Quantum Theory for Mathematicians
Abstract
Ideas from quantum physics play important roles in many parts of modern
mathematics. Many parts of representation theory, for example, are motivated
by quantum mechanics, including the Wigner–Mackey theory of induced
representations, the Kirillov–Kostant orbit method, and, of course,
quantum groups. The Jones polynomial in knot theory, the Gromov–Witten
invariants in topology, and mirror symmetry in algebraic topology are other
notable examples. The awarding of the 1990 Fields Medal to Ed Witten, a
physicist, gives an idea of the scope of the influence of quantum theory in
mathematics.
Despite the importance of quantum mechanics to mathematics, there is
no easy way for mathematicians to learn the subject. Quantum mechanics
books in the physics literature are generally not easily understood by
most mathematicians. There is, of course, a lower level of mathematical
precision in such books than mathematicians are accustomed to. In addition,
physics books on quantum mechanics assume knowledge of classical
mechanics that mathematicians often do not have. And, finally, there is a
subtle difference in “culture”—differences in terminology and notation—
that can make reading the physics literature like reading a foreign language
for the mathematician. There are few books that attempt to translate quantum
theory into terms that mathematicians can understand.
This book is intended as an introduction to quantum mechanics for mathematicians
with little prior exposure to physics. The twin goals of the book
are (1) to explain the physical ideas of quantum mechanics in language
mathematicians will be comfortable with, and (2) to develop the necessary
mathematical tools to treat those ideas in a rigorous fashion.