Bose–Einstein condensation was predicted in 1925 [Ein24, Ein25] but, except for its indirect manifestation in the superfluidity of He4 [Lon38, Kap38, AllMis38], it remained a purely theoretical construct until 1995, when condensation in alkali gases was achieved in the laboratory [CorWie02, Ket02]. Since then, a great deal of the theoretical work in the previous 70 years has been put to the experimental test, while new avenues have been opening up, such as the superfluid-insulator or Mott transition [FWGF89, GMEHB02, CaHuRe06], the BEC-BCS cross-over [Leg06, Reg04] and the Tonks gas regime [Gir60, Par04]. Because of the great experimental control over the relevant parameters and the deep understanding of the fundamental physics, BECs have become a field of choice to perform experiments of interest not just in atomic and molecular physics, but also in quantum optics, condensed matter physics, quantum critical dynamics, even field theory, gravitation, cosmology and black hole physics [CoEnWi99, PetSmi02, Sou02, BonSen04, ParZha93, ParZha95, UnrSch07, BaLiVi05]. Moreover, cold Bose gases on optical lattices have been proposed as a possible implementation of a quantum information processing (QIP) device [JakZo104], boosting new interest in these systems.

As introductions and reviews on this fast evolving subject abound, it is perhaps more fitting for us to focus on certain aspects of the nonequilibrium field theory of BEC, specifically [And04] the application of quantum field theoretic methods described in this book to the description of nonequilibrium evolution of condensed gases in magneto-optical traps [ChCoPh95].

Since the systems described by quantum fields are by definition extended, it is natural to think that in some limit they may reasonably well be approximated as fluids. This means that the state of the system is parameterized by a few locally well-defined fields such as temperature or energy density, obeying a set of hyperbolic equations of motion. A concrete example is the extensive use of fluid models to describe high-energy collisions [BelLan56, CarDuo73, CoFrSc74, Bjo83, CarZac83].

Our earlier derivation of quantum kinetic theory suggests a way to put this insight on a formal basis. Within its range of validity, the Boltzmann equation will drive the one-particle distribution function towards local thermal equilibrium. On scales much larger than the local thermalization scale, we expect to see hydrodynamical behavior [BeCoPa02]. This is, after all, the usual way of deriving hydrodynamics from kinetic theory [Hua87]. Beware, notwithstanding, that even at the level of classical kinetic theory there are still open questions regarding the cross-over from the kinetic to the hydro regime [KarGor02, KarGor03].

If we understand hydrodynamics as stated in the first paragraph of this chapter, then a system defined in terms of a quantum field may not have a hydrodynamic limit. This has been shown in [Elz02] for the case of a free Fermi field.

Perhaps due to its technical complexity, oftentimes one sees in research papers on nonequilibrium quantum field theory (NEqQFT) more emphasis placed on the field-theoretical formalisms than the ideas these sophisticated techniques attempt to capture, or the issues such problems embody. All the more so, we need some basic understanding of the important issues and concepts in nonequilibrium statistical mechanics (NEqSM), and how they are manifested in the context of quantum field theory. Many important advances in this field came from asking such questions and finding out how to answer them in the language of quantum field theory. Because of this somewhat skewed existing emphasis in NEqQFT, and since we do not assume the reader to have had a formal course on NEqSM before, we shall give a brief summary of the basic concepts of NEqSM relevant to the field-theoretical processes discussed in this book. Many fine monographs and reviews written on this subject take a more formal mathematical approach. approach. Since our purpose here is to familiarize readers with these issues and their subtleties, rather than training them to work in the rich field of NEqSM (which includes in addition to the traditional subject matter such as the projection operator formalism and open system concepts, also current topics at their foundation, such as dynamical systems and quantum chaos), we choose to approach these topics in a more intuitive and physical way, sacrificing by necessity rigor and completeness.

One of the major goals in the establishment of a quantum field theory for nonequilibrium systems is to study dynamical problems, following the evolution of the expectation value of a physical variable with respect to an in state. This is different from a scattering problem characterized by the transition amplitude between the in and out states, as is treated in every textbook on quantum field theory. This problem is usually coupled with how one could identify a relevant sector of the theory as the system (light fields vs. heavy fields, slow modes vs. fast modes, long wavelength modes vs. short ones, etc.) and determine the effect of its other sectors as the environment on this system, as we have discussed in the last chapter.

Given a classical nonequilibrium system, described, for example, by a Langevin equation, there are essentially three possible strategies to follow. One may attempt to solve it, usually numerically. In the quantum field case, this gets difficult beyond the linear case. Second, one may try to transform it into an equation for the evolution of a probability distribution function in the system's configuration space. In the quantum field case the relevant object is the reduced density matrix, and the relevant equation is the Liouville-von Neumann equation. This is also infeasible beyond the linear case, unless under restrictive approximations (such as Gaussianity) which in fact reduce this approach to the third and coarsest.

Quantum mechanics as we have discussed it here works remarkably well. When we can figure out how to calculate something using quantum mechanics, we know of no situation where it gives the wrong answer. At a pragmatic level, it therefore works. It does, however, have some aspects about it that are very different from the classical view of the world. Interpreting what quantum mechanics says about reality is a very tricky subject with strange consequences and truly bizarre proposals about how reality actually works. Here, we try to give a short introductory summary of some of those key ideas. It is necessarily brief and incomplete and is certainly not conclusive. Indeed, this field of the interpretation of quantum mechanics can be considered unresolved, even if protagonists of various interpretations might attempt to convince us otherwise. Fundamentally, we do not know how to resolve some of the more philosophical aspects because we have no experiment to discriminate between them, though some points previously believed to be only philosophical have been resolved by experiments, with remarkable consequences.

Hidden variables and Bell's inequalities

Is quantum mechanics truly random? Perhaps the solution to the apparent randomness of quantum mechanics, with states collapsing into eigenstates with only statistical weights, is that quantum mechanics as stated is incomplete, in the sense that classical statistical mechanics is incomplete.

Thus far, we have dealt primarily with energy, position, and linear momentum and have proposed operators for each of these. One other quantity that is important in classical mechanics, angular momentum, is particularly important also in quantum mechanics. Here, we introduce angular momentum, its operators, eigenvalues, and eigenfunctions. If this discussion seems somewhat abstract, the reader can be assured that the concepts of angular momentum will become very concrete in the discussion of the hydrogen atom.

One aspect of angular momentum that is different from the quantities and operators discussed previously is that its operators always have discrete eigenvalues. Whereas linear momentum is associated with eigenfunctions that are functions of position along a specific spatial direction, angular momentum is associated with eigenfunctions that are functions of angle or angles about a specific axis. The fact that the eigenvalues are discrete is associated with the fact that for a single-valued spatial function, once we have gone an angle 2π about a particular axis, we are back to where we started. The wavefunction is presumably continuous and single-valued and, hence, we must therefore have integral numbers of periods of oscillation with angle within this angular range; this requirement of integer numbers of periods leads to the discrete quantization of angular momentum.

Another surprising aspect of angular momentum operators is that the operators corresponding to angular momentum about different orthogonal axes (e.g., and) do not commute with one another (in contrast, e.g., to the linear momentum operators for the different orthogonal coordinate directions).

The “Δ” quoted is the absolute value of the uncertainty in the last two digits of the quoted numerical value corresponding to one standard deviation from the numerical value given. Hence, for example, the possible values of Planck's constant within one standard deviation of the best estimate shown lie between 6.626 068 24 and 6.626 069 28 J s.

The speed of light in vacuum has been chosen to have the exact value shown because the meter is now defined as the length of the path traveled by light in vacuum during the time interval of 1/299 792 458 of a second. The magnetic constant (also known as the permeability of free space) is chosen to have the value shown because it is an arbitrary constant that arises from the choice of the system of units and the electric constant (also known as the permittivity of free space) then follows from it and the (chosen) velocity of light because, by definition, so all three of these quantities have no uncertainty by definition. The Bohr magneton is µB = eħ/2me . The fine structure constant is α = e 2/4πε0 cħ.

These values are the CODATA Internationally recommended values as of 1998. Reference http://physics.nist.gov/cuu/Constants/index.html.

So far, we have introduced quantum mechanics through the example of the Schrödinger equation and the spatial and temporal wavefunctions that are solutions to it. This has allowed us to solve some simple but important problems and to introduce many quantum mechanical concepts by example. Quantum mechanics does, however, go considerably beyond the Schrödinger equation. For example, photons are not described by the kind of Schrödinger equation we have considered so far, though they are undoubtedly very much quantum mechanical.

To prepare for other aspects of quantum mechanics and to make the subject easier to deal with in more complex problems, we need to introduce a more general and extended mathematical formalism. This formalism is actually mostly linear algebra. Readers probably have encountered many of the basic concepts already in subjects such as matrix algebra, Fourier transforms, solutions of differential equations, possibly (though less likely) integral equations, or analysis of linear systems, in general. For this book, we assume that the reader is familiar with, at least, the matrix version of linear algebra – the other examples are not necessary prerequisites. The fact that the formalism is based on linear algebra is because of the basic observation that quantum mechanics is apparently absolutely linear in certain specific ways as we discussed previously.

Thus far, we have dealt with the state of the quantum mechanical system as the wavefunction Ψ(r, t) of a single particle.