An Alternative Perspective on Pauli's Theorem
Zhiyong Wang, Cai-Dong Xiong · arXiv (Cornell University) · 2006
From the point of view of purely mathematics, coordinates and momenta can be imaginary, and then it is not necessary for a Hamiltonian operator to have a semi-bounded spectrum, only if one imposes some physical conditions, a Hilbert space is defined and the spectrum of the Hamiltonian operator is determined. Pauli's Theorem just shows us a mathematical truth: via a time operator, one can find all the possible eigenstates of the Hamiltonian operator, but only a part of them satisfy the given physical conditions and represent the physical solutions, while the remainders represent the redundant solutions, they do not satisfy the given physical conditions and should be discarded. The transition probabilities between the physical solutions and the redundant solutions vanish. Therefore, the existence of a self-adjoint time operator actually presents no problem.