# [time 324] Fisher information and relativity

Tue, 18 May 1999 17:20:41 +0900

Dear Matti,

----- Original Message -----
From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
Sent: Tuesday, May 18, 1999 2:51 PM
Subject: [time 323] Re: [time 320] Re: [time 319] Re: [time 318] correction
to [time317]

>
>
> On Tue, 18 May 1999, Hitoshi Kitada wrote:
>
> > 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)
>
>
> Here I am wondering how the negative sign in (\partial_tp)^2/p
> relateed with time derivative term comes.

This comes from

(\partial p/ \partial_{ict})^2= - c^{-2} (\partial p/ \partial_t)^2

since x_0=ict. Frieden considers the sum

I = \int dx \sum_{\nu=0}^3 (\partial p/ \partial_{x_\nu})^2/p(x)

for the PDF (probability density function) p(x) = p(x_0,x_1,x_2,x_3) that
are functions of multiple variable x=(x_0,...,x_3). This I gives the free
part of Klein-Gordon equation.

As I wrote in the original post [time 321] of [time 320] (that was cut
because I put a period at the top of a line), Frieden makes an ad hoc
assumption that space-time is relativistic. This is because Frieden thinks
the observation only, as his concern is solely with the explanation of how
the relativistic Lagrangian appears in physics and the relativistic
Lagrangian is the quantity that describes how nature looks, at the observer.

In my thought consciousness must be Euclidean because we, inside ourself,
think space Euclidean. Or because what we think as space has been called
Euclidean since Greek age. This I think is a mental property of humans. If
we did not have in mind a Euclidean space as an origin geometry, how could
we imagine the curved geometry?

Is Fisher information
> still in question when one uses imaginery coordinate x0 =it?
>
>
> >
> > If p is expressed by real amplitude function q(x) as
> >
> > p(x)= q(x)^2,
>
>
> Thank You! It was just this step which I did not really
> understand.
>
>
> >
> > 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.
> >
> >
>
> Best,
>
> MP
>

Best wishes,
Hitoshi

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