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An instanton (or pseudoparticle [1] [2] [3]) is a notion appearing in theoretical and mathematical physics.An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory.
In theoretical physics, the BPST instanton is the instanton with winding number 1 found by Alexander Belavin, Alexander Polyakov, Albert Schwarz and Yu. S. Tyupkin. [1] It is a classical solution to the equations of motion of SU(2) Yang–Mills theory in Euclidean space-time (i.e. after Wick rotation), meaning it describes a transition between two different topological vacua of the theory.
Periodic instantons were discovered with the explicit solution of Euclidean-time field equations for double-well potentials and the cosine potential with non-vanishing energy [1] and are explicitly expressible in terms of Jacobian elliptic functions (the generalization of trigonometrical functions). Periodic instantons describe the oscillations ...
Given B 1, B 2, I, J such that = =, an anti-self-dual instanton in a SU gauge theory with instanton number k can be constructed, All anti-self-dual instantons can be obtained in this way and are in one-to-one correspondence with solutions up to a U( k ) rotation which acts on each B in the adjoint representation and on I and J via the ...
The BPST instanton is a solution to the anti-self duality equations, and therefore of the Yang–Mills equations, on R 4. This solution can be extended by Uhlenbeck 's removable singularity theorem to a topologically non-trivial ASD connection on S 4 .
The stability of the instanton configuration in the path integral theory of a scalar field theory with symmetric double-well self-interaction is investigated using the equation of small oscillations about the instanton. One finds that this equation is a Pöschl-Teller equation (i.e. a second order differential equation like the Schrödinger ...
A visual representation of the field strength of a BPST instanton with center z on the compactification S 4 of ℝ 4 (bottom right). The BPST instanton is a classical instanton solution to the Yang–Mills equations on ℝ 4.
These equation imply the Yang–Mills equations in any dimension, and in real dimension four are closely related to the self-dual Yang–Mills equations that define instantons. In particular, when the complex dimension of the Kähler manifold X {\displaystyle X} is 2 {\displaystyle 2} , there is a splitting of the forms into self-dual and anti ...