Symmetries of non-relativistic quantum-mechanical Hamiltonian

Domenico Giulini

Gustavo D. Duarte

Carl M. Bender, Philip D. Mannheim

In nonrelativistic quantum mechanics and in relativistic quantum field theory, the time coordinate $t$ is a parameter and thus the time-reversal operator $\mathcal{T}$ does not actually reverse the sign of $t$. In contrast, in the five-dimensional approach to relativistic quantum mechanics introduced by Feynman, time $t$ is a quantum-mechanical operator. In this paper it is shown how one can use this five-dimensional approach to extend $\mathcal{T}$ and $\mathcal{P}\mathcal{T}$ symmetry from nonrelativistic to relativistic quantum mechanics and implement time-reversal as an operation that effects $\mathcal{T}\text{ }t\mathcal{T}=\ensuremath{-}t$ just as $\mathcal{P}$ effects $\mathcal{P}x\mathcal{P}=\ensuremath{-}x$, with $\mathcal{P}\mathcal{T}$ thus effecting $\mathcal{P}\mathcal{T}\text{ }{x}_{\ensuremath{\mu}}\mathcal{P}\mathcal{T}=\ensuremath{-}{x}_{\ensuremath{\mu}}$. Some illustrative relativistic quantum-mechanical models are constructed whose associated Hamiltonians are non-Hermitian but $\mathcal{P}\mathcal{T}$ symmetric, and it is shown that for each such Hamiltonian the energy eigenvalues are all real.

Andreas Wipf

Andreas Wipf

G. F. Torres del Castillo, J.E. Herrera Flores

It is shown that, in the framework of non-relativistic quantum mechanics, any conserved Hermitian operator (which may depend explicitly on the time) is the generator of a one-parameter group of unitary symmetries of the Hamiltonian and that, conversely, any one-parameter family of unitary symmetries of the Hamiltonian is generated by a conserved Hermitian operator.