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 unit "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 formula_1 may be obtained from real time formula_2 via a Wick rotation by formula_3 in the complex plane: formula_4, where formula_5 Is defined to be formula_6, and formula_7 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, spacetime is represented as a four-dimensional surface or manifold. Its four-dimensional equivalent of a distance in three-dimensional space is called an interval. Assuming that a specific time period is represented as a real number in the same way as a distance in space, an interval formula_8 in relativistic spacetime is given by the usual formula but with time negated:
where formula_10, formula_11 and formula_12 are distances along each spatial axis and formula_13 is a period of time or "distance" along the time axis.
Mathematically this is equivalent to writing 
In this context, formula_7 may be either accepted as a feature of the relationship between space and real time, as above, or it may alternatively be incorporated into time itself, such that the value of time is itself an imaginary number, denoted by formula_16, and the equation rewritten in normalised form:
Similarly its four vector may then be written as
where distances are represented as formula_19, formula_20 is the velocity of light and formula_21.
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 is 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. Penrose has noted that there needs to be a transition from the Riemannian metric (often referred to as "Euclidean" in this context) with imaginary time at the Big Bang to a Lorenzian metric with real time for the evolving Universe. Also, modern observations suggest that the Universe is open and will never shrink back to a Big Crunch. If this proves true then the end-of-time boundary still remains.
In quantum statistical mechanics.
The equations of the quantum field can be obtained by taking the Fourier transform of the equations of statistical mechanics. Since the Fourier transform of a function typically shows up as its inverse, the point particles of statistical mechanics become, under a Fourier transform, the infinitely extended harmonic oscillators of quantum field theory. The Green's function of an inhomogeneous linear differential operator, defined on a domain with specified initial conditions or boundary conditions, is its impulse response, and mathematically we define the point particles of statistical mechanics as Dirac delta functions, which is to say impulses. At finite temperature formula_22, the Green's functions are periodic in imaginary time with a period of formula_23. Therefore, their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies. 
The connection between statistical mechanics and quantum field theory is also seen in the transition amplitude formula_24 between an initial state and a final state , where  is the Hamiltonian of that system. If we compare this with the partition function formula_25 we see that to get the partition function from the transition amplitudes we can replace formula_26, set and sum over . 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.