Quantum mechanical path integrals in curved spaces and the type-A trace anomaly

Abstract

Path integrals for particles in curved spaces can be used to compute trace anomalies in quantum field theories, and more generally to study properties of quantum fields coupled to gravity in first quantization. While their construction in arbitrary coordinates is well understood, and known to require the use of a regularization scheme, in this article we take up an old proposal of constructing the path integral by using Riemann normal coordinates. The method assumes that curvature effects are taken care of by a scalar effective potential, so that the particle lagrangian is reduced to that of a linear sigma model interacting with the effective potential. After fixing the correct effective potential, we test the construction on spaces of maximal symmetry and use it to compute heat kernel coefficients and type-A trace anomalies for a scalar field in arbitrary dimensions up to d = 12. The results agree with expected ones, which are reproduced with great efficiency and extended to higher orders. We prove explicitly the validity of the simplified path integral on maximally symmetric spaces. This simplified path integral might be of further use in worldline applications, though its application on spaces of arbitrary geometry remains unclear.

On the simplified path integral on spheres

“…For d = 14, 16 they read 1 The expression below coincides with Eq. (2.29) of [14], thanks to the Table 2, as one may check.…”

“…We have performed this exercise in [1] to test the correctness and usefulness of the linear sigma model approach. We are now ready to extend those results to identify the trace anomalies in d = 14 and d = 16 dimensions.…”

“…We are now ready to extend those results to identify the trace anomalies in d = 14 and d = 16 dimensions. As reviewed in [1], the trace anomaly of the conformal scalar field can be related to the transition amplitude of a particle in a curved space by…”

“…In particular, path integrals for particles on curved spaces allow one to study gravitationally interacting field theories. In this paper, after reviewing the simplified path integral for a nonrelativistic particle on a sphere that has been introduced recently in [1], by presenting further details of its construction, we extend its perturbative calculation to orders high enough to be able to read off the type-A trace anomalies of a conformal scalar field in dimensions d = 14 and d = 16.…”

“…A simplified version of the path integral for the case of maximally symmetric spaces, like spheres, has been discussed and proved recently in [1]. It builds on an old proposal [12] of constructing the path integral by making use of Riemann normal coordinates.…”