Post Pauli's theorem emerging perspective on time in quantum mechanics

Eric A. Galapon

pp. 25-63

in: Gonzalo Muga, Andreas Ruschhaupt, Adolfo del Campo (eds), Time in quantum mechanics II, Berlin, Springer, 2009

Abstract

In a Hilbert space setting, Pauli's well-known theorem Pauli's theorem asserts that no self-adjoint operator exists that is conjugate to a semibounded or discrete Hamiltonian [58]. Pauli's argument goes as follows. Assume that there exists a self-adjoint operator T conjugate to a given Hamiltonian H, that is, [T,H]=iћI such an operator conjugate to the Hamiltonian is known as a time operator. Since T is self-adjoint, the operator U_ε=exp(–iεT) is unitary for all real number ε. Now if φE is an eigenvector of H with the eigenvalue E, then, according to Pauli, the conjugacy relation [T,H]=iћI implies that T is a generator of energy shifts so that (E+ε)φE+e; this means that H has a continuous spectrum spanning the entire real line because ε is an arbitrary real number. Hence, the "inevitable" conclusion that if the Hamiltonian is semibounded or discrete no self-adjoint time operator T will exist