# Imaginary time

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.

Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary root i. Imaginary time is not imaginary in the sense that it is unreal or made-up, it is simply expressed in terms of what mathematicians call imaginary numbers.

## Origins

Mathematically, imaginary time ${\displaystyle \scriptstyle \tau }$ is obtained from real time ${\displaystyle \scriptstyle t}$ via a Wick rotation by ${\displaystyle \scriptstyle \pi /2}$ in the complex plane: ${\displaystyle \scriptstyle \tau \ =\ it}$.

Stephen Hawking popularized the concept of imaginary time in his book A Brief History of Time.

## In quantum mechanics

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.[clarification needed]

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, where 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.

## In cosmology

In the theory of relativity, time is multiplied by i. This may be incorporated into time itself, as imaginary time, and the equations rewritten accordingly. In physical cosmology, this may be used to describe models of the universe, which must be solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws do not apply, in models of the universe (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity is removed and the Big Bang functions like any other point in spacetime.

## References

### Notes

1. ^ Hawking (2001), p.59.

### Bibliography

• Stephen W. Hawking (1998). A Brief History of Time (Tenth Anniversary Commemorative ed.). Bantam Books. p. 157. ISBN 978-0-553-10953-5.
• 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.