Wick rotation

The Wick rotation (after Gian-Carlo Wick ) is a method for deriving a solution to a problem in Minkowski space from the solution of a related problem in Euclidean space through analytical continuation .

The wick rotation is motivated by considering that the Minkowski metric

${\displaystyle \mathrm {d} s^{2}=-(\mathrm {d} t^{2})+\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}$

and the four-dimensional Euclidean metric

${\displaystyle \mathrm {d} s^{2}=\mathrm {d} t^{2}+\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}}$

are equivalent if the coordinate is allowed to take complex values. The Minkowski metric becomes Euclidean when restricting to imaginary numbers and vice versa. For a problem in the Minkowski space with the coordinates , the substitution is carried out so that the problem is formulated in Euclidean coordinates . The solution to the original problem is obtained through the reverse substitution. ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle x,y,z,t}$ ${\displaystyle w=\mathrm {i} t}$ ${\displaystyle x,y,z,w}$

Quantum Mechanics and Statistical Mechanics

The Wick rotation combines quantum mechanics and statistical mechanics in a surprising way in that it replaces the inverse temperature with imaginary time . A large ensemble of harmonic oscillators at one temperature is given . The relative probability of encountering a particular oscillator with the energy is ${\displaystyle 1/(k_{\mathrm {B} }T)}$ ${\displaystyle \mathrm {i} t/\hbar }$ ${\displaystyle T}$ ${\displaystyle E}$

${\displaystyle \exp \left({\frac {-E}{k_{\mathrm {B} }T}}\right)}$

with the Boltzmann constant . The expectation value of an observable is up to a normalization constant ${\displaystyle k_{\mathrm {B} }}$ ${\displaystyle Q}$

${\displaystyle \sum _{j}Q(j)e^{\frac {-E_{j}}{(k_{\mathrm {B} }T)}}}$

Let us now be a quantum mechanical harmonic oscillator in a superposition of base states and evolve over time with the Hamilton operator . The relative phase change of a base state with energy is ${\displaystyle t}$ ${\displaystyle H}$ ${\displaystyle E}$

${\displaystyle \exp \left({\frac {-E\mathrm {i} t}{\hbar }}\right)}$

with Planck's reduced quantum of action . The probability amplitude that a uniform superposition of the states ${\displaystyle \hbar }$

${\displaystyle |\psi \rangle =\sum _{j}|j\rangle }$

to any state

${\displaystyle |Q\rangle =\sum _{j}Q_{j}|j\rangle }$

is developed, except for a normalization constant

${\displaystyle {\begin{aligned}{\Big \langle }Q{\Big |}e^{-{\frac {\mathrm {i} }{\hbar }}Ht}{\Big |}\psi {\Big \rangle }&=\sum _{j}Q_{j}\exp \left({\frac {-E_{j}\mathrm {i} t}{\hbar }}\right)\langle j|j\rangle \\&=\sum _{j}Q_{j}\exp \left({\frac {-E_{j}\mathrm {i} t}{\hbar }}\right).\end{aligned}}}$

Statics and dynamics

The wick rotation links static problems in dimensions with dynamic problems in dimensions by exchanging a space for a time dimension. A simple example with is a hanging spring in a gravitational field. The shape of the spring is the curve . The spring is in equilibrium when the energy associated with this curve is at a critical point, typically a minimum, so this principle is commonly referred to as that of the least energy. To calculate the energy, we integrate over the energy density at each point: ${\displaystyle n}$ ${\displaystyle n-1}$ ${\displaystyle n=2}$ ${\displaystyle y(x)}$

${\displaystyle E=\int _{x}\left[k\left({\frac {\mathrm {d} y(x)}{\mathrm {d} x}}\right)^{2}+V(x)\right]\mathrm {d} x,}$

with the spring constant and the gravitational potential . ${\displaystyle k}$ ${\displaystyle V(x)}$

The corresponding dynamic problem is that of a stone thrown upwards; its trajectory is a critical point of effect . This is the integral of the Lagrangian ; this critical point is also typically a minimum, which owes the principle to the principle of the smallest effect :

${\displaystyle S=\int _{t}\left[m\left({\frac {\mathrm {d} y(t)}{\mathrm {d} t}}\right)^{2}-V(t)\right]\mathrm {d} t}$

We get the solution of the dynamic problem (except for one factor ) by Wick rotation from the static one by replacing with, by , and the spring constant by the mass of the stone: ${\displaystyle -\mathrm {i} }$ ${\displaystyle x}$ ${\displaystyle t}$ ${\displaystyle \mathrm {d} x}$ ${\displaystyle \mathrm {i} \mathrm {d} t}$ ${\displaystyle k}$ ${\displaystyle m}$

${\displaystyle {\begin{aligned}-\mathrm {i} S&=\int _{t}\left[m\left({\frac {\mathrm {d} y(t)}{\mathrm {i} \mathrm {d} t}}\right)^{2}+V(t)\right]\mathrm {i} \mathrm {d} t\\&=-\mathrm {i} \int _{t}\left[m\left({\frac {\mathrm {d} y(t)}{\mathrm {d} t}}\right)^{2}-V(t)\right]\mathrm {d} t\end{aligned}}}$

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

Combined, the two examples above show how the path integral formulation of quantum mechanics is related to statistical mechanics: the shape of each spring in an ensemble at temperature will deviate from the shape with the lowest energy due to thermal fluctuations; the probability of finding a spring with a given shape falls exponentially with the energy difference to this minimal energy shape. In a similar way, a single quantum particle moving in a potential can be described as a superposition of paths with the phase : The thermal fluctuations of the spring shape across the ensemble are here replaced by a quantum uncertainty in the path of the quantum particle. ${\displaystyle T}$ ${\displaystyle \exp(-\mathrm {i} S)}$

Others

In quantum field theory , the winding rotation is used to circumvent the singularities of Green's functions on the light cone. The Wick rotation also plays an important role in defining the path integral . Quantum field theories in Euclidean space, which can be converted into quantum field theories in Minkowski spacetime by means of Wick rotation, also play an important role in constructive quantum field theory . The Euclidean Green functions have to fulfill a property in particular, which is called reflection positivity, so that meaningful quantum field theories arise in the Minkowski space-time.

The Schrödinger equation and the heat conduction equation are related by the Wick rotation. This relationship continues in thermal quantum field theory , in which the thermodynamics of quantum fields can be described in such a way that the reciprocal of 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 Wick rotation is called rotation because in the complex number plane the multiplication corresponds to a rotation of a vector by an angle of 90 ° or . Note that the Wick rotation cannot be understood as a rotation in the complex vector space ( norm and metric are given by the scalar product ). In this case the rotation would be canceled and have no effect. ${\displaystyle \mathrm {i} }$ ${\displaystyle \pi /2}$

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

Web links

• Wick rotation - a blog introduction