- Imaginary time
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 scriptstyle au is obtained from real time via aWick rotation by scriptstylepi/2: scriptstyle au = it. It can be shown that at finite temperature "T", theGreen's function s are periodic in imaginary time with a period of scriptstyle 2eta = 2/T. Therefore theirFourier transform s 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 scriptstylelangle F,|,e^{-itH}|I angle between an initial state "I" and a final state "F". "H" is theHamiltonian of the system. If we compare this with the partition function scriptstyle Z = operatorname{Tr} e^{-eta H} we see that to get thepartition function from the transition amplitudes we can replace scriptstyle t,=,eta/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 Euclideanquantum field theory in ("D" + 1)-dimensional spacetime is nothing butquantum 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 inphysical 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 number s run perpendicular to thereal number s in thecomplex 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 (seeHartle-Hawking state ). Singularities pose a problem for physicists because these are areas where known physical laws do not apply. TheBig 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.References
cite book
last = Hawking
first = Stephen
authorlink =
coauthors =
title = The Universe in a Nutshell
publisher = Bantam Books
date = 2001
location = United States & Canada
pages = 58-61, 63, 82-85, 90-94, 99, 196
url =
doi =
id =
isbn = 055380202Xee also
*
Wick rotation
*Euclidean quantum gravity
* [http://www.amazon.com/dp/0306463385/ Gerald D. Mahan. Many-Particle Physics, Chapter 3]
* [http://www.amazon.com/dp/0691010196/ A. Zee Quantum field theory in a nutshell, Chapter V.2]External links
* [http://www.hawking.org.uk/text/public/bot.html The Beginning of Time] — Lecture by Stephen Hawking which discusses imaginary time.
* [http://www.pbs.org/wnet/hawking/strange/html/imaginary.html Stephen Hawking's Universe: Strange Stuff Explained] — PBS site on imaginary time
Wikimedia Foundation. 2010.