Alternatives to standard quantum theory...

Date: Wed, 25 Oct 1995 12:05:09 -0700
From: Arkadiusz Jadczyk 
Reply to: quantum-d@teleport.com
To: subscribers of the list 
Subject: QUANTUM-D: Alternatives to standard quantum theory...

There are three alternatives to the standard quantum
theory that employ in addition to the quantum system
also a classical system. These are

(BM) Bohmian mechanics
(NM) Nelson's stochastic mechanics and its descendants
(BB) Bell's idea of local beables

The attractive features of these approaches are well known.
There are however some unattractive features too - and
these are less known. I would like to bring your attention
to these features, as realizing that there is deficiency
at certain points may be useful in the attempts to create
something even better.

The first problem is that of lack of symmetry between Q (quantum)
and C (classical). Q acts on C without being acted upon. There is
action with no reaction. Q is piloting the classical dynamics of C
that is either deterministic (in BM) or stochastic (in NM and BB).
C can be observed and by observing C we may learn about Q. But the
fact that C is coupled to Q does not influence the Schroedinger
evolution of Q.
The very fact of asymmetry may be considered as just an aesthetic
feature. One may say: it is just a feature, not a defect. One can
even say: it is an advantage. Nevertheless action without reaction
is a little bit annoying for a physicist. He may contemplate the idea
of adding some terms to the Schroedinger equation that will restore
the symmetry and which will be experimentally testable.

The second feature is related to the first one. But now, it is
not only the question of aesthetics, it is something more
serious. All three theories are first formulated for pure states.
By solving the Schroedinger equation we find the wave function, and
then we use either to compute classical trajectories - be they
deterministic or random. Then we can go to mixed states, decompose
the into pure ones, and then give extra weights to our classical
solutions. However, one of the main properties that distinguish
quantum theory from any classical one is the fact that the space
of states of a quantum system is not a simplex. That quantum mixtures
decompose into pure states in infinitely many ways. Any new theory
that attempts to enhance the ordinary QM should take this fact into
account. This fact was discussed in the literature often, in particular
N. Gisin made it a strong point in his comment on Weinberg's nonlinear
theory. He argued that the ability to distinguish quantum mixtures
contradicts relativity. Any attempt to go beyond linear Schroedinger 
equation, to go beyond bilinear observables, is dangerous because it 
may lead to contradiction with experiments that confirm quantum 
indistinguishability of mixtures. I do not mean that *any* 
nonlinearity must necessarily violate the quantum indistinguishability
of mixtures. There are known special cases when this does not happen.
However BM, NM and BB are not in this safe class. I did the 
computation for BM and BN. From this I guess that the same will
be true for NM.  

Finally: why do I say that the second feature is related to the
first one? That because there is a no-go theorem (Landsman, Ozawa)
that tells us: it is not possible to couple in a non-trivial
way quantum system to a classical one retaining unitary evolution
of the quantum system! So, if we do want to keep peace with
quantum indistinguishability of mixtures - then terms *must*
be added to the Schroedinger equation, action must cause reaction.


ajad
http://www.ift.uni.wroc.pl/~ajad/qf.htm
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 Arkadiusz Jadczyk              tel (0049)521-106-6199
 BiBoS                          fax (0049)521-106-2961
 University of Bielefeld
 Universitaetstr. 25            e-mail: ajad@physik.uni-bielefeld.de
 D 33615 Bielefeld                      ajad@ift.uni.wroc.pl
 Germany             
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