# [time 329] Re: [time 328] Re: [time 326] Re: Fisher information and relativity

Thu, 20 May 1999 08:32:07 +0900

Dear Stephen,

----- Original Message -----
From: Stephen P. King <stephenk1@home.com>
Sent: Thursday, May 20, 1999 12:42 AM
Subject: [time 328] Re: [time 326] Re: Fisher information and relativity

> Dear Hitoshi,
>
> It is good to see your critique! :) I still wonder about your
> conclusions...
>
> >
> > 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>
> > 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.
>
> I would like to understand the reasoning for using an imaginary valued
> parameter for time myself!

Frieden sometimes uses imaginary space coordinates with real time
coordinate. This is just a problem of Minkowskian metric.

>
> > 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]
>
> Can we think of this treatement of "time as a deterministic, or known,
> coordinate while treating space as random" as the result of the
> uncertainty? Is there a transformation that reverses the situation, viz,
> time coordinates being random and space coordinated deterministic?
> I still will question the a priori status given by the assumption of
> covarience, if I am thinking of it properly! The idea of an LS having
> its own measure of change, e.g. a clock, would imply that the ordering
> of events that are observed by the LS are functions (?) of its clock. I
> see each LS as having its own spacetime of events as center of mass
> interactions of other LSs and do not need to assume a priori that a
> "common spacetime" needs to be defined. The interactions/communications
> between LSs is sufficient to generate the metrics. I am influenced by
> Lee Smolin's discussion of Ideal element free physics, e.g. all
> properties are defined by relations among the components.
> Space is extention and time duration in Leibnitz's thinking and I feel
> that we should not assume an a priori void *in* which events occur; the
> spacetimes are constructed by observation as the background of the
> perceived events. How these events are perceived to change is the
> motivation of construct physics.
>
> > 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
> > 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

Because now I defined in [time 324] by

> > 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).

is independent of time t.

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.)
>
> Does this not complement your thinking about how uncertainty is created
> in LS theory? It is my claim that observations by LSs are measurements,
> and as such manifest a least action aspect. We do need to look carefully
> at this issue. Least action is not definable realistically in a
> universal sense it is <<glocal>> thus we see that one LS's least action
> is not necessarily transformable via a diffeomorphism to anothers! Again
> my statements about Weyl come to mind: we must understand that each of
> us has our own measure of "reality".
> The world that we can "real" is merely that which our common percept
> agree. The aspects of our possible observations that are not observable
> by others are such because they are not communicable to others. This is
> illustrated by how some concepts can not be translated from one language
> to another, the 20 or so different types of snow that are a reality to
> the Inuit people are not expressible to persons from the tropics, but
> their reality is no less diminished. We must understand that what is
> measured or observed in the word is conditioned by the properties of the
> LS making such, it is not a priori given!
> This subtle point is the essense of bisimulation. It seems that I need
> to write up an explicit post on this. Until it is understood my thinking
> will be misunderstood. :(
>
>
> > 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.
>
> Is this not a usefull notion to work with? We solve the problems of
> understanding piece by piece...
>
> > 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.
>
> Would Frieden benefit from using your notion of time intead of the
> usual notion?

Frieden thinks the observational aspect only. Thus space-time is given for
him. Maybe in this sense, my notion or defintion of time is not necessary
for his consideration. It is sufficient that some parameter corresponding to
time is given for him.

>
> > 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.
>
> I believe that Frieden's work is but another piece of the puzzle of
> Quantum gravity, I do not expect his work to completely exhaust the work
> needed.

I agree. But my problem is what the complete understanding is.

I need only to point at the work that went into QM to illustrate
> this! We need to see the big picture!

The problem is we cannot have the picture. We have many pieces, but do not
have any synthetic picture. This seems to be the case at any age, as we see
when reminding the history.

I will write up a review of
> Frieden's book when my copy comes in. Meanwhile I will continue to be a
> philosopher! :)
>
> > Best wishes,
> > Hitoshi
>
> Later,
>
> Stephen
>

Best wishes,
Hitoshi

This archive was generated by hypermail 2.0b3 on Sun Oct 17 1999 - 22:10:32 JST