X-Message-Number: 22016
From: "michaelprice" <>
References: <>
Subject: Infrared Divergences = Physical Infinity
Date: Wed, 18 Jun 2003 00:51:06 +0100

Peter Merel writes:
> Michael C Price writes,
> 
>> When a charged particle accelerates it emits an infinite (alepth_0)
>> number of real photons.  This is known as the infrared divergence,
>> and is an example of where nature realises an infinity.
> 
> It's a matter of choice of representation. 

No, the real[*] photon number is a representation independent
quantity.  This goes back to Planck and Einstein's work on 
the photoelectric effect of 1899-1905.  A photon's energy is 
simply frequency (f) x Planck's constant (h) so that the number 
of photons totalling energy (E) is just E/hf.

When a charge is accelerated the low-frequency energy of the 
Bremsstrahlung radiation emitted is independent of frequency,
so that the number of low energy photons emitted scales with the
inverse of the frequency.  Total photon number is obtained by
integration of the inverse function to yield the logarithmic function.
Since log(0) = infinity we have an infinite number of low-energy 
or "soft" photons, even though the total energy emitted is finite.

Because our sensors always have some threshold of sensitivity, 
only a finite number of soft photons can be measured by a  
particular detector.  Nevertheless the theory is unambiguous
about the existence of an infinite number of soft photons.  The
more sensitive the detector is made the more photons it will
detect.

> Any physical number represents three ontologies - unit, metric, 
> and value. If these are each distinguished as some fraction of 
> the entire universe of description accessible to your sensors, 
> then no infinity would be necessary to characterize the result 
> of an experiment.

Division by zero to produce infinity often occurs in physics.  
In this example the zero is the photon's rest mass.and the 
infinity is the photon number.  These infrared divergences 
occur everywhere in quantum electrodynamics.

[*] In case anyone wonders, the photons I'm talking about
are not "virtual photons", but real photons.

Cheers,
Michael C Price
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http://www.hedweb.com/manworld.htm

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