Dime's generalised matrix theory

The principles of the theory can be completely understood by considering a dynamical system with one degree of freedom; this reduces the detail without affecting the essentials of the argument, and the extension to more degrees of freedom is fairly obvious.

In matrix mechanics a dynamical variable is denoted by a matrix g whose rows and columns refer to stationary states of the system; these are numbered by specified values α, α′, α″, … of some constant of integration x of the dynamical system, so that g (αα′) is a typical constituent of the matrix g. If for example, the constant of integration x is an action variable, the values α, α′, α″, … form a discrete set; if x is a momentum or an energy, they may have a continuous range of values, a discrete set, or both, x is a q-number whose ‘characteristic values’ are α, α′, α″, … or in other words, x is a diagonal matrix whose diagonal terms are α, α′, α″, …

The case where the α's form a continuous range is more general, so that this form will be used.