Hitoshi Kitada (hitoshi@kitada.com)
Tue, 18 May 1999 00:30:07 +0900
Dear Matti,
Although I am still at chapt.3, I will try to give a brief explanation ...
----- Original Message -----
From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
To: <time@kitada.com>
Sent: Monday, May 17, 1999 1:44 PM
Subject: [time 319] Re: [time 318] correction to [time 317]
>
>
> Dear Hitoshi,
>
> could you explain how scalar wave equation results in Frieden's theory.
He assumes a priori that space-time coordinates is
(ict,x,y,z)=(x_0,x_1,x_2,x_3). So Lorentz transformation follows from
(exactly, is sufficient to assure) the invariance of Fisher information
under change of reference frames. In this context his wave equation is
Klein-Gordon one. Shroedinger equation follows from this as an approximation
as c tends to infinity.
Frieden considers the wave function \psi(t,x,y,z) as a complex valued
function. Fisher infromation I for this is
I = const \int_{R^4} dx \nabla \psi* \cdot \nabla \psi,
where
dx = |dx_0| dx_1 dx_2 dx_3,
\nabla=(\partial/\partial x_0, ... ,\partial/\partial x_3).
The bound information J is
J = const \int dr dt \psi* \psi
with r = (x,y,z).
Then the information loss K which should be extrematized (usually minimized)
is
K = I - J
= \int dr dt [ - \nabla\psi*\cdot\nabla\psi +
(1/c^2) (\partial \psi*/\partial t) (\partial \psi/\partial t)
- (m^2 c^2 / \hbar^2) \psi* \psi].
Thus the Euler-Lagrange equation for this variational problem is the
Klein-Gordon equation.
Frieden's points are in that the Fisher information I becomes the free
energy part by some simplification of the expression of the information and
in the argument that the bound information J should be introduced. The
latter point is in chapt. 4 and I have to postpone it. The Fisher
information I for one dimensional case is defined as
I = \int dx (\partial (\log p)/ \partial \tehat)^2 p
for a probability density p=p(x|\theta)=p(x,\theta)/p(\theta) (conditional
probability, here p is assumed real-valued).
In the case of translation invariant p such that p(x|\theta)=p(y-\theta), I
is equal to
I = \int dx p'(x)^2/p(x)
If p is expressed by real amplitude function q(x) as
p(x)= q(x)^2,
then
I = 4 \int dx q'(x)^2.
This is extended to complex valued amplitude function with introducing
multiple probability density functions, which gives the I for \psi above.
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