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, ${\sqrt {-1}}t$ where $t$ is time.

More precisely, in statistical mechanics, the Gibbs measure $\exp(-H/k_{\text{B}}T)$ describes the relative probability of the system to be in any given state at temperature $T$ , where $H$ is a function describing the energy of each state and $k_{\text{B}}$ is Boltzmann's constant. In quantum mechanics, the transformation $\exp(-itH/\hbar )$ describes time evolution, where $H$ is an operator describing the energy (the Hamiltonian) and $\hbar$ is the reduced Planck's constant. The former expression resembles the latter when we replace $it/\hbar$ with $1/k_{\text{B}}T$ , and this replacement is called Wick rotation.^[1]

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 $i = \sqrt -1$ is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude $π /2$ about the origin.^[2]

Overview

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature $(-1, +1, +1, +1)$ 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 $t$ to take on imaginary values. The Minkowski metric becomes Euclidean when $t$ is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates $x, y, z, t$ , and substituting $t = - iτ$ sometimes yields a problem in real Euclidean coordinates $x, y, z, τ$ 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 $1/k_{\text{B}}T$ with $it/\hbar$ , where $t$ is temperature, $k_{B}$ is Boltzmann's constant, $t$ is time, and $\hbar$ is the reduced Planck constant.

For example, consider a quantum system whose Hamiltonian $H$ has eigenvalues $E_{j}$ . When this system is in thermal equilibrium at temperature $T$ , the probability of finding it in its $j$ th energy eigenstate is proportional to $\exp(-E_{j}/k_{\text{B}}T)$ . Thus, the expected value of any observable $Q$ that commutes with the Hamiltonian is, up to a normalizing constant,

\sum _{j}Q_{j}e^{-{\frac {E_{j}}{k_{\text{B}}T}}},

where $j$ runs over all energy eigenstates and $Q_{j}$ is the value of $Q$ in the $j$ th eigenstate.

Alternatively, consider this system in a superposition of energy eigenstates, evolving for a time $t$ under the Hamiltonian $H$ . After time $t$ , the relative phase change of the $j$ th eigenstate is $\exp(-E_{j}it/\hbar ).$ 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 $1/k_{\text{B}}T$ with $it/\hbar$ .

Statics and dynamics

Wick rotation relates statics problems in $n$ dimensions to dynamics problems in $n - 1$ dimensions, trading one dimension of space for one dimension of time. A simple example where $n = 2$ is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve $y (x)$ . 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 $k$ is the spring constant, and $V (y (x))$ 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 $i$ ) from the statics problem by Wick rotation, replacing $y (x)$ by $y (it)$ and the spring constant $k$ by the mass of the rock $m$ :

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 $T$ 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.

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 $β$ to a statistical-mechanical model over the "tube" $R 3 \times S 1$ with the imaginary time coordinate $τ$ being periodic with period $β$ . However, there is a slight difference. Statistical-mechanical $n$ -point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy reflection positivity.^{[further explanation needed]}

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

Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the Osterwalder–Schrader theorem.^[3]

References

^ Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. p. 289. ISBN 978-1-4008-3532-4.
^ Lancaster, Tom; Blundell, Stephen J. (2014-04-17), "Statistical field theory", Quantum Field Theory for the Gifted Amateur, Oxford University Press, pp. 228–229, retrieved 2023-11-12
^ Schlingemann, Dirk (1999). "From Euclidean Field Theory To Quantum Field Theory". Reviews in Mathematical Physics. 11 (9): 1151–78. arXiv:hep-th/9802035. Bibcode:1999RvMaP..11.1151S. doi:10.1142/S0129055X99000362. ISSN 0129-055X. S2CID 9851483.

Wick, G. C. (1954). "Properties of Bethe-Salpeter Wave Functions". Physical Review. 96 (4): 1124–1134. Bibcode:1954PhRv...96.1124W. doi:10.1103/PhysRev.96.1124.

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.

[1] Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. p. 289. ISBN 978-1-4008-3532-4.

[2] Lancaster, Tom; Blundell, Stephen J. (2014-04-17), "Statistical field theory", Quantum Field Theory for the Gifted Amateur, Oxford University Press, pp. 228–229, retrieved 2023-11-12

[3] Schlingemann, Dirk (1999). "From Euclidean Field Theory To Quantum Field Theory". Reviews in Mathematical Physics. 11 (9): 1151–78. arXiv:hep-th/9802035. Bibcode:1999RvMaP..11.1151S. doi:10.1142/S0129055X99000362. ISSN 0129-055X. S2CID 9851483.

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