THE MATHEMATICAL FOUNDATIONS OF QUANTUM THEORY

Publisher Summary This chapter discusses the relation between mathematics and the quantum theory of physics. After the development of noncommutative algebra, it was connected with dynamics by means of an analogy between the commutator and the Poisson bracket of the Hamiltonian form of mechanics and thus, a general quantum mechanics was set up. An important feature of the theory was the wave equation, which had to be linear, and thus treated the time dimension differently from the space coordinates. With the fundamental principle of quantum mechanics—that is, the principle of the superposition of states—the states of any quantum system provide a representation of the Poincare group, which is a mathematical representation of the quantum theories. The study of the group requires working with the infinitesimal operators of which 10 are independent operators where 4 are translation operators and 6 are rotation operators. They satisfy certain definite commutation relations. Any representation of the Poincare group provides the independent operators satisfying these commutation relations. Conversely, any set of 10 operators satisfying these relations gives a representation of the Poincare group.