Multiplication by Infinity: Wick Rotation

This what Stephen Hawking refers to as "imaginary time" in his books. Seems like a pretty cool and a straightforward technique. When is this usually taught to Physics majors? I just found out about it by reading an addendum in Greg Egan's new novel: Orthogonal: The Clockwork Rocket.

In physics, Wick rotation, named after 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. This transformation is also used to find solutions to problems in quantum mechanics and other areas.

Overview

It is motivated by the observation that the Minkowski metric (with (-1,1,1,1) convention for the metric tensor)

d s 2 = - (d t 2) + d x 2 + d y 2 + d z 2

and the four-dimensional Euclidean metric

d s 2 = d τ 2 + d x 2 + d y 2 + d z 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 = i τ

, 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 $1/(k_B T)\,$ with imaginary time $it/\hbar\,$ . Consider a large collection of harmonic oscillators at temperature $T\,$ . The relative probability of finding any given oscillator with energy $E\,$ is $\exp(-E/k_B T)\,$ , where $k_B\,$ is Boltzmann's constant. The average value of an observable $Q\,$ is, up to a normalizing constant,

$\sum_j Q_j e^{-E_j / (k_B T)}.\,$

Now consider a single quantum harmonic oscillator in a superposition of basis states, evolving for a time

t

under a Hamiltonian

H

. The relative phase change of the basis state with energy $E\,$ is $\exp(-E it/ \hbar),\,$ where $\hbar\,$ is Planck's constant. The probability amplitude that a uniform 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,

$\; \langle Q|e^{-iHt/\hbar}|\psi\rangle$

$= \sum_j Q_j e^{-E_j it/ \hbar}\langle j|j\rangle$

$= \sum_j Q_j e^{-E_j it/ \hbar}.$

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; this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate over the energy density at each point:

$E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V(x) \right] dx,$

where

k

is the spring constant and

V (x)

is the gravitational potential.

The corresponding dynamics problem is that of a rock thrown upwards; the path the rock follows is a critical point of the action. Action is the integral of the Lagrangian; as before, this critical point is typically a minimum, so this is called the "principle of least action":

$S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V(t) \right] dt$

We get the solution to the dynamics problem (up to a factor of

- i

) from the statics problem by Wick rotation, replacing

x

t

d x

i d t

, and the spring constant

k

by the mass of the rock

m

$-iS = \int_t \left[ m \left(\frac{dy(t)}{i dt}\right)^2 + V(t) \right] (i dt)$

$= -i \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V(t) \right] dt$

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( - i S)

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

Others

The Schrödinger equation and the heat equation are also related by Wick rotation. However, there is a slight difference. Statistical mechanics n-point functions satisfy positivity whereas Wick-rotated quantum field theories satisfy reflection positivity.

It is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of $\scriptstyle\pi/2$ about theorigin.

When Stephen Hawking wrote about "imaginary time" in his famous book A Brief History of Time, he was referring to Wick rotation.^{[citation needed]}

Wick rotation also relates a QFT at a finite inverse temperature β to a statistical mechanical model over the "tube" R³×S¹ with the imaginary time coordinate τ being periodic with period β.

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 at all.

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

Multiplication by Infinity

Friday, September 2, 2011

Wick Rotation

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