Imaginary Time - in Quantum Mechanics

In Quantum Mechanics

Imaginary time is obtained from real time via a Wick rotation by in the complex plane: . It can be shown that at finite temperature T, the Green's functions are periodic in imaginary time with a period of . Therefore their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies. Another way to see the connection between statistical mechanics and quantum field theory is to consider the transition amplitude between an initial state I and a final state F. H is the Hamiltonian of the system. If we compare this with the partition function we see that to get the partition function from the transition amplitudes we can replace, set F = I = n and sum over n. This way we don't have to do twice the work by evaluating both the statistical properties and the transition amplitudes. Finally by using a Wick rotation one can show that the Euclidean quantum field theory in (D + 1)-dimensional spacetime is nothing but quantum statistical mechanics in D-dimensional space.