Annals of Mathematical Sciences and Applications
Volume 4 (2019)
Orbit space curvature as a source of mass in quantum gauge theory
Pages: 313 – 366
It has long been realized that the natural ‘orbit space’ for non-Abelian Yang–Mills dynamics (i.e., the reduced configuration space of gauge equivalence classes of spatial connections) is a positively curved (infinite dimensional) Riemannian manifold. Expanding upon this result I.M. Singer was led to propose that strict positivity of the corresponding Ricci tensor (computable from the rigorously defined curvature tensor through a suitable zeta function regularization procedure) could play a fundamental role in establishing that the associated Schrödinger operator admits a spectral gap. His argument was based on representing the (suitably regularized) kinetic term in the Schrödinger operator as a Laplace–Beltrami operator on this positively curved orbit space. In this article we revisit Singer’s proposal and show how, when the contribution of the Yang–Mills (magnetic) potential energy is taken into account, the role of the original orbit space Ricci tensor is instead played by a certain ‘Bakry–Émery Ricci tensor’ computable from the ground state wave functional of the quantum theory. We next review the authors’ ongoing Euclidean-signature-semi-classical program for deriving asymptotic expansions for such wave functionals and discuss how, by keeping the dynamical nonlinearities and non-Abelian gauge invariances fully intact at each level of the analysis, our approach surpasses that of conventional perturbation theory for the generation of such approximate wave functionals.
Though our main focus is on Yang–Mills theory we derive the corresponding orbit space curvature for scalar electrodynamics and prove that, whereas the Maxwell factor remains flat, the interaction naturally induces positive curvature in the (charged) scalar factor of the resulting orbit space. This has led us to the conjecture that such orbit space curvature effects could furnish a source of mass for ordinary Klein–Gordon type fields provided the latter are (minimally) coupled to gauge fields, even in the Abelian case.
orbit space curvature, spectral gap, gauge theories
2010 Mathematics Subject Classification
81S10, 81T13, 83C45
Moncrief is grateful to the Swedish Royal Institute of Technology (KTH) for hospitality and support during a visit in March 2015 and especially to Lars Andersson for pointing out the relevance of mathematical literature on Bakry–Émery Ricci tensors to the research discussed herein. The authors are also grateful to the Albert Einstein Institute in Golm, Germany, the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France, the Erwin Schrödinger Institute and the University of Vienna in Vienna, Austria for the hospitality and support extended to several of us during the course of this research.
Received 26 February 2019
Accepted 7 August 2019
Published 2 October 2019