Expanding CCRs, and their underlying meaning

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Discussion Overview

The discussion revolves around the concept of expanding commutation relations (CCRs) in quantum field theory (QFT), particularly focusing on a modification involving Gaussian functions instead of delta functions. Participants explore theoretical implications, potential connections to existing frameworks, and seek further information on the topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant recalls a lecture discussing a form of commutation relations involving a Gaussian function, suggesting a generalization of CCRs leading to a 'smeared' QFT.
  • Another participant proposes a connection to the Epstein-Glaser approach and references a source related to finite QED.
  • A different participant mentions a form of commutation relations with an exponential, providing a link to a paper that discusses this topic.
  • One participant expresses that the referenced materials do not align with their original inquiry about smearing annihilation/creation operators and suggests a connection to Gaussian Processes.
  • Another participant introduces the idea of a Lieb-Robinson bound as a potential related concept, linking it to relativistic QFT and the enforcement of light speed through commuting spacelike observables.
  • One participant concludes that the discussion may relate to Wightman QFT.

Areas of Agreement / Disagreement

Participants express differing views on the connections between the discussed concepts, with no consensus reached on the specific nature of the modifications to the CCRs or their implications.

Contextual Notes

Participants reference various theoretical frameworks and papers, indicating a complex interplay of ideas without resolving the underlying assumptions or definitions related to the modifications of the CCRs.

golanor
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Hi,
I remember seeing a few months ago, at a lecture about statistical signal processing, something which looked similar to commutation relations, only with a gaussian, instead of a delta function. Basically, it looked like this:

$$\left[\phi(x),\phi(y)\right] = ie^{-\alpha(x-y)^2}$$

This reminded me an offhanded remark by my QFT professor, that it is possible to generalise the CCRs, and obtained a `smeared' QFT (or something like that).

Now, for some reason, I can't find anything on this subject, which is a shame, since I find it very interesting.
I'd very much appreciate any info/direction on this subject!
 
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Maybe it's related to the Epstein-Glaser approach. Then a good source is

Scharff, Finite QED, Springer.
 
Last edited:
vanhees71 said:
Maybe it's related to the Epstein-Glaser approach. Then a good source is

Scharff, Finite QED, Springer.

Thanks for the reference. I skimmed over it, seems interesting, but it is definitely not what I was talking about.

atyy said:
There is a form of the commutation relations (but different from what you wrote) that has an exponential, eg. http://rejzner.com/files/QFT-Roma.pdf (Eq 1 and 2).

Not at all what I meant.I tried to look up what I was talking about - apparently the idea was to "smear" the annihilation/creation operators, which modifies the CCR.
Another place this is prevalent, and which made me think of this modification to start with, was the theory of Gaussian Processes.
I think that both theories have the same underlying mathematical structure, and I was wanting to dig deeper into it.
 
Have you found what it is? I have another guess. Is it a Lieb Robinson bound? These make commutation relations that are approximately the same as those as those in relativistic QFT in which enforcing the speed of light is done with commuting spacelike observables, eg. https://arxiv.org/abs/1008.5137 (Eq 10 and the paragraph after, which mentions an approximate light cone) or https://arxiv.org/abs/1412.2970 (which mentions almost local observables).
 
Last edited:
Yes, I think it is Wightman QFT.
 

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