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(Bringing this topic momentarily in from the back-burner.)

From: David Blenkinsop <blenl@sk.sympatico.ca>, Sat, 23 Sep 2000

*>Amara Graps wrote:
*

*>>
*

*>> . . . If one adopts
*

*>> a Bayesian approach to probability, then the Schroedinger wave
*

*>> equation simply becomes a posterior probability describing
*

*>> our incomplete information about the quantum system,
*

*>> rather than wave functions that collapse in reality upon our
*

*>> observation. It could clear up a lot of confusion.
*

*>
*

*>Just a minute, hold on there! As part of your discussion, you yourself
*

*>mention the "amplitude squared" property of quantum waves, where the
*

*>wave just *has* to be considered as have a steadily "rotating" complex
*

*>vector amplitude. Now, it's my understanding that to get a normalized
*

*>probability distribution for the location of the associated electron, or
*

*>whatever, this really *does* involve squaring this complex amplitude, or
*

*>it involves squaring the sum total of any such vectors as may be
*

*>interfering in a given region of space. This is a situation quite unlike
*

*>anything in regular probability theory, something quite different added
*

*>in there, right?
*

*>
*

*>Say you've got a single photon waveform, and say this wave is travelling
*

*>through two slits in a single photon, double slit experiment. The
*

*>probability of the photon hitting a screen is then a kind of
*

*>interference pattern of high and low probabilities as you move across
*

*>the screen. The waveform itself must then actually be a real, extended
*

*>structure of *some* kind, it seems to me, since it directly governs the
*

*>probability of finding the photon in the various bands of the
*

*>interference pattern.
*

In this case, the Bayesian approach is that they are in both states

at once, as a function of the superposition principle, and we are

lacking more variables to describe the scenario. If we interpret the

amplitudes instead, as energies that we are measuring, then the amplitude

squared is the probability that, if we measure its energy, we shall

find the value corresponding to the nth state.

However, I see your confusion. I didn't describe at any length where

Bayes' Theorem applies in Caticha's work, and indeed, I don't see

Bayes theorem in his articles. I apologize for that. I know that

Caticha comes from a Bayes perspective because I met him and we

spoke at a conference on Bayes applications (MaxEnt'98), and I know

that his colleague at the conference, C. Rodriguez (see below), also

researches quantum probabilities from a Bayes perspective.

The Bayes connection in Caticha's work comes in through his

references to R.T. Cox, and E. Jaynes, so it would would have more

accurate for me to say Caticha's subscribes to the "Bayesian

probability view" that probability as a measure of the plausibility

of an event with incomplete knowlege (which is different from the

frequentist or "sampling theory" view that probability as the

long-run frequency of occurrence of an event in a sequence of

repeated (sometimes hypothetical) experiment).

If you read Caticha's papers, you will see that they describe a real

experiment, so it is physical, and all propositions are testable. No

statements about the quantum particle can be made independently of

the experimental context (so you see, "priors" in the Bayesian sense

must be present). He doesn't separate the particle; describing what

it is doing between the source and the detector. He cannot say

whether the particle is a point particle or a wave or both or

neither. He is not saying that it went through either one hole or

through another, or through both holes at the same time. He has very

few assumptions: mostly that the particle is simply capable of being

emitted and detected.

So where does Bayes probability ideas come in? Now I go to a paper

by E.T. Jaynes, titled "Probability in Quantum Theory", and I pull

some things from that paper in my description below.

Much of physical science is information that is organized in

particular ways. We have a basic question, which is:

"To what extent does this information reside in us, and to what

extent is it a property of Nature? Any theory about reality can have

no consequences testable by us unless it can also describe what

humans can see and know." Jaynes says that the proper tool for

incorporating human information into science is simply probability

theory or "Bayesian inference". Probability theory is an extension

of logic, in order to reason in situations where we have incomplete

information.

However, amplitude Psi, according to our usual quantum mechanics

texts is not now to be interpreted in as expressions of human

ignorance of the true physical state. Jaynes says that it is our way

that we interpret the amplitudes that is causing the confusion. The

probabilities that we seek, must be in terms of mutually exclusive

possibilities and must be represented in a "deeper hypothesis space"

that contains more parameters, for example, phases. He doesn't go into

details of what all other parameters must be included, he mainly points

that our perspective has to shift into a wider view, and he says

that we cannot hope to get our probability connections right until we

get some basic points of logic right.

For example, this illogical situation:

Dispersions: (delta F)^2 = <F^2> - <F>^2 are thought by some

physicists to be quantum fluctuations in F that are *real physical

events that take place constantly whether or not any measurement is

being made.*

In basic probability theory, (delta F) represents fundamentally the

accuracy with which we are able to predict the value of F. This does

not deny that it may be also be the variability seen in repeated

measurements of F; but the point is that they need not be the same.

To say that they "must be the same" is what Jaynes calls the "Mind

Projection Fallacy", which is to suppose that creations of our own

imagination are real properties of Nature, or that our own ignorance

signifies some indecision on the part of Nature. If from our

information, we are able to determine F to only 5 percent accuracy,

that does not mean that the object fluctuates by 5 percent!

Jaynes has some interesting things to say in this paper about other

aspects of quantum mechanics such as the Zero-Point Energy and the

Lamb shift. and I encourage you to read his works. Much of it is

available on the Web, mostly here:

http://bayes.wustl.edu/etj/node1.html

Now back to Caticha.

Caticha's colleague, C. Rodriguez, a mathematician at SUNY Albany,

has a paper in the same text as Caticha's

Caticha, Ariel, _Probability and Entropy in Quantum Theory_ in

Maximum Entropy and Bayesian Methods_ (conf proceedings from

MaxEnt'98 conference, Garching, Germany, July 1998), Kluwer Academic

Publ., 1999.

titled:

"Unreal Probabilities": Partial Truth with Clifford Numbers". by C.

Rodriguez. (starting at pg. 247)

I will just quote from the beginning of the paper- it's highly

mathematical, and tough reading, and what follows is a bit abstract,

but straightforward.

{begin quote}

Probability theory was given a firm mathematical foundation in 1933,

when Kolmogorov introduced his axioms. By defining probability as an

"uninterpreted" special case of a positive measure with total unit

mass, the subject exploded with new results and found innumerable

applications, In 1946, Cox showed that the Kolmogorov axioms for

probability are really theorems that follow from basic desiderata

about the representation of particl truth with real numbers. We owe

to Ed Jaynes, to see the importance of Cox's 1946 work. After

Jaynes, it became clear why the calculus of probability is so

successful in the real world. Probability works because its axioms

axiomatize the right thing: partial truth of a logical proposition

given another. Even more, the rules of probability are unique in the

sense that any other set of consistent rules can be brought into the

standard sum and product rules by a change of scale. This is in fact

Cox's main result and it makes futile the enterprise of looking for

alternatives to the calculus of normalized real valued

probabilities. It is only by allowing the partial truth of a

proposition to be encoded by an object other than a real number in

the interval [0,1] that we could find alternatives to the standard

theory of probability.

We seek to find out what happens when standard probobaility theory

is modified by relaxing the axiom that the probability of an event

must be a real number in the interval [0,1]. We show that, by

allowing the measure of a proposition to take a value in a Clifford

Algebra, we automatically find the methods of standard quantum

theory without ever introducing anything specifically related to

nature itself.

The main motivation for this article has come from realizing that

the derivations in Cox still apply if real numbers are replaced by

complex numbers as the encoders of partial truth. This was first

mentioned by Youssef and checked in more detail by Caticha, who also

showed some non-relativistic Quantum theory, as formulated by

Feynman, is the only consistent calculus of probability amplitudes.

By measureing propositions with Clifford numbers we automatically

include the reals, complexes, quaternions, spinors and any

combinations of them as special cases.

{end quote}

Hope that helps explain a bit more.

Now (me) back to dust ....

Amara

*******************************************************************

Amara Graps | Max-Planck-Institut fuer Kernphysik

Interplanetary Dust Group | Saupfercheckweg 1

+49-6221-516-543 | 69117 Heidelberg, GERMANY

Amara.Graps@mpi-hd.mpg.de * http://www.mpi-hd.mpg.de/galileo~graps

*******************************************************************

"Never fight an inanimate object." - P. J. O'Rourke

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