To: huw@mail.usyd.edu.au 10/17 2002
From: "Paul J. Werbos" <
Subject: more on polarizers
Cc:
luda.werbos@verizon.net,Huw.Price@ed.ac.uk,shih@umbc.edu
Bcc:
X-Eudora-Signature: <<No Default>>
Hi, Huw!
Please forgive me for being a bit fuzzy both about polarizers and about MRF modeling yesterday.
Unfortunately, I only have 15 minutes available today, so I should be brief.
The Aharonov calculations are basically just chains of conditional probabilities for events,
a bit like Pr (A,B,C) = sum {D} Pr (A | B,C,D) Pr(B|C,D)Pr(C|D)Pr(D).
He does get a denominator there....
Given a series of NEIGHBORING events A,B,C,D,E,F,
strung in a line,
MRF would give Pr(A,B,C,D,E,F)=P*(A,B)P*(A,B,C)P*(B,C,D)P*(C,D,E)P*(D,E,F)P*(E,F)/Z,
where Z is the "partition function", the sum over all possible combinations of events
of this P*..P* product. A physicist might say "We initially compute P(A...F)
as the product P*....P*(E,F), but then we scale all the probabilities by a scalar factor Z in
order to make them add up to one." The generalization to events forming a network
or relational network or lattice or directed graph (all more or less synonyms) is
essentially obvious. People have given a hundred names to special cases of MRFs in different applications,
like "Ising models" or "spin glass models" or whatever.
-----------------------
If the P* can all be represented IN EFFECT as conditional probabilities, then this all reduces to Aharonov, and we get
time-forwards causality. But that is just a special case.
---
At this moment, I see no problem in representing a polarizer as a time-symmetric argument. On the one hand,
when I consider how thermal effects might give rise to time-asymmetry... it doesn't make sense to me,
physically. On the other hand, I was wavering a bit yesterday based on very simple problems which I literally forgot that
I had already solved. (Sorry about that.)
IN ten minutes I can't go through all the math in the Bell experiment... but I can at least set
up the framework.
If we are looking for a truly realistic picture... we want the SPDC (in any instance of the experiment) to ACTUALLY
output a left channel photon longitudinally polarized with angle thetaL1 (or really just a beam of light
with that polarization), a right channel photon with angle thetaR1. The left polarizer is at thetaLP, the right
thetaRP.
Our problem is to figure out how the joint detection probability for photons coming out of this setup
is essentially just cos**2(thetaLP-thetaRP), the prediction of quantum mechanics, verified in detail.
Bell's book explains well enough why this cannot be done for a forwards-time, Aharonov-style Markhov
process model. (Well.. that picture could use clarification in some quarters. Certainly the
Clauser inequalities are a bit of a distraction; it's basically just common sense -- how can one find
Pr(thetaL1,thetaR1) outgoing from a polarizer in a way which makes the detection probability zero for
ALL possibilities of thetaLP orthogonal to thetaRP).
But even with MRF, getting that equality is not easy. ONe basically must use a FOurier tarnsform to evaluate the convolution integral defined by the joint probabilities...
The P* local joint probability for the polarizer (a node in the network of events)... basically must have the form
a0*delta(thetaL1-thetaLP)+a1*cos**2(thetaL1-thetaLP) for the event of an incoming photon to be passed through.
What I forgot yesterday is that teh alternative event, of photon absorption, also has the SAME form, except with pi/2
added to the arguments. And that easily rpelicates the everyday experiments with polarizers as well as GHZ and Bell.
--
Must run.
Best,
Paul