**Matti Pitkanen** (*matpitka@pcu.helsinki.fi*)

*Fri, 21 May 1999 07:41:31 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 339] Re: Fisher information andrelativity"**Previous message:**Hitoshi Kitada: "[time 337] Re: [time 336] Re: [time 333] Re: [time 332] Re: Big Picture"**In reply to:**Matti Pitkanen: "[time 336] Re: [time 333] Re: [time 332] Re: Big Picture"**Next in thread:**Matti Pitkanen: "[time 339] Re: Fisher information andrelativity"

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
*

*>
*

*>
*

**Next message:**Matti Pitkanen: "[time 339] Re: Fisher information andrelativity"**Previous message:**Hitoshi Kitada: "[time 337] Re: [time 336] Re: [time 333] Re: [time 332] Re: Big Picture"**In reply to:**Matti Pitkanen: "[time 336] Re: [time 333] Re: [time 332] Re: Big Picture"**Next in thread:**Matti Pitkanen: "[time 339] Re: Fisher information andrelativity"

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