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A Expanding CCRs, and their underlying meaning

  1. Dec 18, 2016 #1
    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!
  2. jcsd
  3. Dec 18, 2016 #2


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    Maybe it's related to the Epstein-Glaser approach. Then a good source is

    Scharff, Finite QED, Springer.
  4. Dec 18, 2016 #3


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    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).
    Last edited: Dec 18, 2016
  5. Dec 20, 2016 #4
    Thanks for the reference. I skimmed over it, seems interesting, but it is definitely not what I was talking about.

    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.
  6. Dec 20, 2016 #5


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    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: Dec 20, 2016
  7. Jan 17, 2017 #6
    Yes, I think it is Wightman QFT.
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