**Hitoshi Kitada** (*hitoshi@kitada.com*)

*Tue, 18 May 1999 00:38:54 +0900*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Lester Zick: "[time 322] Universal Angular Momentum"**Previous message:**Hitoshi Kitada: "[time 320] Re: [time 319] Re: [time 318] correction to [time 317]"**In reply to:**Matti Pitkanen: "[time 319] Re: [time 318] correction to [time 317]"

As there seemed some problems in communicating the former mail, I resend

this.

Hitoshi

----- Original Message -----

From: Hitoshi Kitada <hitoshi@kitada.com>

To: <time@kitada.com>

Sent: Tuesday, May 18, 1999 12:30 AM

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

*> 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
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*> Klein-Gordon one. Shroedinger equation follows from this as an
*

approximation

*> as c tends to infinity.
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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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*>
*

*> I = const \int_{R^4} dx \nabla \psi* \cdot \nabla \psi,
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*>
*

*> where
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*>
*

*> dx = |dx_0| dx_1 dx_2 dx_3,
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*>
*

*> \nabla=(\partial/\partial x_0, ... ,\partial/\partial x_3).
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*>
*

*> The bound information J is
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*>
*

*> J = const \int dr dt \psi* \psi
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*>
*

*> with r = (x,y,z).
*

*>
*

*> Then the information loss K which should be extrematized (usually
*

minimized)

*> is
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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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*>
*

*> 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

*> 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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*>
*

*> 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),
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I

*> is equal to
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*>
*

*> I = \int dx p'(x)^2/p(x)
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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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*>
*

*> then
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*>
*

*> I = 4 \int dx q'(x)^2.
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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.
*

*>
*

*> .
*

*> One point which remains obscure in Frieden's argument is when he
*

introduces

*> four vectors is in page 64, where he changes the conditional probability
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*> p(x| t) = p(x,t)/p(t) (t is time) which is Galilean invariant to a
*

possibly

*> covariant (wrt Lorentz transformation) p(x,t)=q(t,x)^2. Here he assumes a
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*> priori Lorentz invariance _ad hoc_, apart from his definition of Fisher
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*> information.
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*>
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*> J describes the prior knowledge of the physical phenomenon or onservation
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*> which is performed. This completely determines the Lagrangian, as I is the
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*> same in all cases describing the free energy.
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*>
*

*> Thus what Friden did is to give an _information theoretic interpretation
*

of

*> physics_, as all what is written above is well-known in physics.
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*>
*

*> > How the possible problems related to the signature of Minkowski metric
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*> > are handled?
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*>
*

*> As explained he assumes the Minkowskian structure of space-time a priori.
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If

*> he chooses Euclidean information I (this is possible by the defintiion),
*

he

*> gets directly Schroedinger equation.
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*>
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*> And how I-J composition appears at the level of
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*> > scalar wave equation.
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*>
*

*> I still do not reach the part J. I have lectures till Thursday, so maybe
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*> after then I will continue ...
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*>
*

*> Best wishes,
*

*> Hitoshi
*

*>
*

*>
*

**Next message:**Lester Zick: "[time 322] Universal Angular Momentum"**Previous message:**Hitoshi Kitada: "[time 320] Re: [time 319] Re: [time 318] correction to [time 317]"**In reply to:**Matti Pitkanen: "[time 319] Re: [time 318] correction to [time 317]"

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