Timeline for Spurious length-scale cutoff emerges in propagator defined in Costello's "Renormalization and EFT"
Current License: CC BY-SA 3.0
13 events
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| Oct 21, 2017 at 20:35 | comment | added | David Roberts | Ah yes, I am wrong. Thanks for pointing this out :) | |
| Oct 16, 2017 at 21:53 | comment | added | Bertram Arnold | Just to be clear, your formula (2) is not correct (as can be easily seen for flat space). There are other expressions for the propagator which do use cutoffs depending on $x$ and $y$, as in Abdelmalek's answer. If you look at the discussion in Paragraph 6.2 (page 49 in my copy), Costello says that the parameter $\tau$ should be thought of as intrinsic to the worldline, which is ``equipped with an “internal clock”; as the particle moves, this clock ticks at a rate independent of the time parameter on space-time.'' | |
| Oct 16, 2017 at 2:54 | comment | added | David Roberts | Very sensible comment. However, according to Abdelmalek's answer, the issue was a little more subtle than that - indeed, the assumption $\tau=\text{len}(\gamma)$ was valid. The problem is that we can make the cutoff disappear by taking $|x-y|\to 0$. A true length-scale cutoff $\epsilon$ in the propagator for a quantum field theory is independent of $x$ and $y$, and in particular remains non-zero. | |
| S Oct 15, 2017 at 18:59 | history | suggested | Ivo Terek | CC BY-SA 3.0 |
improved formatting
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| Oct 15, 2017 at 17:50 | vote | accept | David Roberts | ||
| Oct 15, 2017 at 16:53 | vote | accept | David Roberts | ||
| Oct 15, 2017 at 17:47 | |||||
| Oct 15, 2017 at 16:27 | review | Suggested edits | |||
| S Oct 15, 2017 at 18:59 | |||||
| Oct 15, 2017 at 16:11 | history | edited | Abdelmalek Abdesselam |
edited tags
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| Oct 15, 2017 at 16:09 | answer | added | Abdelmalek Abdesselam | timeline score: 5 | |
| Oct 15, 2017 at 11:28 | comment | added | Bertram Arnold | In the path integral there is no requirement that the path has unit speed, i.e. $\|\mathrm d\gamma\| = 1$. This means that one can go as far as one wants in a given time interval by going "fast enough". I think the confusion stems from the fact that the parameter $\tau$ is not defined in terms of the unparametrized path $\gamma$, whereas you seem to operate under the assumption that we have $\tau = \mathrm{len}(\gamma)$. | |
| Oct 15, 2017 at 10:32 | comment | added | user74900 | frankly, I don't understand why for $\tau<\mathrm{dist}(x, y)$ paths do not exist. Can't we always make a reparametrization of $\gamma$ (if we can't, you probably have some bounds on $|d\gamma|$ that should be written out explicitly)? | |
| Oct 15, 2017 at 6:49 | history | edited | David Roberts | CC BY-SA 3.0 |
added 99 characters in body
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| Oct 15, 2017 at 5:47 | history | asked | David Roberts | CC BY-SA 3.0 |