Matti Pitkanen (matpitka@pcu.helsinki.fi)
Fri, 21 May 1999 07:41:31 +0300 (EET DST)
Dear Hitoshi,
thank you for explanation. I finally understood the decomposition.
There seems however to be a potential paradox in case of free Maxwell
equations. Empty space Maxwell action contains only the I term. Which
would say that
J, which is the total information contained by the state vanishes. It
is not clear whether I, which is the available information and obviously
smaller than J, vanishes or is even non-negative. For planewave
solutions of Maxwell action action density vanishes identifically but what
happens in case general solution which is superposition of Fourier
components representing planewaves? Yes, action vanishes a a sum over
contributions over indidividual planewaves so that also I vanishes.
Empty space Maxwell field contains no information and their is no
information available.
If one considers (in absence of external sources) functional integral
approach, one functionally integrates
over all field configurations, also those which are do satisfy Maxwell
equations. Now J is always zero but I need not vanish: available
information can be larger than total information! Rather peculiar
situation!
Still one possible weak point. J represents the interaction energy of
radiation field with external current and need not be positive definite:
if J is non-negative, one can change the sign of charges and J changes
sign. What does it mean the information contained by system is negative?
One could understand the vanishing of J as expressing the absense of
coupling to observer represented by external currents: in this sense
Frieden's picture makes sense. Situation is however different from
Klein-Gordon for which mass term represents J (as far as I understood
correctly). On basis of Maxwell action one might guess that
in case of General Relativity the coupling to energy momentum tensor
of matter, or actually the action of matter would represent J whereas
curvatures scalar would represent I.
[My earlier picture about I-J decomposition was wrong: there are no
problems covariance.]
MP
On Fri, 21 May 1999, Hitoshi Kitada wrote:
> 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