In
physics
Physics is the scientific study of matter, its Elementary particle, 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 whi ...
, Wick rotation, named after Italian physicist
Gian Carlo Wick, is a method of finding a solution to a mathematical problem in
Minkowski space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model helps show how a ...
from a solution to a related problem in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
and
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
. In this analogy,
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).
Thermodynamic beta has units recipr ...
plays a role in statistical mechanics formally akin to
imaginary time in quantum mechanics: that is, , where is time and is the
imaginary unit
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
().
More precisely, in statistical mechanics, the
Gibbs measure
In physics and mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite ...
describes the relative probability of the system to be in any given state at temperature , where is a function describing the energy of each state and is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
. In quantum mechanics, the transformation describes time evolution, where is an operator describing the energy (the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
) and is the
reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
. The former expression resembles the latter when we replace with , and this replacement is called Wick rotation.
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
The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude about the origin.
Overview
Wick rotation is motivated by the observation that the
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
in natural units (with
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and z ...
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
An imaginary number is the product of a real number and the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is ...
, 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. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
to
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
by replacing
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).
Thermodynamic beta has units recipr ...
with
imaginary time, or more precisely replacing with , where is temperature, is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative thermal energy of particles in a ideal gas, gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin (K) and the ...
, is time, and is the
reduced Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
.
For example, consider a quantum system whose
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
has
eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
. When this system is in
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net flow of thermal energy between them when they are connected by a path permeable to heat. Thermal equilibrium obeys the zeroth law of thermodynamics. A system is said to be in t ...
at
temperature
Temperature is a physical quantity that quantitatively expresses the attribute of hotness or coldness. Temperature is measurement, measured with a thermometer. It reflects the average kinetic energy of the vibrating and colliding atoms making ...
, the probability of finding it in its th
energy
Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
eigenstate
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
is proportional to . Thus, the expected value of any observable that commutes with the Hamiltonian is, up to a normalizing constant,
:
where runs over all energy eigenstates and is the value of in the th eigenstate.
Alternatively, consider this system in a
superposition
In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...
of energy
eigenstates
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
, evolving for a time under the Hamiltonian . After time , the relative phase change of the th eigenstate is . Thus, the
probability amplitude
In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point.
Probability amplitu ...
that a uniform (equally weighted) superposition of states
:
evolves to an arbitrary superposition
:
is, up to a normalizing constant,
:
Note that this formula can be obtained from the formula for thermal equilibrium by replacing with .
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
Action may refer to:
* Action (philosophy), something which is done by a person
* Action principles the heart of fundamental physics
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video gam ...
; as before, this extremum is typically a minimum, so this is called the "
principle of least action
Action principles lie at the heart of fundamental physics, from classical mechanics through quantum mechanics, particle physics, and general relativity. Action principles start with an energy function called a Lagrangian describing the physical sy ...
". 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
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
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 partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
and the
heat equation
In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
are also related by Wick rotation.
Wick rotation also relates a
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
at a finite
inverse temperature
In statistical thermodynamics, thermodynamic beta, also known as coldness, is the reciprocal of the thermodynamic temperature of a system:\beta = \frac (where is the temperature and is Boltzmann constant).
Thermodynamic beta has units recipr ...
to a statistical-mechanical model over the "tube" with the imaginary time coordinate being periodic with period . However, there is a slight difference. Statistical-mechanical
-point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy
reflection positivity.
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, ofte ...
, 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 axioms
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered ''n''-tuples in \mathbb R^d that are pairwise distinct. These functions are called ...
.
See also
*
*
Complex spacetime Complex spacetime is a mathematical framework that combines the concepts of complex numbers and spacetime in physics. In this framework, the usual real-valued coordinates of spacetime are replaced with complex-valued coordinates. This allows for the ...
*
Imaginary time
*
Schwinger function
In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered ''n''-tuples in \mathbb R^d that are pairwise distinct. These functions are called ...
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 on the "Euclidean Gravity" programme.
{{DEFAULTSORT:Wick Rotation
Quantum field theory
Statistical mechanics