Wednesday, December 15, 2010

Quantum Acausal Determinism (4a)

As we saw in the first instalment of this blog series, by 1952 David Bohm resurrected aspects of the deterministic theory which then were threaded into his own "Stochastic Intepretation of Quantum Mechanics" - and which is now often described as "Bohmian Quantum Mechanics". In 1952, Bohm published two papers on the topic. The first basically worked out the consequences for a one-body system using de Broglie's real matter wave basis. The second paper extended the treatment ti the many body system which effectively neutralized many of Pauli's objections from assorted Solvay conferences.

The key insight for Bohm was in terms of the evolution of the probability density:

dP/dt + Sigma (grad _n) * (P grad_n S)/m = 0

where P = U*U is the probability density and the modified Hamilton -Jacobi equation is:

dS/dt + Sigma_n [grad_n S]^2/ 2m + V(x1...xn) _ Q(x1...xn) = 0

From the preceding,. Bohm deduced that each particle would be acted upon not only by a classical potential [1] but also an added quantum potential Q:

Q = - (h-bar)^2/2m {Sigma_n [grad_n ]^2 R/ R

We now examine this in the context of an experimental underpinning. Since Bohm's deterministic theory includes what he calls "hidden variables" - which effectively drive the determinism toward a relative locality- then these must be incorporated. The problem is to invoke an experimental basis which can allow the determinism to be checked.

The original proposal (by Rietdjik and Selleri, [1]) was to show that if a photon is successively transmitted by 2 poalrizers (using appropriate settings or orientations) then the very first transmission must influence a hidden variable which co-determines the second one. "Malus law" was first formulated by Etienne Louis Malus in 1809 and asserts that the intensity of light transiting an analyzer and polarizer is proportional to cos^2(theta) where theta is the angle through which the analyzer is rotated with respect to the polarizer.

One can proceed by first considering (as per the authors[1]) a set-up with 2 orthogonally polarized correlated photons (designated 'gamma 1' and 'gamma 2' in Fig. A of the diagram shown) and these interact with three polarizers denoted A, B and C.

Now if 'E' is the causal event (photon departure from location), then by causal determinism one must find:

AE (less than) CE (less than) BE

In other words, the interaction event denoted 'AE' (whereupon the designated photon for gamma 1 interacts with polarizer A) occurs before CE and CE before BE.

By appropriate computations one is led to a theorem (1):

"A deterministic theory with hidden variable φ reproduces Malus' law for a photon (gamma 2) transmitted by two polarizers C and B, with arbitrarily chosen settings of their axes, only if the hidden variable φ undergoes some change (a redistribution) when gamma 2 crosses the first polarizer. "

Now, to refine the experiment, move the polarizer C from the set up of Fig A. and define p1,2(x - pi/2, y) to be the probability that, in the setup of Fig. B, gamma 1 is transmitted by A, and gamma 2 by B.

[1] C.W. Rietdjik, and F. Selleri: Foundations of Physics, March, 1985, p. 303.
(To be continued)

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