Heat kernel asymptotics, path integrals and infinite-dimensional determinants

JOURNAL OF GEOMETRY AND PHYSICS

卷:131

Article

Ludewig, Matthias¹ 

[1] Max Planck Inst Math, Vivatgasse 7, D-53119 Bonn, Germany

关键词: Path integrals;    Heat kernels;    Heat kernel asymptotics;    Fredholm determinant;    Zeta function;    Zeta determinant;   

DOI  :  10.1016/j.geomphys.2018.04.012

来源: Elsevier

【 摘 要 】

We compare the short-time expansion of the heat kernel on a Riemannian manifold with the formal stationary phase expansion of its representing path integral and prove that these asymptotic expansions coincide. Besides shedding light on the formal properties of quantum mechanical path integrals, this shows that the lowest order term of the heat kernel expansion is given by the Fredholm determinant of the Hessian of the energy functional on the space of finite energy paths. We also relate this to the zeta determinant of thejacobi operator, considering both the near-diagonal asymptotics as well as the behavior at the cut locus. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free