Stephen P. King (stephenk1@home.com)
Wed, 19 May 1999 11:42:43 -0400
Dear Hitoshi,
It is good to see your critique! :) I still wonder about your
conclusions...
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.
I would like to understand the reasoning for using an imaginary valued
parameter for time myself!
> 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 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.)
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?
> 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 need only to point at the work that went into QM to illustrate
this! We need to see the big picture! 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
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