Imaginary-time units in a coherent-state action
= Imaginary-time units in a coherent-state action
Physical imaginary time runs over $[0,\beta\hbar]$ with an action divided by $\hbar$. Rescaling it by $\hbar$ gives a variable in $[0,\beta]$ and the operator $\partial_\tau+\varepsilon$. Matsubara eigenvalues in the latter variable have energy units.