Wick rotation

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 are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, Template:Math, where Template:Math is time and Template:Math is the imaginary unit (Template:Math).

More precisely, in statistical mechanics, the Gibbs measure Template:Math describes the relative probability of the system to be in any given state at temperature Template:Math, where Template:Math is a function describing the energy of each state and Template:Math is the Boltzmann constant. In quantum mechanics, the transformation Template:Math describes time evolution, where Template:Math is an operator describing the energy (the Hamiltonian) and Template:Math is the reduced Planck constant. The former expression resembles the latter when we replace Template:Math with Template:Math, and this replacement is called Wick rotation.<ref>Template:Cite book</ref>

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 Template:Math about the origin.<ref>Template:Citation</ref>

Overview

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature Template:Math convention)

<math>ds^2 = -\left(dt^2\right) + dx^2 + dy^2 + dz^2</math>

and the four-dimensional Euclidean metric

<math>ds^2 = d\tau^2 + dx^2 + dy^2 + dz^2</math>

are equivalent if one permits the coordinate Template:Mvar to take on imaginary values. The Minkowski metric becomes Euclidean when Template:Math is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates Template:Math, Template:Math, Template:Math, Template:Math, and substituting Template:Math sometimes yields a problem in real Euclidean coordinates Template:Math, Template:Math, Template:Math, Template:Math 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 Template:Math with Template:Math, where Template:Math is temperature, Template:Math is the Boltzmann constant, Template:Math is time, and Template:Math is the reduced Planck constant.

For example, consider a quantum system whose Hamiltonian Template:Math has eigenvalues Template:Math. When this system is in thermal equilibrium at temperature Template:Mvar, the probability of finding it in its Template:Mathth energy eigenstate is proportional to Template:Math. Thus, the expected value of any observable Template:Math that commutes with the Hamiltonian is, up to a normalizing constant,

<math>\sum_j Q_j e^{-\frac{E_j}{k_\text{B} T}},</math>

where Template:Mvar runs over all energy eigenstates and Template:Math is the value of Template:Math in the Template:Mathth eigenstate.

Alternatively, consider this system in a superposition of energy eigenstates, evolving for a time Template:Mvar under the Hamiltonian Template:Mvar. After time Template:Math, the relative phase change of the Template:Mvarth eigenstate is Template:Math. Thus, the probability amplitude that a uniform (equally weighted) superposition of states

<math>|\psi\rangle = \sum_j |j\rangle</math>

evolves to an arbitrary superposition

<math>|Q\rangle = \sum_j Q_j |j\rangle</math>

is, up to a normalizing constant,

<math>

 \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}}.

</math> Note that this formula can be obtained from the formula for thermal equilibrium by replacing Template:Math with Template:Math.

Statics and dynamics

Wick rotation relates statics problems in Template:Mvar dimensions to dynamics problems in Template:Math dimensions, trading one dimension of space for one dimension of time. A simple example where Template:Math is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve Template:Math. 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:

<math>E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V\big(y(x)\big) \right] dx,</math>

where Template:Math is the spring constant, and Template:Math 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:

<math>S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V\big(y(t)\big) \right] dt.</math>

We get the solution to the dynamics problem (up to a factor of Template:Mvar) from the statics problem by Wick rotation, replacing Template:Math by Template:Math and the spring constant Template:Mvar by the mass of the rock Template:Mvar:

<math>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).</math>

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 Template:Mvar 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 Template:Math: 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 Template:Math to a statistical-mechanical model over the "tube" Template:Math with the imaginary time coordinate Template:Math being periodic with period Template:Math. However, there is a slight difference. Statistical-mechanical [[N-point function|Template:Mvar-point functions]] satisfy positivity, whereas Wick-rotated quantum field theories satisfy reflection positivity.Template:Explain

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 proof

Template:Needs expansion Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the Osterwalder–Schrader axioms.<ref>Template:Cite journal</ref>

See also

• Template:Section link
• Complex spacetime
• Imaginary time
• Schwinger function

References

• Template:Cite journal

External links

• A Spring in Imaginary Time – a worksheet in Lagrangian mechanics illustrating how replacing length by imaginary time turns the parabola of a hanging spring into the inverted parabola of a thrown particle
• Euclidean Gravity – a short note by Ray Streater on the "Euclidean Gravity" programme.