Caloron
Encyclopedia
In mathematical physics
, a caloron is the finite temperature generalization of an instanton
.
of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime
. They describe tunneling between different topological vacuum states
of the Minkowski theory. One important example of an instanton is the BPST instanton
, discovered in 1975 by Belavin
, Polyakov, Schwartz
and Tyupkin. This is a topologically
stable solution to the four-dimensional SU(2) Yang–Mills field equations in Euclidean spacetime (i.e. after Wick rotation
).
Finite temperatures in quantum field theories are modeled by compactifying the imaginary (Euclidean) time (see thermal quantum field theory
). This changes the overall structure of spacetime, and thus also changes the form of the instanton solutions. At finite temperature, the Euclidean time dimension is periodic, which means that instanton solutions have to be periodic as well.
. The generalization thereof to finite temperature has been found by Harrington and Shepard:
where
is the anti-'t Hooft symbol
, r is the distance from the point x to the center of the caloron, ρ is the size of the caloron, and T is the temperature. This solution was found based on a periodic multi-instanton solution first suggested by 't Hooft
and published by Witten
.
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
, a caloron is the finite temperature generalization of an instanton
Instanton
An instanton is a notion appearing in theoretical and mathematical physics. Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory...
.
Finite temperature and instantons
At zero temperature, instantons are the name given to solutions of the classical equations of motionField equation
A field equation is an equation in a physical theory that describes how a fundamental force interacts with matter...
of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
. They describe tunneling between different topological vacuum states
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
of the Minkowski theory. One important example of an instanton is the BPST instanton
BPST instanton
The BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...
, discovered in 1975 by Belavin
Alexander Belavin
Alexander "Sasha" Abramovich Belavin is a Russian physicist, known for his contributions to string theory.He is a professor at the Independent University of Moscow and is researcher at the Landau Institute of Theoretical Physics....
, Polyakov, Schwartz
Albert Schwarz
Albert S. Schwarz is a mathematician and a theoretical physicist educated in Soviet Union and now a Professor at the University of California-Davis. He is one of the pioneers of Morse theory and brought up the first example of a topological quantum field theory. Schwarz worked on some examples in...
and Tyupkin. This is a topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
stable solution to the four-dimensional SU(2) Yang–Mills field equations in Euclidean spacetime (i.e. after Wick rotation
Wick rotation
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...
).
Finite temperatures in quantum field theories are modeled by compactifying the imaginary (Euclidean) time (see thermal quantum field theory
Thermal quantum field theory
In theoretical physics, thermal quantum field theory or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature....
). This changes the overall structure of spacetime, and thus also changes the form of the instanton solutions. At finite temperature, the Euclidean time dimension is periodic, which means that instanton solutions have to be periodic as well.
In SU(2) Yang–Mills theory
In SU(2) Yang–Mills theory at zero temperature, the instantons have the form of the BPST instantonBPST instanton
The BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. It is a classical solution to the equations of motion of SU Yang-Mills theory in Euclidean space-time , meaning it describes a transition between two...
. The generalization thereof to finite temperature has been found by Harrington and Shepard:
where
is the anti-'t Hooft symbol't Hooft symbol
The t Hooft η symbol is a symbol which allows one to express the generators of the SU Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It was introduced by Gerard 't Hooft...
, r is the distance from the point x to the center of the caloron, ρ is the size of the caloron, and T is the temperature. This solution was found based on a periodic multi-instanton solution first suggested by 't Hooft
Gerardus 't Hooft
Gerardus 't Hooft is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands. He shared the 1999 Nobel Prize in Physics with his thesis advisor Martinus J. G...
and published by Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
.