Manchester Geometry Seminar 2007/2008

It is well known that any quantum system can be described as a classical Hamiltonian system whose phase space is the Hilbert space understood as a symplectic manifold. In particular, the Schrödinger equation takes the form of the Hamilton equation with a quadratic Hamiltonian function and the symplectic form being the imaginary part of the hermitian inner product. In the context of field theory this associated classical system appears to be the classical field theory used as a starting point for the second quantization. I plan to discuss generalization of this construction to the case of constrained Hamiltonian systems quantized in the BRST framework. An interesting subtlety in this case is that the associated classical theory naturally appears in the Lagrangian Batalin-Vilkovisky form. In particular, the even Poisson bracket of the Hamiltonian formalism is replaced by the odd Poisson bracket (antibracket) of the BV quantization. Understanding of this subtlety requires discussing Hamiltonian and Lagrangian BRST quantization in some more details. It especially requires understanding the geometry underlying this relationship and the so-called Alexandrov-Kontsevich-Schwartz-Zaboronsky construction originally proposed in the context of topological field theory.

This study is partially motivated by the possibility to reduce the field theory problems to those of the BRST-quantized constrained system. This allows to apply powerful quantization methods in field theory problems. As an example, I plan to discuss application of the Fedosov quantization in this context.