Expanding CCRs, and their underlying meaning

In summary, the conversation discusses a modification to the commutation relations in quantum field theory by using a Gaussian function instead of a delta function. This idea is prevalent in the theory of Gaussian processes and has similarities to the Lieb Robinson bound and Wightman QFT. The conversation also mentions some resources for further reading on the topic.
  • #1
golanor
59
0
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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  • #2
Maybe it's related to the Epstein-Glaser approach. Then a good source is

Scharff, Finite QED, Springer.
 
  • #3
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).
 
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  • #4
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.
 
  • #5
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).
 
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  • #6
Yes, I think it is Wightman QFT.
 

1. What are CCRs and why are they important?

CCRs, or covenants, conditions, and restrictions, are rules that govern the use and development of a property. They are important because they help maintain the value and integrity of a community by ensuring that all property owners follow the same guidelines.

2. How do CCRs affect property owners?

CCRs can affect property owners in a variety of ways, depending on the specific regulations set forth. Some common examples include restrictions on exterior modifications, limitations on types of businesses that can be run from the property, and guidelines for maintaining the appearance of the property.

3. What is the process for expanding CCRs?

The process for expanding CCRs typically involves gathering support from a majority of property owners in the community, holding a vote to approve the expansion, and then formally amending the existing CCRs to include the new regulations. It is important to carefully follow any procedures outlined in the original CCRs and consult legal counsel if necessary.

4. Can property owners opt out of CCRs?

In most cases, property owners cannot opt out of CCRs that have been approved and recorded with the appropriate authorities. However, there may be instances where a property owner can request a variance or exception to certain regulations, which would need to be approved by the appropriate governing body.

5. How do CCRs impact property values?

CCRs can have a significant impact on property values, as they help maintain a certain level of consistency and quality within a community. Properties located in areas with well-maintained and enforced CCRs may see an increase in value, while properties in areas with lax or non-existent CCRs may see a decrease in value.

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