Imaginary time

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The relationship of real and imaginary time can be visualised as perpendicular axes of direction.



In quantum mechanics[edit]

Imaginary time \scriptstyle\tau is obtained from real time via a Wick rotation by \scriptstyle\pi/2 in the complex plane: \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 \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 \scriptstyle\langle F\mid e^{-itH}\mid I\rangle 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 \scriptstyle Z\ =\ \operatorname{Tr}\ e^{-\beta H} we see that to get the partition function from the transition amplitudes we can replace \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.

In cosmology[edit]


Stephen Hawking


The concept is useful in cosmology because it can help smooth out gravitational singularities in models of the universe (see Hartle–Hawking state). Singularities pose a problem for physicists because these are areas where known physical laws do not apply. The Big Bang, for example, appears as a singularity in "regular time." But when visualized with imaginary time, the singularity is removed and the Big Bang functions like any other point in spacetime.

Interestingly, given the relationship between time and position, moving at twice the speed of light actually gives a value of time equal to the square root of -1; that is, i.

See also[edit]


  • Hawking, Stephen (2001). The Universe in a Nutshell. United States & Canada: Bantam Books. pp. 58–61, 63, 82–85, 90–94, 99, 196. ISBN 0-553-80202-X. 

Further reading[edit]

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