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Suppose I have a two dimensional classical Dirac Hamiltonian with $\Psi=(\psi_1,\psi_2)^T$: $$ H=\int \mathrm{d}x \mathrm{d}y \Psi^\dagger(\sigma^x i\partial_x+\sigma^y i\partial_y+m\sigma^z)\Psi. $$ The partition function is given by $$ Z=\int[\mathrm d\psi^\dagger][\mathrm d\psi]e^{-H}. $$ How can I relate it to a (1+1)-dim quantum system?

A simple replacement $x\rightarrow t$ seems to be wrong. Because in the standard (1+1)-dim Dirac action, the time derivative term must be $\Psi^\dagger\partial_\tau\Psi$, i.e. with identity matrix in front of the time derivative, to ensure the conjugate momenta of $\Psi$ is just $i\Psi^\dagger$.

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