nForum - Discussion Feed (Mellin transform and Schwinger parameterization) 2022-01-22T03:00:41-05:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher zskoda comments on "Mellin transform and Schwinger parameterization" (49710) https://nforum.ncatlab.org/discussion/6237/?Focus=49710#Comment_49710 2014-09-14T12:01:22-04:00 2022-01-22T03:00:40-05:00 zskoda https://nforum.ncatlab.org/account/10/ I just copied the above 2 formulas to the entry Mellin transform.

I just copied the above 2 formulas to the entry Mellin transform.

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Urs comments on "Mellin transform and Schwinger parameterization" (49708) https://nforum.ncatlab.org/discussion/6237/?Focus=49708#Comment_49708 2014-09-13T23:29:20-04:00 2022-01-22T03:00:40-05:00 Urs https://nforum.ncatlab.org/account/4/ What mathematicians call the Mellin transform relating a theta function to its (completed) zeta function &zeta;^(s)=&Integral; 0 &infin;t s&minus;1&theta;^(t)dt \hat \zeta(s) = ...

What mathematicians call the Mellin transform relating a theta function to its (completed) zeta function

$\hat \zeta(s) = \int_0^\infty t^{s-1} \hat \theta(t) \, d t$

is precisely what physicists call the Schwinger parameter-formulation which takes the partition function of the worldline formalism to the zeta-regulated Feynman propagator

$Tr H^{-s} = \int_0^\infty t^{s-1} Tr \exp(- t H) \, d t \,.$

I have tried to briefly mention this relation in relevant entries and to cross-link a bit. But more should be done.

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