arXiv subjects

Rachel Maitra

Publications and source records attributed to Rachel Maitra.

4 recordsLinked to original sources

arXiv2019

Orbit Space Curvature as a Source of Mass in Quantum Gauge Theory

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.

hep-th↗

arXiv2019

A Euclidean Signature Semi-Classical Program

In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schrödinger's equation (for suitable `nonlinear oscillatory' quantum mechanical systems) to the treatment of several physically significant, interacting quantum field theories. Our main focus is on applying these `Euclidean-signature semi-classical' methods to self-interacting (real) scalar fields of renormalizable type in 2, 3 and 4 spacetime dimensions and to Yang-Mills fields in 3 and 4 spacetime dimensions. A central argument in favor of our program is that the asymptotic methods for Schrödinger operators developed in the microlocal literature are far superior, for the quantum mechanical systems to which they naturally apply, to the conventional WKB methods of the physics literature and that these methods can be modified, by techniques drawn from the calculus of variations and the analysis of elliptic boundary value problems, to apply to certain (bosonic) quantum field theories. Unlike conventional (Rayleigh/ Schrödinger) perturbation theory these methods allow one to avoid the artificial decomposition of an interacting system into an approximating `unperturbed' system and its perturbation and instead to keep the nonlinearities (and, if present gauge invariances) of an interacting system intact at every level of the analysis.

math-ph↗

arXiv2016

Euclidean signature semi-classical methods for bosonic field theories: interacting scalar fields

Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schroedinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories. In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of self-interacting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-Jacobi type followed by the integration (for both ground and excited states) of a sequence of linear 'transport' equations along the 'flow' generated by the 'fundamental solution' to the aforementioned Hamilton-Jacobi equation. Using a combination of the direct method of the calculus of variations, elliptic regularity theory and the Banach space version of the implicit function theorem we establish, in a suitable function space setting, the existence, uniqueness and global regularity of this needed 'fundamental solution' to the relevant, Euclidean signature Hamilton-Jacobi equation for the systems under study. Our methods are applicable to (massive) scalar fields with polynomial self-interactions of renormalizable type. They can, as we shall show elsewhere, also be applied to Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.

math-ph↗

arXiv2012

Modified Semi-Classical Methods for Nonlinear Quantum Oscillations Problems

We develop a modified semi-classical approach to the approximate solution of Schrodinger's equation for certain nonlinear quantum oscillations problems. At lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. Under smoothness, convexity and coercivity hypotheses on its potential energy function, we prove, using the calculus of variations together with the Banach space implicit function theorem, the existence of a global, smooth `fundamental solution'. Higher order quantum corrections, for ground and excited states, are computed through the integration of associated systems of linear transport equations, and formal expansions for the corresponding energy eigenvalues obtained by imposing smoothness on the quantum corrections to the eigenfunctions. For linear oscillators our expansions naturally truncate, reproducing the well-known solutions for the energy eigenfunctions and eigenvalues. As an application, we calculate a number of terms in the corresponding expansions for the one-dimensional anharmonic oscillators of quartic, sectic, octic, and dectic types and find that our eigenvalue expansions agree with those of Rayleigh/Schrodinger theory, whereas our wave functions more accurately capture the more-rapid-than-gaussian decay. For the quartic oscillator our results strongly suggest that the ground state energy eigenvalue expansion and its associated wave function expansion are Borel summable to yield natural candidates for the actual exact ground state solution and its energy. Our techniques for proving the existence of the crucial `fundamental solution' to the relevant Hamilton Jacobi equation admit infinite dimensional generalizations. In a parallel project we shall show how this construction can be carried out for the Yang-Mills equations in Minkowski spacetime.

math-ph↗