Wick rotation
Mathematical trick using imaginary numbers to simplify certain formulas in physics / From Wikipedia, the free encyclopedia
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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 imaginarynumber variable for a realnumber variable.
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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=√1 is equivalent to counterclockwise rotating the vector representing that number by an angle of magnitude π/2 about the origin.^{[2]}