In this section a local differential equation for and will be derived from the integral equation (46). As our essential constraint we will use, besides general principles of simplicity (like homogeneity and isotropy of space) the principle of maximal disorder, as realized by the requirement of minimal Fisher information. Using the abbreviation
the general solution of (46) may be written in the form
where the three functions have to vanish upon integration over and are otherwise arbitrary. If we restrict ourselves to an isotropic law, we may write
Then, our problem is to find a function which fulfills the differential equation
and condition (40). The method used in I for a one-dimensional situation, to determine from the requirement of minimal Fisher information, remains essentially unchanged in the present three-dimensional case. The reader is referred to the detailed explanations reported in I.
In I it has been shown that this principle of maximal disorder leads to an anomalous variational problem and to the following conditions for our unknown function :
for the variable . Eq. (53) is a straightforward generalization of the corresponding one-dimensional relation [equation (68) of I] to three spatial dimensions.
Besides (53) a further (consistency) condition exists, which leads to a simplification of the problem. The function may depend on second order derivatives of but this dependence must be of a special form not leading to any terms in the Euler-Lagrange equations [according to (51) our final differential equation for and must not contain higher than second order derivatives of ]. Consequently, the first term in Eq. (53) (as well as the sum of the remaining terms) has to vanish separately and (53) can be replaced by the two equations
where is an arbitrary constant. Eq. (56) presents again the three-dimensional (and isotropic) generalization of the one-dimensional result obtained in I. By means of the identity
it is easily verified that the solution (56) obeys also condition (40). Using the decomposition (3) and renaming according to , the continuity equation (5) and the second differential equation (51) respectively, take the form
is introduced, the two equations (58), (59) may be written in compact form as real and imaginary parts of the linear differential equation
which completes our derivation of Schrödinger’s equation in the presence of a gauge field.
Eq. (61) is in manifest gauge-invariant form. The gauge-invariant derivatives of with respect to and correspond to the two brackets in (61); in particular the canonical momentum corresponds to the momentum operator proportional to . Very frequently, Eq. (61) is written in the form
with the Hamilton operator
This quantity is very useful despite the fact that it contains, if applied to , only one of the two gauge-invariant combinations present in the original time-dependent Schrödinger equation (61). The operator (63) depends on the potentials and and is consequently a non-unique (not gauge-invariant) mathematical object. The same is true for the time-development operator which is an operator function of . This non-uniqueness is a problem if is interpreted as a quantity ruling the time evolution of a single particle. It is no problem from the point of view of the statistical interpretation where and are primarily convenient mathematical objects which occur in a natural way if the time-dependence of statistically relevant (uniquely defined) quantities, like expectation values and transition probabilities, is to be calculated.
A fundamental ’static’ aspect of operators is their role as observables. The spectrum of eigenvalues of the Hamilton operator represents the set of all possible results of energy measurements of a single particle (note that is nevertheless a quantity characterizing an observable property of an ensemble; no single measurement result can be associated with , only the measurement of the complete spectrum of eigenvalues). In this sense the Hamilton operator corresponds to the classical Hamilton function (and analogous relations are postulated for arbitrary classical observables). This correspondence is obvious in the canonical quantization procedure. It is less obvious but nevertheless visible in the present statistical approach as defined in section 2. The classical Hamilton function is implicitly contained in the statistical conditions (6) and (7). The latter comprise the differential structure of the canonical equations, which is itself determined by the Hamilton function. The formal details of the relation between classical observables and operators in the present approach have still to be worked out.