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Bobhawke
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Im trying to learn about instantons at the moments from Cheng and Li's book. It seems to suggest that instantons only occur in theories that are non-abelian. Why is this so?
Bobhawke said:instantons only occur in theories that are non-abelian.
This is not a general rule, in (1+1) dimensions,there are many Abelian models with non-trivial instanton solutions.
Why is this so?
Recall the following results on homotopy groups;
[tex]\pi_{n} (S^{n}) = Z,[/tex]
[tex]\pi_{n} (S^{1}) = 0, \ \ \mbox{for} \ n > 1[/tex]
where [itex]\pi_{n} (S^{m})[/itex] refers to the homotopy group for the mapping [itex]S^{n}\rightarrow S^{m}[/itex] and Z is the group of integers.
If the gauge group is U(1), every mapping of [itex]S^3[/itex] (the boundary of the (3+1)-dimensional Euclidean domain) into [itex]S^1[/itex] (the range of boundary values = U(1) manifold) is continueosly deformable into a single point (the trivial mapping). Thus, in (3+1)-dimensional Euclidean spacetime, Abelian gauge theories have no analog of the winding number, i.e., no non-trivial instanton sectors. This is why peopel choose non-abelian systems to discuss instantons in 4 dimensions.
It is only in (1+1) dimensions that the Abelian instantons exist with integral homotopy indices. Look up the very important Abelian model CP(N).
regards
sam
Instantons are topological solutions of non-abelian gauge theories, which are field theories that describe the interactions between elementary particles. They are localized, finite energy solutions that play a crucial role in understanding the non-perturbative behavior of these theories.
Instantons and monopoles are both topological solutions in non-abelian gauge theories, but they differ in their physical properties. Instantons are localized in space and time, while monopoles are extended objects. Additionally, instantons are associated with the violation of chiral symmetry, while monopoles are associated with the breaking of gauge symmetry.
Instantons play a crucial role in understanding the non-perturbative behavior of non-abelian gauge theories. They provide a framework for studying the strong interactions between elementary particles, and their effects can be observed experimentally in high-energy particle collisions.
Instantons have been used to explain various phenomena in particle physics, such as the strong CP problem and the mass hierarchy of quarks. They also have applications in condensed matter physics, including the study of topological insulators and superconductors.
Yes, there are ongoing research efforts to further understand the properties and applications of instantons in non-abelian gauge theories. Some current research topics include the study of instanton-anti-instanton configurations, the role of instantons in the quark-gluon plasma, and the connection between instantons and supersymmetry.