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

*Tue, 18 May 1999 08:51:20 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Hitoshi Kitada: "[time 324] Fisher information and relativity"**Previous message:**Lester Zick: "[time 322] Universal Angular Momentum"**Next in thread:**Hitoshi Kitada: "[time 324] Fisher information and relativity"

On Tue, 18 May 1999, Hitoshi Kitada wrote:

*> Dear Matti,
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*>
*

*> Although I am still at chapt.3, I will try to give a brief explanation ...
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*>
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*> ----- Original Message -----
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*> From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
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*> To: <time@kitada.com>
*

*> Sent: Monday, May 17, 1999 1:44 PM
*

*> Subject: [time 319] Re: [time 318] correction to [time 317]
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*>
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*>
*

*> >
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*> >
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*> > Dear Hitoshi,
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*> >
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*> > could you explain how scalar wave equation results in Frieden's theory.
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*>
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*> He assumes a priori that space-time coordinates is
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*> (ict,x,y,z)=(x_0,x_1,x_2,x_3). So Lorentz transformation follows from
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*> (exactly, is sufficient to assure) the invariance of Fisher information
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*> under change of reference frames. In this context his wave equation is
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*> Klein-Gordon one. Shroedinger equation follows from this as an approximation
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*> as c tends to infinity.
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*>
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*> Frieden considers the wave function \psi(t,x,y,z) as a complex valued
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*> function. Fisher infromation I for this is
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*>
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*> I = const \int_{R^4} dx \nabla \psi* \cdot \nabla \psi,
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*>
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*> where
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*>
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*> dx = |dx_0| dx_1 dx_2 dx_3,
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*>
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*> \nabla=(\partial/\partial x_0, ... ,\partial/\partial x_3).
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*>
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*> The bound information J is
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*>
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*> J = const \int dr dt \psi* \psi
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*>
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*> with r = (x,y,z).
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*>
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*> Then the information loss K which should be extrematized (usually minimized)
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*> is
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*>
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*> K = I - J
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*>
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*> = \int dr dt [ - \nabla\psi*\cdot\nabla\psi +
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*> (1/c^2) (\partial \psi*/\partial t) (\partial \psi/\partial t)
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*> - (m^2 c^2 / \hbar^2) \psi* \psi].
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*>
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*> Thus the Euler-Lagrange equation for this variational problem is the
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*> Klein-Gordon equation.
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*>
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*> Frieden's points are in that the Fisher information I becomes the free
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*> energy part by some simplification of the expression of the information and
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*> in the argument that the bound information J should be introduced. The
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*> latter point is in chapt. 4 and I have to postpone it. The Fisher
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*> information I for one dimensional case is defined as
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*>
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*> I = \int dx (\partial (\log p)/ \partial \tehat)^2 p
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*>
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*> for a probability density p=p(x|\theta)=p(x,\theta)/p(\theta) (conditional
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*> probability, here p is assumed real-valued).
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*>
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*> In the case of translation invariant p such that p(x|\theta)=p(y-\theta), I
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*> is equal to
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*>
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*> I = \int dx p'(x)^2/p(x)
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Here I am wondering how the negative sign in (\partial_tp)^2/p

relateed with time derivative term comes. Is Fisher information

still in question when one uses imaginery coordinate x0 =it?

*>
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*> If p is expressed by real amplitude function q(x) as
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*>
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*> p(x)= q(x)^2,
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Thank You! It was just this step which I did not really

understand.

*>
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*> then
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*>
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*> I = 4 \int dx q'(x)^2.
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*>
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*> This is extended to complex valued amplitude function with introducing
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*> multiple probability density functions, which gives the I for \psi above.
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*>
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*>
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Best,

MP

**Next message:**Hitoshi Kitada: "[time 324] Fisher information and relativity"**Previous message:**Lester Zick: "[time 322] Universal Angular Momentum"**Next in thread:**Hitoshi Kitada: "[time 324] Fisher information and relativity"

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