From: atems@igor.physics.wayne.edu (Dale Atems)
Subject: Hidden variables, Bell's inequality, and Wiener-Siegel theory
Organization: Wayne State University, Detroit, MI
Date: Sun, 3 Jan 1993 20:36:29 GMT

When I was researching the EPR paradox and Bell's inequality for a
term paper over 10 years ago, one hidden variables theory that I found
especially interesting was Wiener-Siegel (WS) theory [1]. I'm sure
Budnik and others in the field are familiar with it, but I thought I'd
just mention it for everyone else's benefit, and to see if anyone
knows the answer to a question that's puzzled me ever since. Bear in
mind that everything here is culled from my term paper; I haven't
looked at the references in over a decade.

What is WS theory? It's an EPR-complete hidden variables theory that
reproduces the predictions of QM exactly, and of course it violates
Bell's inequality in the process. It's a classical statistical theory
in which quantum states correspond to macrostates, and the hidden
information forbidden to us by the uncertainty principle is contained
in microstates in which noncommuting observables have simultaneous
values. The microstates are defined by introducing an auxiliary state
vector \Xi (not normalized). For any observable O, \Xi together with
the usual state vector \Psi determines the result of measuring O, and
an ad hoc "equilibrium" distribution of the norms of the expansion
coefficients <\phi_k|\Xi> defines the statistics of the theory, where
the {\phi_k} are eigenvectors of O.

What's remarkable about WS theory? If Bell's inequality is violated in
nature, then a certain class of hidden-variables theories, often
called "local realistic", is ruled out. Most current research seems to
focus on pinning down whether nature is or isn't really nonlocal. The
bizarre feature of WS theory is that it throws out realism (on the
quantum level only, of course) instead.

A WS microstate determines which of the complete set of eigenvectors
{\phi_k} the QM state vector will "collapse" to when a measurement is
made. In case of degeneracy, each \phi_k generally corresponds to
simultaneous eigenvalues of two or more commuting observables.  The
choice of basis corresponds to which observables are actually measured
in a given (idealized) experiment. But the catch is that given a WS
microstate, the value obtained for a given observable can depend on
the basis {\phi_k} that you choose.

For example, in a spin-half singlet state, if you expand in the basis
of simultaneous eigenvectors of {S_x(1), S_x(2)}, in a given
microstate S_x(1) might have one value, but if you expand in the basis
{S_x(1), S_y(2)}, in the same microstate S_x(1) might have the
opposite value. (Here 1 and 2 label the particles.) The connection
with "quantum spookies" in this case is clear: the theory is saying
that if I measure S_x(1) and S_x(2) I might find, say, spin up for
S_x(1), but if I had decided to measure S_y(2) rather than S_x(2), I
might have found that S_x(1) was "actually" spin down instead!

But in general the basis-dependence of the information contained in a
WS microstate has nothing to do with action at a distance, since it
shows up in single-particle cases as well [2]. From the WS viewpoint,
Budnik's billboard saying "SOMETHING IS FISHY HERE" is somewhat larger
than Bell's inequality would seem to require. The question I've never
seen answered is whether this is just an artifact of W and S's
formulation, or whether it's a necessary price of making a theory
along the lines of classical statistical mechanics reproduce the
predictions of QM.

References:
 
[1] N. Wiener and A. Siegel, Phys. Rev. 91, 1551 (1953).
 
[2] F. J. Belinfante, _A Survey of Hidden Variables Theories_ (1973),
    pp. 39-43, 112 et seq.

------
Dale Atems
Wayne State University, Detroit, MI
Department of Physics and Astronomy
atems@igor.physics.wayne.edu
Newsgroups: sci.physics