Symmetries in Relativistic Quantum Mechanics

Bojan Tunguz

Andreas Wipf

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.

Kai Johannes Keller, Nikolaos A. Papadopoulos, Andrés Fernando Reyes-Lega

Rodrigo Muñoz-Vega, Alberto García-Quiroz, Ernesto López-Chávez +1 more

The advantages and disadvantages of some pedagogical non-relativistic quantum-mechanical models, used to illustrate spontaneous symmetry breakdown, are discussed. A spinor on the line subject to a magnetostatic interaction is presented as a toy model of the spontaneous breakdown of an internal symmetry.

Michel Le Bellac