Contemporary quantum mechanics meets an explosion of different types
of quantization. Some of these quantization techniques (geometric
quantization, deformation quantization, BRST quantization, noncommutative
geometry, quantum groups, etc.) call into play advanced geometry and
algebraic topology. These techniques possess the following main peculiarities.
• Quantum theory deals with infinite-dimensional manifolds and fibre
bundles as a rule.
• Geometry in quantum theory speaks mainly the algebraic language of
rings, modules, sheaves and categories.
• Geometric and algebraic topological methods can lead to non-
equivalent quantizations of a classical system corresponding to different
values of topological invariants.
Geometry and topology are by no means the primary scope of our book,
but they provide the most effective contemporary schemes of quantization.
At the same time, we present in a compact way all the necessary up to date
mathematical tools to be used in studying quantum problems.
Our book addresses to a wide audience of theoreticians and
mathematicians, and aims to be a guide to advanced geometric and algebraic
topological methods in quantum theory. Leading the reader to these
frontiers, we hope to show that geometry and topology underlie many ideas
in modern quantum physics. The interested reader is referred to extensive
Bibliography spanning mostly the last decade. Many references we quote
are duplicated in E-print arXiv (http://xxx.lanl.gov).
With respect to mathematical prerequisites, the reader is expected to
be familiar with the basics of differential geometry of fibre bundles. For
the sake of convenience, a few relevant mathematical topics are compiled
in Appendixes.