  It has long been realized that the natural orbit space for non-abelian
Yang-Mills dynamics is a positively curved (infinite dimensional) Riemannian
manifold. Expanding on this result I.M. Singer proposed that strict positivity
of the corresponding Ricci tensor (computable through zeta function
regularization) could play a fundamental role in establishing that the
associated Schroedinger operator admits a spectral gap. His argument was based
on representing the (regularized) kinetic term in the Schroedinger operator as
a Laplace-Beltrami operator on this positively curved orbit space. We revisit
Singer's proposal and show how, when the contribution of the Yang-Mills
potential energy is taken into account, the role of the original orbit space
Ricci tensor is instead played by a Bakry-Emery Ricci tensor computable from
the ground state wave functional of the quantum theory. We next review our
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 intact at each level of the
analysis, our approach surpasses that of conventional perturbation theory for
the generation of approximate wave functionals. Though our main focus is on
Yang-Mills theory we derive the 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. Finally we discuss the potential
applicability of our Euclidean-signature program to the Wheeler-DeWitt equation
of canonical quantum gravity.
