Imaginary time

The relationship of real and imaginary time can be visualised as perpendicular axes of direction.

Imaginary time is a concept derived from special relativity and quantum mechanics and is mathematically convenient in connecting quantum mechanics with statistical mechanics.

If we imagine "regular time" as a horizontal line running between "past" in one direction and "future" in the other, then imaginary time would run perpendicular to this line as the imaginary numbers run perpendicular to the real numbers in the complex plane. Imaginary time is not imaginary in the sense that it is unreal or made-up — it simply runs in a direction different from the type of time we experience. In essence, imaginary time is a way of looking at the time dimension as if it were a dimension of space: you can move forward and backward along imaginary time, just like you can move right and left in space.

In quantum mechanics

Imaginary time $\scriptstyle\tau$ is obtained from real time via a Wick rotation by $\scriptstyle\pi/2$ in the complex plane: $\scriptstyle\tau\ =\ it$. It can be shown that at finite temperature T, the Green's functions are periodic in imaginary time with a period of $\scriptstyle 2\beta\ =\ 2/T$. Therefore their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies. Another way to see the connection between statistical mechanics and quantum field theory is to consider the transition amplitude $\scriptstyle\langle F\mid e^{-itH}\mid I\rangle$ between an initial state I and a final state F. H is the Hamiltonian of that system. If we compare this with the partition function $\scriptstyle Z\ =\ \operatorname{Tr}\ e^{-\beta H}$ we see that to get the partition function from the transition amplitudes we can replace $\scriptstyle t\,=\,\beta/i$, 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

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

Imaginary time is also used in cosmology. It is used to describe models of the universe in physical cosmology. Stephen Hawking popularized the concept of imaginary time in his book A Brief History of Time.

The concept is useful in cosmology because it can help smooth out gravitational singularities in models of the universe (see Hartle–Hawking state), where known physical laws do not apply. The Big Bang, for example, appears as a singularity in "regular time." But, when visualized with imaginary time, the singularity is removed and the Big Bang functions like any other point in spacetime.

See also

• Euclidean quantum gravity

Further reading

• Gerald D. Mahan. Many-Particle 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.