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

*Tue, 18 May 1999 17:47:12 +0300 (EET DST)*

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Stephen P. King: "[time 328] Re: [time 326] Re: Fisher information and relativity"**Previous message:**Hitoshi Kitada: "[time 326] Re: [time 325] Re: Fisher information and relativity"**In reply to:**Matti Pitkanen: "[time 325] Re: Fisher information and relativity"**Next in thread:**Hitoshi Kitada: "[time 330] Re: [time 327] Re: [time 326] Re: [time 325] Re: Fisher information andrelativity"

Dear Hitoshi,

thank you very much for seing the trouble of explaining: I will

read your comments carefully.

There is still one question! Frieden considers also

Maxwell action with action density B^2-E^2. My guess

was that this decomposition corresponds to I-J decomposition

of action and would be thus noncovariant. I might well be

wrong. The example of scalar field suggests that I am.

Perhaps you could tell whether my interpretation is

correct or not: if not, does this mean that the term J

is totally absent for Maxwell action?

Thank You in advance,

MP

On Tue, 18 May 1999, Hitoshi Kitada wrote:

*> Dear Matti,
*

*>
*

*> Your question is meaningful. Indeed it cuts the seemingly continuous
*

*> argument of Frieden as I will explain below.
*

*>
*

*> ----- Original Message -----
*

*> From: Matti Pitkanen <matpitka@pcu.helsinki.fi>
*

*> To: Hitoshi Kitada <hitoshi@kitada.com>
*

*> Cc: <time@kitada.com>
*

*> Sent: Tuesday, May 18, 1999 5:48 PM
*

*> Subject: [time 325] Re: Fisher information and relativity
*

*>
*

*> [snip]
*

*>
*

*> > > Is Fisher information
*

*> > > > still in question when one uses imaginary coordinate x0 =it?
*

*> > > >
*

*> >
*

*> > Coordinates correspond to kind of parameters in Fisher
*

*> > information: unfortunately I have not clear picture
*

*> > about what kind of parameters are in question.
*

*>
*

*> Parameters are coordinates in the book at least as I read till now. Other
*

*> examples may be in the book.
*

*>
*

*> What troubled and
*

*> > still troubles me is whether the imaginary
*

*> > value of parameter is indeed consistent with this
*

*> > interpretation.
*

*>
*

*> You seem to point out the gap in Frieden's development of the theory.
*

*>
*

*> Frieden writes in page 64 in section 3.1.2 entitled "On covariance":
*

*>
*

*> [beginning of quotation]
*

*> ... By definition of a conditional probability p(x|t)=p(x,t)/p(t)
*

*> (Frieden, 1991). This implies that the corresponding amplitudes (cf. the
*

*> second Eq. (2.18)) obey q(x|t)= + or - q(x,t)/q(t). The numerator treats x
*

*> and t covariantly, but the denominator, in only depending upon t, does not.
*

*> Thus, principle (3.1) is not covariant. [HK: (3.1) reads: \delta
*

*> I[q(x|t)]=0, q(x|t) = (q_1(x|t), ... , q_N(x|t).]
*

*>
*

*> From a statistical point of view, principle (3.1) is objectionable as
*

*> well, because it treats time as a deterministic, or known, coordinate while
*

*> treating space as random. Why should time be a priori known any more
*

*> accurate than space?
*

*>
*

*> These problems can be remedied if we simply make (3.1) covariant. This
*

*> may readily be done, by replacing it with the more general principle
*

*>
*

*> \delta I[q(x)]=0, q(x)=(q_1(x), ... , q_N(x)). (3.2)
*

*>
*

*> Here I is given by Eq. (2.19) and the q_n(x) are to be varied. Coordinates x
*

*> are, now, any four-vector of coordinates. In the particular case of
*

*> space-time coordinates, x now includes the time.
*

*> [end of quotation]
*

*>
*

*> Here Frieden transforms the Euclidean coordinates to the coordinates
*

*> possibly covariant wrt Lorentz or any other coordinates transformations.
*

*>
*

*> By this transformation of his theory, he misses the I-theorem, which reads
*

*> till he introduces the covariant coordinates:
*

*>
*

*> dI
*

*> ---- (t) < or = 0 for any t.
*

*> dt
*

*>
*

*> This has been assuring that the information I decreases as t increases.
*

*> Hence I takes a minimum value as t goes to infinity (since I > or = 0), and
*

*> this fact has been ensuring the validness of taking the solution of the
*

*> variational problem (3.1) as the physical reality:
*

*>
*

*> \delta I[q(x|t)] = 0. (3.1)
*

*>
*

*> Just when he introduces the covariant coordinates and hence pure imaginary
*

*> time, this I-theorem breaks down and he loses the foundation upon which the
*

*> validity of variational principle has been relying.
*

*>
*

*> He then instead postulates the variational principle as one of his three
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*> axioms for "the measurement process" in pages 70-72. (In fact there is no
*

*> quotation of I-theorem after page 63 till chapter 12 in page 273 entitled
*

*> "Summing up" according to the index.)
*

*>
*

*> This means that the introductory part till page 63 is just an illustration
*

*> which leads to the introduction of his axioms 1 to 3 in pp. 70-72, not a
*

*> justification of the axioms in any sense.
*

*>
*

*> And his axiom 1:
*

*>
*

*> \delta (I - J) =0,
*

*>
*

*> with axiom 2:
*

*>
*

*> I=4 \int dx \sum_n \nabla q_n \cdot \nabla q_n
*

*>
*

*> and
*

*>
*

*> J= \int dx \sum_n j_n(x),
*

*>
*

*> (here n varies from 1 to N, N denoting the number of independent
*

*> measurements done.)
*

*>
*

*> is almost the same requirement as the usual variational principle which
*

*> gives Lagrangian of the system under consideration.
*

*>
*

*> Thus his contribution is just that the free energy part I is given as above
*

*> in his axiom 2. That the form of Fisher information I gives the free energy
*

*> part of Euler-Lagrange equation may be a progress of human knowledge. This
*

*> is but a small calculation which was described in [time 321], and does not
*

*> seem to need a hard covered book.
*

*>
*

*> Frieden's purpose might be in his philosophy. However, he abandons himself
*

*> his philosophy (i.e. I-theorem) as you pointed out:
*

*>
*

*> > whether the imaginary
*

*> > value of parameter is indeed consistent with this
*

*> > interpretation.
*

*>
*

*> The imaginary value of parameters is not consistent with Frieden's own
*

*> philosophy, I-theorem. So he just assumes the principle of the least action
*

*> as axiom 1 in his derivation of Lagrangian. Here is no new thing except for
*

*> an observation that the free energy part follows from the form of the Fisher
*

*> information.
*

*>
*

*> Another point which shows the shallowness of his theory is that he does not
*

*> give any consideration about time. As in the above quotation, he thinks at
*

*> first that time is given. Then he comments that time should be considered an
*

*> inaccurate unknown parameter as other space coordinates, and turns to time
*

*> as a component of the covariant coordinates. This is a too easy way for one
*

*> to construct a unification of physics.
*

*>
*

*> In conclusion, Frieden's theory looks like but a repetition of the principle
*

*> of the least action except for the discovery of the relation between Fisher
*

*> information and the free energy.
*

*>
*

*>
*

*> Best wishes,
*

*> Hitoshi
*

*>
*

*>
*

*>
*

*>
*

**Next message:**Stephen P. King: "[time 328] Re: [time 326] Re: Fisher information and relativity"**Previous message:**Hitoshi Kitada: "[time 326] Re: [time 325] Re: Fisher information and relativity"**In reply to:**Matti Pitkanen: "[time 325] Re: Fisher information and relativity"**Next in thread:**Hitoshi Kitada: "[time 330] Re: [time 327] Re: [time 326] Re: [time 325] Re: Fisher information andrelativity"

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