ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
The regulated propagator G'ε(ξ,ξ') is considered for points ξ and ξ' on the two-dimensional flat torus and with a regularization analogous to the proper-time method with cut-off ε. The inequivalent tori are labeled by a modular parameter τ. In this regularization, the propagator has an expansion in eigenmodes which is a particular Jacobi–Riemann function with beautiful properties and which can be transformed to an integral representation by a Sommerfeld–Watson method. This latter method allows one to derive exact analytic expressions in several domains of the parameters ||ξ−ξ'||, ε, and of the parameter τ. For generic value of τI≡Im(τ) and small value of the cut-off ε we recover the known results relative to short-distance phenomena. At large value of τI(approximately-equal-to)O(cst/4πε), the propagator diverges as τI and the coefficient is computed. A general formula for the Weyl variation of the propagator is derived; again the behavior of the Weyl variation at fixed τI is reproduced, while the behavior at large τI is novel. As a result we find that to O((square root of)1/τI) the Weyl variation of the propagator at large τI is local in the Weyl variation δcursive-phi(ξ).
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.529533
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