C-P-T Fractionalization

Abstract

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as ${\mathbb{Z}}_2$-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 $(-1{)}^{\mathrm{F}}$. 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-$(-1{)}^{\mathrm{F}}$ group structures, e.g., Dirac spinor is in a projective representation of ${\mathbb{Z}}_2^{\mathrm{C}}\times {\mathbb{Z}}_2^{\mathrm{P}}\times {\mathbb{Z}}_2^{\mathrm{T}}$ but in an (anti)linear representation of an order-16 nonabelian 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 T${}^2=(-1{)}^{\mathrm{F}}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

8 pages. Heredity of setups: Spinor theories follow any standard QFT textbook. The 0+1d Majorana zero modes analysis follows arXiv:2011.13921, arXiv:2011.12320. Special thanks to Shing-Tung Yau on "Can C-P-T symmetries be fractionalized more than involutions?" v4: refinement

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Cited by in corpus (4)

• CT or P Problem and Symmetric Gapped Fermion Solution
• Discrete symmetry transformations on non-abelian gauge fields
• Three point interaction of Dirac fermions with higher spin particles and discrete symmetries
• Operator Dimension Parity Fractionalization