Imaginary time

Imaginary time is a concept derived from special relativity and quantum mechanics and is mathematically convenient in connecting quantum mechanics with statistical mechanics.

If we imagine "regular time" as a horizontal line running between "past" in one direction and "future" in the other, then imaginary time would run perpendicular to this line as the imaginary numbers run perpendicular to the real numbers in the complex plane. Imaginary time is not imaginary in the sense that it is unreal or made-up — it simply runs in a direction different from the type of time we experience. In essence, imaginary time is a way of looking at the time dimension as if it were a dimension of space: you can move forward and backward along imaginary time, just like you can move right and left in space.

Imaginary time ${\displaystyle \scriptstyle \tau }$ is obtained from real time via a Wick rotation by ${\displaystyle \scriptstyle \pi /2}$ in the complex plane: ${\displaystyle \scriptstyle \tau \ =\ it}$. It can be shown that at finite temperature T, the Green's functions are periodic in imaginary time with a period of ${\displaystyle \scriptstyle 2\beta \ =\ 2/T}$. 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 ${\displaystyle \scriptstyle \langle F\mid e^{-itH}\mid I\rangle }$ between an initial state I and a final state F. H is the Hamiltonian of that system. If we compare this with the partition function ${\displaystyle \scriptstyle Z\ =\ \operatorname {Tr} \ e^{-\beta H}}$ we see that to get the partition function from the transition amplitudes we can replace ${\displaystyle \scriptstyle t\,=\,\beta /i}$, 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.

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