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In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable.
Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, it, where t is time and i is the imaginary unit (i2 = –1).
More precisely, in statistical mechanics, the Gibbs measure exp(−H/kBT) describes the relative probability of the system to be in any given state at temperature T, where H is a function describing the energy of each state and kB is the Boltzmann constant. In quantum mechanics, the transformation exp(−itH/ħ) describes time evolution, where H is an operator describing the energy (the Hamiltonian) and ħ is the reduced Planck constant. The former expression resembles the latter when we replace it/ħ with 1/kBT, and this replacement is called Wick rotation.[1]
Wick rotation is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by the imaginary unit is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude π/2 about the origin.[2]
- ^ Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. p. 289. ISBN 978-1-4008-3532-4.
- ^ Lancaster, Tom; Blundell, Stephen J. (2014-04-17), "Statistical field theory", Quantum Field Theory for the Gifted Amateur, Oxford University Press, pp. 228–229, retrieved 2023-11-12