From: baez@ucrmath.ucr.edu (john baez)
Newsgroups: sci.physics
Subject: Re: Report on Philosophies of Physicists
Date: 4 Sep 92 23:15:55 GMT
Organization: University of California, Riverside

In article <mcirvin.715559539@husc8> mcirvin@husc8.harvard.edu (Mcirvin) writes:

>Has anybody here worked with quantum logic?  

Yup.

>Does it just include
>an extra truth value to eliminate the excluded middle, 

Nope...

>or is there
>some attempt to turn the mathematics of Hilbert spaces into 
>a system of logic assigning, say, phases and magnitudes to
>various propositions? 

Closer but not quite.  There are various approaches but most of the
better ones work in a roughly similar way.  Propositions in quantum
logic form an orthocomplemented lattice: that is, a set equipped with
operations "and", "or", and "not" satisfying some but not all of the
usual axioms.  The main thing that fails is distributivity of "and" over
"or" and vice versa.  The simplest example of a quantum logic is the
lattice of closed subspaces of a Hilbert space, with intersection as
"and" and closure of the span (=set of all linear combinations) as "or",
and orthogonal complement as "not".  Alternatively we may think in terms
of the orthogonal projections *onto* these subspaces, rather than the
subspaces themselves.   Noncommutativity of the projections gives rise
to the failure of distributivity.  E.g.:

P = the particle is moving to the right (p > 0)
Q = the particle is to the right (q > 0)
R = the particle is to the left (q < 0)

Then 

P & (Q or R) = P

but

(P & Q) or (P & R) = false

"false" being the zero subspace, so distributivity fails.  (The fact
that P & Q = false follows from the fact that the only function 
supported on the positive half-line whose Fourier transform is ALSO supported
on the positive half-line is zero.  I.e., you can never know for SURE
that a particle is both on the right half-line and moving right.)

A state assigns to each proposition an expectation value, that is, a
number between 0 and 1 representing the probability that the proposition
is true.  
 
It's just a nice different way of thinking about quantum theory.  One
can try to see what axioms on the lattice imply that it is the lattice
of subspaces of some Hilbert space, so as to get a "reason" for why
Hilbert spaces are so important in quantum theory.  This project has
been much pursued without definitive success.

There are nice ways to think about superselection sectors and other such
things using quantum logic.  Everyone who likes quantum theory and math
should learn a little quantum logic.  Not because it's an earthshaking
thing but just in order to see how the *noncommutativity* of operator
algebras gives rise to *nondistributivity* of the lattice of projections,
and to see how nondistributivity accounts for all the "quantum weirdnesses"
we know and love.

The nicest intro is:

 Jauch, Josef M. (Josef Maria), 1914-1974.
 Foundations of quantum mechanics [by] Josef M. Jauch.  Reading, Mass.,
 Addison-Wesley Pub. Co. [c1968].
 Series title:  A-W series in advanced physics.

A more technical collection that has some good review articles that
illustrate the various approaches to quantum logic is:

 The Logico-algebraic approach to quantum mechanics / edited by C. A. Hooker.  
 Dordrecht ; Boston : D. Reidel Pub. Co., [1975]-1979.
 Series title:  University of Western Ontario University of Western Ontario
 series in philosophy of science ; v. 5.

If one is willing to drift away from the pure logical (lattice-theoretic)
approach and check out the more algebraic (C*- and W*-algebraic) side of
things, a very readable volume (relatively speaking) is:

 Emch, Gerard G.
 Algebraic methods in statistical mechanics and quantum field theory [by]
 Gerard G. Emch.  New York, Wiley-Interscience [1972].
 Series title:  Interscience monographs and texts in physics and astronomy,
 v. 26.