Imaginary time ($t=-i\tau$)

Analytic continuation from real time to $t=-i\tau$ turns unitary evolution into $e^{-\tau H/\hslash }$. A thermal trace has extent $\beta \hslash$, or $\beta$ when $\hslash =1$. This converts oscillatory action phases into an Euclidean path integral where convergence permits it; it is not the actual laboratory time of a dissipative process.