Terminological suggestions

Date: Mon, 1 Jan 1996 11:50:31 -0800
From: Mitchell Porter <qix@desire.apana.org.au>
To: quantum-d@teleport.com
Subject: QUANTUM-D: Terminological suggestions

I'd like to suggest some terminology, which I for one have found useful:
"world-function", "space-time ontology", "space-time interpretation"
(of quantum mechanics), "Hilbert-space ontology", "Hilbert-space
interpretation". 

A "world-function" is a mathematical function offering a complete
description of the physical properties of a possible world. For
example, think of cosmological models in general relativity.
In numerous gravitation textbooks one may find exact solutions of
Einstein's equations, for a universe containing (say) a homogeneous 
fluid. Those solutions completely describe the universe in question.

Our own universe is somewhat inhomogeneous, but should still be 
completely describable by some very complicated world-function. 
One of the interesting things about quantum theory is that it's by
no means clear what form a world-function should take.  Different 
ontologies have different sorts of world-functions. Some examples:

phi: M -> U, M a space-time manifold, U an "internal manifold of states"
This sort of world-function describes fields on a manifold. Example:
the FRW universes just mentioned.

phi: L -> U, L a lattice of space-time points.
Describes fields on a point lattice. Example: a cellular automaton.

phi: G -> M, G a graph (edges labeled with particle types and quantum numbers)
Describes a set of particle world-lines (or trajectories).

phi: S -> M, S a surface (which may carry internal fields)
Describes string world-sheets or membrane world-volumes. Example: one of
the histories out of a string path-integral.

All of these describe what could be called "space-time ontologies", in
which all actual entities inhabit space-time, in the fashion of classical
particles and fields. I would say that Gell-Mann and Hartle's decoherent
histories, Nelson's stochastic mechanics, and "zigzags in spacetime" are
all "space-time interpretations of quantum mechanics", since they have
space-time ontologies. To these can be contrasted "Hilbert-space 
interpretations", which say that the actual states of systems are defined 
by the quantum state vector. Two examples of Hilbert-space interpretations: 
the Many-Worlds Interpretation as described by Everett; and theories of 
"objective reduction" (the state vector can't be "objectively reduced" 
if it doesn't objectively exist to begin with).

Bohmian mechanics might seem to be a space-time + Hilbert-space theory,
since it has physical entities in space being "guided" by a wavefunction
in Hilbert space, but it seems to me that you could regard each wavefunction
as defining a different law of motion. Thus, in a sense, there would be a
different Bohmian theory for each universal wavefunction. This is the
attitude adopted in a new paper by Durr, Goldstein, and Zanghi,
"Bohmian Mechanics and the Meaning of the Wave Function" (quant-ph/9512031);
see in particular the section entitled "The Wave Function as  LAW".

-mitch
http://desire.apana.org.au/~qix