Regularization and counterterms are some of the most elusive aspects of quantum field theory (QFT), but they already show up in the quantum mechanics (QM) path integral and much can be gained from studying them there. In particular, one-dimensional (1D) QM can be viewed as 1D scalar QFT so in that case the lessons carry over directly. In this note I compute the 1D QM partition function in several different ways, in the process gaining insight into regularization and counterterms.

Scalar Partition Functions

This note studies scalar partition functions in detail, in particular for the sphere and periodic cylinder. It offers a tractable arena to understand the path integral, UV regularization, and conformal anomaly.