Fisher information

5 Fisher information

In this section a local differential equation for $S$ and $ρ$ 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
$( ) 2 ∂ ˜S 1 ∂ S˜ (x, t) L˜ (x, t ) = ----- + ------ ------------- + V (x, t), ∂ t 2m ∂ x$ (47)

the general solution of (46) may be written in the form
$∂ ρ -------˜ L (x, t) = Gk (x, t), ∂ xk$ (48)

where the three functions $Gk (x, t )$ have to vanish upon integration over $3 R$ and are otherwise arbitrary. If we restrict ourselves to an isotropic law, we may write
$∂ ρ G (x, t) = ------L . k ∂ x 0 k$ (49)

Then, our problem is to find a function $L0$ which fulfills the differential equation
$˜L (x, t ) - L = 0, 0$ (50)

and condition (40). The method used in I for a one-dimensional situation, to determine $L 0$ 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 $L0$ :
( )
2
˜ ∂-ρ-- -∂---ρ---
L (x, t ) - L0 ρ, , = 0
∂ x ∂ x ∂ x
[ ( ) ]
∫
∂ ρ ∂ 2 ρ
δ d3x ρ L˜ (x, t) - L ρ, ----, --------- = 0,
0
∂ x ∂ x ∂ x
(51)

(52)
where $L0$ contains only derivatives of $ρ$ up to second order and does not explicitely depend on $x, t$ . If Eq. (51) is taken into account, the Euler-Lagrange equations of the variational problem (52) lead to the following differential equation
$∂ ∂ ∂ β ∂ ∂ β ∂ β β - ----------------(------------)-- + ----------(-------)-- - ----- + ---= 0 ∂ x ∂ x --∂2--ρ---- ∂ x -∂-ρ-- ∂ ρ ρ k i∂ k ∂ ∂x ∂xk ∂xi k$ (53)

for the variable $β = ρL0$ . 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 $L0$ may depend on second order derivatives of $ρ$ 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 $S$ and $ρ$ must not contain higher than second order derivatives of $ρ$ ]. 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
∂ ∂ ∂ β
----------------(------------)-- = 0
∂ x ∂ x ∂2 ρ
k i ∂ -----------
∂xk ∂xi

--∂--------∂-β------ ∂-β-- β--
( ) - + = 0.
∂ x -∂-ρ-- ∂ ρ ρ
k ∂ ∂x
k
(54)

(55)
In I a new derivation of Fisher’s functional has been obtained, using the general solution of the one-dimensional version of (53), as well as the so-called composition law. In the present three-dimensional situation we set ourselves a less ambitious aim. We know that Fisher’s functional describes the maximal amount of disorder. If we are able to find a solution of (54), (55) that agrees with this functional (besides ’null-terms’ giving no contribution to the Euler-Lagrange equations) then we will accept it as our correct solution. It is easy to see that this solution is given by
$⌊ ⌋ ∑ ( )2 ∑ 2 1 ∂ ρ 1 ∂ ρ L0 = B0 ⌈ - ------ ------- + -- ------⌉ , 2 ρ2 ∂ x ρ ∂ x2 j j j j$ (56)

where $B0$ 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
$√ -- √ -- √ -- √ -- √ -- √ -- ∂ ∂ ρ ∂ ρ ∂ ρ ∂ ∂ ρ 1 ∂ ∂ ρ ∂ ρ ---------------------- = ---------------------- + -- ----------------------, ∂ x ∂ x ∂ x ∂ x ∂ x ∂ x 2 ∂ x ∂ x ∂ x i i k k i i k i i$ (57)

it is easily verified that the solution (56) obeys also condition (40). Using the decomposition (3) and renaming $B$ according to $B = ℏ2 ∕4m$ , the continuity equation (5) and the second differential equation (51) respectively, take the form
( )

∂-ρ-- --∂-----ρ-- ∂--S--- e-
+ - Ak = 0,
∂ t ∂ x m ∂ x c
k( k ) √ --
∑ 2 2
∂-S-- -1---- -∂-S--- e- -ℏ----△-----ρ--
+ eφ + - Ak + V - √ -- = 0.
∂ t 2m ∂ xk c 2m ρ
k
(58)

(59)
The function $S$ occurring in (58), (59) is single-valued but not unique (not gauge-invariant). If now the complex-valued variable
$√ -- S- iℏ ψ = ρe ,$ (60)

is introduced, the two equations (58), (59) may be written in compact form as real and imaginary parts of the linear differential equation
$( ℏ ∂ ) 1 ( ℏ ∂ e ) ------ ------ ------- -- ⃗ 2 + e φ ψ + - A ψ + V ψ = 0, i ∂ t 2m i ∂ ⃗x c$ (61)

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 $S˜$ with respect to $t$ and $⃗x$ correspond to the two brackets in (61); in particular the canonical momentum $∂ S ∕ ∂ ⃗x$ corresponds to the momentum operator proportional to $∂ ∕ ∂ ⃗x$ . Very frequently, Eq. (61) is written in the form
$ℏ ∂ - -------ψ = H ψ, , i ∂ t$ (62)

with the Hamilton operator
$( ) -1---- ℏ---∂--- e- ⃗ 2 H = - A + V + e φ, 2m i ∂ ⃗x c$ (63)

This quantity is very useful despite the fact that it contains, if applied to $ψ$ , only one of the two gauge-invariant combinations present in the original time-dependent Schrödinger equation (61). The operator (63) depends on the potentials $Φ$ and $⃗ A$ and is consequently a non-unique (not gauge-invariant) mathematical object. The same is true for the time-development operator $U (H )$ which is an operator function of $H$ [20]. This non-uniqueness is a problem if $U (H )$ 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 $H$ and $U (H )$ 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 $H$ represents the set of all possible results of energy measurements of a single particle (note that $H$ is nevertheless a quantity characterizing an observable property of an ensemble; no single measurement result can be associated with $H$ , 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.

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