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 ...

, periodic instantons are finite energy solutions of Euclidean-time field equations which communicate (in the sense of

quantum tunneling In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...

) between two turning points in the barrier of a potential and are therefore also known as bounces. Vacuum instantons, normally simply called

instanton An instanton (or pseudoparticle) 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. M ...

s, are the corresponding zero energy configurations in the limit of infinite Euclidean time. For completeness we add that

sphaleron A sphaleron ( "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton numbers. Such proces ...

s are the field configurations at the very top of a potential barrier. Vacuum instantons carry a

winding An electromagnetic coil is an electrical conductor such as a wire in the shape of a coil ( spiral or helix). Electromagnetic coils are used in electrical engineering, in applications where electric currents interact with magnetic fields, in ...

(or topological) number, the other configurations do not. Periodic instantons were discovered with the explicit solution of Euclidean-time field equations for

double-well potential The so-called double-well potential is one of a number of quartic potentials of considerable interest in quantum mechanics, in quantum field theory and elsewhere for the exploration of various physical phenomena or mathematical properties since it ...

s and the cosine potential with non-vanishing energy and are explicitly expressible in terms of

Jacobian elliptic function In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum, as well as in the design of electronic elliptic filters. While trigonometric functions are defin ...

s (the generalization of trigonometrical functions). Periodic instantons describe the oscillations between two endpoints of a potential barrier between two potential wells. The frequency

\Omega

of these oscillations or the tunneling between the two wells is related to the bifurcation or level splitting

\Delta E

of the energies of states or wave functions related to the wells on either side of the barrier, i.e.

\Omega = \Delta E/\hbar

. One can also interpret this energy change as the energy contribution to the well energy on either side originating from the integral describing the overlap of the wave functions on either side in the domain of the barrier. Evaluation of

\Delta E

by the path integral method requires summation over an infinite number of widely separated pairs of periodic instantons -- this calculation is therefore said to be that in the dilute gas approximation. Periodic instantons have meanwhile been found to occur in numerous theories and at various levels of complication. In particular they arise in investigations of the following topics. # Quantum mechanics and path integral treatment of periodic and anharmonic potentials.J.-Q. Liang and H.J.W. Müller-Kirsten: Periodic instantons and quantum mechanical tunneling at high energy, Proc. 4th Int. Symposium on Foundations of Quantum Mechanics, Tokyo 1992, Jpn. J. Appl. Phys., Series 9 (1993) 245-250. # Macroscopic spin systems (like ferromagnetic particles) with

phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...

s at certain temperatures. The study of such systems was started by D.A. Garanin and E.M. Chudnovsky in the context of condensed matter physics, where half of the periodic instanton is called a ``thermon´´. # Two-dimensional abelian Higgs model and four-dimensional electro-weak theories. # Theories of

Bose–Einstein condensation Bose–Einstein may refer to: * Bose–Einstein condensate, a phase of matter in quantum mechanics ** Bose–Einstein condensation (network theory), the application of this model in network theory ** Bose–Einstein condensation of polaritons ** B ...

and related topics in which tunneling takes place between weakly-linked macroscopic condensates confined to double-well potential traps.

References

{{reflist Gauge theories Quantum chromodynamics