Re: Does the Uncertainty Relation hold for a photon whose T=0 ?

Firstly, I think some clarification of the question is required. Let me first answer the following question: "If a photon is measured to be an infinitely short pulse in time (i.e. have near zero length), will it have an infinite spread in energy?".

The answer is yes. There is certainly no problem with photons having an infinite energy spectrum.

Now let us go back to the original question...

Let us first consider passage of time:

A photon may certainly experience no passage of time, but if it is a pulse of a certain length then, in the direction that it is propagating in, it will have some components behind itself and some in front. A photon has some sort of "size" in both time and space.

For example, if a photon is 1 microsecond long (a fairly reasonably length for a photon from a laser), it will span this amount of time, regardless of whether it has propagated anywhere. That is to say that the "front end" of the photon will have come out of the laser and will have travelled a long way before the "back end" escapes the laser resonatator.

Similarly, since photons travel at the speed of light, a photon, if you were to take a snapshot of it, would span a certain amount of distance just given by the relation distance = speed * time. In this case the speed is the speed of light (300 000 000 metres per second). A photon with a length of 1 microsecond will span a distance of 300 metres. That is quite a long photon!

Now let us just clarify the zero time point. It doesn't matter at all where you measure zero time from. It could be (and usually is) in the middle of the photon pulse. In which case, some of the photon would exist before zero time and some after. Time always has to be relative to some other time or we can't measure it.

Now let us come back to the Uncertainty Relation. Notice that I sort of avoided the subject in the answer to my own question. Although the Uncertainty Relation for energy and time is discussed extensively in texts on quantum mechanics it is not strictly possible to derive since time is only a parameter in current quantum theory. HOWEVER, the uncertainty relations (certainly in their simple form) can be derived from something called Fourier theory (or Fourier-transforms or -series). This is 19th Century mathematics and is very well understood. For a brief history have a look at

http://en.wikipedia.org/wiki/Fourier_transform

Fourier theory tells you that if you want to make up a pulse of waves (and this is true of any kind of wave, not just quantum particles - i.e. water waves, sound waves, Mexican waves etc.) then you have to use many different frequencies. A single frequency can never make a pulse since a single frequency, by definition, never stops. Maybe you can see now how this relates to the Uncertainty Relation since frequency and energy are the same thing as far as photons are concerned. For the record, the position-momentum relation comes from the description of momentum as a kind of spatial frequency. Let us not go there right now.

As for the "uncertainty" (a poorly chosen word in my opinion) - it is true that if a pulse is short and has a lot of frequency components, and hence a large spread in energy, one cannot define it's energy exactly. Moreover if the energy of the photon is measured with more precision than its spread this will cause a photon pulse to stretch. Simply, if a photon pulse, with a large spread in energy, passes through a narrow energy (or wavelength) filter it will become longer.