C-R-T Fractionalization in the First Quantized Hamiltonian Theory

Recent research has revealed that the CRT symmetry for fermions exhibits a fractionalization distinct from the $\mathbb{Z}2^{\mathcal{C}}\times\mathbb{Z}2^{\mathcal{R}}\times\mathbb{Z}2^{\mathcal{T}}$ for scalar bosons. In fact, the CRT symmetry for fermions can be extended by internal symmetries such as fermion parity, thereby forming a group extension of the $\mathbb{Z}2$ direct product. Conventionally, a Majorana fermion is defined by one Dirac fermion with trivial charge conjugation. However, when the spacetime dimension $d+1=5,6,7\bmod8$, the real dimension of Majorana fermion (dim${\mathbb{R}}χ{\mathcal{C}\ell(d,0)}$) aligns with the real dimension of Dirac fermion (dim${\mathbb{R}}ψ{\mathcal{C}\ell(d)}$), rather than being half, which necessitates the introduction of a symplectic Majorana fermion, defined by two Dirac fermions with trivial charge conjugation. To include these two types of Majorana fermions, we embed the theory in $n_{\mathbb{R}}$ and define the Majorana fermion field as a representation of the real Clifford algebra with 8-fold periodicity. Within the Hamiltonian formalism, we identify the 8-fold CRT-internal symmetry groups across general dimensions. Similarly, Dirac fermion field is defined as a representation of the complex Clifford algebra with 2-fold periodicity. Interestingly, we discover that the CRT-internal symmetry groups exhibit an 8-fold periodicity that is distinct from that of the complex Clifford algebra. In certain dimensions where distinct mass terms can span a mass manifold, the CRT-internal symmetries can act non-trivially upon this mass manifold. Employing domain wall reduction method, we are able to elucidate the relationships between symmetries across different dimensions.

💡 Research Summary

The paper investigates the structure of charge‑conjugation (C), spatial‑reflection (R), and time‑reversal (T) symmetries—collectively called CRT—in fermionic quantum field theories across arbitrary spacetime dimensions. While scalar bosons possess a simple direct‑product symmetry group (G_\phi = \mathbb{Z}_2^C \times \mathbb{Z}_2^R \times \mathbb{Z}_2^T), fermions exhibit a richer pattern because internal symmetries such as fermion parity ((\mathbb{Z}_2^F)), chiral symmetry ((\mathbb{Z}_2^\chi)), and continuous groups can extend the basic (\mathbb{Z}_2) factors. This phenomenon, known as symmetry fractionalization, is naturally described by a group extension \