In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, 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
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
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
Wick rotation is motivated by the observation that the
Minkowski metric in natural units (with
metric signature convention)
:
and the four-dimensional Euclidean metric
:
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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
to
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
by replacing
inverse temperature with
imaginary time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.
M ...
. Consider a large collection of
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
s at
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
. The relative probability of finding any given oscillator with energy is
, where is
Boltzmann's constant. The average value of an observable is, up to a normalizing constant,
:
where the runs over all states,
is the value of in the -th state, and
is the energy of the -th state. Now consider a single
in a
superposition of basis states, evolving for a time under a Hamiltonian . The relative phase change of the basis state with energy is
where
is
reduced Planck's constant.
The
probability amplitude that a uniform (equally weighted) superposition of states
:
evolves to an arbitrary superposition
:
is, up to a normalizing constant,
:
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:
:
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:
:
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 :
:
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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
and the
heat equation
In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
are also related by Wick rotation. However, there is a slight difference. Statistical-mechanical -point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy
reflection positivity.
Wick rotation is called a ''rotation'' because when we represent
complex numbers as a plane, the multiplication of a complex number by is equivalent to rotating the
vector representing that number by an
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles ...
of about the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
.
Wick rotation also relates a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
at a finite
inverse temperature to a statistical-mechanical model over the "tube" with the imaginary time coordinate being periodic with period .
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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, as in this case the rotation would cancel out and have no effect.
Interpretation and rigorous proof
Wick rotations can be seen a useful trick that holds due to the similarity between the equations of two seemingly distinct fields of physics. ''
Quantum Field Theory in a Nutshell
Quantum Field Theory in a Nutshell is a textbook by Anthony Zee covering quantum field theory. The book has been adopted by many universities, including Harvard University, Princeton University, the University of California, Berkeley, the Californ ...
'' by
Anthony Zee discusses Wick rotations, saying that
It has been proven that a more rigorous link between Euclidean and quantum field theory can be constructed using the
Osterwalder–Schrader theorem.
See also
*
Complex spacetime In mathematics and mathematical physics, complex spacetime extends the traditional notion of spacetime described by real-valued space and time coordinates to complex-valued space and time coordinates. The notion is entirely mathematical with no p ...
*
Imaginary time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.
M ...
*
Schwinger function
References
*
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
— a short note by
Ray Streater
Raymond Frederick "Ray" Streater (born 1936) is a British physicist, and professor emeritus of Applied Mathematics at King's College London. He is best known for co-authoring a text on quantum field theory, the 1964 ''PCT, Spin and Statistics ...
on the "Euclidean Gravity" programme.
{{DEFAULTSORT:Wick Rotation
Quantum field theory
Statistical mechanics