Физикада қияли сандармен формулаларды жеңілдету тәсілі

Mathematical trick using imaginary numbers to simplify certain formulas in physics

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/kBT) arising when considering systems in an environment (where t is time, ℏ is the Planck constant, T is temperature and kB is the Boltzmann constant). 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.

Taken together, the previous two examples show how the path integral formulation of quantum mechanics is related to statistical mechanics. From statistical mechanics, the shape of each spring in a collection at temperature T will deviate from the least energy shape due to thermal fluctuations; the probability of finding a spring with a given shape decreases exponentially with the energy difference from the least energy shape. Similarly, a quantum particle moving in a potential can be described by a superposition of paths, each with a phase exp(iS): the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle.

The Schrödinger equation and the heat equation are also related by Wick rotation. However, there is a slight difference. Statistical mechanical n point functions satisfy positivity, whereas Wick rotated quantum field theories satisfy reflection positivity. Wick rotation also relates a quantum field theory at a finite inverse temperature β to a statistical mechanical model over the "tube" 'R'^(3) × S^(1) with the imaginary time coordinate τ being periodic with period β. Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no effect.

Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the Osterwalder–Schrader theorem.