}}}$ is just the canonical Poisson bracket on $\smash {({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$:

(2.32)\begin{equation} {\mathsf{A}} \widehat{\mathcal{L}}_{{\mathsf{x}}} {\mathsf{B}} = \lbrace {\mathsf{A}}, {\mathsf{B}} \rbrace_{{{\mathsf{x}}}} \doteq{-} \frac{\partial {\mathsf{A}}}{\partial t}\frac{\partial {\mathsf{B}}}{\partial \omega} + \frac{\partial {\mathsf{A}}}{\partial \omega}\frac{\partial {\mathsf{B}}}{\partial t} + \frac{\partial {\mathsf{A}}}{\partial x^i}\frac{\partial {\mathsf{B}}}{\partial k_i} - \frac{\partial {\mathsf{A}}}{\partial k_i}\frac{\partial {\mathsf{B}}}{\partial x^i}. \end{equation}

These formulas readily yield

(2.33)\begin{equation} h(\widehat{\,\boldsymbol{{\boldsymbol {\mathsf{x}}}}\,})\widehat{{\mathsf{k}}}_\alpha \leftrightarrow {\mathsf{k}}_\alpha h({\boldsymbol {\mathsf{x}}}) + \frac{\mathrm{i}}{2}\,\partial_\alpha h({\boldsymbol {\mathsf{x}}}), \qquad \widehat{{\mathsf{k}}}_\alpha h(\widehat{\,\boldsymbol{{\boldsymbol {\mathsf{x}}}}\,}) \leftrightarrow {\mathsf{k}}_\alpha h({\boldsymbol {\mathsf{x}}}) - \frac{\mathrm{i}}{2}\,\partial_\alpha h({\boldsymbol {\mathsf{x}}}), \end{equation}

also $h(\widehat {\boldsymbol {{\boldsymbol {\mathsf {k}}}}}) \mathrm {e}^{\mathrm {i}{\boldsymbol {\mathsf {K}}} \cdot \widehat {\boldsymbol {{\boldsymbol {\mathsf {x}}}}}} \leftrightarrow h({\boldsymbol {\mathsf {k}}})\star \mathrm {e}^{\mathrm {i}{\boldsymbol {\mathsf {K}}} \cdot {\boldsymbol {\mathsf {x}}}} = h({\boldsymbol {\mathsf {k}}} + {\boldsymbol {\mathsf {K}}}/2) \mathrm {e}^{\mathrm {i}{\boldsymbol {\mathsf {K}}} \cdot {\boldsymbol {\mathsf {x}}}}$, etc. Another notable formula to be used below, which flows from (2.28) and (2.31), is

(2.34)\begin{equation} \left\langle{{\boldsymbol {\mathsf{x}}}| \widehat{{\mathsf{A}}}\,\widehat{{\mathsf{B}}}\,\widehat{{\mathsf{C}}} | {\boldsymbol {\mathsf{x}}}}\right\rangle = \frac{1}{(2{\rm \pi})^{{{\mathsf{n}}}}}\int \mathrm{d} {\boldsymbol {\mathsf{k}}}\,({\mathsf{A}} \star {\mathsf{B}} \star {\mathsf{C}})({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}). \end{equation}

The Moyal product is particularly handy when $\partial _{{\boldsymbol {\mathsf {x}}}} \partial _{{\boldsymbol {\mathsf {k}}}} \sim \epsilon \ll 1$. Such $\epsilon$ is often called the geometrical-optics parameter. Since $\widehat {\mathcal {L}}_{{\mathsf {x}}} = \mathcal {O}(\epsilon )$, one can express the Moyal product as an asymptotic series in powers of $\epsilon$:

(2.35)\begin{equation} \star = \widehat{{\mathsf{1}}} + \mathrm{i} \widehat{\mathcal{L}}_{{\mathsf{x}}}/2 - \widehat{\mathcal{L}\,}_{{\mathsf{x}}}^2/8 + \ldots . \end{equation}

2.1.5. Weyl expansion of operators

Operators can be approximated by approximating their symbols (McDonald Reference McDonald1988; Dodin et al. Reference Dodin, Ruiz, Yanagihara, Zhou and Kubo2019). If $\smash {\widehat {{\mathsf {A}}}}$ is approximately local in ${\boldsymbol {\mathsf {x}}}$ (i.e. if $\smash {\widehat {{\mathsf {A}}}\psi ({\boldsymbol {\mathsf {x}}})}$ is determined by values $\psi ({\boldsymbol {\mathsf {x}}} + {\boldsymbol {\mathsf {s}}})$ only with small enough ${\boldsymbol {\mathsf {s}}}$), its symbol can be adequately represented by the first few terms of the Taylor expansion in ${\boldsymbol {\mathsf {k}}}$:

(2.36)\begin{equation} {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) = {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{0}}}) + {\boldsymbol{\Theta}}_0({\boldsymbol {\mathsf{x}}}) \cdot {\boldsymbol {\mathsf{k}}} + \ldots, \qquad {\boldsymbol{\Theta}}_0({\boldsymbol {\mathsf{x}}}) \doteq (\partial_{\boldsymbol {\mathsf{k}}} {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}))_{{\boldsymbol {\mathsf{k}}}={\boldsymbol {\mathsf{0}}}}. \end{equation}

Application of $\smash {\text {oper}_{{\mathsf {x}}} }$ to this formula leads to

(2.37)\begin{equation} \widehat{{\mathsf{A}}} \approx {\mathsf{A}}(\widehat{\,\boldsymbol{{\boldsymbol {\mathsf{x}}}}}, {\boldsymbol {\mathsf{0}}}) + \frac{1}{2}\,(\widehat{\boldsymbol{\Theta}}_0 \cdot \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} + \widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}} \cdot \widehat{\boldsymbol{\Theta}}_0) + \ldots, \end{equation}

where $\widehat {\boldsymbol {\Theta }}_0 \doteq {\boldsymbol {\Theta }}_0(\widehat {\,\boldsymbol {{\boldsymbol {\mathsf {x}}}}\,})$. One can also rewrite (2.37) using the commutation property

(2.38)\begin{equation} [\widehat{\boldsymbol{{\boldsymbol {\mathsf{k}}}}}, \widehat{\boldsymbol{\Theta}}_0] ={-}\mathrm{i}(\partial_{{\boldsymbol {\mathsf{x}}}} \cdot {\boldsymbol{\Theta}}_0)(\widehat{\,\boldsymbol{{\boldsymbol {\mathsf{x}}}}\,}). \end{equation}

In the ${\boldsymbol {\mathsf {x}}}$ representation, this leads to

(2.39)\begin{equation} \widehat{{\mathsf{A}}} = {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{0}}}) - \mathrm{i} {\boldsymbol{\Theta}}_0({\boldsymbol {\mathsf{x}}}) \cdot \partial_{{\boldsymbol {\mathsf{x}}}} - \frac{\mathrm{i}}{2}\,(\partial_{{\boldsymbol {\mathsf{x}}}} \cdot {\boldsymbol{\Theta}}_0({\boldsymbol {\mathsf{x}}})) + \ldots . \end{equation}

The effect of a non-local operator on eikonal (monochromatic or quasimonochromatic) fields can be approximated similarly. Suppose $\psi = \mathrm {e}^{\mathrm {i}\theta }{\breve {\psi }}$, where the dependence of $\overline {{\boldsymbol {\mathsf {k}}}} \doteq \partial _{{\boldsymbol {\mathsf {x}}}}\theta$ and ${\breve {\psi }}$ on ${\boldsymbol {\mathsf {x}}}$ is slower than that of $\theta$ by factor $\epsilon \ll 1$. Then, $\widehat {{\mathsf {A}}}\psi = \mathrm {e}^{\mathrm {i}\theta }\widehat {{\mathsf {A}}}' {\breve {\psi }}$, where $\widehat {{\mathsf {A}}}' \doteq \mathrm {e}^{-\mathrm {i}\theta (\widehat {{\mathsf {x}}})}\widehat {{\mathsf {A}}}\mathrm {e}^{\mathrm {i}\theta (\widehat {{\mathsf {x}}})}$, and the symbol of $\widehat {{\mathsf {A}}}'$ can be approximated as follows:

(2.40)\begin{equation} {\mathsf{A}}'({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) = {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, \overline{{\boldsymbol {\mathsf{k}}}}({\boldsymbol {\mathsf{x}}}) + {\boldsymbol {\mathsf{k}}}) + \mathcal{O}(\epsilon^2). \end{equation}

By expanding this in ${\boldsymbol {\mathsf {k}}}$ and applying $\smash {\text {oper}_{{\mathsf {x}}} }$, one obtains

(2.41)\begin{equation} \widehat{{\mathsf{A}}}' = {\mathsf{A}}({\boldsymbol {\mathsf{x}}}, \overline{{\boldsymbol {\mathsf{k}}}}({\boldsymbol {\mathsf{x}}})) - \mathrm{i} {\boldsymbol{\Theta}}({\boldsymbol {\mathsf{x}}}) \cdot \partial_{{\boldsymbol {\mathsf{x}}}} - \frac{\mathrm{i}}{2}\,(\partial_{{\boldsymbol {\mathsf{x}}}} \cdot {\boldsymbol{\Theta}}({\boldsymbol {\mathsf{x}}})) + \mathcal{O}(\epsilon^2), \end{equation}

where ${\boldsymbol {\Theta }}({\boldsymbol {\mathsf {x}}}) \doteq (\partial _{\boldsymbol {\mathsf {k}}} {\mathsf {A}}({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}}))_{{\boldsymbol {\mathsf {k}}}=\overline {{\boldsymbol {\mathsf {k}}}}({\boldsymbol {\mathsf {x}}})}$. Neglecting the $\mathcal {O}(\epsilon ^2)$ corrections in this formula leads to what is commonly known as the geometrical-optics approximation (Dodin et al. Reference Dodin, Ruiz, Yanagihara, Zhou and Kubo2019).

2.1.6. Wigner functions

Any ket ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}$ generates a dyadic ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle} {\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\mathsf{\psi|}}}}}}} \right.}$. In quantum mechanics, such dyadics are known as density operators (of pure states). For our purposes, though, it is more convenient to define the density operator in a slightly different form, namely, as

(2.42)\begin{equation} \widehat{{\mathsf{W}}}_\psi \doteq (2{\rm \pi})^{-{{\mathsf{n}}}} {\left. {\boldsymbol {\mathsf {|x}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {x|}}}} \right.}. \end{equation}

The symbol of this operator, $\smash {{\mathsf {W}}_{\psi } = \text {symb}_{{\mathsf {x}}} \widehat {{\mathsf {W}}}_\psi }$, is a real function called the Wigner function. It is given by

(2.43)\begin{align} {\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) & = \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}} \int \mathrm{d}{\boldsymbol {\mathsf{s}}} \left\langle{{\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2 | \psi}\right\rangle \left\langle{\psi | {\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2}\right\rangle \mathrm{e}^{-\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}} \nonumber\\ & = \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}} \int \mathrm{d}{\boldsymbol {\mathsf{s}}}\, \psi({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2) \psi^*({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2)\,\mathrm{e}^{-\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}}, \end{align}

which is manifestly real and can be understood as the (inverse) Fourier image of

(2.44)\begin{equation} {\mathsf{C}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{s}}}) \doteq \psi({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2) \psi^*({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2) = \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,{\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}}. \end{equation}

Any function bilinear in $\smash {\psi }$ and $\smash {\psi ^*}$ can be expressed through $\smash {{\mathsf {W}}_\psi }$. Specifically, for any operators $\smash {\widehat {{\mathsf {L}}}}$ and $\smash {\widehat {{\mathsf {R}}}}$, one has

(2.45)\begin{align} (\widehat{{\mathsf{L}}}\psi({\boldsymbol {\mathsf{x}}}))(\widehat{{\mathsf{R}}}\psi({\boldsymbol {\mathsf{x}}}))^* & = \left\langle{{\boldsymbol {\mathsf{x}}}|\widehat{{\mathsf{L}}}|\psi}\right\rangle \left\langle{\psi|\widehat{{\mathsf{R}}}^{{\dagger}} |{\boldsymbol {\mathsf{x}}}}\right\rangle \nonumber\\ & = (2{\rm \pi})^{{\mathsf{n}}} \left\langle{{\boldsymbol {\mathsf{x}}}|\widehat{{\mathsf{L}}}\,\widehat{{\mathsf{W}}}_\psi\widehat{{\mathsf{R}}}^{{\dagger}}|{\boldsymbol {\mathsf{x}}}}\right\rangle \nonumber\\ & = \textstyle \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,{\mathsf{L}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \star {\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \star {\mathsf{R}}^*({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}), \end{align}

where ${\mathsf {L}}$ and ${\mathsf {R}}$ are the corresponding symbols and (2.28) was used along with (2.31). As a corollary, and as also seen from (2.28), one has

(2.46)\begin{equation} |\psi({\boldsymbol {\mathsf{x}}})|^2 = \int \mathrm{d} {\boldsymbol {\mathsf{k}}}\,{\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}), \qquad |\mathring{\psi}({\boldsymbol {\mathsf{k}}})|^2 = \int \mathrm{d} {\boldsymbol {\mathsf{x}}}\,{\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}). \end{equation}

As a reminder, $\psi ({\boldsymbol {\mathsf {x}}}) = \left\langle {{\boldsymbol {\mathsf {x}}}|\psi }\right\rangle$ and $\mathring {\psi }({\boldsymbol {\mathsf {k}}}) \doteq \left\langle {{\boldsymbol {\mathsf {k}}}|\psi }\right\rangle$ is the Fourier image of $\psi$ (2.23), so $\smash {|\psi ({\boldsymbol {\mathsf {x}}})|^2}$ and $\smash {|\mathring {\psi }({\boldsymbol {\mathsf {k}}})|^2}$ can be loosely understood as the densities of quanta (associated with the field $\smash {\psi }$) in the $\smash {{\boldsymbol {\mathsf {x}}}}$-space and the $\smash {{\boldsymbol {\mathsf {k}}}}$-space, respectively. Because of (2.46), $\smash {{\mathsf {W}}_\psi }$ is commonly attributed as a quasiprobability distribution of wave quanta in phase space. (The prefix ‘quasi’ is added because $\smash {{\mathsf {W}}_\psi }$ can be negative.) In case of real fields, which satisfy $\left\langle {{\boldsymbol {\mathsf {x}}}|\psi }\right\rangle = \left\langle {\psi |{\boldsymbol {\mathsf {x}}}}\right\rangle$, one also has

(2.47)\begin{equation} {\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) = {\mathsf{W}}_\psi({\boldsymbol {\mathsf{x}}}, -{\boldsymbol {\mathsf{k}}}). \end{equation}

Of particular importance are Wigner functions averaged over a sufficiently large phase-space volume $\varDelta {\boldsymbol {\mathsf {x}}}\,\varDelta {\boldsymbol {\mathsf {k}}} \gtrsim 1$. The average Wigner function $\smash {\overline {{\mathsf {W}}}_\psi }$ is a local property of the field (as opposed to, say, the field's global Fourier spectrum) and satisfies (Appendix A)

(2.48)\begin{equation} \overline{{\mathsf{W}}}_\psi \geqslant 0. \end{equation}

2.1.7. Generalization to vector fields

In case of vector (tuple) fields ${\boldsymbol {\psi }} = (\psi ^1, \psi ^2, \ldots, \psi ^M)^{\intercal }$, kets are column vectors, ${\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle} = ({\left. {\boldsymbol {\mathsf {|\psi ^1}}} \right\rangle} , {\left. {\boldsymbol {\mathsf {|\psi ^2}}} \right\rangle}, \ldots, {\left. {\boldsymbol {\mathsf {|\psi ^M}}} \right\rangle})^{\intercal }$, and bras are row vectors, ${\left\langle {{\boldsymbol {\mathsf {{\boldsymbol {\psi |}}}}}} \right.} = ({\left\langle {{\boldsymbol {\mathsf {{\psi ^1}|}}}} \right.}, {\left\langle {{\boldsymbol {\mathsf {{\psi ^2}|}}}} \right.}, \ldots, {\left\langle {{\boldsymbol {\mathsf {{\psi ^M}|}}}} \right.})^\intercal$. The operators acting on such kets and bras are matrices of operators. The Weyl symbol of a matrix operator is defined as the matrix of the corresponding symbols. As a result, the symbol of a Hermitian adjoint of a given operator is the Hermitian adjoint of the symbol of that operator:

(2.49)\begin{equation} \smash{\widehat{\boldsymbol{{\boldsymbol {\mathsf{A}}}}}}^{{\dagger}} \leftrightarrow \smash{{{\boldsymbol {\mathsf{A}}}}}^{{\dagger}}, \end{equation}

and as a corollary, the symbol of a Hermitian matrix operator is a Hermitian matrix.

In particular, the density operator of a given vector field ${\boldsymbol {\psi }}$ is a matrix operator

(2.50)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{W}}}}}_{{\boldsymbol{\psi}}} \doteq (2{\rm \pi})^{-{{\mathsf{n}}}} {\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {\psi |}}}} \right.}. \end{equation}

The symbol of this operator, $\smash {{{\boldsymbol {\mathsf {W}}}}_{{\boldsymbol {\psi }}} = \text {symb}_{{\mathsf {x}}} \widehat {\boldsymbol {{\boldsymbol {\mathsf {W}}}}}_{{\boldsymbol {\psi }}}}$, is a Hermitian matrix functionFootnote ⁸

(2.51)\begin{equation} {{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) = \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}} \int \mathrm{d}{\boldsymbol {\mathsf{s}}}\, {\boldsymbol{\psi}}({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2) {\boldsymbol{\psi}}^{{\dagger}}({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2)\,\mathrm{e}^{-\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}}\end{equation}

called the Wigner matrix. (It is also called the ‘Wigner tensor’ when $\smash {{\boldsymbol {\psi }}}$ is a true vector rather than a tuple.) It can be understood as the (inverse) Fourier image of

(2.52)\begin{equation} {{\boldsymbol {\mathsf{C}}}}_{{\boldsymbol{\psi}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{s}}}) \doteq {\boldsymbol{\psi}}({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2) {\boldsymbol{\psi}}^{{\dagger}}({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2) = \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,{{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol {\mathsf{k}}} \cdot {\boldsymbol {\mathsf{s}}}}. \end{equation}

The analogue of (2.45) is (Appendix B.1)

(2.53a)\begin{equation} (\widehat{\boldsymbol{{\boldsymbol {\mathsf{L}}}}}{\boldsymbol{\psi}}({\boldsymbol {\mathsf{x}}}))(\widehat{\boldsymbol{{\boldsymbol {\mathsf{R}}}}}{\boldsymbol{\psi}}({\boldsymbol {\mathsf{x}}}))^{{\dagger}} = \textstyle \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,{{\boldsymbol {\mathsf{L}}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \star {{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) \star {{\boldsymbol {\mathsf{R}}}}^{{\dagger}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}). \end{equation}

The Wigner matrix averaged over a sufficiently large phase-space volume $\varDelta {\boldsymbol {\mathsf {x}}}\,\varDelta {\boldsymbol {\mathsf {k}}} \gtrsim 1$ is a local property of the field, and it is positive-semidefinite (Appendix A).

For real fields, one also has

(2.53b)\begin{equation} {{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}({\boldsymbol {\mathsf{x}}}, {\boldsymbol {\mathsf{k}}}) = {{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}^\intercal({\boldsymbol {\mathsf{x}}}, -{\boldsymbol {\mathsf{k}}}) = {{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}^*({\boldsymbol {\mathsf{x}}}, -{\boldsymbol {\mathsf{k}}}), \end{equation}

and (2.53) yields the following corollary at $\smash {\epsilon \to 0}$, when $\smash {\star }$ becomes the usual product (Appendix B.1):

(2.53c)\begin{equation} (\widehat{\boldsymbol{{\boldsymbol {\mathsf{L}}}}}{\boldsymbol{\psi}})^{{\dagger}} \widehat{\boldsymbol{{\boldsymbol {\mathsf{R}}}}}{\boldsymbol{\psi}} = \int \mathrm{d}{\boldsymbol {\mathsf{k}}}\,\operatorname{tr}\big({{\boldsymbol {\mathsf{W}}}}_{{\boldsymbol{\psi}}}({{\boldsymbol {\mathsf{L}}}}^{{\dagger}}{{\boldsymbol {\mathsf{R}}}})_\text{H}\big). \end{equation}

The generalizations of the other formulas from the previous sections are obvious.

2.2. Weyl symbol calculus on phase space

2.2.1. Notation

Consider a Hamiltonian system with coordinates ${\boldsymbol {x}} \equiv (x^1, x^2, \ldots, x^n)$ and canonical momenta ${\boldsymbol {p}} \equiv (p_1, p_2, \ldots, p_n)$. Together, these variables comprise the phase-space coordinates ${\boldsymbol {z}} \equiv ({\boldsymbol {x}}, {\boldsymbol {p}})$, i.e.

(2.54)\begin{equation} {\boldsymbol{z}} \equiv (z^1, \ldots, z^{2n}) = (x^1, \ldots, x^n, p_1, \ldots, p_n). \end{equation}

Components of ${\boldsymbol {z}}$ will be denoted with Greek indices ranging from 1 to $2n$.Footnote ⁹

Hamilton's equations for $z^\alpha$ can be written as $\dot {z}^{\alpha } = \lbrace z^{\alpha }, H \rbrace$, or equivalently, as

(2.55)\begin{equation} \dot{z}^{\alpha} = J^{\alpha\beta}\,\partial_\beta H. \end{equation}

Here, $H = H(t, {\boldsymbol {z}})$ is a Hamiltonian, $\partial _\beta \doteq \partial /\partial z^\beta$,

(2.56)\begin{equation} \lbrace A, B \rbrace \doteq J^{\alpha\beta}\, (\partial_\alpha A) (\partial_\beta B) \end{equation}

is the Poisson bracket on $\smash {{\boldsymbol {z}}}$, and $J^{\alpha \beta }$ is the canonical Poisson structure:

(2.57)\begin{equation} {\boldsymbol{J}} ={-}{\boldsymbol{J}}^\intercal = \left( \begin{array}{cc} {\boldsymbol{0}}_n & {\boldsymbol{1}}_n \\ -{\boldsymbol{1}}_n & {\boldsymbol{0}}_n \\ \end{array} \right), \end{equation}

where ${\boldsymbol {0}}_n$ is an $n$-dimensional zero matrix, and ${\boldsymbol {1}}_n$ is an $n$-dimensional unit matrix. The corresponding equation for the probability distribution $f(t, {\boldsymbol {z}})$ is

(2.58)\begin{equation} \partial_t f = \lbrace H, f \rbrace. \end{equation}

Solutions of (2.58) and other functions of the extended-phase-space coordinates ${\boldsymbol {X}} \equiv (t, {\boldsymbol {z}})$ can be considered as vectors in the Hilbert space $\mathscr {H}_X$ with the usual inner productFootnote ¹⁰

(2.59)\begin{equation} {\left\langle{\xi|\psi}\right\rangle} \doteq \int \mathrm{d}{\boldsymbol{X}}\, \xi^*({\boldsymbol{X}}) \psi({\boldsymbol{X}}). \end{equation}

Assuming the notation $N \doteq \dim {\boldsymbol {X}} = 2n + 1$, one has

(2.60)\begin{equation} \mathrm{d}{\boldsymbol{X}} \doteq \mathrm{d} X^1\, \mathrm{d} X^2 \ldots \mathrm{d} X^N = \mathrm{d} t\, \mathrm{d} x^1 \ldots \mathrm{d} x^n\,\mathrm{d} p_1, \ldots , \mathrm{d} p_n. \end{equation}

Let us introduce the position operator on ${\boldsymbol {z}}$,

(2.61)\begin{equation} \widehat{\boldsymbol{z}} \doteq ( \underbrace{x^1, \ldots, x^n}_{\widehat{\boldsymbol{x}}}, \underbrace{p_1, \ldots, p_n}_{\widehat{\boldsymbol{p}}} ), \end{equation}

and the momentum operator on ${\boldsymbol {z}}$,

(2.62)\begin{equation} \widehat{\boldsymbol{q}} \equiv ( \underbrace{-\mathrm{i} \partial_1, \ldots, -\mathrm{i}\partial_n}_{\widehat{\boldsymbol{k}}}, \underbrace{-\mathrm{i}\partial^1, \ldots, -\mathrm{i}\partial^n}_{\widehat{\boldsymbol{r}}} ), \end{equation}

where $\partial _i \doteq \partial /\partial x^i$ but $\partial ^i \doteq \partial /\partial p_i$; that is, $\widehat {\boldsymbol {z}} = (\widehat {\boldsymbol {x}}, \widehat {\boldsymbol {p}})$, $\widehat {\boldsymbol {q}} = (\widehat {\boldsymbol {k}}, \widehat {\boldsymbol {r}}\,)$ and

(2.63)\begin{equation} \widehat{z}^{\kern1.5pt\alpha}\doteq z^{\alpha}, \qquad \widehat{q}_{\alpha}\doteq{-}\mathrm{i}\partial_{\alpha}. \end{equation}

Then, much like in § 2.1, one can also introduce the position and momentum operators on the extended phase space ${\boldsymbol {X}}$:

(2.64)\begin{equation} \widehat{\boldsymbol{X}} = (\;\widehat{t}, \widehat{\boldsymbol{z}\,}) = (\;\widehat{t}, \widehat{\boldsymbol{x}}, \widehat{\boldsymbol{p}\,}), \qquad \widehat{\boldsymbol{K}} = (-\widehat{\omega}, \widehat{\boldsymbol{q}}\,) = (-\widehat{\omega}, \widehat{\boldsymbol{k}}, \widehat{\boldsymbol{r}\,}). \end{equation}

Assuming the convention that Latin indices from the beginning of the alphabet $(a, b, c, \ldots )$ range from 0 to $2n$, and $\partial _a \doteq \partial /\partial X^a$, one can compactly express this as

(2.65)\begin{equation} \widehat{X}^a = X^a, \qquad \widehat{K}_a ={-} \mathrm{i}\partial_a. \end{equation}

The eigenvectors of these operators will be denoted ${\left. {\boldsymbol {\mathsf {|X}}} \right\rangle}$ and ${\left. {\boldsymbol {\mathsf {|K}}} \right\rangle}$:

(2.66)\begin{equation} \widehat{\boldsymbol{X}} {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle} = {\boldsymbol{X}} {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle}, \qquad \widehat{\boldsymbol{K}} {\left. {\boldsymbol {\mathsf {|K}}} \right\rangle} = {\boldsymbol{K}} {\left. {\boldsymbol {\mathsf {|K}}} \right\rangle}, \end{equation}

and we assume the usual normalization

(2.67)\begin{equation} \left\langle{{\boldsymbol{X}}_1|{\boldsymbol{X}}_2}\right\rangle = \delta({\boldsymbol{X}}_1 - {\boldsymbol{X}}_2), \qquad \left\langle{{\boldsymbol{K}}_1|{\boldsymbol{K}}_2}\right\rangle = \delta({\boldsymbol{K}}_1 - {\boldsymbol{K}}_2). \end{equation}

Both sets $\lbrace {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle}, {\boldsymbol {X}} \in \mathbb {R}^N \rbrace$ and $\lbrace {\left. {\boldsymbol {\mathsf {|K}}} \right\rangle}, {\boldsymbol {K}} \in \mathbb {R}^N \rbrace$ form a complete basis on $\mathscr {H}_X$, and the eigenvalues of these operators form a real extended phase space $({\boldsymbol {X}}, {\boldsymbol {K}})$, where

(2.68)\begin{equation} {\boldsymbol{X}} \equiv (t, {\boldsymbol{z}}), \qquad {\boldsymbol{K}} \equiv (-\omega, {\boldsymbol{q}}). \end{equation}

Particularly note the following formula, which will be used below:

(2.69)\begin{equation} J^{\alpha\beta}\widehat{q}_\alpha q_\beta =\left( \begin{array}{cc} \widehat{\boldsymbol{k}} & \widehat{\boldsymbol{r}} \end{array} \right) \left( \begin{array}{cc} {\boldsymbol{0}}_n & {\boldsymbol{1}}_n \\ -{\boldsymbol{1}}_n & {\boldsymbol{0}}_n \\ \end{array} \right) \left( \begin{array}{c} {\boldsymbol{k}}\\ {\boldsymbol{r}}\\ \end{array} \right) = \widehat{\boldsymbol{k}} \cdot {\boldsymbol{r}} - \widehat{\boldsymbol{r}} \cdot {\boldsymbol{k}}. \end{equation}

2.2.2. Wigner–Weyl transform

One can construct the Weyl symbol calculus on the extended phase space ${\boldsymbol {X}}$ just like it is done on spacetime ${\boldsymbol {\mathsf {x}}}$ in § 2.1, with an obvious modification of the notation. The Wigner–Weyl transform is defined as

(2.70)\begin{gather} \displaystyle A({\boldsymbol{X}}, {\boldsymbol{K}}) = \int \mathrm{d}{\boldsymbol{S}} \left\langle{{\boldsymbol{X}} + {\boldsymbol{S}}/2 | \widehat{A} | {\boldsymbol{X}} - {\boldsymbol{S}}/2}\right\rangle \mathrm{e}^{-\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}} \equiv \text{symb}_X \widehat{A}, \end{gather}

(2.71)\begin{gather}\displaystyle \widehat{A} = \frac{1}{(2{\rm \pi})^N} \int \mathrm{d}{\boldsymbol{X}}\,\mathrm{d}{\boldsymbol{K}}\,\mathrm{d}{\boldsymbol{S}} {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle}+{{\boldsymbol{S}}/2} A({\boldsymbol{X}}, {\boldsymbol{K}}) {\left\langle {{\boldsymbol {\mathsf {X|}}}} \right.} - {{\boldsymbol{S}}/2} \mathrm{e}^{\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}} \equiv \text{oper}_X A. \end{gather}

(Notice the change in the font and in the index compared with (2.26) and (2.25).) The corresponding Moyal product is denoted $\bigstar$ (as opposed to $\star$ introduced earlier) and is given by

(2.72)\begin{equation} A \,\bigstar\, B = A({\boldsymbol{X}}, {\boldsymbol{K}})\,\mathrm{e}^{\mathrm{i}\widehat{\mathcal{L}}_X/2}B({\boldsymbol{X}}, {\boldsymbol{K}}), \end{equation}

where $\widehat {\mathcal {L}}_X \doteq \overset {{\scriptscriptstyle \leftarrow }}{\partial }_{\boldsymbol {X}} \cdot \overset {{\scriptscriptstyle \rightarrow }}{\partial }_{\boldsymbol {K}} - \overset {{\scriptscriptstyle \leftarrow }}{\partial }_{\boldsymbol {K}} \cdot \overset {{\scriptscriptstyle \rightarrow }}{\partial }_{\boldsymbol {X}}$ can be expressed as follows:

(2.73)\begin{equation} A \widehat{\mathcal{L}}_X B \doteq{-} \frac{\partial A}{\partial t}\frac{\partial B}{\partial \omega} + \frac{\partial A}{\partial \omega}\frac{\partial B}{\partial t} + \frac{\partial A}{\partial x^i}\frac{\partial B}{\partial k_i} - \frac{\partial A}{\partial k_i}\frac{\partial B}{\partial x^i} + \frac{\partial A}{\partial p^i}\frac{\partial B}{\partial r_i} - \frac{\partial A}{\partial r_i}\frac{\partial B}{\partial p^i}. \end{equation}

If an operator $\smash {\widehat {A}}$ is local in $\smash {{\boldsymbol {p}}}$, its $\smash {{\boldsymbol {X}}}$ representation and $\smash {{\boldsymbol {\mathsf {x}}}}$ representation satisfy

(2.74)\begin{equation} \left\langle{t, {\boldsymbol{x}}, {\boldsymbol{p}}|\widehat{\boldsymbol{A}}|t', {\boldsymbol{x}}', {\boldsymbol{p}}'}\right\rangle = \left\langle{t, {\boldsymbol{x}}|\widehat{\boldsymbol{A}}|t', {\boldsymbol{x}}'}\right\rangle \delta({\boldsymbol{p}} - {\boldsymbol{p}}'), \end{equation}

and therefore the Weyl symbol of $\smash {\widehat {A}}$ is the same irrespective of whether the operator is considered on $\smash {\mathscr {H}_X}$ or on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$. In this case, we will use a unifying notation $\smash {\text {symb}\,\widehat {A}}$ instead of $\smash {\text {symb}_X \widehat {A}}$ and $\smash {\text {symb}_{{\mathsf {x}}} \widehat {A}}$.

2.2.3. Wigner functions and Wigner matrices

The density operator of a given scalar field $\psi$ is given by

(2.75)\begin{equation} \widehat{W}_{\psi} \doteq (2{\rm \pi})^{{-}N} {\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {\psi |}}}} \right.}. \end{equation}

The symbol of this operator, $\smash {W_{\psi } = \text {symb}_X \widehat {W}_\psi }$, is a real function called the Wigner function. It is given by

(2.76)\begin{equation} W_{\psi}({\boldsymbol{X}}, {\boldsymbol{K}}) = \frac{1}{(2{\rm \pi})^N} \int \mathrm{d}{\boldsymbol{S}}\, \psi({\boldsymbol{X}} + {\boldsymbol{S}}/2) \psi^{*}({\boldsymbol{X}} - {\boldsymbol{S}}/2)\,\mathrm{e}^{-\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}, \end{equation}

which can be understood as the (inverse) Fourier image of

(2.77)\begin{equation} C_\psi({\boldsymbol{X}}, {\boldsymbol{S}}) \doteq \psi({\boldsymbol{X}} + {\boldsymbol{S}}/2) \psi^*({\boldsymbol{X}} - {\boldsymbol{S}}/2) = \int \mathrm{d}{\boldsymbol{K}}\,W_\psi({\boldsymbol{X}}, {\boldsymbol{K}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}. \end{equation}

In particular, one has

(2.78)\begin{equation} \int \mathrm{d}{\boldsymbol{r}}\,\text{symb}_X \widehat{W}_\psi = \text{symb}_{{\mathsf{x}}} \widehat{{\mathsf{W}}}_\psi({\boldsymbol{p}}), \end{equation}

where the right-hand side is $\smash {{\mathsf {W}}_\psi }$ given by (2.43), with ${\boldsymbol {p}}$ treated as a parameter. Also, for real fields,

(2.79)\begin{equation} W_\psi({\boldsymbol{X}}, {\boldsymbol{K}}) = W_\psi({\boldsymbol{X}}, -{\boldsymbol{K}}). \end{equation}

The density operator of a given vector field ${\boldsymbol {\psi }} = (\psi ^1, \psi ^2, \ldots, \psi ^M)$ is a matrix operator

(2.80)\begin{equation} \widehat{\boldsymbol{W}}_{{\boldsymbol{\psi}}} \doteq (2{\rm \pi})^{{-}N} {\left. {\boldsymbol {\mathsf {|\psi}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {\psi |}}}} \right.}. \end{equation}

The symbol of this operator, or the Wigner matrix, is a Hermitian matrix function

(2.81)\begin{equation} {\boldsymbol{W}}_{{\boldsymbol{\psi}}}({\boldsymbol{X}}, {\boldsymbol{K}}) = \frac{1}{(2{\rm \pi})^N} \int \mathrm{d}{\boldsymbol{S}}\, {\boldsymbol{\psi}}({\boldsymbol{X}} + {\boldsymbol{S}}/2) {\boldsymbol{\psi}}^{{\dagger}}({\boldsymbol{X}} - {\boldsymbol{S}}/2)\,\mathrm{e}^{-\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}, \end{equation}

which can be understood as the (inverse) Fourier image of

(2.82)\begin{equation} {\boldsymbol{C}}_{{\boldsymbol{\psi}}}({\boldsymbol{X}}, {\boldsymbol{S}}) \doteq {\boldsymbol{\psi}}({\boldsymbol{X}} + {\boldsymbol{S}}/2) {\boldsymbol{\psi}}^{{\dagger}}({\boldsymbol{X}} - {\boldsymbol{S}}/2) = \int \mathrm{d}{\boldsymbol{K}}\,{\boldsymbol{W}}_{{\boldsymbol{\psi}}}({\boldsymbol{X}}, {\boldsymbol{K}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}. \end{equation}

In particular, one has

(2.83)\begin{equation} \int \mathrm{d}{\boldsymbol{r}}\,\text{symb}_X \widehat{\boldsymbol{W}}_{{\boldsymbol{\psi}}} = \text{symb}_{{\mathsf{x}}} \widehat{\boldsymbol{{\boldsymbol {\mathsf{W}}}}}_{{\boldsymbol{\psi}}}({\boldsymbol{p}}), \end{equation}

where the right-hand side is $\smash {{{\boldsymbol {\mathsf {W}}}}_{{\boldsymbol {\psi }}}}$ given by (2.51), with ${\boldsymbol {p}}$ treated as a parameter. Also, for real fields,

(2.84)\begin{equation} {\boldsymbol{W}}_{{\boldsymbol{\psi}}}({\boldsymbol{X}}, {\boldsymbol{K}}) = {\boldsymbol{W}}_{{\boldsymbol{\psi}}}^\intercal({\boldsymbol{X}}, -{\boldsymbol{K}}) = {\boldsymbol{W}}_{{\boldsymbol{\psi}}}^*({\boldsymbol{X}}, -{\boldsymbol{K}}). \end{equation}

Like those on $\smash {({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$, the Wigner matrices (Wigner functions) on $\smash {({\boldsymbol {X}}, {\boldsymbol {K}})}$ become positive-semidefinite (non-negative), and characterize local properties of the corresponding fields, when averaged over a sufficiently large phase-space volume $\varDelta {\boldsymbol {X}}\,\varDelta {\boldsymbol {K}} \gtrsim 1$.

2.3. Summary of § 2

In summary, we have introduced a generic $\smash {n}$-dimensional physical space $\smash {{\boldsymbol {x}}}$, the dual $\smash {n}$-dimensional wavevector space $\smash {{\boldsymbol {k}}}$, the corresponding $\smash {{{\mathsf {n}}}}$-dimensional ($\smash {{{\mathsf {n}}} = n + 1}$) spacetime $\smash {{\boldsymbol {\mathsf {x}}} \equiv (t, {\boldsymbol {x}})}$ and the dual $\smash {{{\mathsf {n}}}}$-dimensional wavevector space $\smash {{\boldsymbol {\mathsf {k}}} \equiv (-\omega, {\boldsymbol {k}})}$. We have also introduced an $\smash {n}$-dimensional momentum space $\smash {{\boldsymbol {p}}}$, the corresponding $\smash {2n}$-dimensional phase space $\smash {{\boldsymbol {z}} \equiv ({\boldsymbol {x}}, {\boldsymbol {p}})}$, the $\smash {N}$-dimensional ($\smash {N = 2n + 1}$) extended space $\smash {{\boldsymbol {X}} \equiv (t, {\boldsymbol {z}}) \equiv (t, {\boldsymbol {x}}, {\boldsymbol {p}})}$ and the dual $\smash {N}$-dimensional wavevector space $\smash {{\boldsymbol {K}} \equiv (-\omega, {\boldsymbol {q}}) \equiv (-\omega, {\boldsymbol {k}}, {\boldsymbol {r}})}$, where $\smash {{\boldsymbol {r}}}$ is the $\smash {n}$-dimensional wavevector space dual to $\smash {{\boldsymbol {p}}}$. We have also introduced the $\smash {2N}$-dimensional phase space $\smash {({\boldsymbol {X}}, {\boldsymbol {K}})}$. Each of the said variables has a corresponding operator associated with it, which is denoted with a caret. For example, $\smash {\widehat {\boldsymbol {x}}}$ is the operator of position in the $\smash {{\boldsymbol {x}}}$ space, and $\smash {\widehat {\boldsymbol {k}} = -\mathrm {i}\partial _{{\boldsymbol {x}}}}$ is the corresponding wavevector operator.

Functions on $\smash {{\boldsymbol {\mathsf {x}}}}$ form a Hilbert space $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$, and the corresponding bra–ket notation is introduced as usual. Any operator $\smash {\widehat {{\mathsf {A}}}}$ on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$ can be represented by its Weyl symbol $\smash {{\mathsf {A}}({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$. The correspondence between operators and their symbols, $\smash {\widehat {{\mathsf {A}}} \leftrightarrow {\mathsf {A}}}$, is determined by the Wigner–Weyl transform and is natural in the sense that (2.29) is satisfied. In particular, $\smash {\widehat {{\mathsf {A}}}\,\widehat {{\mathsf {B}}} \leftrightarrow {\mathsf {A}} \star {\mathsf {B}}}$, where $\smash {\star }$ is the Moyal product on $\smash {({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$. When the geometrical-optics parameter is negligible ($\smash {\epsilon \to 0}$), one has $\smash {\widehat {{\mathsf {A}}} = {\mathsf {A}}(\,\widehat {\boldsymbol {{\boldsymbol {\mathsf {x}}}}}, \widehat {\boldsymbol {{\boldsymbol {\mathsf {k}}}}})}$ and the Moyal product becomes the usual product. Similarly, functions on $\smash {{\boldsymbol {X}}}$ form a Hilbert space $\smash {\mathscr {H}_X}$, the corresponding bra–ket notation is also introduced as usual, any operator $\smash {\widehat {A}}$ on $\smash {\mathscr {H}_X}$ can be represented by its Weyl symbol $\smash {A({\boldsymbol {X}}, {\boldsymbol {K}})}$, and $\smash {\widehat {A\,}\widehat {B} \leftrightarrow A \,\bigstar \, B}$. An operator that is local in $\smash {{\boldsymbol {p}}}$ has the same symbol irrespective of whether it is considered on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$ or on $\smash {\mathscr {H}_X}$.

Any given field $\smash {\psi }$ generates the corresponding density operator and its symbol called the Wigner function (Wigner matrix, if the field is a vector). If the density operator is considered on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$, it is denoted $\smash {\widehat {{\mathsf {W}}}_\psi }$ and the corresponding Wigner function is denoted $\smash {{\mathsf {W}}_\psi ({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$. If the density operator is considered on $\smash {\mathscr {H}_X}$, it is denoted $\smash {\widehat {W}_\psi }$ and the corresponding Wigner function is denoted $\smash {W_\psi ({\boldsymbol {X}}, {\boldsymbol {K}})}$. The two Wigner functions are related via $\smash {\int \mathrm {d}{\boldsymbol {r}}\,W_\psi (t, {\boldsymbol {x}}, {\boldsymbol {p}}, \omega, {\boldsymbol {k}}, {\boldsymbol {r}}) = {\mathsf {W}}_\psi (t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}}; {\boldsymbol {p}})}$, where $\smash {{\boldsymbol {p}}}$ enters $\smash {{\mathsf {W}}_\psi }$ as a parameter, if at all. If averaged over a sufficiently large phase-space volume, the Wigner functions (matrices) are non-negative (positive-semidefinite) and characterize local properties of the corresponding fields.

3.1. Basic assumptions

3.1.1. Ordering

Let us consider particles governed by a Hamiltonian $H = \overline {H} + \widetilde {H}$ such that

(3.1)\begin{equation} \widetilde{H} = \mathcal{O}(\varepsilon) \ll \overline{H} = \mathcal{O}(1). \end{equation}

In other words, $\widetilde {H}$ serves as a small perturbation to the leading-order Hamiltonian $\overline {H}$. The system will be described in canonical variables $\smash {{\boldsymbol {z}} \equiv ({\boldsymbol {x}}, {\boldsymbol {p}}) \in \mathbb {R}^{2n}}$. Let us also assume that the system is close to being homogeneous in ${\boldsymbol {x}}$. This includes two conditions. First, we require that the external fields are weak (yet see § 3.1.2), meaning

(3.2)\begin{equation} \partial_{{\boldsymbol{x}}}\overline{H} \sim \kappa_x\overline{H} = \mathcal{O}(\epsilon), \qquad \partial_{{\boldsymbol{p}}}\overline{H} \sim \kappa_p\overline{H} = \mathcal{O}(1), \end{equation}

where $\epsilon \ll 1$ is a small parameter, $\kappa _x$ and $\kappa _p$ are the characteristic inverse scales in the ${\boldsymbol {x}}$ and ${\boldsymbol {p}}$ spaces, respectively, and the bar denotes local averaging.Footnote ¹¹ Hence, the particle momenta ${\boldsymbol {p}}$ are close to being local invariants. Second, the statistical properties of $\smash {\widetilde {H}}$ are also assumed to vary in ${\boldsymbol {x}}$ slowly. These properties can be characterized using the density operator of the perturbation Hamiltonian,

(3.3)\begin{equation} \widehat{W} \doteq (2{\rm \pi})^{{-}N} {\left. {\boldsymbol {\mathsf {|\widetilde{H}}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {\widetilde{H} |}}}} \right.}, \end{equation}

and its symbol, the (real) Wigner function, as in (2.43):

(3.4)\begin{equation} W({\boldsymbol{X}}, {\boldsymbol{K}}) = \frac{1}{(2{\rm \pi})^N} \int \mathrm{d}{\boldsymbol{S}}\,\widetilde{H}({\boldsymbol{X}} + {\boldsymbol{S}}/2) \widetilde{H}({\boldsymbol{X}} - {\boldsymbol{S}}/2)\,\mathrm{e}^{-\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}. \end{equation}

Specifically, we will use the average Wigner function, $\overline {W}$, which represents the Fourier spectrum of the symmetrized autocorrelation function of $\widetilde {H}$:

(3.5)\begin{equation} \overline{C}({\boldsymbol{X}}, {\boldsymbol{S}}) \doteq \overline{\widetilde{H}({\boldsymbol{X}} + {\boldsymbol{S}}/2) \widetilde{H}({\boldsymbol{X}} - {\boldsymbol{S}}/2)} = \int \mathrm{d}{\boldsymbol{K}}\,\overline{W}({\boldsymbol{X}}, {\boldsymbol{K}})\,\mathrm{e}^{\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}. \end{equation}

The averaging is performed over sufficiently large volume of ${\boldsymbol {x}}$ to eliminate rapid oscillations and also over phase-space volumes $\varDelta {\boldsymbol {X}}\,\varDelta {\boldsymbol {K}} \gtrsim 1$, which guarantees $\overline {W}$ to be non-negative and local (§ 2.2.3). The function $\overline {W}$ can be understood as a measure of the phase-space density of wave quanta when the latter is well defined (§ 7).

We will assumeFootnote ¹²

(3.6)\begin{equation} \partial_t \overline{W} = \mathcal{O}(\epsilon), \qquad \partial_{\boldsymbol{x}} \overline{W} = \mathcal{O}(\epsilon), \qquad \partial_{\boldsymbol{p}} \overline{W} = \mathcal{O}(1). \end{equation}

That said, we will also allow (albeit not require) for oscillations to be constrained by a dispersion relation. In this case, $\overline {W}\, \propto \, \delta (\omega -\overline {\omega }(t, {\boldsymbol {x}}))$, so (3.6) per se is not satisfied; then we assume a similar ordering for $\int \mathrm {d}\omega \,\overline {W}$ instead. Also note that in application to the standard QLT of homogeneous turbulence (Stix Reference Stix1992, chapter 16), $\epsilon$ is understood as the geometrical-optics parameter characterizing the smallness of the linear-instability growth rates. (We discuss the ordering further in the end of § 3.3.)

3.1.2. Quasilinear approximation

The particle-motion equations can be written as

(3.7)\begin{equation} \dot{z}^{\alpha} = \lbrace z^{\alpha}, \overline{H} + \widetilde{H} \rbrace = v^{\alpha} + u^{\alpha}, \end{equation}

where $v^{\alpha }$ and $u^{\alpha }$ are understood as the unperturbed phase-space velocity and the perturbation to the phase-space velocity, respectively:

(3.8)\begin{equation} v^{\alpha} \doteq J^{\alpha\beta} \partial_{\beta}\overline{H}, \qquad u^{\alpha} \doteq J^{\alpha\beta}\partial_{\beta}\widetilde{H}. \end{equation}

The notation $v^i$ (with $i = 1, 2, \ldots n$) will also be used for the spatial part of the phase-space velocity $v^{\alpha }$, i.e. for the true velocity per se. Likewise, ${\boldsymbol {v}}$ will be used to denote either the phase-space velocity vector or the spatial velocity vector depending on the context. Also note that a slightly different definition of ${\boldsymbol {v}}$ will be used starting from § 5.6.

The corresponding Klimontovich equation for the particle distribution $f(t, {\boldsymbol {z}})$ is

(3.9)\begin{equation} \partial_t f = \lbrace \overline{H} + \widetilde{H}, f \rbrace. \end{equation}

(If collisions are not of interest, (3.9) can as well be understood as the Vlasov equation. Also, a small collision term can be included ad hoc; see the comment in the end of § 3.3.) Let us search for $f$ in the form

(3.10)\begin{equation} f = \overline{f} + \widetilde{f}, \qquad \overline{\widetilde{f}} = 0. \end{equation}

The equations for $\smash {\overline {f}}$ and $\smash {\widetilde {f}}$ are obtained as the average and oscillating parts of (3.9), and we neglect the nonlinearity in the equation for $\smash {\widetilde {f}}$, following the standard QL approximation (Stix Reference Stix1992, chapter 16). Then, one obtains

(3.11)\begin{gather} \partial_t\overline{f} = \lbrace \overline{H}, \overline{f} \rbrace + \overline{\lbrace \widetilde{H}, \widetilde{f} \rbrace}, \end{gather}

(3.12)\begin{gather}\partial_t\widetilde{f} = \lbrace \overline{H}, \widetilde{f} \rbrace + \lbrace \widetilde{H}, \overline{f} \rbrace. \end{gather}

A comment is due here regarding plasmas in strong fields and magnetized plasmas in particular. Our formulation can be applied to such plasmas in canonical angle–action variables $\smash {({{\phi }}, \boldsymbol{\mathsf {J}})}$. For fast angle variables, the ordering (3.2) is not satisfied and the Weyl symbol calculus is inapplicable as is (see the footnote in § 2.1.3). Such systems can be accommodated by representing the distribution function as a Fourier series in $\smash {{{\phi }}}$ and treating the individual-harmonic amplitudes separately as slow functions of the remaining coordinates. Then, our averaging procedure subsumes averaging over $\smash {{{\phi }}}$, so the averaged quantities are $\smash {{{\phi }}}$-independent and (3.2) is reinstated. In particular, magnetized plasmas can be described using guiding-centre variables. Although not canonical by default (Littlejohn Reference Littlejohn1983), they can always be cast in a canonical form, at least in principle (Littlejohn Reference Littlejohn1979). Examples of canonical guiding-centre variables are reviewed in Cary & Brizard (Reference Cary and Brizard2009). To make the connection with the homogeneous-plasma theory, one can also order the canonical pairs of guiding-centre variables such that they would describe the gyromotion, the parallel motion, and the drifts separately (Wong Reference Wong2000). This readily leads to results similar to those in Catto et al. (Reference Catto, Lee and Ram2017). Further discussions on this topic are left to future papers.

3.2. Equation for ${\widetilde {f}}$

Let us consider solutions of (3.12) as a subclass of solutions of the more general equation

(3.13)\begin{equation} \partial_\tau\widetilde{f} = \widehat{L}\widetilde{f} + \mathscr{F}, \qquad \mathscr{F}({\boldsymbol{X}}) \doteq \lbrace \widetilde{H}, \overline{f} \rbrace. \end{equation}

Here, we have introduced an auxiliary second ‘time’ $\smash {\tau }$, the operator

(3.14)\begin{equation} \widehat{L} \doteq{-}\partial_t + \lbrace \overline{H}, {\unicode{x25AA}} \rbrace ={-}\partial_t + J^{\alpha\beta}(\partial_{\alpha}\overline{H})\partial_{\beta} ={-}\partial_t - v^{\lambda}\partial_{\lambda} ={-}V^a \partial_a \end{equation}

(here and further, ${\unicode{x25AA}}$ denotes a placeholder), and ${\boldsymbol {V}}({\boldsymbol {X}}) \equiv (1, {\boldsymbol {v}}(t, {\boldsymbol {z}}))$ is the unperturbed velocity in the ${\boldsymbol {X}}$ space. Note that

(3.15)\begin{equation} \partial_a V^a = \partial_{\lambda}v^{\lambda} = 0 \end{equation}

due to the incompressibility of the phase flow. Hence, $\smash {[\partial _a, V^a] = 0}$, so $\smash {\widehat {L}}$ is anti-Hermitian.

Let us search for a solution of (3.13) in the formFootnote ¹³

(3.16)\begin{equation} \widetilde{f}(\tau, {\boldsymbol{X}}) = \mathrm{e}^{\widehat{L} \tau} \xi (\tau, {\boldsymbol{X}}). \end{equation}

Then, $\partial _\tau \widetilde {f} = \widehat {L\,}\,\widetilde {f} + \mathrm {e}^{\widehat {L} \tau }\partial _\tau \xi$, so $\partial _\tau \xi = \mathrm {e}^{-\widehat {L} \tau }\mathscr {F}({\boldsymbol {X}})$ and therefore

(3.17)\begin{equation} \xi(\tau, {\boldsymbol{X}}) = \mathrm{e}^{-\widehat{L} \tau_0} \xi_0({\boldsymbol{X}}) + \int_{\tau_0}^\tau \mathrm{d} \tau'\,\mathrm{e}^{-\widehat{L} \tau'} \mathscr{F}({\boldsymbol{X}}), \end{equation}

where $\smash {\xi _0({\boldsymbol {X}}) \doteq \widetilde {f}(\tau _0, {\boldsymbol {X}})}$. Hence, one obtains

(3.18)\begin{equation} \widetilde{f}(\tau, {\boldsymbol{X}}) = \mathrm{e}^{\widehat{L}(\tau - \tau_0)} \xi_0({\boldsymbol{X}}) + \int_{\tau_0}^\tau \mathrm{d} \tau'\,\mathrm{e}^{-\widehat{L} (\tau' - \tau)} \mathscr{F}({\boldsymbol{X}}), \end{equation}

or equivalently, using $\tau '' \doteq \tau - \tau '$,

(3.19)\begin{equation} \widetilde{f}(\tau, {\boldsymbol{X}}) = g_0(\tau, {\boldsymbol{X}}) + \int_0^{\tau - \tau_0} \mathrm{d}\tau''\,\widehat{T}_{\tau''} \mathscr{F}({\boldsymbol{X}}). \end{equation}

Here, $\smash {g_0}$ is a solution of $\partial _\tau g_0 = \widehat {L} g_0$, specifically,

(3.20)\begin{equation} g_0(\tau, {\boldsymbol{X}}) \doteq \widehat{T}_{\tau - \tau_0} \xi_0({\boldsymbol{X}}), \qquad g_0(\tau_0, {\boldsymbol{X}}) = \widetilde{f}(\tau_0, {\boldsymbol{X}}), \end{equation}

and we have also introduced

(3.21)\begin{equation} \widehat{T}_\tau \doteq \mathrm{e}^{\widehat{L} \tau} = \mathrm{e}^{-\tau V^a\partial_a}. \end{equation}

Because $\smash {\widehat {L}}$ is anti-Hermitian, the operator $\smash {\widehat {T}_\tau }$ is unitary, and comparison with (2.19) shows that it can be recognized as a shift operator. For further details, see § 4.1.

Using $\smash {\widehat {T}_\tau }$, one can express (3.19) as

(3.22)\begin{equation} \widetilde{f} = g_0 + \widehat{\mathscr{G}}\mathscr{F}, \qquad \textstyle \widehat{\mathscr{G}} \doteq \int_0^{\tau-\tau_0} \mathrm{d}\tau'\,\widehat{T}_{\tau'}, \end{equation}

where $\widehat {\mathscr {G}}$ is the Green's operator understood as the right inverse of the operator $\smash {\partial _\tau - \widehat {L}}$, or on the space of $\smash {\tau }$-independent functions, $\smash {\partial _t - \lbrace \overline {H}, {\unicode{x25AA}} \rbrace }$. Let us rewrite this operator as $\smash {\widehat {\mathscr {G}} = \widehat {\mathscr {G}}_< + \widehat {\mathscr {G}}_>}$, where

(3.23)\begin{equation} \widehat{\mathscr{G}}_<{=} \int_0^{\tau-\tau_0} \mathrm{d}\tau'\,\mathrm{e}^{-\nu \tau'}\widehat{T}_{\tau'}, \qquad \widehat{\mathscr{G}}_>{=} \int_0^{\tau-\tau_0} \mathrm{d}\tau'\,(1 - \mathrm{e}^{-\nu \tau'})\widehat{T}_{\tau'}, \end{equation}

and $\nu$ is a positive constant. Note that $\widehat {\mathscr {G}}_<$ is well defined at $\tau _0 \to -\infty$, meaning that $\widehat {\mathscr {G}}_<\mathscr {F}$ is well defined for any physical (bounded) field $\mathscr {F}$.Footnote ¹⁴ Thus, so is $\smash {g_0 + \widehat {\mathscr {G}}_>\mathscr {F}}$. Let us take $\tau _0 \to -\infty$ and then take $\nu \to 0+$. (Here, $0+$ denotes that $\nu$ must remain positive, i.e. the upper limit is taken.) Then, (3.22) can be expressed as

(3.24)\begin{equation} \widetilde{f} = g + \widehat{G}\mathscr{F}, \qquad g \doteq \lim_{\nu \to 0+} \lim_{\tau_0 \to -\infty} (g_0 + \widehat{\mathscr{G}}_>\mathscr{F}). \end{equation}

Here, we introduced an ‘effective’ Green's operator $\smash {\widehat {G} \doteq \lim _{\nu \to 0+} \lim _{\tau _0 \to -\infty } \widehat {\mathscr {G}}_<}$, i.e.

(3.25)\begin{equation} \widehat{G} \doteq \lim_{\nu \to 0+} \int_0^\infty \mathrm{d}\tau\,\mathrm{e}^{-\nu \tau}\widehat{T}_{\tau}. \end{equation}

This operator will be discussed in § 4.2, and $g$ will be discussed in § 4.3. Meanwhile, note that because $\smash {\tau }$ is just an auxiliary variable, we will be interested in solutions independent of $\tau$. In particular, this means that $\smash {\widetilde {f}(\tau _0, {\boldsymbol {X}}) = \widetilde {f}({\boldsymbol {X}})}$, so $\smash {\xi _0({\boldsymbol {X}}) = \widetilde {f}({\boldsymbol {X}})}$, so (3.20) leads to

(3.26)\begin{equation} g_0(\tau, {\boldsymbol{X}}) = \widehat{T}_{\tau - \tau_0} \widetilde{f}({\boldsymbol{X}}). \end{equation}

3.3. Equation for ${\overline {f}}$

Using (3.22), one can rewrite (3.12) for $\overline {f}$ as follows:

(3.27)\begin{equation} \partial_t\overline{f} = \lbrace \overline{H}, \overline{f} \rbrace + \overline{\lbrace \widetilde{H}, g \rbrace} + \overline{\lbrace \widetilde{H}, \widehat{G}\lbrace \widetilde{H}, \overline{f} \rbrace \rbrace} .\end{equation}

Notice that

(3.28)\begin{equation} \lbrace \widetilde{H}, g \rbrace ={-} \lbrace g, \widetilde{H} \rbrace ={-}\partial_{\alpha} (J^{\alpha\beta}\, g \partial_{\beta} \widetilde{H}) ={-}\partial_{\alpha} (u^\alpha g), \end{equation}

and also

(3.29)\begin{align} \lbrace \widetilde{H}, \widehat{G}\lbrace \widetilde{H}, \overline{f} \rbrace \rbrace & = \partial_{\beta}( J^{\alpha\beta} (\partial_{\alpha}\widetilde{H}) \widehat{G}\lbrace \widetilde{H}, \overline{f} \rbrace )\nonumber\\ & = \partial_{\beta}(J^{\alpha\beta} (\partial_{\alpha}\widetilde{H}) \widehat{G} (J^{\mu \nu} (\partial_{\mu}\widetilde{H})(\partial_{\nu}\overline{f})))\nonumber\\ & = \partial_{\beta}(u^{\beta}\widehat{G}(u^{\nu}\partial_{\nu}\overline{f})). \end{align}

The field $u^\alpha$ enters here as a multiplication factor and can be considered as an operator:

(3.30)\begin{equation} \widehat{u\,\,}^\alpha\psi({\boldsymbol{X}}) \doteq u^\alpha({\boldsymbol{X}})\psi({\boldsymbol{X}}). \end{equation}

Then, (3.29) can be compactly represented as

(3.31)\begin{equation} \lbrace \widetilde{H}, \widehat{G}\lbrace \widetilde{H}, \overline{f} \rbrace \rbrace = \partial_{\alpha} (\,\widehat{u}^{\kern1.5pt\alpha} \widehat{G}\, \widehat{u}^{\kern1.5pt\beta} \partial_{\beta} \overline{f}). \end{equation}

We will also use the notation

(3.32)\begin{equation} \mathrm{d}_t \doteq \partial_t + v^{\gamma}\partial_{\gamma} = \partial_t - \lbrace \overline{H}, {\unicode{x25AA}} \rbrace. \end{equation}

This leads to the following equation for $\overline {f}$:

(3.33)\begin{equation} \mathrm{d}_t \overline{f} = \partial_{\alpha}(\widehat{D}^{\alpha\beta} \partial_{\beta}\overline{f}) + \varGamma, \end{equation}

where we introduced the following average quantities:

(3.34)\begin{equation} \widehat{D}^{\alpha\beta} \doteq \overline{\widehat{u\,}^{\alpha}\widehat{G\,}\widehat{u}^{\beta}}, \qquad \varGamma \doteq{-} \partial_\alpha(\overline{u^{\alpha} g}). \end{equation}

Our goal is to derive explicit approximate expressions for the quantities (3.34) and to rewrite (3.33) in a more tractable form using the assumptions introduced in § 3.1. We will useFootnote ¹⁵

(3.35)\begin{equation} \partial_t\overline{f} \sim \lbrace \overline{H}, \overline{f} \rbrace = \mathcal{O}(\epsilon), \qquad \mathrm{d}_t\overline{f} = \mathcal{O}(\varepsilon^2), \end{equation}

and we will keep terms of order $\epsilon$, $\varepsilon ^2$ and $\smash {\epsilon \varepsilon ^2}$ in the equation for $\smash {\overline {f}}$, while terms of order $\smash {\varepsilon ^4}$, $\smash {\epsilon ^2 \varepsilon ^2}$ and higher will be neglected. This implies the ordering

(3.36)\begin{equation} \varepsilon^2 \ll \epsilon \ll \varepsilon \ll 1. \end{equation}

As a reminder, $\smash {\varepsilon }$ is a linear measure of the characteristic amplitude of oscillations, and $\smash {\epsilon }$ is the geometrical-optics parameter, which is proportional to the inverse scale of the plasma inhomogeneity in spacetime. As usual then, linear dissipation is assumed to be of order $\smash {\epsilon }$. This model implies the assumption that collisionless dissipation is much stronger than collisional dissipation, which is to emerge as an effect quadratic in $\smash {\widetilde {f}}$ (§ 6). Furthermore, the inverse plasma parameterFootnote ¹⁶ will be assumed to be of order $\smash {\epsilon }$, so the collision operator for $\smash {\overline {f}}$ (§ 6.8) will be of order $\smash {\epsilon \varepsilon ^2}$. Within the assumed accuracy, this operator must be retained, while the dynamics of $\smash {\widetilde {f}}$ is considered linear and therefore collisionless. Alternatively, one can switch from the Klimontovich description to the Vlasov–Boltzmann description and introduce an ad hoc order-$\smash {\epsilon }$ collision operator directly in (3.9). This will alter the Green's operator, but the conceptual formulation would remain the same, so it will not be considered separately in detail.

3.4. Summary of § 3

Our QL model is defined as usual, except: (i) we allow for a general particle Hamiltonian $\smash {H}$; (ii) we use the Klimontovich equation rather than the Vlasov equation to retain collisions; (iii) we use local averaging (denoted with overbar) and allow for weak inhomogeneity of all averaged quantities; (iv) we retain the initial conditions $\smash {g}$ for the oscillating part of the distribution function (defined as in (3.24) but yet to be calculated explicitly). Then, the average part of the distribution function satisfies

(3.37)\begin{equation} \partial_t\overline{f} - \lbrace \overline{H}, \overline{f} \rbrace = \partial_{\alpha}(\widehat{D}^{\alpha\beta} \partial_{\beta}\overline{f}) + \varGamma, \end{equation}

where $\smash {\widehat {D}^{\alpha \beta } \doteq \overline {\widehat {u}^{\alpha }\widehat {G\,}\widehat {u}^{\beta }}}$, $\smash {\vphantom{\frac{f^l}{f^l}}\varGamma \doteq - \partial _\alpha (\overline {u^{\alpha } g})}$, $\smash {u^{\alpha }}$ is the wave-driven perturbation of the phase-space velocity (see (3.8)), $\smash {\widehat {u\,\,}^{\alpha }}$ is the same quantity considered as an operator on $\smash {\mathscr {H}_X}$ (see (3.30)) and $\smash {\widehat {G} }$ is the ‘effective’ Green's operator given by

(3.38)\begin{equation} \widehat{G} \doteq \lim_{\nu \to 0+} \int_0^\infty \mathrm{d}\tau\,\mathrm{e}^{-\nu \tau-\tau V^a\partial_a}. \end{equation}

Also, $\smash {\partial _{\alpha } \equiv \partial /\partial z^\alpha }$, and $\smash {\lbrace {\unicode{x25AA}}, {\unicode{x25AA}} \rbrace }$ is the Poisson bracket on the particle phase space $\smash {{\boldsymbol {z}}}$. The equation for $\smash {\overline {f}}$ used in the standard QLT is recovered from (3.37) by neglecting $\smash {\varGamma }$ and the spatial gradients (in particular, the whole Poisson bracket) and also by approximating the operator $\smash {\widehat {D}^{\alpha \beta }}$ with a local function of $\smash {{\boldsymbol {z}}}$.

4.1. Shift operator

Here, we derive some properties of the shift operator $\smash {\widehat {T}_\tau }$ introduced in § 3.2.

4.1.1. ${\widehat {T}_\tau }$ as a shift

Here, we formally prove (an admittedly obvious fact) that

(4.1)\begin{equation} \widehat{T}_\tau \psi({\boldsymbol{X}}) = \psi({\boldsymbol{X}} - {\boldsymbol{\ell}}_\tau({\boldsymbol{X}})), \qquad \textstyle \ell_{\tau}^{a}({\boldsymbol{X}}) \doteq \int_0^{\tau} \mathrm{d} t\, V^{a}({\boldsymbol{Y}}(t, {\boldsymbol{X}})), \end{equation}

where the ‘characteristics’ $\smash {Y^a}$ solveFootnote ¹⁷

(4.2)\begin{equation} \frac{\mathrm{d} Y^{a}}{\mathrm{d}\tau} ={-}V^{a}({\boldsymbol{Y}}), \qquad Y^a(\tau = 0) = X^a, \end{equation}

and thus $\ell _{\tau }^{a}$ can be Taylor-expanded in $\tau$ as

(4.3)\begin{equation} \ell_{\tau}^{a}({\boldsymbol{X}}) = \tau V^{a} - \frac{1}{2}\,\tau^2 V^{b} \partial_{b} V^{a} + \ldots, \qquad V^{a} \equiv V^{a}({\boldsymbol{X}}). \end{equation}

As the first step to proving (4.1), let us Taylor-expand $V^{a}$ around a fixed point ${\boldsymbol {X}}_1$:

(4.4)\begin{equation} V^{a} = V^{a}_1 + (\partial_{b} V^{a}_1)\,\delta X^{b} + \ldots, \qquad \delta X^{a}\doteq X^{a} - X_1^{a}, \end{equation}

where $V^a_1 \equiv V^a({\boldsymbol {X}}_1)$. If one neglects the first and higher derivatives of $V^{a}$, one obtains

(4.5)\begin{equation} \widehat{T}_{\tau}\psi({\boldsymbol{X}}) \approx \mathrm{e}^{-\tau V^{a}_1\partial_{a}}\psi({\boldsymbol{X}}) = \psi({\boldsymbol{X}} - \tau {\boldsymbol{V}}_1). \end{equation}

By taking the limit ${\boldsymbol {X}}_1 \to {\boldsymbol {X}}$, which corresponds to ${\boldsymbol {V}}_1 \to {\boldsymbol {V}}$, one obtains

(4.6)\begin{equation} \widehat{T}_{\tau}\psi({\boldsymbol{X}}) = \psi({\boldsymbol{X}} - \tau {\boldsymbol{V}}) + \mathcal{O}(\tau^2). \end{equation}

Similarly, if one neglects the second and higher derivatives of $V^{a}$, one obtainsFootnote ¹⁸

(4.7)\begin{align} \widehat{T}_{\tau}\psi({\boldsymbol{X}}) & = \mathrm{e}^{ - \tau (V^{a}_1 + (\partial_{b}V^{a}_1) \delta X^{b}+\ldots) \partial_{a} } \psi({\boldsymbol{X}}) \nonumber\\ & \approx \mathrm{e}^{- \tau (\partial_{b} V^{a}_1) \delta X^{b} \partial_{a}} \,\mathrm{e}^{-\tau V^{a}_1 \partial_{a}} \,\mathrm{e}^{-\frac{1}{2}[ -\tau(\partial_{b}V^{a}_1) \delta X^{b}\partial_{a}, -\tau V^{c}_1\partial_{c} ]} \psi({\boldsymbol{X}}) \nonumber\\ & \approx \mathrm{e}^{ -\tau (\partial_{b} V^{a}_1) \delta X^{b} \partial_{a}} \,\mathrm{e}^{-\tau V^{a}_1 \partial_{a}} \,\mathrm{e}^{ \frac{1}{2}\tau^2 V^{c}_1(\partial_{b}V^{a}_1) [\partial_{c}, \delta X^{b}\partial_{a}] }\psi({\boldsymbol{X}}) \nonumber\\ & \approx \mathrm{e}^{-\tau (\partial_{b}V^{a}_1)\delta X^{b}\partial_{a}} \,\mathrm{e}^{-\tau V^{a}_1 \partial_{a}} \,\mathrm{e}^{\frac{1}{2}\tau^2 V^{b}_1 (\partial_{b}V^{a}_1) \partial_{a}} \psi({\boldsymbol{X}}) \nonumber\\ & \approx \mathrm{e}^{-\tau (\partial_{b}V^{a}_1) \delta X^{b} \partial_{a}} \,\mathrm{e}^{-\tau V^{a}_1\partial_{a} + \frac{1}{2}\tau^2 V^{b}_1(\partial_{b}V^{a}_1)\partial_{a}}\psi({\boldsymbol{X}}) \nonumber\\ & \approx \mathrm{e}^{-\tau (\partial_{b}V^{a}_1)\delta X^{b}\partial_{a}} \psi({\boldsymbol{X}} - \tau {\boldsymbol{V}}_1 + \textstyle\frac{1}{2}\,\tau^2 V^{b}_1 \partial_{b}{\boldsymbol{V}}_1). \end{align}

In the limit ${\boldsymbol {X}}_1 \to {\boldsymbol {X}}$, when $\smash {\mathrm {e}^{-\tau (\partial _{b}V^{a}_1)\delta X^{b} \partial _{a}} \to 1}$ and ${\boldsymbol {V}}_1 \to {\boldsymbol {V}}$, one obtains

(4.8)\begin{equation} \widehat{T}_{\tau}\psi({\boldsymbol{X}}) = \psi\left({\boldsymbol{X}} - \tau {\boldsymbol{V}}({\boldsymbol{X}}) + \frac{1}{2}\,\tau^2 ({\boldsymbol{V}} \cdot \partial_{{\boldsymbol{X}}}) {\boldsymbol{V}}\right) + \mathcal{O}(\tau^3). \end{equation}

In conjunction with (4.3), equations (4.6) and (4.8) show agreement with the sought result (4.3) within the assumed accuracy. One can also retain ${\mathsf {m}}$ derivatives of ${\boldsymbol {V}}$ and derive the corresponding approximations similarly. Then the error will be $\smash {\mathcal {O}(\tau ^{{\mathsf {m}} + 2})}$.

For an order-one time interval $\tau$, one can split this interval on $N_\tau \gg 1$ subintervals of small duration $\tau /N_\tau$ and apply finite-${\mathsf {m}}$ formulas (for example, (4.6) or (4.8)) to those. Then the total error scales as $\smash {\mathcal {O}(N_\tau ^{-{\mathsf {m}} - 1})}$ and the exact formula (4.1) is obtained at $N_\tau \to \infty$.

4.1.2. Symbol of ${\widehat {T}_\tau }$

Using the bra–ket notation, (4.1) can be written as

(4.9)\begin{equation} \left\langle{{\boldsymbol{X}} |\widehat{T}_{\tau} | \psi}\right\rangle = \left\langle{{\boldsymbol{X}} - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}})|\psi}\right\rangle. \end{equation}

Thus, $ {{\left\langle {{\boldsymbol {\mathsf {X|}}}} \right.}} \widehat {T}_{\tau } = {\left\langle {{\boldsymbol {\mathsf {X|}}}} \right.} - {\boldsymbol {\ell }}_{\tau }({\boldsymbol {X}})$, so

(4.10)\begin{equation} \left\langle{{\boldsymbol{X}}_1|\widehat{T}_{\tau}|{\boldsymbol{X}}_2}\right\rangle = \left\langle{{\boldsymbol{X}}_1 - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}}_1)|{\boldsymbol{X}}_2}\right\rangle = \delta ({\boldsymbol{X}}_1 - {\boldsymbol{X}}_2 - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}}_1)). \end{equation}

Using (2.70), one obtains the Weyl symbol of $\smash {\widehat {T}_{\tau }}$ in the form

(4.11)\begin{equation} T_{\tau}({\boldsymbol{X}}, {\boldsymbol{K}}) = \int \mathrm{d}{\boldsymbol{S}}\, \mathrm{e}^{-\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}\, \delta ({\boldsymbol{S}} - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}} + {\boldsymbol{S}}/2)). \end{equation}

From (4.3), one has

(4.12)\begin{align} \ell^a_{\tau}({\boldsymbol{X}} + {\boldsymbol{S}}/2) & = \tau V^{a}({\boldsymbol{X}} + {\boldsymbol{S}}/2) - (\tau^2/2)\,V^{b} \partial_{b} V^{a} + \mathcal{O}(\epsilon^2) \nonumber\\ & = \tau V^{a} + (\tau/2) (\partial_b V^{a}) S^b - (\tau^2/2)\,V^{b} \partial_{b} V^{a} + \mathcal{O}(\epsilon^2) \nonumber\\ & = {M^a}_b V^{b}\tau + {m^a}_b S^b + \mathcal{O}(\epsilon^2), \end{align}

where we introduced a matrix $\smash {{\boldsymbol {M}} \doteq {\boldsymbol {1}} - {\boldsymbol {m}}}$, or explicitly,

(4.13)\begin{equation} {M^a}_b \doteq \delta^a_b - {m^a}_b, \qquad {m^a}_b \doteq (\tau/2)(\partial_b V^{a}). \end{equation}

Let us express the term $\smash {\mathcal {O}(\epsilon ^2)}$ in (4.12) as $\smash {-{M^a}_b\mu ^b}$. Then,

(4.14)\begin{align} \delta ({\boldsymbol{S}} - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}} + {\boldsymbol{S}}/2)) & = \delta ({\boldsymbol{S}} - {\boldsymbol{M}}{\boldsymbol{V}}\tau - {\boldsymbol{m}}{\boldsymbol{S}} + {\boldsymbol{M}}{\boldsymbol{\mu}}) \nonumber\\ & = \delta ({\boldsymbol{M}}({\boldsymbol{S}} - {\boldsymbol{V}}\tau + {\boldsymbol{\mu}})) \nonumber\\ & = \delta ({\boldsymbol{S}} - {\boldsymbol{V}}\tau + {\boldsymbol{\mu}})/|\det {\boldsymbol{M}}|. \end{align}

Because $\smash {{\boldsymbol {m}} = \mathcal {O}(\epsilon )}$, the well-known formula yields $\smash {\det {\boldsymbol {M}} = 1 + \operatorname {tr}{\boldsymbol {m}} + \mathcal {O}(\epsilon ^2)}$. But $\operatorname {tr}{\boldsymbol {m}} = 0$ by (3.15), so

(4.15)\begin{equation} \delta ({\boldsymbol{S}} - {\boldsymbol{\ell}}_{\tau}({\boldsymbol{X}} + {\boldsymbol{S}}/2)) = \delta ({\boldsymbol{S}} - {\boldsymbol{V}}\tau + {\boldsymbol{\mu}}) + \mathcal{O}(\epsilon^2). \end{equation}

The last term $\smash {\mathcal {O}(\epsilon ^2)}$ is insignificant and can be neglected right away, so (4.11) leads to

(4.16)\begin{equation} T_{\tau}({\boldsymbol{X}}, {\boldsymbol{K}}) \approx \exp(\mathrm{i} \tau \varOmega({\boldsymbol{X}}, {\boldsymbol{K}}) + \mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{\mu}}), \end{equation}

where we have introduced the following notation:

(4.17)\begin{equation} \varOmega({\boldsymbol{X}}, {\boldsymbol{K}}) \doteq{-} {\boldsymbol{K}} \cdot {\boldsymbol{V}}({\boldsymbol{X}}) = \omega - q_{\alpha} v^{\alpha} = \omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}} + \mathcal{O}(\epsilon). \end{equation}

By definition, $\smash {{\boldsymbol {\mu }}}$ is a polynomial of $\smash {\tau }$ with coefficients that are of order $\smash {\epsilon ^2}$ and therefore small. But because $\smash {\tau }$ can be large, and because $\smash {{\boldsymbol {\mu }}}$ is under the exponent, this makes $\smash {T_{\tau }}$ potentially sensitive to this term, so we retain it (for now).

4.2. Effective Green's operator

The effective Green's operator (3.25) can be understood as the right inverse of the operator (cf. § 3.2)

(4.18)\begin{equation} \widehat{L}_\text{eff} \doteq \lim_{\nu \to 0+} (\partial_t - \lbrace \overline{H}, {\unicode{x25AA}} \rbrace + \nu), \end{equation}

so we denote it also as $\widehat {G} = \widehat {L}^{-1}_\text {eff}$ (which is admittedly abuse of notation). Because $\smash {\widehat {L}_\text {eff}}$ has real ${\boldsymbol {X}}$ representation by definition, the ${\boldsymbol {X}}$ representation of $\smash {\widehat {G}}$ is real too. In particular, $\smash {\left\langle {{\boldsymbol {X}} + {\boldsymbol {S}}/2 | \widehat {G} | {\boldsymbol {X}} - {\boldsymbol {S}}/2}\right\rangle}$ is real, hence

(4.19)\begin{equation} G({\boldsymbol{X}}, -{\boldsymbol{K}}) = G^*({\boldsymbol{X}}, {\boldsymbol{K}}) \end{equation}

by definition of the Weyl symbol (2.70). As a corollary, the derivative of $G({\boldsymbol {X}}, {\boldsymbol {K}})$ with respect to the $a$th component of the whole second argument, denoted $G^{|a}$, satisfies

(4.20)\begin{equation} (G^{|a}({\boldsymbol{X}}, {\boldsymbol{K}}))^* ={-}G^{|a}({\boldsymbol{X}}, -{\boldsymbol{K}}). \end{equation}

Also note that $\widehat {G}$ can be expressed as

(4.21)\begin{equation} \widehat{G} = \lim_{\nu \to 0+} \mathrm{i} (\widehat{\omega} - \dot{x}^i \widehat{k}_i - \dot{p}_i \widehat{r\,}^i + \mathrm{i} \nu)^{{-}1} \end{equation}

(the notation ‘$\smash {\lim _{\nu \to 0+} A(\omega + \mathrm {i} \nu )}$’ will also be shortened as ‘$\smash {A(\omega + \mathrm {i} 0)}$’), whence

(4.22)\begin{equation} \frac{\partial G}{\partial r^i} = \mathcal{O}(\dot{p}_i) = \mathcal{O}(\epsilon). \end{equation}

Due to (4.16), the leading-order approximation of the symbol of the operator (3.25) is $G({\boldsymbol {X}}, {\boldsymbol {K}}) = G_0(\varOmega ({\boldsymbol {X}}, {\boldsymbol {K}}))$, where

(4.23)\begin{equation} G_0(\varOmega) \doteq \lim_{\nu \to 0+}\int_0^{\infty}\mathrm{d}\tau\, \mathrm{e}^{-\nu \tau + \mathrm{i}\varOmega \tau} = {\rm \pi}\,\delta(\varOmega) + \mathrm{i}\,\operatorname{pv}\frac{1}{\varOmega} \end{equation}

and the (standard) notation $\operatorname {pv} (1/\varOmega )$ is defined as follows:

(4.24)\begin{equation} \operatorname{pv}\frac{1}{\varOmega} \doteq \lim_{\nu \to 0+} \frac{\varOmega}{\nu^2 + \varOmega^2}. \end{equation}

This means, in particular, that for any $A$, one has

(4.25)\begin{align} \mathcal{J}[A, G_0] & \doteq \int \mathrm{d}{\boldsymbol{K}}\,A({\boldsymbol{X}}, {\boldsymbol{K}}) G_0(\varOmega({\boldsymbol{X}}, {\boldsymbol{K}})) \nonumber\\ & ={\rm \pi} \int \mathrm{d}{\boldsymbol{K}}\,A({\boldsymbol{X}}, {\boldsymbol{K}}) \delta(\varOmega ({\boldsymbol{X}}, {\boldsymbol{K}})) + \mathrm{i} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}}\,\frac{A({\boldsymbol{X}}, {\boldsymbol{K}})}{\varOmega({\boldsymbol{X}}, {\boldsymbol{K}})}, \end{align}

where ${\unicode{x2A0F}}$ is a principal-value integral. Also usefully, $\smash {\overline {G}_0 = G_0}$ and

(4.26)\begin{align} \partial_a \mathcal{J}[A, G_0] & = \int \mathrm{d}{\boldsymbol{K}}\,A({\boldsymbol{X}}, {\boldsymbol{K}}) G_0'(\varOmega({\boldsymbol{X}}, {\boldsymbol{K}}))\,\partial_a\varOmega({\boldsymbol{X}}, {\boldsymbol{K}}) \nonumber\\ & ={-}(\partial_a V^b({\boldsymbol{X}})) \int \mathrm{d}{\boldsymbol{K}}\,K_b A({\boldsymbol{X}}, {\boldsymbol{K}}) G_0'(\varOmega({\boldsymbol{X}}, {\boldsymbol{K}})) \nonumber\\ & ={-}(\partial_a V^b({\boldsymbol{X}}))\,\frac{\eth}{\partial \varOmega} \int \mathrm{d}{\boldsymbol{K}}\,K_b A({\boldsymbol{X}}, {\boldsymbol{K}}) G_0(\varOmega ({\boldsymbol{X}}, {\boldsymbol{K}})), \end{align}

where the notation $\eth /\partial \lambda \equiv \eth _\lambda$ is defined, for any $\smash {\lambda }$ and $Q$, as follows:

(4.27)\begin{equation} \frac{\eth}{\partial \lambda} \int Q(\lambda) \doteq \left(\frac{\partial}{\partial \vartheta}\,\int Q(\lambda + \vartheta)\right)_{\vartheta = 0}. \end{equation}

Now let us reinstate the term ${\boldsymbol {\mu }}$ in (4.16). It is readily seen (Appendix B.2) that, although ${\boldsymbol {\mu }}$ may significantly affect $\smash {T_{\tau }}$ per se, its effect on $\mathcal {J}[A, G]$ is small, namely,

(4.28)\begin{equation} \mathcal{J}[A, G] - \mathcal{J}[A, G_0] = \mathcal{O}(\epsilon^2). \end{equation}

Below, we apply this formulation to $\smash {A = \mathcal {O}(\varepsilon ^2)}$, in which case (4.28) becomes $\smash {\mathcal {J}[A, G] - \mathcal {J}[A, G_0] = \mathcal {O}(\epsilon ^2\varepsilon ^2)}$. Such corrections are negligible within our model, so from now on we adopt

(4.29)\begin{equation} G({\boldsymbol{X}}, {\boldsymbol{K}}) \approx \overline{G}({\boldsymbol{X}}, {\boldsymbol{K}}) \approx G_0(\varOmega({\boldsymbol{X}}, {\boldsymbol{K}})). \end{equation}

4.3. Initial conditions

Consider the function $g$ from (3.24). Using (3.26), the latter can be written as follows:

(4.30)\begin{equation} g = \lim_{\nu \to 0+} \lim_{\tau_0 \to -\infty} \left(\widehat{T}_{\tau - \tau_0} \widetilde{f}({\boldsymbol{X}}) + \int_0^{\tau-\tau_0} \mathrm{d}\tau'\,(1 - \mathrm{e}^{-\nu \tau'})\widehat{T}_{\tau'}\mathscr{F}({\boldsymbol{X}})\right). \end{equation}

Because $(1 - \mathrm {e}^{-\nu \tau })$ is smooth and $\smash {\widehat {T}_{\tau }\mathscr {F}}$ is rapidly oscillating, the second term in the external parenthesis is an oscillatory function of $\tau _0$ with the average negligible at $\smash {\nu \to 0}$. But the whole expression in these parenthesis is independent of $\tau _0$ at large $\tau _0$ (§ 3.2). Thus, it can be replaced with its own average over $\tau _0$, denoted $\smash {\langle \ldots \rangle _{\tau _0}}$. Because there is no $\smash {\nu }$-dependence left in this case, one can also omit $\smash {\lim _{\nu \to 0+}}$. That gives

(4.31)\begin{equation} g = \lim_{\tau_0 \to -\infty} \langle \widehat{T}_{\tau - \tau_0} \widetilde{f}({\boldsymbol{X}}) \rangle_{\tau_0}. \end{equation}

Using

(4.32)\begin{equation} f({\boldsymbol{X}}) = \sum_\sigma \delta({\boldsymbol{z}} - {\boldsymbol{z}}_\sigma(t)), \qquad \mathfrak{f}({\boldsymbol{X}}) \doteq \sum_\sigma \delta({\boldsymbol{z}} - \overline{{\boldsymbol{z}}}_\sigma(t)), \end{equation}

where the sum is taken over individual particles, one can writeFootnote ¹⁹

(4.33)\begin{equation} \textstyle f({\boldsymbol{X}}) \approx \mathfrak{f}({\boldsymbol{X}}) - \sum_\sigma \widetilde{{\boldsymbol{z}}}_\sigma({\boldsymbol{X}}) \partial_{{\boldsymbol{z}}}\delta({\boldsymbol{z}} - \overline{{\boldsymbol{z}}}_\sigma({\boldsymbol{X}})), \end{equation}

where $\smash {{\boldsymbol {z}}_\sigma \doteq {\boldsymbol {z}}_\sigma - \overline {{\boldsymbol {z}}}_\sigma }$ are the $\smash {\widetilde {H}}$-driven small deviations from the particle unperturbed trajectories $\smash {\overline {{\boldsymbol {z}}}_\sigma }$. Then, $\smash {\overline {f} = \overline {\mathfrak {f}}({\boldsymbol {X}})}$, and the linearized perturbation $\smash {\widetilde {f} \doteq f - \overline {f}}$ is given by

(4.34)\begin{equation} \textstyle \widetilde{f}({\boldsymbol{X}}) = \underbrace{\, \mathfrak{f}({\boldsymbol{X}}) -\overline{\mathfrak{f}}({\boldsymbol{X}}) }_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{f}}} \underbrace{\textstyle - \sum_\sigma \widetilde{{\boldsymbol{z}}}_\sigma({\boldsymbol{X}}) \partial_{{\boldsymbol{z}}}\delta({\boldsymbol{z}} - \overline{{\boldsymbol{z}}}_\sigma({\boldsymbol{X}})) }_{\underline{\widetilde{f}}}. \end{equation}

By definition, the unperturbed trajectories $\smash {\overline {{\boldsymbol {z}}}_\sigma }$ satisfy $\smash {\widehat {L} \delta ({\boldsymbol {z}} - \overline {{\boldsymbol {z}}}_\sigma ({\boldsymbol {X}})) = 0}$, where $\smash {\widehat {L}}$ as in (3.14); thus,

(4.35)\begin{equation} \widehat{T}_{\tau - \tau_0} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{f}} = \mathrm{e}^{\widehat{L} (\tau - \tau_0)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{f}} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{f}}. \end{equation}

Also, $\smash {\langle \widehat {T}_{\tau - \tau _0} \underline {\widetilde {f}} \rangle _{\tau _0} = 0}$, because $\smash {\widetilde {{\boldsymbol {z}}}_\sigma }$ are oscillatory functions of ${\boldsymbol {X}}$ that is slowly evolved by $\smash {\widehat {T}_{\tau - \tau _0}}$. Hence, $\smash {g}$ is the microscopic part of the unperturbed distribution function:

(4.36)\begin{equation} g = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{f}} = \mathfrak{f}({\boldsymbol{X}}) - \overline{\mathfrak{f}}({\boldsymbol{X}}). \end{equation}

This indicates that the term $\smash {\varGamma }$ defined in (3.34) is due to collisional effects. We postpone discussing these effects until § 6, so $\smash {\varGamma }$ will be ignored for now.

4.4. Summary of § 4

The main result of this section is that the Weyl symbol of the effective Green's operator $\smash {\widehat {G}}$ can be approximated within the assumed accuracy as follows:

(4.37)\begin{equation} G({\boldsymbol{X}}, {\boldsymbol{K}}) \approx G_0(\varOmega({\boldsymbol{X}}, {\boldsymbol{K}})), \qquad \varOmega({\boldsymbol{X}}, {\boldsymbol{K}}) \doteq{-} {\boldsymbol{K}} \cdot {\boldsymbol{V}}({\boldsymbol{X}}). \end{equation}

Here, $\smash {{\boldsymbol {V}}}$ is the unperturbed velocity in the $\smash {{\boldsymbol {X}}}$ space, so $\smash {\varOmega ({\boldsymbol {X}}, {\boldsymbol {K}}) = \omega - {\boldsymbol {k}} \cdot {\boldsymbol {v}} + \mathcal {O}(\epsilon )}$, where $\smash {{\boldsymbol {v}}}$ is the unperturbed velocity in the $\smash {{\boldsymbol {x}}}$ space, and

(4.38)\begin{equation} G_0(\varOmega) = {\rm \pi}\,\delta(\varOmega) + \mathrm{i}\,\operatorname{pv}\frac{1}{\varOmega}, \qquad \operatorname{pv}\frac{1}{\varOmega} \doteq \lim_{\nu \to 0+} \frac{\varOmega}{\nu^2 + \varOmega^2}. \end{equation}

We also show that the term $\smash {\varGamma }$ defined in (3.34) is due to collisional effects. We postpone discussing these effects until § 6, so $\smash {\varGamma }$ will be ignored for now.

5.1. Expansion of the dispersion operator

The (effective) Green's operator can be represented through its symbol $G$ using (2.71):

(5.1)\begin{equation} \widehat{G} = \frac{1}{(2{\rm \pi})^N} \int \mathrm{d}{\boldsymbol{X}}\,\mathrm{d}{\boldsymbol{K}}\,\mathrm{d}{\boldsymbol{S}} {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle} + {{\boldsymbol{S}}/2} G({\boldsymbol{X}}, {\boldsymbol{K}}) {\left\langle {{\boldsymbol {\mathsf {X|}}}} \right.} - {{\boldsymbol{S}}/2} \mathrm{e}^{\mathrm{i} {\boldsymbol{K}} \cdot {\boldsymbol{S}}}. \end{equation}

The corresponding representation of $\widehat {u\,\,}^\alpha$ is even simpler, because the symbol of $\widehat {u\,\,}^\alpha$ is independent of ${\boldsymbol {K}}$:Footnote ²⁰

(5.2)\begin{equation} \widehat{u\,}^\alpha = \int \mathrm{d}{\boldsymbol{X}} {\left. {\boldsymbol {\mathsf {|X}}} \right\rangle} u^\alpha({\boldsymbol{X}}) {\left\langle {{\boldsymbol {\mathsf {X|}}}} \right.}. \end{equation}

Let us also introduce the Wigner matrix of $u^\alpha$, denoted $\smash {W_{{\boldsymbol {u}}}^{\alpha \beta }}$, and its inverse Fourier transform $\smash {C_{{\boldsymbol {u}}}^{\alpha \beta }}$ as in § 2.2.3. Using these together with (2.67), one obtains

(5.3)

Then, by taking $\smash {\text {symb}_X }$ of (

), one finds that the symbol of $\smash {\widehat {D}^{\alpha \beta }}$ is a convolution of $\smash {\overline {W}_{{\boldsymbol {u}}}^{\alpha \beta }}$ and $\smash {G}$ (Appendix B.3):

(5.4)\begin{equation} D^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}) = \int \mathrm{d}{\boldsymbol{K}}'\, \overline{W}_{{\boldsymbol{u}}}^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}') G({\boldsymbol{X}}, {\boldsymbol{K}} - {\boldsymbol{K}}'). \end{equation}

Let us Taylor-expand the symbol (5.4) in ${\boldsymbol {K}}$:

(5.5)\begin{align} D^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}) &\approx \int \mathrm{d}{\boldsymbol{K}}'\, \overline{W}_{{\boldsymbol{u}}}^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}') G({\boldsymbol{X}}, -{\boldsymbol{K}}')\nonumber\\ &\quad + K_c \int \mathrm{d}{\boldsymbol{K}}'\,\overline{W}_{{\boldsymbol{u}}}^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}') G^{|c}({\boldsymbol{X}}, -{\boldsymbol{K}}') +\mathcal{O}(K_a K_b G^{|ab}). \end{align}

As a reminder, $G^{|a}({\boldsymbol {X}}, -{\boldsymbol {K}}) = - \partial ^a G({\boldsymbol {X}}, -{\boldsymbol {K}})$ denotes the derivative of $G$ with respect to (the $a$th component of) the whole second argument, $-K_a$, and

(5.6)\begin{equation} K_a\,\frac{\partial G}{\partial K_a} = \omega\,\frac{\partial G}{\partial \omega} + k_i\,\frac{\partial G}{\partial k_i} + r^i\,\frac{\partial G}{\partial r^i}. \end{equation}

Upon application of $\smash {\text {oper}_X }$, $\omega$ gets replaced (roughly) with $\mathrm {i} \partial _t = \mathcal {O}(\epsilon )$ and $k_i$ gets replaced (also roughly) with $-\mathrm {i}\partial _i = \mathcal {O}(\epsilon )$. By (4.22), the last term in (5.6) is of order $\epsilon$ too. This means that the contribution of the whole $\smash {K_a \partial ^a G}$ term to the equation for $\smash {\overline {f}}$ is of order $\epsilon$. The standard QLT neglects this contribution entirely, i.e. adopts $\smash {D^{\alpha \beta }({\boldsymbol {X}}, {\boldsymbol {K}}) \approx D^{\alpha \beta }({\boldsymbol {X}}, {\boldsymbol {0}})}$, in which case the diffusion operator becomes just a local function of phase-space variables, $\smash {\widehat {D}^{\alpha \beta } \approx D^{\alpha \beta }({\boldsymbol {X}}, {\boldsymbol {0}})}$. In this work, we retain corrections to the first order in ${\boldsymbol {X}}$, i.e. keep the second term in (5.5) as well, while neglecting the higher-order terms as usual.

Within this model, one can rewrite (5.5) as follows:

(5.7)\begin{equation} D^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}) \approx D_0^{\alpha\beta}({\boldsymbol{X}}) + K_c \varTheta^{\alpha \beta c}({\boldsymbol{X}}). \end{equation}

Here, we used (4.19) and introduced

(5.8)\begin{gather} \displaystyle D_0^{\alpha\beta}({\boldsymbol{X}}) \doteq \int \mathrm{d}{\boldsymbol{K}}\,\overline{W}_{{\boldsymbol{u}}}^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}})G^*({\boldsymbol{X}}, {\boldsymbol{K}}), \end{gather}

(5.9)\begin{gather}\displaystyle \varTheta^{\alpha \beta c}({\boldsymbol{X}}) \doteq{-}\int \mathrm{d}{\boldsymbol{K}}\, \overline{W}_{{\boldsymbol{u}}}^{\alpha\beta}({\boldsymbol{X}}, {\boldsymbol{K}}) (G^{|c}({\boldsymbol{X}}, {\boldsymbol{K}}))^*, \end{gather}

which satisfy (Appendix B.4)

(5.10)\begin{equation} D_0^{\alpha\beta}({\boldsymbol{X}}) = (D_0^{\alpha\beta}({\boldsymbol{X}}))^*, \qquad \varTheta^{\alpha\beta c}({\boldsymbol{X}}) ={-} (\varTheta^{\alpha\beta c}({\boldsymbol{X}}))^*. \end{equation}

The first-order Weyl expansion of $\smash {\widehat {D}^{\alpha \beta }}$ is obtained by applying $\smash {\text {oper}_X }$ to (5.7). Namely, for any $\psi$, one has (cf. § 2.1.5)

(5.11)\begin{equation} \widehat{D}^{\alpha\beta}\psi \approx D_0^{\alpha\beta} \psi - \mathrm{i} \varTheta^{\alpha \beta c} \partial_c\psi - \frac{\mathrm{i}}{2}\,(\partial_c\varTheta^{\alpha \beta c})\psi. \end{equation}

What remains now is to calculate the functions $\smash {D_0^{\alpha \beta }}$ and $\smash {\varTheta ^{\alpha \beta c}}$ explicitly.

5.2. Wigner matrix of the velocity oscillations

To express $\smash {\widehat {D}^{\alpha \beta }}$ through the Wigner function $W$ of the perturbation Hamiltonian (§ 3.1.1), we need to express $\smash {W_{{\boldsymbol {u}}}^{\alpha \beta }}$ through $W$. Recall that $\smash {W_{{\boldsymbol {u}}}^{\alpha \beta }}$ is the symbol of the density operator (§ 2.2.3). By definition (3.8), one has $\smash {u^{\alpha } = \mathrm {i} J^{\alpha \mu } \widehat {q}_{\mu } \widetilde {H}}$, where $\smash {\widehat {q}_\alpha \doteq - \mathrm {i}\partial _\alpha }$ (§ 2.2.1). Then,

(5.12)\begin{equation} \widehat{W}_{{\boldsymbol{u}}}^{\alpha\beta} = (2{\rm \pi})^{{-}N} J^{\alpha \mu}\widehat{q}_{\mu}{\left. {\boldsymbol {\mathsf {|{\widetilde{H}}}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {{\widetilde{H}}|}}}} \right.}\widehat{q}_{\nu} J^{\beta \nu} = J^{\alpha\mu} J^{\beta \nu}\widehat{q}_{\mu} \widehat{W} \widehat{q}_{\nu}, \end{equation}

where $\widehat {W}$ is the density operator whose symbol is $W$. By applying $\smash {\text {symb}_X }$, one obtains

(5.13)\begin{equation} W_{{\boldsymbol{u}}}^{\alpha\beta} = J^{\alpha \mu}J^{\beta \nu} (q_{\mu} \,\bigstar \,W \,\bigstar\, q_{\nu}), \end{equation}

where $\bigstar$ is the Moyal product (2.72). Using formulas analogous to (2.33) in the $\smash {({\boldsymbol {X}}, {\boldsymbol {K}})}$ space, one obtains

(5.14)\begin{align} q_{\mu} \,\bigstar\, W \,\bigstar \,q_{\nu} & = \left(q_{\mu} W -\frac{\mathrm{i}}{2} \frac{\partial W}{\partial z^{\mu}} \right) \,\bigstar \,q_{\nu} \nonumber\\ & = q_{\mu}q_{\nu} W - \frac{\mathrm{i}}{2}\,q_{\nu}\,\frac{\partial W}{\partial z^{\mu}} + \frac{\mathrm{i}}{2} \frac{\partial}{\partial z^{\nu}} \left(q_{\mu} W_h - \frac{\mathrm{i}}{2}\frac{\partial W}{\partial z^{\mu}}\right) \nonumber\\ & = q_{\mu}q_{\nu} W_h + \frac{\mathrm{i}}{2} \left(q_{\mu}\frac{\partial W}{\partial z^{\nu}} - q_{\nu}\frac{\partial W}{\partial z^{\mu}}\right) + \frac{1}{4}\frac{\partial^2 W}{\partial z^{\mu}\partial z^{\nu}}. \end{align}

Hence, $W_{{\boldsymbol {u}}}^{\alpha \beta }$ and $W$ are connected via the following exact formula:

(5.15)\begin{equation} W_{{\boldsymbol{u}}}^{\alpha\beta} = J^{\alpha \mu}J^{\beta \nu} \left( q_{\mu} q_{\nu} W - \frac{\mathrm{i}}{2} \left( q_{\nu}\,\frac{\partial W}{\partial z^{\mu}} -q_{\mu}\,\frac{\partial W}{\partial z^{\nu}} \right) + \frac{1}{4} \frac{\partial^2 W}{\partial z^{\mu} \partial z^{\nu}} \right). \end{equation}

5.3. Nonlinear potentials

Due to (5.10), one has $D_0^{\alpha \beta } = \operatorname {re} D_0^{\alpha \beta }$. Using this together with (5.8), (5.15), (4.29), and (4.23), one obtains

(5.16)\begin{align} D_0^{\alpha\beta} = J^{\alpha \mu}J^{\beta \nu} \operatorname{re} &\int \mathrm{d}{\boldsymbol{K}} \left({\rm \pi}\,\delta(\varOmega) - \mathrm{i}\, \operatorname{pv}\frac{1}{\varOmega}\right). \nonumber\\ & \qquad \times\left( q_{\mu} q_{\nu} \overline{W} - \frac{\mathrm{i}}{2}\left( q_{\nu}\,\frac{\partial\overline{W}}{\partial z^{\mu}} - q_{\mu}\,\frac{\partial\overline{W}}{\partial z^{\nu}} \right) + \frac{1}{4} \frac{\partial^2\overline{W}}{\partial z^{\mu}\partial z^{\nu}} \right), \end{align}

with notation as in (2.10). This can be written as $\smash {D_0^{\alpha \beta } = {\mathsf {D}}^{\alpha \beta } + \varrho ^{\alpha \beta } + \varsigma ^{\alpha \beta }}$, where

(5.17)\begin{equation} {\mathsf{D}}^{\alpha\beta} \doteq J^{\alpha\mu}J^{\beta\nu} \int \mathrm{d}{\boldsymbol{K}}\, {\rm \pi}\,\delta(\varOmega)\, q_{\mu}q_{\nu}\overline{W}, \end{equation}

and we also introduced

(5.18)\begin{gather} \displaystyle \varrho^{\alpha\beta} \doteq{-} \frac{1}{2}\,J^{\alpha\mu}J^{\beta\nu} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}} \left( q_{\nu}\,\frac{\partial \overline{W}}{\partial z^{\mu}} - q_{\mu}\,\frac{\partial \overline{W}}{\partial z^{\nu}} \right) \frac{1}{\varOmega}, \end{gather}

(5.19)\begin{gather}\displaystyle \varsigma^{\alpha\beta} \doteq \frac{1}{4}\, J^{\alpha\mu} J^{\beta\nu} \int \mathrm{d}{\boldsymbol{K}}\,{\rm \pi}\,\delta(\varOmega)\,\frac{\partial^2 \overline{W}}{\partial z^{\mu} \partial z^{\nu}}. \end{gather}

As shown in Appendix B.5, the contributions of these two functions to (3.33) are

(5.20)\begin{equation} \frac{\partial}{\partial z^{\alpha}} \left(\varrho^{\alpha\beta}\, \frac{\partial\overline{f}}{\partial z^{\beta}} \right) = \mathcal{O}(\epsilon\varepsilon^2), \qquad \frac{\partial}{\partial z^{\alpha}} \left( \varsigma^{\alpha\beta}\,\frac{\partial\overline{f}}{\partial z^{\beta}} \right) = \mathcal{O}(\epsilon^2\varepsilon^2). \end{equation}

Thus, $\smash {\varrho ^{\alpha \beta }}$ must be retained and $\smash {\varsigma ^{\alpha \beta }}$ must be neglected, which leads to

(5.21)\begin{equation} D_0^{\alpha\beta} \approx {\mathsf{D}}^{\alpha\beta} + \varrho^{\alpha\beta}. \end{equation}

The function $\smash {\varTheta ^{\alpha \beta c} = \mathrm {i}\, \operatorname {im} \varTheta ^{\alpha \beta c}}$ can be written as follows:

\[\varTheta^{\alpha \beta c}({\boldsymbol{X}}) = \mathrm{i} J^{\alpha \mu}J^{\beta \nu} \int \mathrm{d}{\boldsymbol{K}}\,q_{\mu}q_{\nu}\overline{W}({\boldsymbol{X}}, {\boldsymbol{K}})\,\frac{\partial}{\partial K_c}\,\operatorname{pv} \frac{1}{\varOmega ({\boldsymbol{X}}, {\boldsymbol{K}})} ={-}\mathrm{i} V^c({\boldsymbol{X}})\, \Theta^{\alpha\beta}({\boldsymbol{X}}), \]

where we introduced

(5.22)\begin{equation} \Theta^{\alpha\beta} \doteq J^{\alpha \mu}J^{\beta \nu} \frac{\eth}{\partial \varOmega} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}}\,\frac{q_{\mu}q_{\nu}\overline{W}}{\varOmega} \end{equation}

and $\smash {\eth }$ is defined as in (4.27). Then finally, one can rewrite (5.11) as follows:

(5.23)\begin{align} \widehat{D}^{\alpha\beta}\psi & \approx ({\mathsf{D}}^{\alpha\beta} + \varrho^{\alpha\beta})\psi -\Theta^{\alpha\beta} V^c \partial_c\psi - \frac{1}{2}\,V^c(\partial_c\Theta^{\alpha\beta})\psi \nonumber\\ & = ({\mathsf{D}}^{\alpha\beta} + \varrho^{\alpha\beta})\psi - \Theta^{\alpha\beta} (\partial_t + v^{\lambda}\partial_{\lambda})\psi -\frac{1}{2}\,((\partial_t+v^{\lambda}\partial_{\lambda})\Theta^{\alpha\beta})\psi, \end{align}

where we used (3.15). With some algebra (Appendix B.6), and assuming the notation

(5.24)\begin{equation} \varPhi ={-} J^{\mu \nu}\,\frac{\partial}{\partial z^{\mu}} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}}\,\frac{q_{\nu}\overline{W}}{2\varOmega}, \end{equation}

one finds that (5.23) leads to

(5.25)\begin{equation} \partial_{\alpha}(\widehat{D}^{\alpha\beta}\partial_{\beta}\overline{f}) = \partial_{\alpha}({\mathsf{D}}^{\alpha\beta}\partial_{\beta}\overline{f}) - \frac{1}{2}\,\mathrm{d}_t\partial_{\alpha}(\Theta^{\alpha\beta} \partial_{\beta}\overline{f}) + \lbrace \varPhi, \overline{f} \rbrace. \end{equation}

Hence, (3.33) becomes (to the extent that $\varGamma$ is negligible; see § 6.7)

(5.26)\begin{equation} \mathrm{d}_t \overline{f} + \frac{1}{2}\,\mathrm{d}_t\partial_{\alpha}(\Theta^{\alpha\beta} \partial_{\beta}\overline{f}) - \lbrace \varPhi, \overline{f} \rbrace = \partial_{\alpha}({\mathsf{D}}^{\alpha \beta} \partial_{\beta} \overline{f}). \end{equation}

The functions $\smash {\Theta ^{\alpha \beta }}$, $\smash {\varPhi }$ and $\smash {{\mathsf {D}}^{\alpha \beta }}$ that determine the coefficients in this equation are fundamental and, for the lack of a better term, will be called nonlinear potentials.

5.4. Oscillation-centre distribution

Let us introduce

(5.27)\begin{equation} F \doteq \overline{f} + \frac{1}{2}\,\partial_{\alpha}(\Theta^{\alpha\beta} \partial_{\beta}\overline{f}). \end{equation}

Then, using (5.25), one can rewrite (5.26) asFootnote ²¹

(5.28)\begin{equation} \partial_t F - \lbrace \mathcal{H}, F \rbrace = \partial_{\alpha}({\mathsf{D}}^{\alpha \beta} \partial_{\beta} F), \end{equation}

where corrections $\mathcal {O}(\varepsilon ^4)$ have been neglected and we introduced $\smash {\mathcal {H} \doteq \overline {H} + \varPhi }$. As a reminder, the nonlinear potentials in (5.28) are as follows:

(5.29)\begin{align} {\mathsf{D}}^{\alpha\beta} &= J^{\alpha\mu}J^{\beta\nu} \int \mathrm{d}{\boldsymbol{K}}\, {\rm \pi}\,\delta(\varOmega)\, q_{\mu}q_{\nu}\overline{W}, \end{align}

(5.30)\begin{align} \Theta^{\alpha\beta} &= J^{\alpha \mu}J^{\beta \nu} \frac{\eth}{\partial \varOmega} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}}\,\frac{q_{\mu}q_{\nu}\overline{W}}{\varOmega}, \end{align}

(5.31)\begin{align} \varPhi & ={-} J^{\mu \nu}\,\frac{\partial}{\partial z^{\mu}} {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{K}}\,\frac{q_{\nu}\overline{W}}{2\varOmega}. \end{align}

Equations (5.27)–(5.31) form a closed model that describes the evolution of the average distribution $\smash {\overline {f}}$ in turbulence with prescribed $\overline {W}$. In particular, (5.28) can be interpreted as a Liouville-type equation for $F$ as an effective, or ‘dressed’, distribution. The latter can be understood as the distribution of ‘dressed’ particles called OCs. Then, $\mathcal {H}$ serves as the OC Hamiltonian, $\smash {{\mathsf {D}}^{\alpha \beta }}$ is the phase-space diffusion coefficient, $\smash {\varPhi }$ is the ponderomotive energy, $\smash {\varOmega = \omega - q_\alpha v^\alpha }$ and $\smash {v^\alpha \doteq J^{\alpha \beta }\partial _\beta \overline {H}}$. Within the assumed accuracy, one can redefine $\smash {v^\alpha }$ to be the OC velocity rather than the particle velocity; specifically,Footnote ²²

(5.32)\begin{equation} v^\alpha \doteq J^{\alpha\beta}\partial_\beta\mathcal{H} = J^{\alpha\beta}\partial_\beta\overline{H} + \mathcal{O}(\epsilon^2). \end{equation}

Then, the presence of $\smash {\delta (\varOmega )}$ in (5.29) signifies that OCs diffuse in phase space in response to waves they are resonant with. Below, we use the terms ‘OCs’ and ‘particles’ interchangeably except where specified otherwise.

That said, the interpretation of OCs as particle-like objects is limited. Single-OC motion equations are not introduced in our approach. (They would have been singular for resonant interactions.) Accordingly, the transformation (5.27) of the distribution function $\smash {\overline {f} \mapsto F}$ is not derived from a coordinate transformation but rather is fundamental. As a result, particles and OCs live in the same phase space, but the ‘dynamics of OCs’ can be irreversible (§ 5.5). This qualitatively distinguishes our approach from the traditional OC theory (Dewar Reference Dewar1973) and from the conceptually similar gyrokinetic theory (Littlejohn Reference Littlejohn1981; Cary & Brizard Reference Cary and Brizard2009), where coordinate transformations are central.

5.5. ${H}$-theorem

Because $\smash {\overline {W}}$ is non-negative (§ 2.1.6), $\smash {{\mathsf {D}}^{\alpha \beta }}$ is positive-semidefinite; that is,

(5.33)\begin{equation} {\mathsf{D}}^{\alpha\beta} \psi_\alpha \psi_\beta = \int \mathrm{d}{\boldsymbol{K}}\, {\rm \pi}\,\delta(\varOmega)\, a^2 \overline{W} \geqslant 0, \qquad a \doteq J^{\alpha\mu} \psi_\alpha q_{\mu} \end{equation}

for any real $\smash {\psi }$. This leads to the following theorem. Consider the OC entropy defined as

(5.34)\begin{equation} \mathscr{S} \doteq{-} \int \mathrm{d}{\boldsymbol{z}}\,F(t, {\boldsymbol{z}})\ln F(t, {\boldsymbol{z}}). \end{equation}

According to (5.28), $\smash {\mathscr {S}}$ satisfies

(5.35)\begin{align} \frac{\mathrm{d}\mathscr{S}}{\mathrm{d} t} & ={-} \int \mathrm{d}{\boldsymbol{z}}\,\frac{\mathrm{d}(F \ln F)}{\mathrm{d} F}\left( \lbrace \mathcal{H}, F \rbrace + \partial_{\alpha}({\mathsf{D}}^{\alpha \beta} \partial_{\beta} F) \right)\nonumber\\ & ={-} \int \mathrm{d}{\boldsymbol{z}}\,\frac{\mathrm{d}(F \ln F)}{\mathrm{d} F}\,J^{\alpha\beta}(\partial_\alpha \mathcal{H})(\partial_\beta F) - \int \mathrm{d}{\boldsymbol{z}}\,(1 + \ln F)\,\partial_{\alpha}({\mathsf{D}}^{\alpha \beta} \partial_{\beta} F) \nonumber\\ & ={-} \int \mathrm{d}{\boldsymbol{z}}\,J^{\alpha\beta}(\partial_\alpha \mathcal{H}) \partial_\beta(F \ln F) - \int \mathrm{d}{\boldsymbol{z}}\,\ln F\,\partial_{\alpha}({\mathsf{D}}^{\alpha \beta} \partial_{\beta} F) \nonumber\\ & = \int \mathrm{d}{\boldsymbol{z}}\,(J^{\alpha\beta} \partial^2_{\alpha\beta} \mathcal{H})\,F \ln F + \int \mathrm{d}{\boldsymbol{z}}\,{\mathsf{D}}^{\alpha \beta} (\partial_\alpha\ln F) (\partial_{\beta} \ln F) F. \end{align}

The first integral vanishes due to $\smash {J^{\alpha \beta } \partial ^2_{\alpha \beta } = 0}$. The second integral is non-negative due to (5.33). Thus,

(5.36)\begin{equation} \frac{\mathrm{d}\mathscr{S}}{\mathrm{d} t} \geqslant 0, \end{equation}

which is recognized as the $\smash {H}$-theorem (Lifshitz & Pitaevskii Reference Lifshitz and Pitaevskii1981, § 4) for QL OC dynamics.

5.6. Summary of § 5

From now on, we assume that the right-hand side of (5.28) scales not as $\smash {\mathcal {O}(\varepsilon ^2)}$ but as $\smash {\mathcal {O}(\epsilon \varepsilon ^2)}$, either due to the scarcity of resonant particles or, for QL diffusion driven by microscopic fluctuations (§ 6), due to the plasma parameter's being large. Also, the spatial derivatives can be neglected within the assumed accuracy in the definition of $F$ (5.27) and on the right-hand side of (5.28). Using this together with (2.69), and with (2.56) for the Poisson bracket, our results can be summarized as follows.

QL evolution of a particle distribution in a prescribed wave field is governed byFootnote ²³

(5.37)\begin{equation} \frac{\partial F}{\partial t} - \frac{\partial \mathcal{H}}{\partial {\boldsymbol{x}}} \cdot \frac{\partial F}{\partial {\boldsymbol{p}}} + \frac{\partial \mathcal{H}}{\partial {\boldsymbol{p}}} \cdot \frac{\partial F}{\partial {\boldsymbol{x}}} = \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol {\mathsf{D}}}\,\frac{\partial F}{\partial {\boldsymbol{p}}}\right). \end{equation}

The OC distribution $\smash {F}$ is defined as

(5.38)\begin{equation} F = \overline{f} + \frac{1}{2}\,\frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol{\Theta}}\, \frac{\partial \overline{f}}{\partial {\boldsymbol{p}}}\right), \end{equation}

so the density of OCs is the same as the locally averaged density of the true particles:

(5.39)\begin{equation} \mathcal{N} \doteq \int \mathrm{d}{\boldsymbol{p}}\,F = \int \mathrm{d}{\boldsymbol{p}}\,\overline{f}. \end{equation}

The function $\smash {\mathcal {H}}$ is understood as the OC Hamiltonian. It is given by

(5.40)\begin{equation} \mathcal{H} \doteq \overline{H} + \varPhi, \end{equation}

where $\smash {\overline {H}}$ is the average Hamiltonian, which may include interaction with background fields, and $\smash {\varPhi }$ is the ponderomotive potential. The nonlinear potentials that enter (5.37) can be calculated to the zeroth order in $\epsilon$ and are given byFootnote ²⁴

(5.41)\begin{align} {{\boldsymbol {\mathsf{D}}}} &= \int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, {\rm \pi}\,{\boldsymbol{k}}{\boldsymbol{k}} \overline{{\mathsf{W}}}(t, {\boldsymbol{k}} \cdot {\boldsymbol{v}}, {\boldsymbol{k}}; {\boldsymbol{p}}) , \end{align}

(5.42)\begin{align} {\boldsymbol{\Theta}} &=\frac{\partial}{\partial \vartheta} {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, \left. \frac{{\boldsymbol{k}} {\boldsymbol{k}} \overline{{\mathsf{W}}}}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}} + \vartheta} \right|_{\vartheta=0}, \end{align}

(5.43)\begin{align} \varPhi &= \frac{1}{2}\frac{\partial}{\partial {\boldsymbol{p}}} \cdot {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\frac{{\boldsymbol{k}} \overline{{\mathsf{W}}}}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}}, \end{align}

where $\smash {{\boldsymbol {k}} {\boldsymbol {k}}}$ is a dyadic matrix with two lower indices, and the same conventions apply as in § 2.1.2. Also, $\smash {{\boldsymbol {v}}}$ is hereby redefined as the OC spatial velocity, namely,

(5.44)\begin{equation} {\boldsymbol{v}} \doteq \partial_{\boldsymbol{p}}\mathcal{H} = \partial_{\boldsymbol{p}}\overline{H} + \mathcal{O}(\epsilon^2). \end{equation}

The function $\smash {\overline {{\mathsf {W}}}}$ is defined as

(5.45)\begin{equation} \overline{{\mathsf{W}}}(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}}) \doteq \int \mathrm{d}{\boldsymbol{r}}\,\overline{W}(t, {\boldsymbol{x}}, {\boldsymbol{p}}, \omega, {\boldsymbol{k}}, {\boldsymbol{r}}), \end{equation}

where $\overline {W}$ is the average Wigner function (3.4) of the perturbation Hamiltonian, i.e. the spectrum of its symmetrized autocorrelation function (3.5). Due to (2.78), it can be understood as the average of $\smash {{\mathsf {W}} \doteq \text {symb}_{{\mathsf {x}}} \widehat {{\mathsf {W}}}}$ (where $\smash {\widehat {{\mathsf {W}}}}$ is defined in (3.3)), i.e. as the Wigner function of the perturbation Hamiltonian with $\smash {{\boldsymbol {p}}}$ treated as a parameter. As such, $\smash {\overline {{\mathsf {W}}}}$ is non-negative, so $\smash {{{\boldsymbol {\mathsf {D}}}}}$ is positive-semidefinite. This leads to an $\smash {H}$-theorem (proven similarly to (5.36)) for the entropy density $\smash {\sigma \doteq - \int \mathrm {d}{\boldsymbol {p}}\,F\ln F}$:

(5.46)\begin{equation} \left(\frac{\mathrm{d}\sigma}{\mathrm{d} t}\right)_{{\mathsf{D}}} \geqslant 0, \qquad \left(\frac{\partial \psi}{\partial t}\right)_{{\mathsf{D}}} \doteq \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol {\mathsf{D}}}\,\frac{\partial \psi}{\partial {\boldsymbol{p}}}\right). \end{equation}

Also note that for homogeneous turbulence in particular, where $\smash {\overline {{\mathsf {W}}}}$ is independent of $\smash {{\boldsymbol {x}}}$, (2.46) yields that

(5.47)\begin{equation} \int \mathrm{d}\omega \,\overline{{\mathsf{W}}}(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}}) = \frac{1}{\mathscr{V}_n} \int \mathrm{d}\omega \,\mathrm{d}{\boldsymbol{x}}\,\overline{{\mathsf{W}}}(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}}) = \frac{1}{\mathscr{V}_n}\,\overline{|\mathring{\widetilde{H}}(t, {\boldsymbol{k}}, {\boldsymbol{p}})|^2}, \end{equation}

where $\smash {\mathscr {V}_n}$ is the plasma volume (the index $\smash {n}$ denotes the number of spatial dimensions) and $\smash {\mathring {\widetilde {H}}}$ is the spatial spectrum of $\smash {\widetilde {H}}$ as defined in (2.23).

Equation (5.37) can be used to calculate the ponderomotive force $\smash {\partial _t\int \mathrm {d}{\boldsymbol {p}}\,\overline {f}}$ that a given wave field imparts on a plasma. This potentially resolves the controversies mentioned in Kentwell & Jones (Reference Kentwell and Jones1987). We will revisit this subject for on-shell waves in § 7.5.

6.1. Interaction model

Let us assume that particles interact via an $M$-component real field ${\boldsymbol {\varPsi }} \equiv (\varPsi ^1, \varPsi ^2, \ldots, \varPsi ^M)^\intercal$. It is treated below as a column vector; hence the index $\smash {^\intercal }$. (A complex field can be accommodated by considering its real and imaginary parts as separate components.) We split this field into the average part $\smash {\overline {{\boldsymbol {\varPsi }}}}$ and the oscillating part $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$. The former is considered given. For the latter, we assume the action integral of this field without plasma in the form

(6.1)\begin{equation} S_0 = \int \mathrm{d}{\boldsymbol {\mathsf{x}}}\,\mathfrak{L}_0, \qquad \mathfrak{L}_0 = \frac{1}{2}\,\smash{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}}\widehat{\boldsymbol{\varXi}}_0\widetilde{{\boldsymbol{\varPsi}}} \end{equation}

(see § 9 for examples), where $\smash {\widehat {\boldsymbol {\varXi }}_0}$ is a Hermitian operatorFootnote ²⁵ and $\smash {\smash {\widetilde {{\boldsymbol {\varPsi }}}}^{{\dagger}} = \smash {\widetilde {{\boldsymbol {\varPsi }}}}^\intercal }$ is a row vector dual to $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$. Plasma is allowed to consist of multiple species, henceforth denoted with index $\smash {s}$. Because $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ is assumed small, the generic Hamiltonian for each species $\smash {s}$ can be Taylor-expanded in $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ and represented in a generic form

(6.2)\begin{equation} H_s(t, {\boldsymbol{x}}, {\boldsymbol{p}}) \approx H_{0s} + \widehat{\boldsymbol{\alpha}}_s^{{\dagger}} \widetilde{{\boldsymbol{\varPsi}}} + \frac{1}{2}\,(\widehat{\boldsymbol{L}}_s\widetilde{{\boldsymbol{\varPsi}}})^{{\dagger}} (\widehat{\boldsymbol{R}}_s\widetilde{{\boldsymbol{\varPsi}}}) \end{equation}

(see § 9 for examples), which can be considered as a second-order Taylor expansion of the full Hamiltonian in $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$. Here, $\smash {H_{0s} \equiv H_{0s}(t, {\boldsymbol {x}}, {\boldsymbol {p}})}$ is independent of $\smash {\widetilde {{\boldsymbol {\varPsi }}} \equiv \widetilde {{\boldsymbol {\varPsi }}}(t, {\boldsymbol {x}})}$, $\smash {\widehat {\boldsymbol {\alpha }}_s \equiv (\widehat {\alpha }_{s,1}, \widehat {\alpha }_{s,2}, \ldots, \widehat {\alpha }_{s,M})^\intercal }$ is a column vector whose elements $\smash {\widehat {\alpha }_{s,i}}$ are linear operators on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$. The dagger is added so that $\smash {\widehat {\boldsymbol {\alpha }}_s^{{\dagger}} }$ could be understood as a row vector whose elements $\smash {\widehat {\alpha }_{s,i}^{{\dagger}} }$ act on the individual components of the field; i.e. $\smash {\widehat {\boldsymbol {\alpha }}_s^{{\dagger}} \widetilde {{\boldsymbol {\varPsi }}} \equiv \widehat {\alpha }^{{\dagger}} _{s,i}\widetilde {\varPsi }^i}$. We let $\smash {\widehat {\boldsymbol {\alpha }}_s}$ be non-local in $t$ and ${\boldsymbol {x}}$ (for example, $\smash {\widehat {\boldsymbol {\alpha }}_s}$ can be a spacetime derivative or a spacetime integral), and we also let $\smash {\widehat {\boldsymbol {\alpha }}_s}$ depend on $\smash {{\boldsymbol {p}}}$ parametrically so

(6.3)\begin{equation} \text{symb}\, \widehat{\boldsymbol{\alpha}}_s = {\boldsymbol{\alpha}}_s(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}}). \end{equation}

The matrix operators $\smash {\widehat {\boldsymbol {L}}_s}$ and $\smash {\widehat {\boldsymbol {R}}_s}$ and their symbols $\smash {{\boldsymbol {L}}_s}$ and $\smash {{\boldsymbol {R}}_s}$ are understood similarly.

The Lagrangian density of the oscillating-field–plasma system is

(6.4)\begin{equation} \mathfrak{L}_p = \frac{1}{2}\,\smash{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}}\widehat{\boldsymbol{\varXi}}_0 \widetilde{{\boldsymbol{\varPsi}}} + \sum_s \sum_{\sigma_s} ({\boldsymbol{p}}_{\sigma_s} \cdot \dot{{\boldsymbol{x}}}_{\sigma_s} - H_s(t, {\boldsymbol{x}}_{\sigma_s}, {\boldsymbol{p}}_{\sigma_s})) \delta({\boldsymbol{x}} - {\boldsymbol{x}}_{\sigma_s}(t)), \end{equation}

where the sum is taken over individual particles. Note that

(6.5)\begin{align} \sum_{\sigma_s} H_s(t, {\boldsymbol{x}}_{\sigma_s}, {\boldsymbol{p}}_{\sigma_s}) \delta({\boldsymbol{x}} - {\boldsymbol{x}}_{\sigma_s}(t)) & = \int \mathrm{d}{\boldsymbol{p}}\, \sum_{\sigma_s} \delta({\boldsymbol{z}} - {\boldsymbol{z}}_{\sigma_s}(t)) H_s(t, {\boldsymbol{x}}, {\boldsymbol{p}}) \nonumber\\ & = \int \mathrm{d}{\boldsymbol{p}}\, f_s(t, {\boldsymbol{x}}, {\boldsymbol{p}}) H_s(t, {\boldsymbol{x}}, {\boldsymbol{p}}), \end{align}

so the $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$-dependent part of the system action can be written as $\smash {S = \int \mathrm {d}{\boldsymbol {\mathsf {x}}}\,\mathfrak {L}}$ with

(6.6)\begin{gather} \displaystyle \mathfrak{L} = \frac{1}{2}\,\smash{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}} \widehat{\boldsymbol{\varXi}}_p \widetilde{{\boldsymbol{\varPsi}}} - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widetilde{f}_s\widehat{\boldsymbol{\alpha}}_s^{{{\dagger}}}\widetilde{{\boldsymbol{\varPsi}}}, \end{gather}

(6.7)\begin{gather}\displaystyle \widehat{\boldsymbol{\varXi}}_p \doteq \widehat{\boldsymbol{\varXi}}_0 - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\smash{\widehat{\boldsymbol{L}}}_s^{{\dagger}} f_s \widehat{\boldsymbol{R}}_s. \end{gather}

(The contribution of $\smash {\widetilde {f}_s}$ to the second term in (6.6) has been omitted because it averages to zero at integration over spacetime and thus does not contribute to $\smash {S}$.) This ‘abridged’ action is not sufficient to describe the particle motion, but it is sufficient to describe the dynamics of $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ at given $\smash {f_s}$, as discussed below. The operator $\smash {\widehat {\boldsymbol {\varXi }}_p}$ can be considered Hermitian without loss of generality, because its anti-Hermitian part does not contribute to $\smash {S}$. Also, we assume that unless either of $\smash {\widehat {\boldsymbol {L}}}$ and $\smash {\widehat {\boldsymbol {R}}}$ is zero, the high-frequency field has no three-wave resonances, so terms cubic in $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ can be neglected in $\smash {S}$;Footnote ²⁶ then,

(6.8)\begin{equation} \widehat{\boldsymbol{\varXi}}_p \approx \widehat{\boldsymbol{\varXi}}_0 - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,(\smash{\widehat{\boldsymbol{L}}}_s^{{\dagger}} F_s \widehat{\boldsymbol{R}}_s)_\text{H}. \end{equation}

Using the same assumption, one can also adopt

(6.9)\begin{equation} \overline{H}_s = H_{0s} + \frac{1}{2}\,\overline{(\widehat{\boldsymbol{L}}_s\widetilde{{\boldsymbol{\varPsi}}})^{{\dagger}} (\widehat{\boldsymbol{R}}_s\widetilde{{\boldsymbol{\varPsi}}})}, \qquad \widetilde{H}_s \approx \widehat{\boldsymbol{\alpha}}_s^{{\dagger}} \widetilde{{\boldsymbol{\varPsi}}}, \end{equation}

because in the absence of three-wave resonances, the oscillating part of $\smash {(\widehat {\boldsymbol {L}}_s\widetilde {{\boldsymbol {\varPsi }}})^{{\dagger}} (\widehat {\boldsymbol {R}}_s\widetilde {{\boldsymbol {\varPsi }}})}$ contributes only $\smash {\mathcal {O}(\varepsilon ^4)}$ terms to the equation for $\smash {F_s}$.

6.2. Field equations

The Euler–Lagrange equation for $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ derived from (6.6) is

(6.10)\begin{equation} \widehat{\boldsymbol{\varXi}}_p \widetilde{{\boldsymbol{\varPsi}}} = \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s \widetilde{f}_s. \end{equation}

Then, to the extent that the linear approximation for $\smash {\tilde {f}_s}$ is sufficient (see below), one finds that the oscillating part of the field satisfies

(6.11)\begin{equation} \widehat{\boldsymbol{\varXi}}_p \widetilde{{\boldsymbol{\varPsi}}} - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s \widehat{G}_s \lbrace \widehat{\boldsymbol{\alpha}}_s^{{\dagger}} \widetilde{{\boldsymbol{\varPsi}}}, \overline{f}_s \rbrace = \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s g_s, \end{equation}

where we used (3.24). Note that the right-hand side of (6.11) is determined by microscopic fluctuations $g_s(t, {\boldsymbol {x}}, {\boldsymbol {p}})$ (§ 4.3). Equation (6.11) can also be expressed as

(6.12)\begin{equation} \widehat{\boldsymbol{\varXi}}\widetilde{{\boldsymbol{\varPsi}}} = \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s g_s, \end{equation}

where $\smash {\widehat {\boldsymbol {\varXi }}}$ is understood as the plasma dispersion operator and is given by

(6.13)\begin{equation} \widehat{\boldsymbol{\varXi}} \doteq \widehat{\boldsymbol{\varXi}}_p - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s \widehat{G}_s \lbrace \widehat{\boldsymbol{\alpha}}_s^{{\dagger}} \,{\unicode{x25AA}}\,, \overline{f}_s \rbrace, \end{equation}

where $\smash {{\unicode{x25AA}} }$ is a placeholder. The general solution of (6.12) can be written as

(6.14)\begin{equation} \widetilde{{\boldsymbol{\varPsi}}} = \underline{\widetilde{{\boldsymbol{\varPsi}}}} + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}}, \qquad \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}} = \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\smash{\widehat{\boldsymbol{\varXi}}}^{{-}1}\widehat{\boldsymbol{\alpha}}_s g_s. \end{equation}

Here, $\smash {\smash {\widehat {\boldsymbol {\varXi }}}^{-1}}$ is the right inverse of $\smash {\widehat {\boldsymbol {\varXi }}}$ (specifically, $\smash {\widehat {\boldsymbol {\varXi }} \smash {\widehat {\boldsymbol {\varXi }}}^{-1} = \widehat {\boldsymbol {1}}}$ yet $\smash {\smash {\widehat {\boldsymbol {\varXi }}}^{-1}\widehat {\boldsymbol {\varXi }} \ne \widehat {\boldsymbol {1}}}$) such that $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}$ vanishes at zero $g$.Footnote ²⁷ The rest of the solution, $\smash {\underline {\widetilde {{\boldsymbol {\varPsi }}}}}$, is the macroscopic field that satisfies

(6.15)\begin{equation} \widehat{\boldsymbol{\varXi}}\underline{\widetilde{{\boldsymbol{\varPsi}}}} = {\boldsymbol{0}}. \end{equation}

In the special case when the dispersion operator is Hermitian ($\smash {\widehat {\boldsymbol {\varXi }} = \widehat {\boldsymbol {\varXi }}_\text {H}}$), (6.15) also flows from the ‘adiabatic’ macroscopic part of the action $\smash {S}$, namely,

(6.16)\begin{equation} S_{\text{ad}} \doteq \frac{1}{2}\int \mathrm{d}{\boldsymbol {\mathsf{x}}}\,\smash{\underline{\widetilde{{\boldsymbol{\varPsi}}}}}^{{\dagger}}\widehat{\boldsymbol{\varXi}}_\text{H}\underline{\widetilde{{\boldsymbol{\varPsi}}}}. \end{equation}

Because we have assumed a linear model for $\smash {f_s}$ in (6.11), $\smash {\underline {\widetilde {{\boldsymbol {\varPsi }}}}}$ is decoupled from $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}$, and hence the dynamics of $\smash {\underline {\widetilde {{\boldsymbol {\varPsi }}}}}$ turns out to be collisionless. This is justified, because collisional dissipation is assumed to be much slower that collisionless dissipation (§ 3.3). One can reinstate collisions in (6.15) by modifying $\smash {\widehat {G}_s}$ ad hoc, if necessary. Alternatively, one can avoid separating $\smash {\underline {\widetilde {{\boldsymbol {\varPsi }}}}}$ and $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}$ and, instead, derive an equation for the average Wigner matrix of the whole $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ (McDonald Reference McDonald1991). However, this approach is beyond QLT, so it is not considered in this paper.

6.3. Dispersion matrix

As readily seen from the definition (6.13), the operator $\smash {\widehat {\boldsymbol {\varXi }}}$ can be expressed as

(6.17)\begin{equation} \widehat{\boldsymbol{\varXi}} = \widehat{\boldsymbol{\varXi}}_p - \mathrm{i}\widehat{k}_j \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\widehat{\boldsymbol{\alpha}}_s \widehat{G}_s \widehat{\boldsymbol{\alpha}}_s^{{\dagger}} \,\frac{\partial F_s}{\partial p_j} + \mathcal{O}(\epsilon, \varepsilon^2). \end{equation}

The corrections caused by non-zero $\epsilon$ and $\varepsilon$ in this formula will be insignificant for our purposes, so they will be neglected. In particular, this means that $\smash {G_s \doteq \text {symb}_X \widehat {G}_s}$ can be adopted in the form independent of $\smash {{\boldsymbol {r}}}$ (§ 4.2):

(6.18)\begin{equation} G_s \approx \frac{\mathrm{i}}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s + \mathrm{i} 0} = {\rm \pi}\,\delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s) + \mathrm{i}\,\operatorname{pv}\frac{1}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s}. \end{equation}

Then, $\smash {\widehat {G}_s}$ can be considered as an operator on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$ with ${\boldsymbol {p}}$ as a parameter, and $\smash {\text{symb}\, \widehat {G}_s = G_s(t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}}; {\boldsymbol {p}})}$. Also,

(6.19)\begin{equation} \text{symb}\,(\widehat{\boldsymbol{\alpha}}_s \widehat{G}_s \widehat{\boldsymbol{\alpha}}_s^{{\dagger}}) = {\boldsymbol{\alpha}}_s \star G_s \star {\boldsymbol{\alpha}}_s^{{\dagger}} \approx {\boldsymbol{\alpha}}_s G_s {\boldsymbol{\alpha}}_s^{{\dagger}}. \end{equation}

This readily yields the ‘dispersion matrix’ $\smash {{\boldsymbol {\varXi }} \doteq \text {symb}_{{\mathsf {x}}} \widehat {\boldsymbol {\varXi }}}$:

(6.20)\begin{gather} \displaystyle {\boldsymbol{\varXi}}(\omega, {\boldsymbol{k}}) \approx {\boldsymbol{\varXi}}_p(\omega, {\boldsymbol{k}}) + \sum_s \int \mathrm{d}{\boldsymbol{p}}\, \frac{{\boldsymbol{\alpha}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}){\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s + \mathrm{i} 0}\, {\boldsymbol{k}} \cdot \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}, \end{gather}

(6.21)\begin{gather}\displaystyle {\boldsymbol{\varXi}}_p(\omega, {\boldsymbol{k}}) \approx {\boldsymbol{\varXi}}_0(\omega, {\boldsymbol{k}}) - \sum_s \int \mathrm{d}{\boldsymbol{p}}\,{\boldsymbol{\wp}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}})F_s({\boldsymbol{p}}) \end{gather}

(see § 9 for examples). Here, $\smash {{\boldsymbol {\alpha }}_s {\boldsymbol {\alpha }}_s^{{\dagger}} }$ is a dyadic matrix, and the arguments $t$ and ${\boldsymbol {x}}$ are henceforth omitted for brevity. Also, we introduced the operators $\smash {\widehat {\boldsymbol {\wp }}_s = \widehat {\boldsymbol {\wp }}_s^{{\dagger}} }$ and their symbols $\smash {{\boldsymbol {\wp }}_s = {\boldsymbol {\wp }}_s^{{\dagger}} }$ as

(6.22)\begin{equation} \widehat{\boldsymbol{\wp}}_s \doteq (\smash{\widehat{\boldsymbol{L}}}_s^{{\dagger}}\widehat{\boldsymbol{R}})_\text{H}, \qquad {\boldsymbol{\wp}}_s \doteq \text{symb}\, \widehat{\boldsymbol{\wp}}_s \approx ({\boldsymbol{L}}_s^{{\dagger}} {\boldsymbol{R}}_s)_\text{H}. \end{equation}

The appearance of $+ \mathrm {i} 0$ in the denominator in (6.20) is related to the Landau rule. (Remember that as arguments of Weyl symbols, $\omega$ and ${\boldsymbol {k}}$ are real by definition.) The Hermitian and anti-Hermitian parts of the dispersion matrix are

(6.23)\begin{align} {\boldsymbol{\varXi}}_\text{H}(\omega, {\boldsymbol{k}}) & \approx {\boldsymbol{\varXi}}_p(\omega, {\boldsymbol{k}}) + \sum_s {\unicode{x2A0F}} \mathrm{d}{\boldsymbol{p}}\, \frac{{\boldsymbol{\alpha}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}){\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s}\, {\boldsymbol{k}} \cdot \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}, \end{align}

(6.24)\begin{align} {\boldsymbol{\varXi}}_\text{A}(\omega, {\boldsymbol{k}}) & \approx{-}{\rm \pi} \sum_s \int \mathrm{d}{\boldsymbol{p}}\, {\boldsymbol{\alpha}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}){\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) \delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s)\, {\boldsymbol{k}} \cdot \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}. \end{align}

Assuming the notation $\smash {{\boldsymbol {\varXi }}^{-{{\dagger}} } \doteq ({\boldsymbol {\varXi }}^{{\dagger}} )^{-1}}$, the inverse dispersion matrix can be expressed as

(6.25)\begin{equation} {\boldsymbol{\varXi}}^{{-}1} = {\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}^{{{\dagger}}} {\boldsymbol{\varXi}}^{-{{\dagger}}} = {\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}_\text{H} {\boldsymbol{\varXi}}^{-{{\dagger}}} - \mathrm{i} {\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}_\text{A} {\boldsymbol{\varXi}}^{-{{\dagger}}}. \end{equation}

Because $\smash {{\boldsymbol {\varXi }}^{-{{\dagger}} } = ({\boldsymbol {\varXi }}^{-1})^{{\dagger}} }$, this leads to the following formulas, which we will need later:

(6.26)\begin{equation} ({\boldsymbol{\varXi}}^{{-}1})_\text{H} = {\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}_\text{H} {\boldsymbol{\varXi}}^{-{{\dagger}}}, \qquad ({\boldsymbol{\varXi}}^{{-}1})_\text{A} ={-}{\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}_\text{A} {\boldsymbol{\varXi}}^{-{{\dagger}}}. \end{equation}

6.4. Spectrum of microscopic fluctuations

Other objects to be used below are the density operators of the oscillating fields:

(6.27)

and the corresponding average Wigner matrices on $\smash {({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$. The former, $\smash {{{\boldsymbol {\mathsf {U}}}} \doteq \overline {{{\boldsymbol {\mathsf {W}}}}}_{\underline {\widetilde {{\boldsymbol {\varPsi }}}}}}$, is readily found by definition (2.51), and the latter, $\smash {{\boldsymbol {\mathfrak {W}}} \doteq \overline {{{\boldsymbol {\mathsf {W}}}}}_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}}$, is calculated as follows. Let us consider $\smash {g_s(t, {\boldsymbol {x}}, {\boldsymbol {p}})}$ as a ket in $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$, with $\smash {{\boldsymbol {p}}}$ as a parameter. Then, (6.14) readily yields

(6.28)\begin{equation} \widehat{\boldsymbol{{\boldsymbol {\mathsf{W}}}}}_{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}}} = \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}}\sum_{s,s'} \int \mathrm{d}{\boldsymbol{p}}\,\mathrm{d}{\boldsymbol{p}}'\, \smash{\widehat{\boldsymbol{\varXi}}}^{{-}1}\widehat{\boldsymbol{\alpha}}_s({\boldsymbol{p}}) {\left. {\boldsymbol {\mathsf {|{g_s({\boldsymbol{p}})}}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {{g_{s'}({\boldsymbol{p}}')}|}}}} \right.} \widehat{\boldsymbol{\alpha}}^{{\dagger}}_{s'}({\boldsymbol{p}}')\smash{\widehat{\boldsymbol{\varXi}}}^{-{{\dagger}}}. \end{equation}

By applying $\smash {\text {symb}_{{\mathsf {x}}} }$ to this, one obtains

(6.29)\begin{equation} {\boldsymbol{\mathfrak{W}}}= \sum_{s,s'} \int \mathrm{d}{\boldsymbol{p}}\,\mathrm{d}{\boldsymbol{p}}'\, {\boldsymbol{\varXi}}^{{-}1} \star {\boldsymbol{\alpha}}_s({\boldsymbol{p}}) \star {\boldsymbol{\mathfrak{G}}}_{ss'}({\boldsymbol{p}}, {\boldsymbol{p}}') \star {\boldsymbol{\alpha}}^{{\dagger}}_{s'}({\boldsymbol{p}}') \star {\boldsymbol{\varXi}}^{-{{\dagger}}}, \end{equation}

where most arguments are omitted for brevity and (Appendix B.7)

(6.30)\begin{align} \mathfrak{G}_{ss'}({\boldsymbol{p}}, {\boldsymbol{p}}') & \doteq \frac{1}{(2{\rm \pi})^{{\mathsf{n}}}} \int \mathrm{d}{\boldsymbol {\mathsf{s}}}\,\mathrm{e}^{- \mathrm{i}{\boldsymbol {\mathsf{k}}}\cdot{\boldsymbol {\mathsf{s}}}}\, \overline{g_{s}({\boldsymbol {\mathsf{x}}} + {\boldsymbol {\mathsf{s}}}/2, {\boldsymbol{p}})\, g_{s'}({\boldsymbol {\mathsf{x}}} - {\boldsymbol {\mathsf{s}}}/2, {\boldsymbol{p}}')} \nonumber\\ & \approx \frac{1}{(2{\rm \pi})^n}\,\delta_{ss'}\delta({\boldsymbol{p}} - {\boldsymbol{p}}') \delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s)F_s({\boldsymbol{p}}), \end{align}

assuming corrections due to inter-particle correlations are negligible. Then, (6.29) gives

(6.31)\begin{align} {\boldsymbol{\mathfrak{W}}}(\omega, {\boldsymbol{k}}) = \frac{1}{(2{\rm \pi})^{n}} \sum_{s'}\int \mathrm{d}{\boldsymbol{p}}'\,&\delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})F_{s'}({\boldsymbol{p}}') \nonumber\\ & \times {\boldsymbol{\varXi}}^{{-}1}(\omega, {\boldsymbol{k}}) ({\boldsymbol{\alpha}}_{s'}{\boldsymbol{\alpha}}_{s'}^{{\dagger}})(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}') {\boldsymbol{\varXi}}^{-{{\dagger}}}(\omega, {\boldsymbol{k}}), \end{align}

where $\smash {{\boldsymbol {v}}'_{s'} \doteq {\boldsymbol {v}}_{s'}(t, {\boldsymbol {x}}, {\boldsymbol {p}}')}$. It is readily seen from (6.31) that $\smash {{\boldsymbol {\mathfrak {W}}}}$ is positive-semidefinite. One can also recognize (6.31) as a manifestation of the dressed-particle superposition principle (Rostoker Reference Rostoker1964). Specifically, (6.31) shows that the contributions of individual OCs to $\smash {{\boldsymbol {\mathfrak {W}}}}$ are additive and affected by the plasma collective response, i.e. by the difference between $\smash {{\boldsymbol {\varXi }}}$ and the vacuum dispersion matrix $\smash {{\boldsymbol {\varXi }}_0}$.

Using (6.31), one can also find other averages quadratic in the field via (cf. (2.53a))

(6.32)\begin{equation} \overline{(\widehat{\boldsymbol{{\boldsymbol {\mathsf{L}}}}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}})(\widehat{\boldsymbol{{\boldsymbol {\mathsf{R}}}}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}})^{{\dagger}}} \approx \int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, ({{\boldsymbol {\mathsf{L}}}} {\boldsymbol{\mathfrak{W}}} {{\boldsymbol {\mathsf{R}}}}^{{\dagger}})(\omega, {\boldsymbol{k}}), \end{equation}

where $\smash {\widehat {\boldsymbol {{\boldsymbol {\mathsf {L}}}}}}$ and $\smash {\widehat {\boldsymbol {{\boldsymbol {\mathsf {R}}}}}}$ are any linear operators and $\smash {{{\boldsymbol {\mathsf {L}}}}}$ and $\smash {{{\boldsymbol {\mathsf {R}}}}}$ are their symbols; for example,

(6.33)\begin{equation} \overline{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}}(t, {\boldsymbol{x}})\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}}(t, {\boldsymbol{x}})} \approx \int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,{\boldsymbol{\mathfrak{W}}}(\omega, {\boldsymbol{k}}). \end{equation}

Because of this, we loosely attribute $\smash {{\boldsymbol {\mathfrak {W}}}}$ as the spectrum of microscopic oscillations, but see also § 8.2, where an alternative notation is introduced and a fluctuation–dissipation theorem is derived from (6.31) for plasma in thermal equilibrium. See also § 9 for specific examples.

6.5. Nonlinear potentials

From (6.14), the oscillating part of the Hamiltonian (6.9) can be split into the macroscopic part and the microscopic part as $\smash {\widetilde {H}_s = \underline {\widetilde {H}}_s + \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {H}}_s}$, $\smash {\underline {\widetilde {H}}_s = \widehat {\boldsymbol {\alpha }}_s^{{\dagger}} \underline {\widetilde {{\boldsymbol {\varPsi }}}}}$, and

(6.34)\begin{equation} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{\widetilde{H}}_s({\boldsymbol{p}}) = \sum_{s'} \int \mathrm{d}{\boldsymbol{p}}'\,\widehat{\mathcal{X}}_{ss'}({\boldsymbol{p}}, {\boldsymbol{p}}')g_{s'}({\boldsymbol{p}}'). \end{equation}

Here, $\smash {\widehat {\mathcal {X}}_{ss'}}$ is an operator on $\smash {\mathscr {H}_{{{\mathsf {x}}}}}$ given by

(6.35)\begin{equation} \widehat{\mathcal{X}}_{ss'}({\boldsymbol{p}}, {\boldsymbol{p}}') \doteq \widehat{\boldsymbol{\alpha}}_s^{{\dagger}}({\boldsymbol{p}})\smash{\widehat{\boldsymbol{\varXi}}}^{{-}1} \widehat{\boldsymbol{\alpha}}_{s'}({\boldsymbol{p}}'), \end{equation}

with the symbol

(6.36)\begin{equation} \mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') \approx {\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\varXi}}^{{-}1}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}') \end{equation}

(see § 9 for examples). The corresponding average Wigner functions on $\smash {({\boldsymbol {\mathsf {x}}}, {\boldsymbol {\mathsf {k}}})}$ are $\smash {\overline {{\mathsf {W}}}_s = \overline {{\mathsf {W}}}_s^{\text {(m)}} + \overline {{\mathsf {W}}}_s^{(\mu )}}$, where the index ‘m’ stands for ‘macroscopic’ and the index ‘$\smash {\mu }$’ stands for ‘microscopic’. Because the dependence on $\smash {t}$ and $\smash {{\boldsymbol {x}}}$ is slow, one can approximate them as follows:

(6.37a)\begin{gather} \overline{{\mathsf{W}}}_s^{\text{(m)}} \approx {\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {{\boldsymbol {\mathsf{U}}}}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}), \end{gather}

(6.37b)\begin{gather}\overline{{\mathsf{W}}}_s^{(\mu)} \approx {\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\mathfrak{W}}}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_s(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}). \end{gather}

The matrix $\smash {{{\boldsymbol {\mathsf {U}}}}}$ is positive-semidefinite as an average Wigner tensor (§ 2.1.7), and so is $\smash {{\boldsymbol {\mathfrak {W}}}}$ (§ 6.4). Hence, both $\smash {\overline {{\mathsf {W}}}_s^{\text {(m)}}}$ and $\smash {\overline {{\mathsf {W}}}_s^{(\mu )}}$ are non-negative. Using (6.31), one can also rewrite the Wigner function of $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {H}}_s}$ more compactly as

(6.38)\begin{equation} \overline{{\mathsf{W}}}_s^{(\mu)} (\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) = (2{\rm \pi})^{{-}n} \sum_{s'}\int \mathrm{d}{\boldsymbol{p}}'\, \delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'}) |\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2\,F_{s'}({\boldsymbol{p}}'). \end{equation}

Now we can represent the nonlinear potentials (5.41)–(5.43) as

(6.39)\begin{equation} {{\boldsymbol {\mathsf{D}}}}_s = {{\boldsymbol {\mathsf{D}}}}_s^{\text{(m)}} + {{\boldsymbol {\mathsf{D}}}}_s^{(\mu)}, \qquad {\boldsymbol{\Theta}}_s = {\boldsymbol{\Theta}}_s^{\text{(m)}} + {\boldsymbol{\Theta}}_s^{(\mu)}, \qquad \varPhi_s = \varPhi_s^{\text{(m)}} + \varPhi_s^{(\mu)}. \end{equation}

Here, the index $\smash {^{\text {(m)}}}$ denotes contributions from $\smash {\overline {{\mathsf {W}}}_s^{\text {(m)}}}$ and the index $\smash {^{(\mu )}}$ denotes contributions from $\smash {\overline {{\mathsf {W}}}_s^{(\mu )}}$. Specifically,

(6.40)\begin{align} {{\boldsymbol {\mathsf{D}}}}_s^{\text{(m)}} & = \int \mathrm{d}{\boldsymbol{k}}\, {\rm \pi}\, {\boldsymbol{k}} {\boldsymbol{k}} \overline{{\mathsf{W}}}_s^{\text{(m)}}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}), \end{align}

(6.41)\begin{align} {\boldsymbol{\Theta}}_s^{\text{(m)}} & =\frac{\partial}{\partial \vartheta} {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, \left. \frac{{\boldsymbol{k}} {\boldsymbol{k}} \overline{{\mathsf{W}}}_s^{\text{(m)}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s + \vartheta} \right|_{\vartheta=0}, \end{align}

(6.42)\begin{align} \varPhi_s^{\text{(m)}} & = \frac{1}{2}\frac{\partial}{\partial {\boldsymbol{p}}} \cdot {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\frac{{\boldsymbol{k}} \overline{{\mathsf{W}}}_s^{\text{(m)}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s}. \end{align}

Here, $\smash {\overline {{\mathsf {W}}}_s^{\text {(m)}}}$ is a non-negative function (6.37a), so $\smash {{{\boldsymbol {\mathsf {D}}}}_s^{\text {(m)}}}$ is positive-semidefinite and leads to an $\smash {H}$-theorem similar to (5.46). One also has

(6.43)\begin{align} {{\boldsymbol {\mathsf{D}}}}_s^{(\mu)} & = \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\,{\rm \pi}\, {\boldsymbol{k}} {\boldsymbol{k}} \, \delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\, |\mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_{s}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2\,F_{s'}({\boldsymbol{p}}'), \end{align}

(6.44)\begin{align} {\boldsymbol{\Theta}}_s^{(\mu)} &= \sum_{s'} \frac{\partial}{\partial \vartheta} {\unicode{x2A0F}} \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, \left. \frac{{\boldsymbol{k}} {\boldsymbol{k}} F_{s'}({\boldsymbol{p}}')}{{\boldsymbol{k}} \cdot ({\boldsymbol{v}}'_{s'} - {\boldsymbol{v}}_s) + \vartheta}\, |\mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') |^2 \right|_{\vartheta=0}, \end{align}

(6.45)\begin{align} \varPhi_s^{(\mu)} & = \sum_{s'}\frac{\partial}{\partial {\boldsymbol{p}}} \cdot {\unicode{x2A0F}} \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, \frac{{\boldsymbol{k}} F_{s'}({\boldsymbol{p}}')}{2{\boldsymbol{k}} \cdot ({\boldsymbol{v}}'_{s'} - {\boldsymbol{v}}_s)} \, |\mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') |^2. \end{align}

The functions $\smash {{\boldsymbol {\Theta }}_s^{(\mu )}}$ and $\smash {\varPhi _s^{(\mu )}}$ scale as $\smash {\overline {{\mathsf {W}}}_s^{(\mu )}}$, i.e. as $\smash {\epsilon \varepsilon ^2}$ (§ 3.3). Their contribution to (5.37) is of order $\smash {\epsilon {\boldsymbol {\Theta }}_s^{(\mu )}}$ and $\smash {\epsilon \varPhi _s^{(\mu )}}$, respectively, so it scales as $\smash {\epsilon ^2\varepsilon ^2}$ and therefore is negligible within our model. In contrast, $\smash {{{\boldsymbol {\mathsf {D}}}}_s^{(\mu )}}$ must be retained alongside with $\smash {{{\boldsymbol {\mathsf {D}}}}_s^{\text {(m)}}}$. This is because although weak, macroscopic fluctuations can resonate with particles from the bulk distribution, while the stronger macroscopic fluctuations are assumed to resonate only with particles from the tail distribution, which are few.

6.6. Oscillation-centre Hamiltonian

Within the assumed accuracy, the OC Hamiltonian is $\smash {\mathcal {H}_s = \overline {H}_s + \varPhi _s^{\text {(m)}}}$, and $\smash {\overline {H}_s}$ is given by (6.9). Combined with the general theorem (2.53c), the latter readily yields $\smash {\overline {H}_s = H_{0s} + \phi _s}$, where

(6.46)\begin{equation} \phi_s \approx \frac{1}{2}\int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, \operatorname{tr}\big({{\boldsymbol {\mathsf{U}}}}{\boldsymbol{\wp}}_s\big) \end{equation}

and the contribution of $\smash {{\boldsymbol {\mathfrak {W}}}}$ has been neglected. Because both $\smash {\varPhi _s^{\text {(m)}}}$ and $\smash {\phi _s}$ are quadratic in $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ and enter $\smash {\mathcal {H}_s}$ only in the combination $\smash {\varDelta _s \doteq \varPhi _s^{\text {(m)}} + \phi _s}$, it is convenient to attribute the latter as the ‘total’ ponderomotive energy. Using (6.42) in combination with (6.37a), one can express it as follows:

(6.47)\begin{equation} \varDelta_s = \frac{1}{2}\frac{\partial}{\partial {\boldsymbol{p}}} \cdot {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,{\boldsymbol{k}}\, \frac{{\boldsymbol{\alpha}}_s^{{\dagger}} {{\boldsymbol {\mathsf{U}}}} {\boldsymbol{\alpha}}_s}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s} + \frac{1}{2}\int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\operatorname{tr}\big({{\boldsymbol {\mathsf{U}}}}{\boldsymbol{\wp}}_s\big) \end{equation}

(see § 9 for examples). Notably,

(6.48)\begin{equation} \varDelta_s ={-}\frac{1}{2} \frac{\delta}{\delta F_s}{\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, \operatorname{tr}({\boldsymbol{\varXi}}_\text{H} {{\boldsymbol {\mathsf{U}}}}) ={-}\frac{\delta S_{\text{ad}}}{\delta F_s}, \end{equation}

where $\smash {\delta /\delta F_s}$ denotes a functional derivative and $\smash {S_{\text {ad}}}$ is the adiabatic action defined in (6.16). Equation (6.48) is a generalization of the well-known ‘$\smash {K}$–$\smash {\chi }$ theorem’ (Kaufman & Holm Reference Kaufman and Holm1984; Kaufman Reference Kaufman1987). Loosely speaking, it says that the coefficient connecting $\smash {\varDelta _s}$ with $\smash {{{\boldsymbol {\mathsf {U}}}}}$ is proportional to the linear polarizability of an individual particle of type $\smash {s}$ (Dodin et al. Reference Dodin, Zhmoginov and Ruiz2017; Dodin & Fisch Reference Dodin and Fisch2010a) (‘$\smash {K}$’ in the name of this theorem is the same as our $\smash {\varDelta _s}$, and ‘$\smash {\chi }$’ is the linear susceptibility). Also, the OC Hamiltonian and the OC velocity can be expressed as

(6.49)\begin{equation} \mathcal{H}_s = H_{0s} + \varDelta_s, \qquad {\boldsymbol{v}} = \partial_{\boldsymbol{p}} H_{0s} + \partial_{\boldsymbol{p}}\varDelta_s. \end{equation}

6.7. Polarization drag

Within the assumed accuracy, the OC distribution can be expressed as

(6.50)\begin{equation} F_s = \overline{f}_s + \frac{1}{2}\,\frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol{\Theta}}_s^{\text{(m)}}\, \frac{\partial \overline{f}_s}{\partial {\boldsymbol{p}}}\right), \end{equation}

and (5.37) becomes

(6.51)\begin{gather} \displaystyle \frac{\partial F_s}{\partial t} - \frac{\partial \mathcal{H}_s}{\partial {\boldsymbol{x}}} \cdot \frac{\partial F_s}{\partial {\boldsymbol{p}}} + \frac{\partial \mathcal{H}_s}{\partial {\boldsymbol{p}}} \cdot \frac{\partial F_s}{\partial {\boldsymbol{x}}} = \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol {\mathsf{D}}}_s^{\text{(m)}}\,\frac{\partial F_s}{\partial {\boldsymbol{p}}}\right) + \mathcal{C}_s, \end{gather}

(6.52)\begin{gather}\displaystyle \mathcal{C}_s \doteq \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol {\mathsf{D}}}_s^{(\mu)}\,\frac{\partial F_s}{\partial {\boldsymbol{p}}}\right) + \varGamma_s, \end{gather}

where we have reinstated the term $\smash {\varGamma _s}$ introduced in § 3.3. As a collisional term, $\smash {\varGamma _s}$ is needed only to the zeroth order in $\smash {\epsilon }$, so

(6.53)\begin{equation} \displaystyle \varGamma_s = \overline{\lbrace \widetilde{H}_s, g_s \rbrace} \approx \partial_{{\boldsymbol{p}}} \cdot (\overline{g_s \partial_{{\boldsymbol{x}}}\widetilde{H}_s}) \equiv \partial_{{\boldsymbol{p}}} \cdot {\boldsymbol{\zeta}}_s, \qquad {\boldsymbol{\zeta}}_s = \mathrm{i} \overline{\left\langle{{\boldsymbol {\mathsf{x}}}|\widehat{\boldsymbol{k}} \widetilde{H}_s}\right\rangle \left\langle{{\boldsymbol {\mathsf{x}}}|g_s}\right\rangle}. \end{equation}

Correlating with $\smash {g_s}$ is only the microscopic part of $\smash {\widetilde {H}_s}$, so using (6.34) one obtains

(6.54)\begin{equation} {\boldsymbol{\zeta}}_s = \mathrm{i} \sum_{s'} \int \mathrm{d}{\boldsymbol{p}}'\,\langle{\boldsymbol {\mathsf{x}}}|\widehat{\boldsymbol{k}}\widehat{\mathcal{X}}_{ss'}({\boldsymbol{p}}, {\boldsymbol{p}}') \overline{{\left. {\boldsymbol {\mathsf {|{g_{s'}({\boldsymbol{p}}')}}}} \right\rangle}{\left\langle {{\boldsymbol {\mathsf {{g_s({\boldsymbol{p}})}|}}}} \right.}} {\boldsymbol {\mathsf{x}}}\rangle. \end{equation}

Next, let us use (2.28) and $\smash {{\boldsymbol {\zeta }} = \operatorname {re} {\boldsymbol {\zeta }}}$ to express this result as follows:

(6.55)\begin{align} {\boldsymbol{\zeta}}_s & = \mathrm{i} \sum_{s'} \int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\mathrm{d}{\boldsymbol{p}}'\,{\boldsymbol{k}} \star \mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}, {\boldsymbol{p}}, {\boldsymbol{p}}') \star {\boldsymbol{\mathfrak{G}}}_{s's}(\omega, {\boldsymbol{k}}, {\boldsymbol{p}}', {\boldsymbol{p}})\nonumber\\ & \approx \mathrm{i} \int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\mathrm{d}{\boldsymbol{p}}'\,{\boldsymbol{k}} \mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}, {\boldsymbol{p}}, {\boldsymbol{p}}') \,(2{\rm \pi})^{{-}n}\delta({\boldsymbol{p}} - {\boldsymbol{p}}')\,\delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s)\,F_s({\boldsymbol{p}}) \nonumber\\ & ={-} \int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,{\boldsymbol{k}} \operatorname{im} \mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}, {\boldsymbol{p}}, {\boldsymbol{p}})\,F_s({\boldsymbol{p}}), \end{align}

where we have approximated $\smash {\star }$ with the usual product and substituted (6.30). Hence,

(6.56)\begin{equation} \varGamma_s \approx{-}\partial_{\boldsymbol{p}} \cdot ({\boldsymbol{\mathfrak{F}}}_s F_s), \end{equation}

where $\smash {{\boldsymbol {\mathfrak {F}}}_s}$ can be interpreted as the polarization drag (i.e. the average force that is imposed on an OC by its dress) and is given by

(6.57)\begin{equation} {\boldsymbol{\mathfrak{F}}}_s = \int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,{\boldsymbol{k}} \operatorname{im} \mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}, {\boldsymbol{p}}, {\boldsymbol{p}}). \end{equation}

Using (6.36), one also rewrite this as follows:

(6.58a)\begin{align} {\boldsymbol{\mathfrak{F}}}_s & \approx \int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,{\boldsymbol{k}}\, ({\boldsymbol{\alpha}}_s^{{\dagger}} ({\boldsymbol{\varXi}}^{{-}1})_\text{A} {\boldsymbol{\alpha}}_s)({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}) \end{align}

(6.58b)\begin{align} & \approx{-}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,{\boldsymbol{k}}\, ({\boldsymbol{\alpha}}_s^{{\dagger}} {\boldsymbol{\varXi}}^{{-}1} {\boldsymbol{\varXi}}_\text{A} {\boldsymbol{\varXi}}^{-{{\dagger}}}{\boldsymbol{\alpha}}_s)({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}), \end{align}

where we have substituted (6.26) for $\smash {({\boldsymbol {\varXi }}^{-1})_\text {A}}$. With (6.24) for $\smash {{\boldsymbol {\varXi }}_\text {A}}$, this yields

(6.59)\begin{align} {\boldsymbol{\mathfrak{F}}}_s \approx \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, &{\rm \pi}\,\delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\,{\boldsymbol{k}}{\boldsymbol{k}}\cdot\frac{\partial F_{s'}({\boldsymbol{p}}')}{\partial {\boldsymbol{p}}'} \nonumber\\ & \times {\boldsymbol{\alpha}}_s^{{\dagger}}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\varXi}}^{{-}1}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}') \nonumber\\ & \times {\boldsymbol{\alpha}}_{s'}^{{\dagger}}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}') {\boldsymbol{\varXi}}^{-{{\dagger}}}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_s({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}). \end{align}

The product of the last two lines equals $\smash {|\mathcal {X}_{ss'}({\boldsymbol {k}} \cdot {\boldsymbol {v}}_{s}, {\boldsymbol {k}}; {\boldsymbol {p}}, {\boldsymbol {p}}')|^2}$. Hence,

(6.60)\begin{equation} {\boldsymbol{\mathfrak{F}}}_s = \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, {\rm \pi}\,\delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\, |\mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2\,{\boldsymbol{k}}{\boldsymbol{k}}\cdot \frac{\partial F_{s'}({\boldsymbol{p}}')}{\partial {\boldsymbol{p}}'}. \end{equation}

6.8. Collision operator

By combining (6.56) for $\smash {\varGamma _s}$ with (6.43) for $\smash {{{\boldsymbol {\mathsf {D}}}}_s^{(\mu )}}$, one can express $\smash {\mathcal {C}_s}$ as

(6.61)\begin{align} \mathcal{C}_s = \frac{\partial}{\partial {\boldsymbol{p}}}\cdot \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, &{\rm \pi} \,\delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\, |\mathcal{X}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_{s}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 \nonumber\\ &\times {\boldsymbol{k}}{\boldsymbol{k}} \cdot \left( \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}\,F_{s'}({\boldsymbol{p}}') - F_s({\boldsymbol{p}})\,\frac{\partial F_{s'}({\boldsymbol{p}}')}{\partial {\boldsymbol{p}}'} \right), \end{align}

where $\mathcal {X}_{ss'}$ is given by (6.36). One can recognize this as a generalization of the Balescu–Lenard collision operator (Krall & Trivelpiece Reference Krall and Trivelpiece1973, § 11.11) to interactions via a general multi-component field ${\boldsymbol {\varPsi }}$. Specific examples can be found in § 9.

It is readily seen that $\smash {\mathcal {C}_s}$ conserves particles, i.e.

(6.62)\begin{equation} \int \mathrm{d}{\boldsymbol{p}}\,\mathcal{C}_s = 0, \end{equation}

and vanishes in thermal equilibrium (§ 8.1). Other properties of $\smash {\mathcal {C}_s}$ are determined by the properties of the coupling coefficient $\smash {\mathcal {X}_{ss'}}$, which are as follows. Note that

(6.63)\begin{equation} |\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 = \mathcal{Q}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') + \mathcal{R}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')/2, \end{equation}

where we introduced

(6.64)\begin{align} \mathcal{Q}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') & \doteq (|\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 + |\mathcal{X}_{s's}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}', {\boldsymbol{p}})|^2)/2, \end{align}

(6.65)\begin{align} \mathcal{R}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') & \doteq |\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 - |\mathcal{X}_{s's}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}', {\boldsymbol{p}})|^2. \end{align}

To calculate $\smash {\mathcal {R}_{ss'}}$, note that (6.36) yields

(6.66)\begin{align} |\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 & \approx |{\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\varXi}}^{{-}1}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}')|^2,\nonumber\\ |\mathcal{X}_{s's}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}', {\boldsymbol{p}})|^2 & \approx |{\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\varXi}}^{-{{\dagger}}}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}')|^2, \end{align}

whence one obtains

(6.67)\begin{align} \mathcal{R}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') \approx 4\operatorname{im}\big(&{\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) ({\boldsymbol{\varXi}}^{{-}1})_\text{H}(\omega, {\boldsymbol{k}}){\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}')\nonumber\\ & {\boldsymbol{\alpha}}_{s'}^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}') ({\boldsymbol{\varXi}}^{{-}1})_\text{A}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}') \big). \end{align}

The operators $\smash {({\boldsymbol {\varXi }}^{-1})_\text {H}}$, $\smash {({\boldsymbol {\varXi }}^{-1})_\text {A}}$ and $\smash {\widehat {\boldsymbol {\alpha }}_s}$ (for all $\smash {s}$) have been introduced for real fields, so their matrix elements in the coordinate representation are real. Then, the corresponding symbols satisfy $\smash {{\boldsymbol {A}}(-\omega, -{\boldsymbol {k}}) = {\boldsymbol {A}}^*(\omega, {\boldsymbol {k}})}$, where $\smash {{\boldsymbol {A}}}$ is any of the three symbols. This gives

(6.68)\begin{equation} \mathcal{R}_{ss'}({\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') \doteq \mathcal{R}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') ={-}\mathcal{R}_{ss'}(-{\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}'). \end{equation}

Because the rest of the integrand in (6.61) is even in $\smash {{\boldsymbol {k}}}$, (6.68) signifies that $\smash {\mathcal {R}_{ss'}}$ does not contribute to $\smash {\mathcal {C}_s}$. Thus, $\smash {\mathcal {X}_{ss'}}$ in (6.61) can also be replaced with $\smash {\mathcal {Q}_{ss'}}$:

(6.69)\begin{align} \mathcal{C}_s = \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, &{\rm \pi} \,\delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\, \mathcal{Q}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_{s}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') \nonumber\\ & \times {\boldsymbol{k}}{\boldsymbol{k}} \cdot \left( \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}\,F_{s'}({\boldsymbol{p}}') - F_s({\boldsymbol{p}})\,\frac{\partial F_{s'}({\boldsymbol{p}}')}{\partial {\boldsymbol{p}}'} \right). \end{align}

In this representation, the coupling coefficient in $\smash {\mathcal {C}_s}$ is manifestly symmetric,

(6.70)\begin{equation} \mathcal{Q}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') = \mathcal{Q}_{s's}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}', {\boldsymbol{p}}), \end{equation}

which readily leads to momentum and energy conservation (Appendix C):Footnote ²⁸

(6.71)\begin{equation} \sum_s \int \mathrm{d}{\boldsymbol{p}}\,{\boldsymbol{p}}\mathcal{C}_s = 0, \qquad \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\mathcal{H}_s\mathcal{C}_s = 0. \end{equation}

The collision operator $\smash {\mathcal {C}_s}$ also satisfies the $\smash {H}$-theorem (Appendix C.3):

(6.72)\begin{equation} \left(\frac{\mathrm{d}\sigma}{\mathrm{d} t}\right)_{\text{coll}} \geqslant 0, \end{equation}

where the entropy density $\smash {\sigma }$ is defined as

(6.73)\begin{equation} \sigma \doteq{-}\sum_s \int \mathrm{d}{\boldsymbol{p}}\,F_s({\boldsymbol{p}})\ln F_s({\boldsymbol{p}}), \end{equation}

and $\smash {(\partial _t F_s)_{\text {coll}} \doteq \mathcal {C}_s}$. Note that these properties are not restricted to any particular $\smash {\mathcal {H}_s}$. Also note that if applied in proper variables (§ 3.1.2), our formula (6.69) can describe collisions in strong background fields. This topic, including comparison with the relevant literature, is left to future work.

6.9. Summary of § 6

Let us summarize the above general results (for examples, see § 9). We consider species $\smash {s}$ governed by a Hamiltonian of the form

(6.74)\begin{equation} H_s = H_{0s} + \widehat{\boldsymbol{\alpha}}_s^{{\dagger}}({\boldsymbol{p}}) \widetilde{{\boldsymbol{\varPsi}}} + \frac{1}{2}\,(\widehat{\boldsymbol{L}}_s\widetilde{{\boldsymbol{\varPsi}}})^{{\dagger}} (\widehat{\boldsymbol{R}}_s\widetilde{{\boldsymbol{\varPsi}}}), \end{equation}

where $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ is a real oscillating field (of any dimension $\smash {M}$), which generally consists of a macroscopic part $\smash {\underline {\widetilde {{\boldsymbol {\varPsi }}}}}$ and a microscopic part $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}$. The term $\smash {H_{0s}}$ is independent of $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$, and the operators $\smash {\smash {\widehat {\boldsymbol {\alpha }}}_s^{{\dagger}} }$, $\smash {\widehat {\boldsymbol {L}}_s}$ and $\smash {\widehat {\boldsymbol {R}}_s}$ may be non-local in $t$ and ${\boldsymbol {x}}$ and may depend on the momentum ${\boldsymbol {p}}$ parametrically. The dynamics of this system averaged over the fast oscillations can be described in terms of the OC distribution function

(6.75)\begin{equation} F_s = \overline{f}_s + \frac{1}{2}\,\frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol{\Theta}}_s\, \frac{\partial \overline{f}_s}{\partial {\boldsymbol{p}}}\right) \end{equation}

(the index $\smash {^{\text {(m)}}}$ is henceforth omitted for brevity), which is governed by the following equation of the Fokker–Planck type:

(6.76)\begin{equation} \frac{\partial F_s}{\partial t} - \frac{\partial \mathcal{H}_s}{\partial {\boldsymbol{x}}} \cdot \frac{\partial F_s}{\partial {\boldsymbol{p}}} + \frac{\partial \mathcal{H}_s}{\partial {\boldsymbol{p}}} \cdot \frac{\partial F_s}{\partial {\boldsymbol{x}}} = \frac{\partial}{\partial {\boldsymbol{p}}} \cdot \left({\boldsymbol {\mathsf{D}}}_s\,\frac{\partial F_s}{\partial {\boldsymbol{p}}}\right) + \mathcal{C}_s. \end{equation}

Here, $\smash {\mathcal {H}_s = H_{0s} + \varDelta _s}$ is the OC Hamiltonian, $\smash {{\boldsymbol {\Theta }}_s}$ is the dressing function and $\smash {\varDelta _s}$ is the total ponderomotive energy (i.e. the part of the OC Hamiltonian that is quadratic in $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$), so $\smash {{\boldsymbol {v}}_s(t, {\boldsymbol {x}}, {\boldsymbol {p}}) \doteq \partial _{\boldsymbol {p}}\mathcal {H}_s}$ is the OC velocity. Specifically,

(6.77a)\begin{align} {{\boldsymbol {\mathsf{D}}}}_s & = \int \mathrm{d}{\boldsymbol{k}}\,{\rm \pi} \,{\boldsymbol{k}}{\boldsymbol{k}} \overline{{\mathsf{W}}}_s(t, {\boldsymbol{x}}, {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s, {\boldsymbol{k}}; {\boldsymbol{p}}), \end{align}

(6.77b)\begin{align} {\boldsymbol{\Theta}}_s & =\frac{\partial}{\partial \vartheta} {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,{\boldsymbol{k}}{\boldsymbol{k}} \left. \frac{\overline{{\mathsf{W}}}_s(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s + \vartheta} \right|_{\vartheta=0}, \end{align}

(6.77c)\begin{align} \varDelta_s & = \frac{1}{2}\frac{\partial}{\partial {\boldsymbol{p}}} \cdot {\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,{\boldsymbol{k}}\,\frac{\overline{{\mathsf{W}}}_s(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s} \nonumber\\ & \quad +\frac{1}{2}\int \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\,\operatorname{tr}\big({{\boldsymbol {\mathsf{U}}}}{\boldsymbol{\wp}}_s\big)(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}; {\boldsymbol{p}}). \end{align}

Here, $\smash {\overline {{\mathsf {W}}}_s = {\boldsymbol {\alpha }}_s^{{\dagger}} {{\boldsymbol {\mathsf {U}}}} {\boldsymbol {\alpha }}_s}$ is a scalar function, the average Wigner matrix $\smash {{{\boldsymbol {\mathsf {U}}}}}$ is understood as the Fourier spectrum of the symmetrized autocorrelation matrix of the macroscopic oscillations:

(6.78)\begin{equation} {{\boldsymbol {\mathsf{U}}}}(t, {\boldsymbol{x}}, \omega, {\boldsymbol{k}}) = \int \frac{\mathrm{d}\tau}{2{\rm \pi}}\,\frac{\mathrm{d}{\boldsymbol{s}}}{(2{\rm \pi})^{n}}\,\, \overline{ {\underline{\widetilde{{\boldsymbol{\varPsi}}}}}(t + \tau/2, {\boldsymbol{x}} + {\boldsymbol{s}}/2)\, \smash{\underline{\widetilde{{\boldsymbol{\varPsi}}}}}^{{\dagger}}(t - \tau/2, {\boldsymbol{x}} - {\boldsymbol{s}}/2) } \,\mathrm{e}^{\mathrm{i}\omega \tau - \mathrm{i} {\boldsymbol{k}} \cdot {\boldsymbol{s}}}, \end{equation}

with $\smash {n \doteq \dim {\boldsymbol {x}}}$. Also, the vector $\smash {{\boldsymbol {\alpha }}_s(t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}}; {\boldsymbol {p}})}$ is the Weyl symbol of $\smash {\widehat {\boldsymbol {\alpha }}_s}$ as defined in (2.26), $\smash {{\boldsymbol {\wp }}_s(t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}}; {\boldsymbol {p}}) \approx ({\boldsymbol {L}}_s^{{\dagger}} {\boldsymbol {R}}_s)_\text {H}}$, $\smash {{\boldsymbol {L}}_s}$ and $\smash {{\boldsymbol {R}}_s}$ are the Weyl symbols of $\smash {\widehat {\boldsymbol {L}}_s}$ and $\smash {\widehat {\boldsymbol {R}}_s}$, respectively, and $\smash {_\text {H}}$ denotes the Hermitian part. The matrix $\smash {{{\boldsymbol {\mathsf {D}}}}_s}$ is positive-semidefinite and satisfies an $\smash {H}$-theorem of the form (5.46). Also, $\smash {\varDelta _s}$ satisfies the ‘$\smash {K}$–$\smash {\chi }$ theorem’

(6.79)\begin{equation} \varDelta_s ={-}\frac{1}{2} \frac{\delta}{\delta F_s}{\unicode{x2A0F}} \mathrm{d}\omega\,\mathrm{d}{\boldsymbol{k}}\, \operatorname{tr}({\boldsymbol{\varXi}}_\text{H} {{\boldsymbol {\mathsf{U}}}}). \end{equation}

The matrix ${\boldsymbol {\varXi }}$ characterizes the collective plasma response to the field $\widetilde {{\boldsymbol {\varPsi }}}$ and is given by

(6.80)\begin{equation} {\boldsymbol{\varXi}}\approx {\boldsymbol{\varXi}}_0 + \sum_s \int \mathrm{d}{\boldsymbol{p}}\, \Bigg( \frac{{\boldsymbol{\alpha}}_s({\boldsymbol{p}})\,{\boldsymbol{\alpha}}_s^{{\dagger}}({\boldsymbol{p}})}{\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}_s({\boldsymbol{p}}) + \mathrm{i} 0}\, {\boldsymbol{k}}\cdot \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}} - {\boldsymbol{\wp}}_s({\boldsymbol{p}})F_s({\boldsymbol{p}}) \Bigg). \end{equation}

Here, the arguments $\smash {(t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}})}$ are omitted for brevity, $\smash {{\boldsymbol {\alpha }}_s{\boldsymbol {\alpha }}_s^{{\dagger}} }$ is a dyadic matrix and $\smash {{\boldsymbol {\varXi }}_0}$ is the symbol of the Hermitian dispersion operator $\smash {\widehat {\boldsymbol {\varXi }}_0}$ that governs the field $\smash {\widetilde {{\boldsymbol {\varPsi }}}}$ in the absence of plasma. Specifically, $\smash {\widehat {\boldsymbol {\varXi }}_0}$ is defined such that the field Lagrangian density without plasma is $\smash {\mathfrak {L}_0 = \smash {\widetilde {{\boldsymbol {\varPsi }}}}^{{\dagger}} \widehat {\boldsymbol {\varXi }}_0 \widetilde {{\boldsymbol {\varPsi }}}/2}$.

The spectrum of microscopic fluctuations (specifically, the spectrum of the symmetrized autocorrelation function of the microscopic field $\smash {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}} {\widetilde {{\boldsymbol {\varPsi }}}}}$) is a positive-semidefinite matrix function and given by

(6.81)\begin{align} {\boldsymbol{\mathfrak{W}}}(\omega, {\boldsymbol{k}}) = \frac{1}{(2{\rm \pi})^{n}} \sum_{s'}\int \mathrm{d}{\boldsymbol{p}}'\,&\delta(\omega - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})F_{s'}({\boldsymbol{p}}') \nonumber\\ &\times {\boldsymbol{\varXi}}^{{-}1}(\omega, {\boldsymbol{k}}) ({\boldsymbol{\alpha}}_{s'}{\boldsymbol{\alpha}}_{s'}^{{\dagger}})(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}') {\boldsymbol{\varXi}}^{-{{\dagger}}}(\omega, {\boldsymbol{k}}), \end{align}

where $\smash {{\boldsymbol {v}}'_{s'} \doteq {\boldsymbol {v}}_{s'}({\boldsymbol {p}}')}$. (The dependence on t and $\boldsymbol{x}$ is assumed too but not emphasized.) The microscopic fluctuations give rise to a collision operator of the Balescu–Lenard type:

(6.82)\begin{align} \mathcal{C}_s = \frac{\partial}{\partial {\boldsymbol{p}}}\cdot \sum_{s'}\int \frac{\mathrm{d}{\boldsymbol{k}}}{(2{\rm \pi})^n}\,\mathrm{d}{\boldsymbol{p}}'\, &{\rm \pi} \,\delta({\boldsymbol{k}} \cdot {\boldsymbol{v}}_s - {\boldsymbol{k}} \cdot {\boldsymbol{v}}'_{s'})\, \mathcal{Q}_{ss'}({\boldsymbol{k}} \cdot {\boldsymbol{v}}_{s}, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')\nonumber\\ &\times {\boldsymbol{k}}{\boldsymbol{k}} \cdot \left( \frac{\partial F_s({\boldsymbol{p}})}{\partial {\boldsymbol{p}}}\,F_{s'}({\boldsymbol{p}}') - F_s({\boldsymbol{p}})\,\frac{\partial F_{s'}({\boldsymbol{p}}')}{\partial {\boldsymbol{p}}'} \right), \end{align}

where the coupling coefficient $\smash {\mathcal {Q}_{ss'}(\omega, {\boldsymbol {k}}; {\boldsymbol {p}}, {\boldsymbol {p}}') = \mathcal {Q}_{s's}(\omega, {\boldsymbol {k}}; {\boldsymbol {p}}', {\boldsymbol {p}})}$ is given by

(6.83)\begin{align} \mathcal{Q}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') &= (|\mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}')|^2 + |\mathcal{X}_{s's}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}', {\boldsymbol{p}})|^2)/2, \end{align}

(6.84)\begin{align} \mathcal{X}_{ss'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}, {\boldsymbol{p}}') &\approx {\boldsymbol{\alpha}}_s^{{\dagger}}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}) {\boldsymbol{\varXi}}^{{-}1}(\omega, {\boldsymbol{k}}) {\boldsymbol{\alpha}}_{s'}(\omega, {\boldsymbol{k}}; {\boldsymbol{p}}'). \end{align}

The operator $\smash {\mathcal {C}_s}$ satisfies the $\smash {H}$-theorem and conserves particles, momentum and energy

\begin{equation*} \int \mathrm{d}{\boldsymbol{p}}\,\mathcal{C}_s = 0, \qquad \sum_s \int \mathrm{d}{\boldsymbol{p}}\,{\boldsymbol{p}}\mathcal{C}_s = 0, \qquad \sum_s \int \mathrm{d}{\boldsymbol{p}}\,\mathcal{H}_s\mathcal{C}_s = 0. \end{equation*}

7.1. Monochromatic waves

Conservative (non-dissipative) waves can be described using the least-action principle $\delta S = 0$. Assuming the notation as in § 6.2, the action integral can be expressed as $S = \int \mathrm {d}{\boldsymbol {\mathsf {x}}}\,\mathfrak {L}$ with the Lagrangian density given by

(7.1)\begin{equation} \mathfrak{L} = \frac{1}{2}\,\smash{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}}\widehat{\boldsymbol{\varXi}}_\text{H}\widetilde{{\boldsymbol{\varPsi}}}. \end{equation}

First, let us assume a homogeneous stationary medium, so $\smash {{\boldsymbol {\varXi }}_\text {H}(t, {\boldsymbol {x}}, \omega, {\boldsymbol {k}}) = {\boldsymbol {\varXi }}_\text {H}(\omega, {\boldsymbol {k}})}$. Because we assume real fields,Footnote ²⁹ $\smash {\left\langle {t_1, {\boldsymbol {x}}_1 |\widehat {\boldsymbol {\varXi }}|t_2, {\boldsymbol {x}}_2}\right\rangle}$ is real for all $\smash {(t_1, {\boldsymbol {x}}_1, t_2, {\boldsymbol {x}}_2)}$, one also has

(7.2)\begin{equation} {\boldsymbol{\varXi}}_\text{H}(-\omega, -{\boldsymbol{k}}) = {\boldsymbol{\varXi}}_\text{H}^*(\omega, {\boldsymbol{k}}) = {\boldsymbol{\varXi}}_\text{H}^\intercal(\omega, {\boldsymbol{k}}), \end{equation}

where the latter equality is due to $\smash {{\boldsymbol {\varXi }}_\text {H}^{{\dagger}} (\omega, {\boldsymbol {k}}) = {\boldsymbol {\varXi }}_\text {H}(\omega, {\boldsymbol {k}})}$.

Because ${\boldsymbol {\varXi }}_\text {H}(\omega, {\boldsymbol {k}})$ is Hermitian, it has $M \doteq \dim {\boldsymbol {\varXi }}_\text {H}$ orthonormal eigenvectors ${\boldsymbol {\eta }}_b$:

(7.3)\begin{equation} {\boldsymbol{\varXi}}_\text{H}(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_b(\omega, {\boldsymbol{k}}) = \varLambda_b(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_b(\omega, {\boldsymbol{k}}), \qquad {\boldsymbol{\eta}}_b^{{\dagger}}(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_{b'}(\omega, {\boldsymbol{k}}) = \delta_{b,b'}. \end{equation}

Here, $\smash {\varLambda _b}$ are the corresponding eigenvalues, which are real and satisfy

(7.4)\begin{equation} \varLambda_b(\omega, {\boldsymbol{k}}) = {\boldsymbol{\eta}}_b^{{\dagger}}(\omega, {\boldsymbol{k}}){\boldsymbol{\varXi}}_\text{H}(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_b(\omega, {\boldsymbol{k}}). \end{equation}

Due to (7.2), one has

(7.5)\begin{equation} \varLambda_b(-\omega, -{\boldsymbol{k}}) = \text{eigv}_b\, ({\boldsymbol{\varXi}}_\text{H}(\omega, {\boldsymbol{k}}))^\intercal = \text{eigv}_b\, {\boldsymbol{\varXi}}_\text{H}(\omega, {\boldsymbol{k}}) = \varLambda_b(\omega, {\boldsymbol{k}}), \end{equation}

where $\text {eigv}_b$ stands for the $b$th eigenvalue. Using this together with (7.2), one obtains from (7.3) that

(7.6)\begin{equation} {\boldsymbol{\varXi}}_\text{H}^*(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_b(-\omega, -{\boldsymbol{k}}) = \varLambda_b(\omega, {\boldsymbol{k}}){\boldsymbol{\eta}}_b(-\omega, -{\boldsymbol{k}}), \end{equation}

whence

(7.7)\begin{equation} {\boldsymbol{\eta}}_b(-\omega, -{\boldsymbol{k}}) = {\boldsymbol{\eta}}_b^*(\omega, {\boldsymbol{k}}). \end{equation}

Let us consider a monochromatic wave of the form

(7.8)\begin{equation} \widetilde{{\boldsymbol{\varPsi}}}(t, {\boldsymbol{x}}) = \operatorname{re}(\mathrm{e}^{-\mathrm{i} \overline{\omega}t + \mathrm{i} \overline{{\boldsymbol{k}}} \cdot {\boldsymbol{x}}} \,{\breve{{\boldsymbol{\varPsi}}}}), \end{equation}

with real frequency $\overline {\omega }$, real wavevector $\overline {{\boldsymbol {k}}}$ and complex amplitude ${\breve {{\boldsymbol {\varPsi }}}}$. For such a wave, the action integral can be expressed as $\smash {S = \int \mathrm {d}{\boldsymbol {\mathsf {x}}}\,\overline {\mathfrak {L}}}$, where the average Lagrangian density $\overline {\mathfrak {L}}$ is given byFootnote ³⁰

(7.9)\begin{equation} \overline{\mathfrak{L}} = \frac{1}{2}\,\overline{\smash{\widetilde{{\boldsymbol{\varPsi}}}}^{{\dagger}} \widehat{\boldsymbol{\varXi}}_\text{H} \widetilde{{\boldsymbol{\varPsi}}}} = \frac{1}{4}\,\operatorname{re}(\smash{{\breve{{\boldsymbol{\varPsi}}}}}^{{\dagger}} {\boldsymbol{\varXi}}_\text{H}(\overline{\omega}, \overline{{\boldsymbol{k}}}) {\breve{{\boldsymbol{\varPsi}}}}) = \frac{1}{4}\,\smash{{\breve{{\boldsymbol{\varPsi}}}}}^{{\dagger}} {\boldsymbol{\varXi}}_\text{H}(\overline{\omega}, \overline{{\boldsymbol{k}}}) {\breve{{\boldsymbol{\varPsi}}}}. \end{equation}

Let us decompose ${\breve {{\boldsymbol {\varPsi }}}}$ in the basis formed by the eigenvectors $\smash {{\boldsymbol {\eta }}_b}$, that is, as

(7.10)\begin{equation} {\breve{{\boldsymbol{\varPsi}}}} = \sum_b {\boldsymbol{\eta}}_b {\breve{a}}^b. \end{equation}

Then, (7.9) becomes

(7.11)\begin{equation} \overline{\mathfrak{L}} = \frac{1}{4}\sum_b \varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}})\,|{\breve{a}}^b|^2. \end{equation}

The real and imaginary parts of the amplitudes ${\breve {a}}^b$ can be treated as independent variables. This is equivalent to treating ${\breve {a}}^{b*}$ and ${\breve {a}}^b$ as independent variables, so one arrives at the following Euler–Lagrange equations:

(7.12)\begin{equation} 0 = \frac{\delta S[{\breve{{\boldsymbol{a}}}}^*, {\breve{{\boldsymbol{a}}}}]}{\delta {\breve{a}}^{b*}} = \frac{1}{4}\,\varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}}){\breve{a}}^b, \qquad 0 = \frac{\delta S[{\breve{{\boldsymbol{a}}}}^*, {\breve{{\boldsymbol{a}}}}]}{\delta {\breve{a}}^b} = \frac{1}{4}\,{\breve{a}}^{b*}\varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}}). \end{equation}

Hence the $b$th mode with a non-zero amplitude ${\breve {a}}^b$ satisfies the dispersion relation

(7.13)\begin{equation} 0 = \varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}}) = \varLambda_b(-\overline{\omega}, -\overline{{\boldsymbol{k}}}). \end{equation}

Equation (7.13) determines a dispersion surface in the ${\boldsymbol {\mathsf {k}}}$ space where the waves can have non-zero amplitude. This surface is sometimes called a shell, so waves constrained by a dispersion relation are called on-shell. Also note that combining (7.13) with (7.3) yields that on-shell waves satisfy

(7.14)\begin{equation} {\boldsymbol{\varXi}}_\text{H}(\overline{\omega}, \overline{{\boldsymbol{k}}}){\boldsymbol{\eta}}_b(\overline{\omega}, \overline{{\boldsymbol{k}}}) = {\boldsymbol{0}}, \qquad {\boldsymbol{\eta}}_b^{{\dagger}}(\overline{\omega}, \overline{{\boldsymbol{k}}}){\boldsymbol{\varXi}}_\text{H}(\overline{\omega}, \overline{{\boldsymbol{k}}}) = {\boldsymbol{0}}, \end{equation}

which are two mutually adjoint representations of the same equation.

Below, we consider the case when (7.13) is satisfied only for one mode at a time, so summation over $b$ and the index $b$ itself can be omitted. (A more general case is discussed, for example, in Dodin et al. (Reference Dodin, Ruiz, Yanagihara, Zhou and Kubo2019).) Then, $\smash {{\breve {{\boldsymbol {\varPsi }}}} = {\boldsymbol {\eta }}(\overline {\omega }, \overline {{\boldsymbol {k}}}){\breve {a}}}$,

(7.15)\begin{equation} \overline{\mathfrak{L}} = \frac{1}{4}\,\varLambda(\overline{\omega}, \overline{{\boldsymbol{k}}})|{\breve{a}}|^2, \end{equation}

and $\overline {\omega }$ is connected with $\overline {{\boldsymbol {k}}}$ via $\smash {\overline {\omega } = w(\overline {{\boldsymbol {k}}})}$, where $w({\boldsymbol {k}}) = -w(-{\boldsymbol {k}})$ is the function that solves $\varLambda (w({\boldsymbol {k}}), {\boldsymbol {k}}) = 0$. Also importantly, (7.14) ensures that

(7.16)\begin{align} \partial_{\unicode{x25AA}} \varLambda(\overline{\omega}, \overline{{\boldsymbol{k}}}) & = ((\partial_{\unicode{x25AA}} {\boldsymbol{\eta}}^{{\dagger}}) {\boldsymbol{\varXi}}_\text{H}{\boldsymbol{\eta}} + {\boldsymbol{\eta}}^{{\dagger}}(\partial_{\unicode{x25AA}} {\boldsymbol{\varXi}}_\text{H}){\boldsymbol{\eta}} + {\boldsymbol{\eta}}^{{\dagger}}{\boldsymbol{\varXi}}_\text{H}(\partial_{\unicode{x25AA}} {\boldsymbol{\eta}}))\big|_{(\omega, {\boldsymbol{k}}) = (\overline{\omega}, \overline{{\boldsymbol{k}}})} \nonumber\\ & = ({\boldsymbol{\eta}}^{{\dagger}}(\partial_{\unicode{x25AA}} {\boldsymbol{\varXi}}_\text{H}){\boldsymbol{\eta}})\big|_{(\omega, {\boldsymbol{k}}) = (\overline{\omega}, \overline{{\boldsymbol{k}}})}, \end{align}

where $\smash {{\unicode{x25AA}} }$ can be replaced with any variable.

7.2. Conservative eikonal waves

7.2.1. Basic properties

In case of a quasimonochromatic eikonal wave and, possibly, inhomogeneous non-stationary plasma, one can apply the same arguments as in § 7.1 except the above equalities are now satisfied up to $\smash {\mathcal {O}(\epsilon )}$. For a single-mode wave, one has

(7.17)\begin{equation} \widetilde{{\boldsymbol{\varPsi}}}(t, {\boldsymbol{x}}) = \operatorname{re}(\widetilde{{\boldsymbol{\varPsi}}}_{\text{c}}(t, {\boldsymbol{x}})) + \mathcal{O}(\epsilon), \qquad \widetilde{{\boldsymbol{\varPsi}}}_{\text{c}} = \mathrm{e}^{\mathrm{i} \theta(t, {\boldsymbol{x}})} {\boldsymbol{\eta}}(t, {\boldsymbol{x}}) {\breve{a}}(t, {\boldsymbol{x}}), \end{equation}

where the local frequency and the wavevector,

(7.18)\begin{equation} \overline{\omega} \doteq{-} \partial_t\theta, \qquad \overline{{\boldsymbol{k}}} \doteq \partial_{{\boldsymbol{x}}} \theta \end{equation}

are slow functions of $\smash {(t, {\boldsymbol {x}})}$, and so is $\smash {{\boldsymbol {\eta }}(t, {\boldsymbol {x}}) \doteq {\boldsymbol {\eta }}(t, {\boldsymbol {x}}, \overline {\omega }(t, {\boldsymbol {x}}), \overline {{\boldsymbol {k}}}(t, {\boldsymbol {x}}))}$, which satisfies (7.3). Then,

(7.19)\begin{equation} \overline{\mathfrak{L}} = \frac{1}{4}\,\varLambda(t, {\boldsymbol{x}}, \overline{\omega}, \overline{{\boldsymbol{k}}})\,|{\breve{a}}(t, {\boldsymbol{x}})|^2 + \mathcal{O}(\epsilon). \end{equation}

Within the leading-order theory, the term $\smash {\mathcal {O}(\epsilon )}$ is neglected.Footnote ³¹ Then, the least-action principle

(7.20)\begin{equation} 0 = \frac{\delta S[\theta, {\breve{{\boldsymbol{a}}}}^*, {\breve{{\boldsymbol{a}}}}]}{\delta {\breve{a}}^{b*}} \approx \frac{1}{4}\,\varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}}){\breve{a}}^b, \qquad 0 = \frac{\delta S[\theta, {\breve{{\boldsymbol{a}}}}^*, {\breve{{\boldsymbol{a}}}}]}{\delta {\breve{a}}^b} \approx \frac{1}{4}\,{\breve{a}}^{b*}\varLambda_b(\overline{\omega}, \overline{{\boldsymbol{k}}}) \end{equation}

leads to the same (but now local) dispersion relation as for monochromatic waves, $\smash {\varLambda (t, {\boldsymbol {x}}, \overline {\omega }, \overline {{\boldsymbol {k}}}) = 0}$. This shows that quasimonochromatic waves are also on-shell, and thus they satisfy (7.16) as well. Also notice that the dispersion relation can now be understood as a Hamilton–Jacobi equation for the eikonal phase $\smash {\theta }$:

(7.21)\begin{equation} \varLambda(t, {\boldsymbol{x}}, -\partial_t \theta, \partial_{{\boldsymbol{x}}}\theta) = 0. \end{equation}

Like in the previous section, let us introduce the function $\smash {w}$ that solves

(7.22)\begin{equation} \varLambda(t, {\boldsymbol{x}}, w(t, {\boldsymbol{x}}, {\boldsymbol{k}}), {\boldsymbol{k}}) = 0 \end{equation}

and therefore satisfies

(7.23)\begin{equation} w(t, {\boldsymbol{x}}, {\boldsymbol{k}}) ={-}w(t, {\boldsymbol{x}}, -{\boldsymbol{k}}). \end{equation}

Differentiating (7.22) with respect to $t$, ${\boldsymbol {x}}$ and ${\boldsymbol {k}}$ leads to

(7.24a)\begin{align} & \partial_{t} \varLambda + (\partial_\omega \varLambda)\partial_{t}w = 0, \end{align}

(7.24b)\begin{align} & \partial_{{\boldsymbol{x}}} \varLambda + (\partial_\omega \varLambda)\partial_{{\boldsymbol{x}}}w = 0, \end{align}

(7.24c)\begin{align} & \partial_{{\boldsymbol{k}}} \varLambda + (\partial_\omega \varLambda)\partial_{{\boldsymbol{k}}}w = 0, \end{align}

where the derivatives of $\varLambda$ are evaluated at $\smash {(t, {\boldsymbol {x}}, w(t, {\boldsymbol {x}}, {\boldsymbol {k}}), {\boldsymbol {k}})}$. In particular, (7.24c) gives

(7.25)\begin{equation} {\boldsymbol{v}}_{\text{g}} \doteq \frac{\partial w}{\partial {\boldsymbol{k}}} ={-} \frac{\partial_{{\boldsymbol{k}}} \varLambda}{\partial_\omega \varLambda}, \end{equation}

for the group velocity $\smash {{\boldsymbol {v}}_{\text {g}}}$, whose physical meaning is to be specified shortly.

Because $\theta$ is now an additional dynamical variable, one also obtains an additional Euler–Lagrange equation

(7.26)\begin{equation} 0 = \delta_{\theta} S[\theta, {\breve{a}}^*, {\breve{a}}] = \partial_t\mathcal{I} + \partial_{{\boldsymbol{x}}} \cdot {\boldsymbol{\mathcal{J}}}, \end{equation}

where $\mathcal {I}$ is called the action density and ${\boldsymbol {\mathcal {J}}}$ is the action flux density:

(7.27)\begin{align} & \mathcal{I} \doteq \frac{\partial \mathfrak{L}}{\partial \omega} = \frac{|{\breve{a}}|^2}{4}\frac{\partial \varLambda}{\partial \omega} = \frac{|{\breve{a}}|^2}{4}\,{\boldsymbol{\eta}}^{{\dagger}}\,\frac{\partial {\boldsymbol{\varXi}}_\text{H}}{\partial \omega}\, {\boldsymbol{\eta}}, \end{align}

(7.28)\begin{align} & \mathcal{J}^i \doteq{-}\frac{\partial \mathfrak{L}}{\partial k_i} ={-} \frac{|{\breve{a}}|^2}{4}\frac{\partial \varLambda}{\partial k_i} ={-} \frac{|{\breve{a}}|^2}{4}\,{\boldsymbol{\eta}}^{{\dagger}}\,\frac{\partial {\boldsymbol{\varXi}}_\text{H}}{\partial k_i}\,{\boldsymbol{\eta}}, \end{align}

where we used (7.16) and the derivatives are evaluated on $\smash {(t, {\boldsymbol {x}}, w(t, {\boldsymbol {x}}, \overline {{\boldsymbol {k}}}(t, {\boldsymbol {x}})), \overline {{\boldsymbol {k}}}(t, {\boldsymbol {x}}))}$. Using (7.25), one can also rewrite (7.28) as

(7.29)\begin{equation} {\boldsymbol{\mathcal{J}}} = \overline{{\boldsymbol{v}}}_{\text{g}} \mathcal{I}, \qquad \overline{{\boldsymbol{v}}}_{\text{g}}(t, {\boldsymbol{x}}) \doteq {\boldsymbol{v}}_{\text{g}}(t, {\boldsymbol{x}}, \overline{{\boldsymbol{k}}}(t, {\boldsymbol{x}})). \end{equation}

(The arguments $(t, {\boldsymbol {x}})$ will be omitted from now on for brevity. We will also use $({\boldsymbol {k}})$ as a shorthand for $(w({\boldsymbol {k}}), {\boldsymbol {k}})$ where applicable.) Then, (7.26) becomes

(7.30)\begin{equation} \partial_t\mathcal{I} + \partial_{{\boldsymbol{x}}} \cdot (\overline{{\boldsymbol{v}}}_{\text{g}}\mathcal{I}) = 0,\end{equation}

which can be a understood as a continuity equation for quasiparticles (‘photons’ or, more generally, ‘wave quanta’) with density $\smash {\mathcal {I}}$ and fluid velocity $\smash {\overline {{\boldsymbol {v}}}_{\text {g}}}$ (see also § 7.2.2). Thus, if an eikonal wave satisfies the least-action principle, its total action $\smash {\int \mathrm {d}{\boldsymbol {x}}\,\mathcal {I}}$ (‘number of quanta’) is an invariant. This conservation law can be attributed to the fact that the wave Lagrangian density $\smash {\overline {\mathfrak {L}}}$ depends on derivatives of $\smash {\theta }$ but not on $\smash {\theta }$ per se.

Also notice the following. By expanding (7.19) in $\smash {\partial _t \theta }$ around $\smash {\partial _t \theta = -w(t, {\boldsymbol {x}}, \partial _{{\boldsymbol {x}}}\theta )}$, which is satisfied on any solution, one obtains

(7.31)\begin{equation} \overline{\mathfrak{L}} \approx{-}\frac{1}{4}\,(\partial_t \theta + w(t, {\boldsymbol{x}}, \partial_{{\boldsymbol{x}}}\theta))\partial_\omega \varLambda\,|{\breve{a}}|^2 ={-}(\partial_t \theta + w(t, {\boldsymbol{x}}, \partial_{{\boldsymbol{x}}}\theta))\mathcal{I}, \end{equation}

where we used that $\smash {\overline {\mathfrak {L}}(t, {\boldsymbol {x}}, -\partial _t\theta, \partial _{{\boldsymbol {x}}}\theta ) = 0}$ due to (7.21). Then, one arrives at the canonical form of the action integral (Hayes Reference Hayes1973)

(7.32)\begin{equation} S[\mathcal{I}, \theta] ={-} \int \mathrm{d} t\,\mathrm{d}{\boldsymbol{x}}\,(\partial_t \theta + w(t, {\boldsymbol{x}}, {\boldsymbol{k}}))\mathcal{I}. \end{equation}

From here, $\smash {\delta _{\mathcal {I}} S = 0}$ yields the dispersion relation in the Hamilton–Jacobi form $\smash {\partial _t \theta + w(t, {\boldsymbol {x}}, {\boldsymbol {k}}) = 0}$, and $\smash {\delta _{\theta} S = 0}$ yields the action conservation (7.30).