Wick rotation

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature convention) : ds^2 = -\left(dt^2\right) + dx^2 + dy^2 + dz^2 and the four-dimensional Euclidean metric : ds^2 = d\tau^2 + dx^2 + dy^2 + dz^2 are equivalent if one permits the coordinate to take on imaginary values. The Minkowski metric becomes Euclidean when is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates , , , , and substituting sometimes yields a problem in real Euclidean coordinates , , , which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem. Statistical and quantum mechanics Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature with imaginary time, or more precisely replacing with , where is temperature, is the Boltzmann constant, is time, and is the reduced Planck constant. For example, consider a quantum system whose Hamiltonian has eigenvalues . When this system is in thermal equilibrium at temperature , the probability of finding it in its th energy eigenstate is proportional to . Thus, the expected value of any observable that commutes with the Hamiltonian is, up to a normalizing constant, : \sum_j Q_j e^{-\frac{E_j}{k_\text{B} T}}, where runs over all energy eigenstates and is the value of in the th eigenstate. Alternatively, consider this system in a superposition of energy eigenstates, evolving for a time under the Hamiltonian . After time , the relative phase change of the th eigenstate is . Thus, the probability amplitude that a uniform (equally weighted) superposition of states : |\psi\rangle = \sum_j |j\rangle evolves to an arbitrary superposition : |Q\rangle = \sum_j Q_j |j\rangle is, up to a normalizing constant, : \left\langle Q \left| e^{-\frac{iHt}{\hbar}} \right| \psi \right\rangle = \sum_j Q_j e^{-\frac{E_j it}{\hbar}} \langle j|j\rangle = \sum_j Q_j e^{-\frac{E_j it}{\hbar}}. Note that this formula can be obtained from the formula for thermal equilibrium by replacing with . == Statics and dynamics ==

Wick rotation relates statics problems in dimensions to dynamics problems in dimensions, trading one dimension of space for one dimension of time. A simple example where is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve . The spring is in equilibrium when the energy associated with this curve is at a critical point (an extremum); this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate the energy spatial density over space: : E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V\big(y(x)\big) \right] dx, where is the spring constant, and is the gravitational potential. The corresponding dynamics problem is that of a rock thrown upwards. The path the rock follows is that which extremalizes the action; as before, this extremum is typically a minimum, so this is called the "principle of least action". Action is the time integral of the Lagrangian: : S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V\big(y(t)\big) \right] dt. We get the solution to the dynamics problem (up to a factor of ) from the statics problem by Wick rotation, replacing by and the spring constant by the mass of the rock : : iS = \int_t \left[ m \left(\frac{dy(it)}{dt}\right)^2 + V\big(y(it)\big) \right] dt = i \int_t \left[ m \left(\frac{dy(it)}{dit}\right)^2 - V\big(y(it)\big) \right] d(it). == Both thermal/quantum and static/dynamic ==

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 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 : the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle. == Further details ==

The Schrödinger equation and the heat equation are also related by Wick rotation. Wick rotation also relates a quantum field theory at a finite inverse temperature to a statistical-mechanical model over the "tube" with the imaginary time coordinate being periodic with period . However, there is a slight difference. Statistical-mechanical -point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy reflection positivity. 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. == Rigorous results ==

The Osterwalder-Schrader theorem states that, in a Minkowski-space quantum field theory that satisfies the Wightman axioms, all correlation functions admit an analytic continuation to Euclidean space. In addition, if a Euclidean QFT satisfies both the Euclidean-space Wightman axioms and a growth condition on the correlation functions, it admits an analytic continuation to Minkowski space. The same correspondence has also been shown in the context of the Haag-Kastler axioms. Although the Wightman axioms have not been shown to hold for general quantum field theories, they have been verified for free field theories and for several special cases in low dimensions. == See also ==