# 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 ${\displaystyle \scriptstyle \tau }$ may be 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 ${\displaystyle i^{2}}$ Is defined to be ${\displaystyle -1}$, and is known as the imaginary unit.

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

In fact, the names "real" and "imaginary" for numbers are just a historical accident, much like the names "rational" and "irrational":

## In cosmology

In the Minkowski spacetime model adopted by the theory of relativity, time can be understood as multiplied by i.[citation needed].

Spacetime may be represented as a four-dimensional surface or manifold. The equivalent of a distance in space is called an interval. Given an assumption that time is real, an interval ${\displaystyle d}$ in relativistic spacetime is given by the usual formula but with time negated:

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}-t^{2}}$

where ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are distances along each spatial axis and ${\displaystyle t}$ a period of time. Mathematically this is equivalent to writing

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+(it)^{2}}$

In this context, ${\displaystyle i}$ may be either accepted as a feature of the relationship between space and real time, as above, or it may be incorporated into time itself, as imaginary time, and the equation rewritten in normalised form:

${\displaystyle d^{2}=x^{2}+y^{2}+z^{2}+t^{2}}$

Similarly its four vector may then be written as

${\displaystyle (x_{0},x_{1},x_{2},x_{3})}$

where distances are represented as ${\displaystyle x_{n}}$, ${\displaystyle c}$ is the velocity of light and ${\displaystyle x_{0}=ict}$.

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 unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.[3]

## In quantum statistical mechanics

It can be shown that at finite temperature ${\displaystyle T}$, 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.

## References

### Notes

1. ^ Hawking (2001), p.59.
2. ^ Coxeter, H.S.M.; The Real Projective Plane, 3rd Edn, Springer 1993, p. 210 (footnote).
3. ^ 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.