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Imaginary time in quantum mechanics
Imaginary time is a concept derived from quantum mechanics and is essential in connecting quantum mechanics with statistical mechanics. Imaginary time <math>\scriptstyle\tau</math> is obtained from real time via a Wick rotation by <math>\scriptstyle\pi/2</math>: <math>\scriptstyle\tau\ =\ it</math>. It can be shown that at finite temperature T, the Green's functions are periodic in imaginary time with a period of <math>\scriptstyle 2\beta\ =\ 2/T</math>. 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 is to consider the transition amplitude <math>\scriptstyle\langle F\,|\,e^{-itH}|I\rangle </math> between an initial state I and a final state F. H is the Hamiltonian of the system. If we compare this with the partition function <math>\scriptstyle Z\ =\ \operatorname{Tr}\ e^{-\beta H}</math> we see that to get the partition function from the transition amplitudes we can replace <math>\scriptstyle t\,=\,\beta/i</math>, 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.
Imaginary time in cosmology
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. Imaginary time is difficult to visualize. If we imagine "regular time" as a horizontal line with "past" on one side and "future" on 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. However, 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. The concept is useful in cosmology because it can help smooth out gravitational singularities in models of the universe (see Hartle-Hawking state). Singularities pose a problem for physicists because they are areas 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
- Wick rotation
- Euclidean quantum gravity
- 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
