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Heat kernel for higher-order differential operators and generalized exponential functions | | A. O. Barvinsky
; P. I. Pronin
; W. Wachowski
; | | Date: |
6 Aug 2019 | | Abstract: | We consider the heat kernel for higher-derivative and nonlocal operators in
$d$-dimensional Euclidean space-time and its asymptotic behavior. As a building
block for operators of such type, we consider the heat kernel of the minimal
operator - generic power of the Laplacian - and show that it is given by the
expression essentially different from the conventional exponential WKB ansatz.
Rather it is represented by the generalized exponential function (GEF) directly
related to what is known in mathematics as the Fox-Wright $varPsi$-functions
and Fox $H$-functions. The structure of its essential singularity in the proper
time parameter is different from that of the usual exponential ansatz, which
invalidated previous attempts to directly generalize the Schwinger-DeWitt heat
kernel technique to higher-derivative operators. In particular, contrary to the
conventional exponential decay of the heat kernel in space, we show the
oscillatory behavior of GEF for higher-derivative operators. We give several
integral representations for the generalized exponential function, find its
asymptotics and semiclassical expansion, which turns out to be essentially
different for local operators and nonlocal operators of non-integer order.
Finally, we briefly discuss further applications of the GEF technique to
generic higher-derivative and pseudo-differential operators in curved
space-time, which might be critically important for applications of
Horava-Lifshitz and other UV renormalizable quantum gravity models. | | Source: | arXiv, 1908.2161 | | Services: | Forum | Review | PDF | Favorites | |
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