C-P-T fractionalization

Discrete spacetime symmetries of parity P or reflection R, and time reversal T, act naively as Z2 involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C -P -R -T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity (-1)F. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1=2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang -Mills (QCD) gauge theories. We discover various C-P-R-T-(-1)F group structures, e.g., Dirac spinor is in a projective representation of ZC2 x ZP2 x ZT2 but in an (anti)linear representation of an order-16 non-Abelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T2 = (-1)F. The general theme may be coined as C -P -T or C -R -T fractionalization.