Temperature in Fermion Systems and the Chiral Fermion Determinant

We give an interpretation to the issue of the chiral determinant in the heat-kernel approach. The extra dimension (5-th dimension) is interpreted as temperature. The 1+4 dim Dirac equation is naturally derived by the Wick rotation for the temperature. In order to define a ``good'' temperature, we choose those solutions of the Dirac equation which propagate in a fixed direction in the extra coordinate. This choice fixes the regularization of the fermion determinant. The 1+4 dimensional Dirac mass ($M$) is naturally introduced and the relation: $|$4 dim electron momentum$|$ $\ll$ $|M|$ $\ll$ ultraviolet cut-off, naturally appears. The chiral anomaly is explicitly derived for the 2 dim Abelian model. Consistent and covariant anomalies appear depending on the choice of propagators.