Wick rotation

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 can be seen as a useful trick that works because of the similarity between the equations of two seemingly distinct fields of physics. This can be seen by the similarity between two central objects in quantum mechanics and statistical mechanics, where H is the Hamiltonian relating to conserved energy: The transformation $exp(- iHt /ℏ)$ derived from the Schrödinger equation and the Gibbs measure $exp(H / k B T)$ arising when considering systems in an environment (where t is time, ℏ is the Planck constant, T is temperature and k_B is the Boltzmann constant).^[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 $i = \sqrt -1$ 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

[1] Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. p. 289. ISBN 978-1-4008-3532-4.

[2] 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

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