Wick-Rotation

The Wick rotation ( by Gian- Carlo Wick ) is a method for the derivation of a solution to a problem in Minkowski space from the solution of a related problem in Euclidean space by analytic continuation.

The Wick rotation is motivated by the observation that the Minkowski metric

And the four-dimensional Euclidean metric

Are equivalent if one allows the coordinate t takes complex values ​​. The Minkowski metric is Euclidean, when t is restricted to imaginary numbers, and vice versa. A problem in Minkowski space with the coordinates x, y, z, t is the substitution is w = it performed. This sometimes gives a problem in Euclidean coordinates x, y, z, w, which can be solved easily. For the original problem it is obtained by the reverse substitution of the solution.

Quantum Mechanics and Statistical Mechanics

The Wick rotation combines quantum mechanics and statistical mechanics, in a surprising manner by replacing the inverse temperature by the imaginary time. Consider a large ensemble of harmonic oscillators at a temperature. The relative probability to encounter a certain oscillator in the energy,

With the Boltzmann constant. The expectation value of an observable is up to a normalization constant

Let now a quantum mechanical harmonic oscillator in a superposition of base states and develop during the time with the Hamiltonian. The relative phase change of a basis state with the energy

With the reduced Planck 's constant. The probability amplitude, that a uniform superposition of the states

To an arbitrary state

Developed up to a normalizing constant

Statics and Dynamics

The Wick rotation combined static problems in dimensions with dynamic problems in dimensions by a room - by exchanges a time dimension. A simple example is a hanging spring in a gravitational field. The shape of the spring curve. The spring is in equilibrium when the energy associated with this curve, there is a critical point, typically a minimum, so that this principle is generally referred to as the least energy. To calculate the energy, we integrate over the energy density at each point:

With the spring constants and the gravitation potential.

The corresponding dynamic problem is the one thrown up Steins; its trajectory is a critical point of the effect. This is the integral of the Lagrangian; also, this critical point is typically a minimum, the principle owes the name principle of least action:

We obtain the solution of the dynamic problem ( up to a factor ) by Wick rotation from the static, by replacing by, through, and the spring constant by the mass of stone:

Combination of pairs of thermodynamics / quantum mechanics and statics / dynamics

Combined, the two upper examples of how the path integral formulation of quantum mechanics is related to the statistical mechanics: the shape of each spring in an ensemble at the temperature will differ from the shape of the lowest energy due to thermal fluctuations; the probability of finding a spring having given shape falls exponentially with the energy difference to this minimum energy shape. In a similar way, a single quantum particle moving in a potential, described as a superposition of paths each with the phase: the thermal fluctuations of the spring form across the ensemble are here replaced by a quantum uncertainty in the way of the quantum particle.

Others

In quantum field theory the Wick rotation is used to avoid the singularities of the Green's functions on the light cone. For the definition of the path integral of the winding rotation plays an important role. Quantum field theories in Euclidean space, which can be converted by Wick rotation in quantum field theories in Minkowski space-time, play an important role in the constructive quantum field theory. The Euclidean Green's functions must in particular satisfy a property that Reflexionspositivität is, so that meaningful quantum field theories arise in the Minkowski space-time.

The Schrödinger equation and the heat equation are related by the Wick rotation. This relationship is also reflected in the quantum field theory in which the thermodynamics of quantum fields can be described such that the reciprocal of the temperature is treated as imaginary time. A precise definition of thermodynamic states by means of such an imaginary time is given in the form of the KMS states. The winding rotation is called rotation in the complex plane as the multiplication by i corresponds to a rotation of a vector by an angle of 90 ° or. Note that the Wick rotation rather than rotation in the complex vector space (standard and metric are given by the scalar product ) can be considered. In this case, the rotation would be canceled and have no effect.

When Stephen Hawking wrote in his book A Brief History of Time about " imaginary time ," he was referring to the Wick rotation.