Subject: "I only wish I could have acted on
all suggestions!" (arXiv:0708.2192v1
[quant-ph]) Date: Fri, 17 Aug 2007 15:22:35 +0300 From: Dimi Chakalov <[email protected]> To: Jeremy Nicholas Butterfield <[email protected]> Dear Jeremy, RE your arXiv:0708.2192v1 [quant-ph] and 'The moral in general relativity', arXiv:0708.2189v1 [quant-ph], perhaps you may wish to read http://god-does-not-play-dice.net/Trautman.html#pedestrians http://god-does-not-play-dice.net/Trautman.html#Zeh_note Since you endorse the "block-universe", I suppose you wouldn't like to comment on the ideas outlined above, as you haven't done it in the past seven years. It would be nice if you at least acknowledge the receipt of this email. As ever, Dimi ---- Note: Regarding
arXiv:0708.2192v1
[quant-ph], Sec. "Events and regions", and Reichenbach's Principle of the
Common Cause ( But what is the dynamics of this Common Cause,
or 'third event', Jeremy Nicholas Butterfield has never made any
comment on the proposal that
Subject: From regions to points
Hi Jeremy, I spotted your name in Sec. Acknowledgments of Chris Isham's quant-ph/0206090 [Ref. 1]. Perhaps you or John could be so kind as to help me understand the basic premise in your approach. May I ask two questions. In relative-frequency interpretation of probability, you can get "points" IFF you set N --> [infinity], "as is appropriate for a hypothetical 'infinite ensemble'" [Ref. 1]. 1. Is there anything which you set to approach infinity? If yes, how does it 'get there' so that you can get "points"? These are very old questions, http://members.aon.at/chakalov/Magueijo.html 2. How do you get points "determined in some way by the regions" [Ref. 1] while keeping the regions 'alive and kicking' for a finite time interval, as measured with your wristwatch? Regarding the latter, please see my recent email to Bob Wald, http://members.aon.at/chakalov/Wald.html I believe you know about my proposal, http://members.aon.at/chakalov/dimi.html I'm just trying to understand yours. Best regards, Dimi --
"However, one of the key questions of interest in the
present paper is not how the standard ideas of space and time (and probability)
might *emerge* from a different formalism; but rather how one might proceed
to construct a quantum theory *ab initio* in which whatever fundamental
spatio-temporal concepts are present are definitely not the familiar continuum
ones: for example, if one is given a finite causal set as a background
structure. The first step, and the only one taken in the present paper,
is to sound a cautionary note by emphasizing how strongly the continuum
ideas of space and time are implicitly embedded in the standard formulation
of quantum theory.
"My concern is that the use of these numbers may be problematic in the context of a quantum gravity theory whose underlying notion of space and time is different from that of a smooth manifold. The danger is that by imposing a continuum structure in the quantum theory *a priori*, one may be creating a theoretical system that is fundamentally unsuitable for the incorporation of spatio-temporal concepts of a non-continuum nature: this would be the theoretical-physics analogue of what a philosopher might call a 'category error'. For this reason, it is important to consider carefully the origin, and role, in standard quantum theory of this particular facet of the real (and complex) numbers. In general terms, the real numbers arise in three ways
in physical theories: (i) as the values of physical quantities; (ii) to
model space and time; and (iii) as the values of probabilities. Our present
task is to consider more precisely the use real numbers in quantum theory
in these terms.
"It is thus pertinent to ask *why* physical quantities -- classical or quantum -- are taken to be real-valued. Many will doubtless say that the answer is obvious, or that it is even part of the definition of a physical quantity, but I would challenge these assertions as being over-hasty. One reason why the values of physical quantities are assumed to be real numbers is undoubtedly the operational one that -- at least, in the pre-digital age -- physical quantities are ultimately measured with rulers and pointers, and so it is the assumed continuum nature of physical space that comes into play. However, it is by no means obvious that physical quantities should necessarily be real-valued in, for example, a quantum gravity theory in which it is not appropriate to think of space as a smooth manifold, and where, therefore, there is no place for operational considerations that presuppose a continuum nature for space and/or time. Of course, it is a totally open question as to what should
replace R as the value space of a physical quantity in these
circumstances -- it could be something as obvious as a finite number field,
but it could also be something far more radical. In any event, a key role
in deciding this issue should be played by any underlying spatio-temporal
concepts (albeit, non-standard) that are present.
"In the context of standard physics,
it is clear why probabilities are required to lie in the interval [0,1].
As physicists, we most commonly employ a relative-frequency interpretation
of probability in which an experiment is repeated a large number, say N,
times, and the probability associated with a particular result is then
defined to be the ratio N_i/N, where N_i is the number of experiments in
which that result was obtained. The rational numbers N_i/N necessarily
lie between 0 and 1, and if we take the limit as N --> [infinity]
, as is appropriate for a hypothetical 'infinite ensemble', we get real
numbers in the closed interval [0, 1].
p. 9: "Under such circumstances it might be more natural to follow Aristotle, Heisenberg and Popper in adopting a *propensity* interpretation of probability, perhaps within the context of a 'post-Everett' form of quantum theory, such as consistent-histories theory. "However, if probability is viewed in this more realist
way, there is no overwhelming reason for assigning its values to be real
numbers lying in the interval [0, 1]. The minimal requirement is presumably
only that the value space should be a partially ordered set (V,<_and_=)
so that it makes sense to say that certain events are more, or less, probable
(in the sense of the partial-ordering operation <_and_=) than others.
"In standard general relativity -- and, indeed, in all
classical physics -- space (and similarly time) is modelled by a set, and
the elements of that set correspond to points in space. However, it is
often claimed that the notion of a spatial (or temporal) point has no real
physical meaning, and this motivates trying to construct a theory in which
'regions' are the primary concept. In such a theory, 'points' -- if they
exist at all -- would play a secondary role in which they are determined
in some way by the regions (rather than regions being collections of points,
as in standard set theory).
"This could give valuable insight into what is perhaps
the hardest task of all: to construct a quantum formalism for use in situations
where there are no prima facie spatio-temporal concepts at all -- a situation
that could well arise in a quantum gravity theory in which all of what
we might want to call "spatio-temporal concepts" emerge from the basic
formalism only in some limiting sense.
"Acknowledgements "Some of the material presented here is a development
of earlier work [1] with Jeremy Butterfield. I am grateful to him for permission
to include this material."
====== Subject: Panta rei conditio sine qua
non est
Dear Jeremy, I read last night some of the papers you sent me three years ago, and, as it always happens with reading papers from you, Chris, and Karel, I found some new ideas, which I typed a few minutes ago and posted at http://God-does-not-play-dice.net/Miller.html#note Thank you all very much! No need to reply, please, I know you're very busy. Best regards, |