Hitoshi Kitada (hitoshi@kitada.com)
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>
To: Hitoshi Kitada <hitoshi@kitada.com>
Cc: Time List <time@kitada.com>
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
This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:10:32 JST