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.
Highlights
JHEP04(2017)050 as generating function for the type-A trace anomalies of a scalar field in arbitrary d dimensions
We are led to provide an explicit proof of the validity of the simplified path integral on maximally symmetric spaces, which is presented in appendix A, while appendix B is left for details on our linear sigma-model worldline calculations
In the previous section we have reviewed the quantum mechanical path integral for a nonlinear sigma model, that describes a particle moving in a curved space by using arbitrary coordinates
Summary
Where gij(x) is the metric in an arbitrary coordinate system. It is the action of a nonlinear sigma model in one dimension, and the corresponding equations of motion are the geodesic equations written in terms of the affine parameter t, the time used in the definition of the velocity xi dxi dt. In the present case the counterterms are finite if one includes the local terms arising from the general coordinate invariant path integral measure. Somewhat confusing status of the subject present in previous literature With this tool at hand, more general applications of the path integral were possible, in particular in the first quantized approach to quantum fields [28] coupled to gravitational backgrounds, such as the worldline description of fields of spin 0, 1/2 and 1 coupled to gravity [29,30,31,32], the analysis of amplitudes in Einstein-Maxwell theory [33,34,35,36], the study of photon-graviton conversion in strong magnetic fields [37, 38], the description of higher spin fields in first quantization [39], as well as worldline approaches to perturbative quantum gravity [40]
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