Matti Pitkanen (matpitka@pcu.helsinki.fi)
Tue, 18 May 1999 17:47:12 +0300 (EET DST)
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
> 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
>
>
>
>
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