# [time 335] Re: [time 331] Re: [time 327] Re: [time 326] Re: [time 325] Re: Fisherinformation andrelativity

Fri, 21 May 1999 00:55:12 +0900

Dear Matti,

I found time to see Frieden's book more closely. I will try to explain the
derivation of Maxwell's equations.

----- Original Message -----
From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
Sent: Thursday, May 20, 1999 1:09 PM
Subject: [time 331] Re: [time 327] Re: [time 326] Re: [time 325] Re:
Fisherinformation andrelativity

[snip]

> > Frieden considers J, but I cannot understand why such a messy treatment
of J
> > is necessary.
>
> What I thought was that integral of B^2 might correspond to
> I and integral of E^2 might correspond J. But it seems that this
> kind of interpretation is not possible. Thank You in any case.

Frieden's construction of Fisher information I for classical electrodynamics
is

I = 4c \int dr dt \sum_{n=1}^4 [\nabla q_n \cdot \nabla q_n -c^{-2}
(\partial q_n / \partial t)^2],

where

(q_1,q_2,q_3)=(A_1,A_2,A_3) = A is the vector potential

and q_4 = \phi is a scalar potential. This I has the same form as that for
QM in [time 321].

J is

J=4c \int dr dt \sum_{n=1}^4 E_n J_n(q, j, \rho),

where J_n are functions (of q, j, \rho) determined by using Frieden's
axioms, j is the current, and \rho is the charge density.

His variational axiom (which is equivalent with the principle of the least
action except for that the action is replaced by I-J) is

\delta (I - J) = 0.

By the above definitions of I and J, the Euler-Lagrange equation for this
case is

( \nabla^2 - c^{-2} (\partial / \partial t)^2 ) q_n = - 2^{-1}\sum_m
(\partial J_m / \partial q_n).

>From this and some other considerations by the use of his axioms Frieden
derives the equation

( \nabla^2 - c^{-2} (\partial / \partial t)^2 ) q_n = - (4 \pi / c) J_s,

where J_s = (j, c\rho).

The fields E and B are then defined by

B = \nabla x A,

and

E = - \nabla \phi - c^{-1} (\partial A / \partial t).

Maxwell's equations follows from these.

Fundamentally J should be equal to I as information, but in classical case,
this is not the case according to Frieden, but

I= J/2.

Frieden considers this as an expression of the incompleteness of classical
mechanics. In QM, J is obtained as the Fourier transform of I, thus I=J with
J being expressed in momentum-energy space. The treatment of J is clear in
this case, and I think this tells some truth. The classical case looks
different and complicated, which let me write "messy" in the former post.

Best wishes,
Hitoshi

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