On the group generated by $\mathbf{C}$, $\mathbf{P}$ and $\mathbf{T}$: ${\mathbf{I}}^2={\mathbf{T}}^2={\mathbf{P}}^2=\mathbf{ITP}=-1$, with applications to pseudo-scalar mesons

Abstract

We study faithful representations of the discrete Lorentz symmetry operations of parity $\mathbf{P}$ and time reversal $\mathbf{T}$, which involve complex phases when acting on fermions. If the phase of $\mathbf{P}$ is a rational multiple of $\pi$ then ${\mathbf{P}}^{2n}=1$ for some positive integer $n$ and it is shown that, when this is the case, $\mathbf{P}$ and $\mathbf{T}$ generate a discrete group, a dicyclic group (also known as a generalised quaternion group) which are generalisations of the dihedral groups familiar from crystallography. Charge conjugation $\mathbf{C}$ introduces another complex phase and, again assuming rational multiples of $\pi$ for complex phases, $\mathbf{TC}$ generates a cyclic group of order $2m$ for some positive integer $m$.There is thus a doubly infinite series of possible finite groups labelled by $n$ and $m$. Demanding that $\mathbf{C}$ commutes with $\mathbf{P}$ and $\mathbf{T}$ forces $n=m=2$ and the group generated by $\mathbf{P}$ and $\mathbf{T}$ is uniquely determined to be the quaternion group. Neutral pseudo-scalar mesons can be simultaneous $\mathbf{C}$ and $\mathbf{P}$ eigenstates. $\mathbf{T}$ commutes with $\mathbf{P}$ and $\mathbf{C}$ when acting on fermion bi-linears so neutral pseudo-scalar mesons can also be $\mathbf{T}$ eigenstates. The $\mathbf{T}$-parity should therefore be experimentally observable and the $\mathbf{CPT}$ theorem dictates that $T=CP$.

21 pages of text plus an appendix; in v3 the discussion is expanded to include both choices of sign for the Minkowski metric and the final discussion is updated

References in corpus (4)

• Precise Measurements of Direct CP Violation, CPT Symmetry, and Other Parameters in the Neutral Kaon System
• Observation of Time Reversal Violation in the B0 Meson System
• Time Reversal Violation from the entangled B0-antiB0 system
• CPT violation and B-meson oscillations

Cited by in corpus (2)

• C-P-T Fractionalization
• Generalized imaginary units in quantum mechanics