Fisher information as the hallmark of quantum theory

The remaining nontrivial task is the derivation of a local differential equation for $S$ and $ρ$ from the integral equation (37). 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

(39)

the general solution of (37) may be written in the form

-∂--ρ--¯ L (x, t) = Gk (x, t), ∂ xk

(40)

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

(41)

Then, our problem is to find a function $L0$ which fulfills the differential equation

¯L (x, t ) - L = 0, 0

(42)

and condition (31). 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 $L0$ :

( ) ∂ ρ ∂ 2ρ ¯ ----- --------- L (x, t ) - L0 ρ, , = 0 (43) ∂ x ∂ x ∂ x [ ( ) ] ∫ 2 3 ∂ ρ ∂ ρ δ d x ρ L¯ (x, t ) - L ρ, ----, --------- = 0, (44) 0 ∂ x ∂ x ∂ x

for the variable $β = ρL0$ . Eq. (45) is a straightforward generalization of the corresponding one-dimensional relation [equation (68) of I] to three spatial dimensions.

Besides (45) 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 (43) our final differential equation for $S$ and $ρ$ must not contain higher than second order derivatives of $ρ$ ]. Consequently, the first term in Eq. (45) (as well as the sum of the remaining terms) has to vanish separately and (45) can be replaced by the two equations

--∂------∂----------∂-β--------- ( 2 ) = 0 (46) ∂ xk ∂ xi --∂---ρ---- ∂ ∂x ∂x k i ∂ ∂ β ∂ β β ----------(-------)--- ----- + --- = 0. (47) ∂ ρ ∂ xk ∂ ------ ∂ ρ ρ ∂xk

where $B0$ is an arbitrary constant. Eq. (48) 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

(49)

it is easily verified that the solution (48) obeys also condition (31). Using the decomposition (16) and renaming $B$ according to $B = ℏ2 ∕4m$ , the continuity equation (3) and the second differential equation (43) respectively, take the form

( ) ∂-ρ-- --∂----ρ--- -∂-S--- e- + - Ak = 0, (50) ∂ t ∂ xk m ∂ xk c ( ) √ -- ∂ S 1 ∑ ∂ S e 2 ℏ2 △ ρ ----- ------ ------- -- --------------- + e ϕ + - Ak + V - √ -- = (501.) ∂ t 2m ∂ xk c 2m ρ k

is introduced, the two equations (50), (51) may be written in compact form as real and imaginary parts of the linear differential equation

ℏ ∂ 1 ℏ ∂ e ( -------+ e ϕ ) ψ + ------(--------- -- ⃗A )2 ψ + V ψ = 0, i ∂ t 2m i ∂ ⃗x c

(53)

which completes our derivation of Schrödinger’s equation in the presence of a gauge field.

Eq. (53) is in manifest gauge-invariant form. The gauge-invariant derivatives of $S˜$ with respect to $t$ and $⃗x$ correspond to the two brackets in (53). In particular, the canonical momentum $∂ S ∕ ∂ ⃗x$ corresponds to the momentum operator proportional to $∂ ∕ ∂ ⃗x$ . Very frequently, Eq. (53) is written in the form

ℏ ∂ - -------ψ = H ψ, , i ∂ t

(54)

with the Hamilton operator

1 ℏ ∂ e 2 H = ------(--------- --A⃗ ) + V + e ϕ, 2m i ∂ ⃗x c

(55)

Our final result, Eqs. (54), (55), agrees with the result of the conventional quantization procedure. In its simplest form, the latter starts from the classical relation $H (x, p ) = E$ , where $H (x, p )$ is the Hamiltonian of a classical particle in a conservative force field, and $E$ is its energy. To perform a ”canonical quantization” means to replace $p$ and $E$ by differential expressions according to (1) and let then act both sides of the equation $H (x, p ) = E$ on states $ψ$ of a function space. The ’black magic’ involved in this process has been eliminated, or at least dramatically reduced, in the present approach, where Eqs. (54), (55) have been derived from a set of assumptions which can all be interpreted in physical terms.

The Hamiltonian (55) depends on the potentials $Φ$ and $A⃗$ 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 $H$ , see e.g. [34]. 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 SI 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.

5 Fisher information as the hallmark of quantum theory