Question

Extra $i$ in grand canonical partition function: why the Wick rotation?

Answer 1

Hey mate ^_^

I think that you are missing is the fact that the action S appearing in the exponent is the Euclidean action, not the plane action one encounters in classical mechanics.....

The Euclidean action is precisely the plane action but after Wick rotation, i.e. in imaginary times....

#Be Brainly❤️

Answer 2

the action S appearing in the exponent is the Euclidean action, not the plane action one encounters in classical mechanics. The Euclidean action is precisely the plane action but after Wick rotation, i.e. in imaginary times. The time integral, thus, acquires an overall factor of i and the kinetic term changes sign. Eventually, the Euclidean action is (almost) the Hamiltonian of the system.

HOPE HELPS ✌️