I wish to know if generalized versions of the Clifford algebra exists such that
$$\{\gamma^{\dagger}_{\mu}, \gamma_{\nu}\} = 2\eta_{\mu\nu}$$
In physics, we usually work with the simplification $\gamma^{\dagger}_{\mu} = \gamma_{\mu}$. However, are there any fields of physics where such a generalized version of the Clifford algebra is required?
For what it's worth, if $$c_j~=~a_j+ib_j, \qquad a_j^{\dagger}~=~a_j,\qquad b_j^{\dagger}~=~b_j,\tag{1}$$ are complex fermionic operators satisfying a standard CAR $$ \{c_j,c_k^{\dagger}\}_+~=~2\eta_{jk}\hat{\bf 1}, \qquad \{c_j,c_k\}_+~=~0~=\{c_j^{\dagger},c_k^{\dagger}\}_+, \tag{2}$$ then $$ \{a_j,a_k\}_+~=~\eta_{jk}\hat{\bf 1}~=~\{b_j,b_k\}_+, \qquad \{a_j,b_k\}_+~=~0, \tag{3}$$ is a usual Clifford algebra.
The above is the fermionic analogue of rewriting the bosonic CCR in terms of bosonic creation & annihilation operators.
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