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arXiv · 2402.18633

The Collective Coordinate Fix

Arindam Bhattacharya·Jordan Cotler·Aurélien Dersy·Matthew D. Schwartz
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Abstract

Collective coordinates are frequently employed in path integrals to manage divergences caused by fluctuations around saddle points that align with classical symmetries. These coordinates parameterize a manifold of zero modes and more broadly provide judicious coordinates on the space of fields. However, changing from local coordinates around a saddle point to more global collective coordinates is remarkably subtle. The main complication is that the mapping from local coordinates to collective coordinates is generically multi-valued. Consequently one is forced to either restrict the domain of path integral in a delicate way, or otherwise correct for the multi-valuedness by dividing the path integral by certain intersection numbers. We provide a careful treatment of how to fix collective coordinates while accounting for these intersection numbers, and then demonstrate the importance of the fix for free theories. We also provide a detailed study of the fix for interacting theories and show that the contributions of higher intersections to the path integral can be non-perturbatively suppressed. Using a variety of examples ranging from single-particle quantum mechanics to quantum field theory, we explain and resolve various pitfalls in the implementation of collective coordinates.

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Arindam Bhattacharya, Jordan Cotler, Aurélien Dersy, Matthew D. Schwartz. 2024-02-28. The Collective Coordinate Fix. https://arxiv.org/abs/2402.18633

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