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 Wentzel-Kramers-Brillouin (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 noninteger order. Finally, we briefly discuss further applications of the GEF technique to generic higher-derivative and pseudodifferential operators in curved space-time, which might be critically important for applications of Horava-Lifshitz and other UV renormalizable quantum gravity models.
Highlights
Physical phenomena in higher derivative and nonlocal field theories are essentially different from conventional local quantum field theory (QFT) with the wave operators of second order in space-time derivatives
Rather it is represented by the generalized exponential function (GEF) directly related to what is known in mathematics as the Fox–Wright Ψ-functions and Fox H-functions
The heat kernel method is one of the most powerful tools in mathematical physics, that has a wide range of applications extending from pure mathematics to the analysis of financial markets
Summary
Physical phenomena in higher derivative and nonlocal field theories are essentially different from conventional local quantum field theory (QFT) with the wave operators of second order in space-time derivatives. To understand the nature of the generalization of (1.1) for minimal higher-derivative operators it is enough to consider the case of a flat space-time of the Euclidean signature with the world function σðx; yÞ 1⁄4 ðx − yÞ2=2 and the operator Fð∇Þ 1⁄4 ð−ΔÞν—the νth power of the Laplacian Δ 1⁄4 δab∂a∂b, so that the standard heat kernel takes the translationally invariant form eτΔδðx; yÞ 1⁄4 eτΔδðx − yÞ with eτΔδðxÞ. III we discuss the properties of the generalized exponential functions Eν;αðzÞ and consider their Mellin– Barnes integral representation generating their asymptotic behavior at z → ∞, which is responsible for the short time, τ → 0, or large jxj → ∞ limit of the heat kernel (1.3) This limiting behavior turns out to be different for fractional and integer powers ν. In concluding section we briefly discuss further application of GEF to generic minimal and nonminimal higher-derivative operators in curved space-time, which will allow us to build a solvable recurrent equations for the full set of generalized HaMiDeW-coefficients
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