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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 13th 2014
   • (edited Sep 13th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 14th 2014
   • PermaLink
   Author: zskoda
   Format: MarkdownItexI just copied the above 2 formulas to the entry [[Mellin transform]].

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