Subsidiary condition

9 Subsidiary condition

It will be convenient in the course of the following calculations to write the differential equation (26) in the form
$′ ′′ L (x, t) - L0 (ρ, ρ , ρ ) = 0,$ (53)

where $L (x, t)$ is given by (25) and $L 0$ is defined by
$′ ′ ′′ ′ ′′ Q----(ρ,--ρ--, ρ--)- L0 (ρ, ρ , ρ ) = . ρ ′$ (54)

In (54) it has been assumed that $L 0$ does not depend explicitely on $x, t$ , that the problem is basically of a time-independent nature, and that no higher derivatives with respect to $x$ than $ρ ′′$ occur. This last assumption is in agreement with the mathematical form of all fundamental differential equations of physics; we shall come back to this point later. Our task is to determine the functional form of $L0$ , with respect to the variables $′ ′′ ρ, ρ , ρ$ , using a general statistical extremal principle. As a consequence of the general nature of this problem we do not expect the solution to depend on the particular form of $L (x, t )$ . For the same reason $L 0$ does not depend on $S$ .

We tentatively formulate a principle of maximal disorder of the form (47) and identify $I [ρ ]$ with the Fisher functional (48). Then, the next step is to find a proper constraint $C [ρ ]$ . In accord with general statistical principles the prescribed quantity should have the form of a statistical average. A second condition is that our final choice should be as similar to the classical requirement (45) as possible. Adopting these criteria one is led more or less automatically to the constraint
$C [ρ ] = 0,$ (55)

where
$∫ [ ] C [ρ ] = dx ρ L (x, t ) - L (ρ, ρ ′, ρ ′′) . 0$ (56)

Our guideline in setting up this criterion has been the idea of a prescribed value of the average energy; the new term $L0$ plays the role of an additional contribution to the energy. For $S (x, t ) = - E t + S0 (x )$ and $L0 = 0$ the constraint (55) agrees with (45) provided the ‘classical‘ identification of $p$ with the gradient of $S (x ) 0$ is performed [see (8)]. The most striking difference between (45) and (55) is the fact that the quantity $L - L 0$ in (56), whose expectation value yields the constraint, is not defined independently from the statistics [like $E$ in (45)] but depends itself on $ρ$ (and its derivatives up to second order). This aspect of non-classical theories has already been discussed in section 5.

Let us try to apply the mathematical apparatus of variational calculus [49] to the constraint problem (47) with the ”‘entropy”’ functional $I [ρ ]$ defined by (48) and a single constraint defined by (55) and (56) (there is no normalization condition here because we do not want to exclude potentially meaningful non-normalizable states from the consideration). Here we encounter immediately a first problem which is due to the fact that our problem consists in the determination of an unknown function $L0$ of $′ ′′ ρ, ρ , ρ$ . This function appears in the differential equation and in the subsidiary condition for the variational problem. Thus, our task is to identify from a variational problem the functional form of a constraint defining this variational problem. Variational calculus, starts, of course, from constraints whose functional forms are fixed; these fixed functionals are used to derive differential equations for the variable $ρ$ . Thus, whenever the calculus of variations is applied, the function $′ ′′ L0 ( ρ, ρ , ρ )$ must be considered as unknown but fixed. We shall have to find a way to ’transform’ the condition for the variation of $ρ (x )$ in a corresponding condition for the variation of $L ( ρ, ρ ′, ρ ′′) 0$ .

The variational calculation defined above belongs to a class of ’isoperimetric’ variational problems which can be solved using the standard method of Lagrange multipliers, provided certain mathematical conditions are fulfilled [49]. Analyzing the situation we encounter here a second problem, which is in fact related to the first. Let us briefly recall the way the variational problem (47) is solved, in particular with regard to the role of the Lagrange multipliers $λi$ [49]. Given the problem to find an extremal $ρ0 (x )$ of $I [ρ ]$ under $m$ constraints of the form $Ci [ρ ] = 0$ and two prescribed values of $ρ$ at the boundaries, one proceeds as follows. The Euler-Lagrange equation belonging to the functional (47) is solved. The general solution for $ρ$ depends (besides on $x$ ) on two integration constants, say $C 1$ and $C2$ , and on the $m$ Lagrange multipliers $λ1, ..., λm$ . To obtain the final extremal $ρ0 (x )$ , these $m + 2$ constants have to be determined from the two boundary values and the $m$ constraints (which are differential equations for isoperimetric problems). This is exactly the way the calculation has been performed (even though a simpler form of the constraints has been used) in the classical case. For the present problem, however, this procedure is useless, since we do not want the constraints to determine the shape of individual solutions but rather the functional form of a term in the differential equation, which is then the same for all solutions. For that reason the ‘normal‘ variational problem (47) does not work (we shall come back to a mathematical definition of ‘normal‘ and ‘abnormal‘ shortly). This means that the classical principle of maximal entropy, as discussed in section 7, cannot be taken over literally to the non-classical domain.

For the same reason, no subsidiary conditions can be taken into account in the calculations reported by Frieden [10] and by Reginatto [36]. In these works, a different route is chosen to obtain Schrödingers equation; in contrast to the present work (see below) the Fisher functional is added as a new term to a classical Lagrangian and the particular form of this new term is justified by introducing a new ”‘principle of extreme physical information”’ [12].

A variational problem is called ‘normal‘ if an extremal of the functional $I [ρ ] + λ1C [ρ ]$ (here we restrict ourselves to the present case of a single constraint) exists which is not at the same time an extremal of the constraint functional $C [ρ ]$ . If this is not the case, i.e. if the extremal is at the same time an extremal of $C [ρ ]$ , then the problem is called ‘abnormal‘ [49]. Then, the usual derivation becomes invalid and the condition (47) must be replaced by the condition of extremal $C [ρ ]$ alone,
$λ C [ρ ] → extremum, 1$ (57)

which then yields $δC [ρ ] = 0$ as only remaining condition to determine the extremal. This type of problem is also sometimes referred to as ”rigid”; the original formulation (47) may be extended to include the abnormal case by introducing a second Lagrange multiplier [49].

We conclude that our present problem should be treated as an abnormal variational problem since we thereby get rid of our main difficulty, namely the unwanted dependence of individual solutions on Lagrange multipliers [ $λ1$ drops actually out of Eq. (57)]. A somewhat dissatisfying (at first sight) feature of this approach is the fact that the Fisher functional $I [ρ ]$ itself does no longer take part in the variational procedure; the original idea of implementing maximal statistical disorder seems to have been lost. But it turns out that we shall soon recover the Fisher $I$ in the course of the following calculation. The vanishing of the first variation of $C [ρ ]$ , written explicitely as
$∫ [ ′ ′′ ] δC [ρ ] = δ dx ρ L (x, t ) - L0 (ρ, ρ , ρ ) = 0,$ (58)

means that (for fixed $L0$ ) the spatial variation of $ρ$ should extremize (minimize) the average value of the deviation from $L (x, t)$ . This requirement is [as a condition for $ρ (x )$ ] in agreement with the principle of minimal Fisher information as a special realization of the requirement of maximal disorder. Eq. (58) defines actually a Lagrangian for $ρ$ and yields as Euler-Lagrange equations a differential equation for $ρ$ . When this equation is derived the task of variational calculus is finished. On the other hand, we know that $ρ$ obeys also Eq. (53). Both differential equations must agree and this fact yields a condition for our unknown function $L 0$ . Eq. (53) also guarantees that the original constraint (55) is fulfilled. In this way we are able to ’transform’ the original variational condition for $ρ (x )$ in a condition for $′ ′′ L0 (ρ, ρ , ρ )$ . In the next section this condition will be used to calculate $L0$ and to recover the form of the Fisher $I$ .

It should be mentioned that Eq. (58) has been used many times in the last eighty years to derive Schrödinger’s equation from the Hamilton-Jacobi equation. The first and most important of these works is Schrödinger’s ”‘Erste Mitteilung”’ [39]. In all of these papers $L0$ is not treated as an unknown function but as a given function, constructed with the help of the following procedure. First, a transformation from the variable $S$ to a complex variable $ψ ′ = exp [iS ∕ ℏ ]$ is performed. Secondly, a new variable $ρ$ is introduced by means of the formal replacement $ψ ′ ⇒ ψ = ρ ψ ′$ . This creates a new term in the Lagrangian, which has exactly the form required to create quantum mechanics. More details on the physical motivations underlying this replacement procedure may be found in a paper by Lee and Zhu [29]. It is interesting to note that the same formal replacement may be used to perform the transition from the London theory of superconductivity to the Ginzburg-Landau theory [25]. There, the necessity to introduce a new variable is obvious, in contrast to the present much more intricate situation.

[next] [prev] [prev-tail] [front] [up]