Nonlocality from space-time zigzags

Date: Fri, 29 Dec 1995 23:30:30 -0800
From: Mitchell Porter <qix@desire.apana.org.au>
To: quantum-d@teleport.com
Subject: Nonlocality from space-time zigzags

In Chapter 9 of _The Undivided Universe_, Bohm and Hiley discuss
stochastic models and conclude that they must be nonlocal. In
Section 9.7 they say that Nelson reached the same conclusion.

The idea that one can obtain effective nonlocality from a local
physics through space-time zigzags strikes me as very interesting,
but in need of quantitative expression. Something like a law of
motion needs to be proposed, from which one can ultimately derive
quantum-mechanical transition probabilities.

There are apparently relationships between quantum field theory in n 
(space) dimensions and the statistical mechanics of Euclidean field 
theory in n+1 dimensions, so perhaps one could construct such a 
derivation starting with a "random Euclidean field". I don't know how 
to go about doing that for the continuum case, but I can see a way to 
work on a lattice of space-time points.

A possible history would consist of an assignment of field values to
the lattice points (or of a set of particle trajectories along the edges
connecting them). One would assign probabilities to each possible 
field-value (or trajectory component) - I shall call these the
"elementary probabilities" - and calculate the total probability of a 
history or history segment by multiplying the elementary
probabilities of its constituents, in the usual fashion of
probability theory. (Note that these are ordinary probabilities, not 
complex probability amplitudes.) 

Finally, one can obtain a transition probability between an initial
condition on one space-like surface, and a final condition on some
other surface, by simply summing the probabilities of all possible 
connecting histories. (I am assuming that the "elementary probabilities"
are chosen so we have unitarity/normalization.)

If one can recover quantum-mechanical transition probabilities from
such a starting point, an obvious interpretation suggests itself.
Ontologically, the world consists of a random Euclidean field -
"random" in that it doesn't follow a law of motion, but admitting
of a simple characterization in terms of those elementary probabilities.
Probabilities here would admit of relative frequency interpretation:
the probabilities assigned to the fundamental processes above would
simply be the relative frequency of those processes in space-time.
Unfortunately, as I said, I don't have a precise idea regarding
the continuum case.

Just to reemphasize: the notion here is that one could get spacelike
correlations out of locally-defined elementary probabilities because
of the existence of back-and-forth-in-time event-chains affecting
spacelike conditional probabilities.

Rafael Sorkin also has some ideas in "Quantum Measure Theory and
Its Interpretation" (gr-qc/9507057). He proposes three principles
("Realism", "the Space-time Character of Reality", and "the
Single World") which are consistent with this approach.

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

PS A recent realization concerning Gell-Mann and Hartle's "decoherent
histories": since their scheme of assigning "a priori probabilities"
to each of a set of histories works only if the histories are
coarse-grained, I had assumed that GM&H were supposing a
"coarse-grained" _ontology_ as well - e.g., a universe in which
there are fields taking values only on some discrete lattice of
space-time points. But in several places (for example, a recent
preprint on "Strong decoherence", gr-qc/9509054), they define a
coarse-grained history as an equivalence class of fine-grained 
histories (namely all those f-g histories which pass within the limits
defining the c-g history). So when one assigns a probability to
a coarse-grained history, one is really stating the probability
that the universe belongs to a particular set of possible
fine-grained histories.



---
this document at
http://www.teleport.com/~rhett/quantum-d/posts/qix_12-29.html


  (see further:
   http://www.teleport.com/~rhett/quantum-d/posts/vjs_12-19.html
   and related postings...
  
    &
      http://www.phys.hawaii.edu/vjs/www/visual.ps
      http://www.phys.hawaii.edu/vjs/www/visual.txt

   - rhett)