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$_{ \mathbb{ R}}\chi_{ \mathcal{ C}\ell(d,0)}$) aligns with the real dimension of Dirac fermion (dim$_{ \mathbb{ R}}\psi_{ \mathcal{ C}\ell(d)}$), 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.
Categories: cond-mat.str-el hep-ph hep-th math-ph math.MP quant-ph