Imaginary time
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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 madeup, it is simply expressed in terms of what mathematicians call imaginary numbers.
Contents
Origins
Mathematically, imaginary time is obtained from real time via a Wick rotation by in the complex plane: .
Stephen Hawking popularized the concept of imaginary time in his book A Brief History of Time.
“  One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?  ” 
— Stephen Hawking^{[1]} 
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 . 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 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 we see that to get the partition function from the transition amplitudes we can replace , 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 Ddimensional 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.
See also
References
Notes
 ^ Hawking (2001), p.59.
Bibliography
 Stephen W. Hawking (1998). A Brief History of Time (Tenth Anniversary Commemorative ed.). Bantam Books. p. 157. ISBN 9780553109535.
 Hawking, Stephen (2001). The Universe in a Nutshell. United States & Canada: Bantam Books. pp. 58–61, 63, 82–85, 90–94, 99, 196. ISBN 055380202X.
Further reading
 Gerald D. Mahan. ManyParticle Physics, Chapter 3
 A. Zee Quantum field theory in a nutshell, Chapter V.2
External links
 The Beginning of Time — Lecture by Stephen Hawking which discusses imaginary time.
 Stephen Hawking's Universe: Strange Stuff Explained — PBS site on imaginary time.