# 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 (any more than, say, irrational numbers defy logic), 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: ${\displaystyle \scriptstyle \tau \ =\ it}$, where ${\displaystyle \scriptstyle i}$ is the square root of minus one, ${\displaystyle \scriptstyle i={\sqrt {(}}-1)}$

Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell.

## 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 accepted as a feature of the relationship between space and time, or it may be incorporated into time itself, as imaginary time, and the equations rewritten accordingly.

In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four-dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down. With all such singularities removed from the Universe it thus can have no boundary and Stephen Hawking has speculated that "the boundary condition to the Universe may be that it has no boundary".

However the unclear nature of the relationship between real time and imaginary time required by such models has cast doubt on them.[2]

## References

### Notes

1. ^ Hawking (2001), p.59.
2. ^ Robert J. Deltete & Reed A. Guy; "Emerging from Imaginary Time", Synthese, Vol. 108, No. 2 (Aug., 1996), pp. 185-203.

### 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.