X-Message-Number: 22094
Date: Sat, 28 Jun 2003 08:07:47 -0700
Subject: No Physical Infinity.
From: Peter Merel <>

Michael Price writes,

> Peter Merel and myself seem to be in agreement about whether
> an infinite number of photons may exist and that they can't be
> directly measured, since only a finite number will exceed the
> sensitivity threshold of any detector. Our disagreement seems
> to be whether this means the (quantum field) theory is "ill-framed"
> or not.

Perhaps I haven't made myself clear. I don't think an infinite number 
of photons, or of anything, may exist. I think the infinities you're 
talking about are purely artifacts of your notation with no existing 
counterparts at all.

Look at it this way: there's something going on. We can't describe it 
except by a language. Any language is constructed by making 
distinctions on a space of patterns of relationship observed in the 
behavior of empirical apparatus.

All physics is built on such a language. All math is built on such a 
language. It is the language of Zermelo-Fraenkel set theory.

We construct a complex hierarchy of distinctions within this language, 
but we always make a finite number of distinctions. Where infinities 
creep in is with the theoretical explanation of causes of our observed 
behaviors. The question here is whether we need to let them in, or 
whether we're simply lazy about our terms of reference.

Since all theoretics have some maximum "size of the universe" and some 
minimum "limit of resolution", it seems that is possible to reformulate 
any theoretic notation in terms of finite, fractional subintervals on 
some universal unit interval.

If this reforumulation is done then no infinitely large collection of 
infinitely small subintervals will be necessary. Except in the 
procedure of taking limits, but we know any limit we take does not 
correspond to physical reality, but to our interpolation.

To be more positive, without an empirical necessity we should assume 
nothing about the physical universe. I do not say let's just have a 
gentleman's agreement not to discuss infinitely many infinitely small 
faerie folk. I say there are no such infinite faeries in existence 
unless we empirically require them. I think we've simply been taught a 
choice of notations that began by assuming them.

>   Attempts to reformulate theories with infinite quantities
> in terms of finite measurable observables is called "renormalisation"
> and is a well studied and active area of research.   However the
> numerical infinities from the infrared divergences are very mild
> (they don't involve infinite energies, for instance) and the common
> view seems to be that they are acceptable in theories precisely
> because they don't lead to any measurable paradoxes.

The popularity of a theory has nothing to do with its necessity. Thank 
goodness! As to "measurable paradoxes", I take that to mean 
"theoretical failures" - which obviously don't bear on the matter here.

> I was using "representation independent" in a narrow technical
> sense.  In quantum mechanics / quantum field theory a "representation"
> refers to the choice of basis or axes within the Hilbert space.  
> Physical
> or measurable quantities are unaffected by such a choice and are thus
> said to be scalar or representation independent .  Real photon number
> is such a scalar quantity, even though the actual number measured is
> also dependent on detector sensitivity.

I prefer to use "representation" in the universal sense of ZF. If we 
forget to adequately teach the set-theoretic basis of our number 
systems, they must fall into uncontrolled abuse. It rather seems there 
are no "real numbers" in observable reality ...

Peter Merel.

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