BRST Extension of Geometric Quantization

Consider a physical system for which a mathematically rigorous geometric quantization procedure exists. Now subject the system to a finite set of irreducible first class (bosonic) constraints. It is shown that there is a mathematically rigorous BRST quantization of the constrained system whose cohomology at ghost number zero recovers the constrained quantum states. Moreover this space of constrained states has a well-defined Hilbert space structure inherited from that of the original system. Treatments of these ideas in the Physics literature are more general but suffer from having states with infinite or zero "norms" and thus are not admissible as states. Also the BRST operator for many systems require regularization to be well-defined. In our more restricted context we show that our treatment does not suffer from any of these difficulties. This work was submitted for publication March 21,2006

Functionals and the Quantum Master Equation

The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of a functional. Both of these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither this Laplacian nor the anti-bracket is well-defined for sections of an arbitrary vector bundle. We show that if the functionals are permitted to have their values in an appropriate graded tensor algebra whose factors are the dual of the space of smooth functions on M, then both the anti-bracket and the Laplace operator can be invariantly defined. Additionally, one obtains a new anti-bracket for ordinary functionals.Comment: 21 pages, Late

Infinite dimensional super Lie groups

A super Lie group is a group whose operations are $G^{\infty}$ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are $G^{\infty}$ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of supernumbers is a Banach Grassmann algebra with a countably infinite set of generators. In this context, we prove that if \hfrak is a closed, split sub-super Lie algebra of the super Lie algebra of a super Lie group \Gcal, then \hfrak is the super Lie algebra of a sub-super Lie group of \Gcal. Additionally, we show that if \gfrak is a Banach super Lie algebra satisfying certain natural conditions, then there is a super Lie group \Gcal such that the even part of \gfrak is the even part of the super Lie algebra of \Gcal. In general, the module structure on \gfrak is required to obtain \Gcal, but the "structure constants" involving the odd part of \gfrak can not be recovered without further restrictions. We also show that if \Hcal is a closed sub-super Lie group of a super Lie group \Gcal, then \Gcal \rar \Gcal/\Hcal is a principal fiber bundle. Finally, we show that if \gfrak is a graded Lie algebra over $C,$ then there is a super Lie group whose super Lie algebra is the Grassmann shell of \gfrak. We also briefly relate our theory to techniques used in the physics literature.Comment: 46 page