# QED: redshifting light and infrared divergence

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• Ken Gallock

#### Ken Gallock

I am looking for some resources describing the following content:

A light with wavelength ##\lambda## is propagating in flat spacetime. The light redshifts as its wavelength gets larger and larger. In quantum field theory, this causes an infrared divergence of the field.
What I want to know is the way to tame the divergence. More specifically, I want to calculate a cutoff wavelength ##\Lambda_{\mathrm{cut}}##.
Are there any articles or textbooks cover this topic? If any, pedagogical ones are appreciated.

Thanks.

A light with wavelength ##\lambda## is propagating in flat spacetime. The light redshifts as its wavelength gets larger and larger. In quantum field theory, this causes an infrared divergence of the field.

No, that's not what "infrared divergence" means.

First, the "wavelength" of light is not a property of the light by itself. It's an observable, which depends on both the light and the state of motion of the thing doing the observing. Two observers in relative motion will measure one and the same light ray to have different wavelengths.

Second, light does not "redshift" as it propagates in flat spacetime; since spacetime is flat, there is no gravity, no expanding universe, nothing like that that could cause a redshift that is not just a simple Doppler shift based on the state of motion of the observer (at least heuristically--there are plenty of technical complications lurking here that I don't think we need to dig into, although in an "A" level thread we might end up doing so).

So before even trying to think about the quantum field theory of light, you need to understand clearly the two things I said above; the way you posed your question indicates that you don't fully grasp those things yet.

All that said, the term "infrared divergence" for the quantum field theory of light refers (heuristically speaking) to the fact that, since light is massless, there is no lower limit to the amount of energy and momentum that a single quantum of light (photon) can carry. So a finite amount of energy could in principle be carried by an infinite number of photons according to the theory; but physically speaking, "an infinite number of photons" doesn't seem reasonable. This issue is not considered as serious as an ultraviolet divergence, since it still allows a finite amount of energy to be contained in the field; whereas a UV divergence causes the theory to predict that the quantum field, even in its vacuum state, carries an infinite amount of energy, which is worse, physically speaking, than just predicting an infinite number of particles.

No, that's not what "infrared divergence" means.

First, the "wavelength" of light is not a property of the light by itself. It's an observable, which depends on both the light and the state of motion of the thing doing the observing. Two observers in relative motion will measure one and the same light ray to have different wavelengths.

Second, light does not "redshift" as it propagates in flat spacetime; since spacetime is flat, there is no gravity, no expanding universe, nothing like that that could cause a redshift that is not just a simple Doppler shift based on the state of motion of the observer (at least heuristically--there are plenty of technical complications lurking here that I don't think we need to dig into, although in an "A" level thread we might end up doing so).

So before even trying to think about the quantum field theory of light, you need to understand clearly the two things I said above; the way you posed your question indicates that you don't fully grasp those things yet.

All that said, the term "infrared divergence" for the quantum field theory of light refers (heuristically speaking) to the fact that, since light is massless, there is no lower limit to the amount of energy and momentum that a single quantum of light (photon) can carry. So a finite amount of energy could in principle be carried by an infinite number of photons according to the theory; but physically speaking, "an infinite number of photons" doesn't seem reasonable. This issue is not considered as serious as an ultraviolet divergence, since it still allows a finite amount of energy to be contained in the field; whereas a UV divergence causes the theory to predict that the quantum field, even in its vacuum state, carries an infinite amount of energy, which is worse, physically speaking, than just predicting an infinite number of particles.

So, I think I should know about more basic concepts. Let me ask the following:
The situation is that there is a light source in flat spacetime which emits a light of wavelength ##\lambda##. And an observer, who detects the light, is in the same frame of reference of the light source. In this case, the observer sees the source as a stationary object.
Now, let us assume that the wavelength ##\lambda## gets larger and larger. The observer detects the light for a while but eventually he/she might not be able to detect it since the wavelength is too long to observe in principle.

The question is
(1) Is the statement "he/she might not be able to detect it since the wavelength is too long to observe" correct? I believe this statement has to do with divergences occur in quantum field theory (someone told me).

(2) If the statement in question (1) holds, then what is the problem called? Are there any articles that cover this topic?

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And an observer, who detects the light, is in the same frame of reference of the light source.
There is no such thing as ”being in a frame”. All observers in SR exist in all frames. What you likely want to say is that the source and detector are at relative rest. It may seem minor, but this distinction can cause serious misunderstandings.

Now, let us assume that the wavelength λλ\lambda gets larger and larger.
How? You cannot just assume this and so your meaning is unclear. Do you mean to say that the source is emitting at ever increasing wavelength or that due to some magical interference the wavelength of the light gets longer as it propagates?

(1) Is the statement "he/she might not be able to detect it since the wavelength is too long to observe" correct? I believe this statement has to do with divergences occur in quantum field theory (someone told me).
Non-observability is not due to IR divergence. Quite the opposite actually, the implication goes the other way. You can argue about observability to regulate IR divergences, but non-observability is not due to IR divergences.

Also ”someone told me” is not an appropriate reference as it is impossible to know exactly what you were told and deduce whether the statement itself was faulty or your understanding of it.

I suggest that you cement your understanding of classical relativity before you attempt QFT.

What you likely want to say is that the source and detector are at relative rest.
Yes, that is what I meant. Sorry.

How? You cannot just assume this and so your meaning is unclear. Do you mean to say that the source is emitting at ever increasing wavelength or that due to some magical interference the wavelength of the light gets longer as it propagates?
For example, we can use a device like a laser; The wavelength of the light can be changed as we turn a dial of the device. By doing so, the wavelength of the light can be changed (I assume nothing affects the wavelength while the light propagates).

Non-observability is not due to IR divergence. Quite the opposite actually, the implication goes the other way. You can argue about observability to regulate IR divergences, but non-observability is not due to IR divergences.
Could you tell me what is responsible for non-observability?

Is the statement "he/she might not be able to detect it since the wavelength is too long to observe" correct?

It's correct as a statement about the practical limitations of any real detector, yes. Any real detector will have an upper limit to how long a wavelength it can detect. But there is no theoretical upper limit in QED (see further comments below); it's just a practical matter of building more capable detectors.

I believe this statement has to do with divergences occur in quantum field theory (someone told me).

Whoever told you this was wrong. Theoretically speaking, QED allows detection of light of arbitrarily long wavelengths. The "infrared divergences" in QED have nothing to do with it.

If the statement in question (1) holds, then what is the problem called?

Um, the fact that real detectors are not idealized theoretical detectors?

But there is no theoretical upper limit in QED
I was thinking about the theoretical upper limit. So, you mean that even a very long wavelength, such as an infinitely long one, can be detected in principle?

So, you mean that even a very long wavelength, such as an infinitely long one, can be detected in principle?

There is no such thing as an infinitely long wavelength. But in principle, according to QED, any finite wavelength can be detected, yes.