Solitons and Instantons

Autumn '06

General references: "Aspects of Symmetry" by Sidney Coleman, "Solitons and Instantons" by R. Rajaraman.

Schab Robinger
Abstract - I will talk about how the WKB method in quantum mechanics can be reformulated in terms of saddle point expansions of functional integrals and classical solutions to the Euclidean action equation of motion (instantons). I will begin by discussing the relevance of instanton physics to quantum mechanics and developing some aspects of the general theory. I will then specialize to a double-well potential, illustrate the equivalence of functional integral methods to the usual WKB approach, and calculate the partition function for a single instanton solution in this potential. I will then argue that it is straight-forward to generalize this result to study a dilute instanton gas in the same potential and conclude with the calculation of the energy splitting between the ground and 1st excited state, a quantity which is not accessible via canonical perturbation theory.
Refs -
- Rajaraman, (above), 10.1
- Forkel, "A Primer on Instantons in QCD", hep-ph/0009136

Steve Paik
Abstract - My aim is to define soliton solutions in classical field theory and discuss what happens when the theory is canonically quantized. We will study the simplest possible renormalizable field theories in 1+1 dimensions, in particular the phi^4 theory in the spontaneously broken phase. Here we shall look at some calculations supporting the claim that the particle spectrum of the quantum theory contains heavy solitonic particles in addition to the ordinary mesonic particles we are familiar with. An effort will be made to import classical intuition and reasoning over to quantum mechanics.
Refs -
- Jackiw, "Quantum meaning of classical field theory", Rev. Mod. Phys. 49 (1977) 681
- Coleman, (above), 185-194
- Dashen, Hasslacher, and Neveu, "Nonperturbative methods and extended-hadron models in field theory II", Phys. Rev. D10 (1974) 4130

Kristan Jensen
Abstract - Our current goal is to find classical soliton solutions to an arbitrary classical field theory. Pure scalar theories in more than (1+1) dimensions will be shown to be irrelevant, thereby motivating a search in classical gauge theories. To illustrate the general algorithm, we will examine in detail how SU(2) acts on a scalar triplet in (3+1) dimensions. Here we will see the origin of topological conservation laws - statements we can use to both classify and find soliton solutions. We will then find an equivalence in the general case between these laws and homotopy groups. Armed with this knowledge, we will elicit the boundary behaviour of a nontrivial soliton solution in SU(2) and exhibit its monopole nature. Time providing, we will then formally extend the algorithm in this example to the general case.
Refs -
- Coleman, (above), 6.2.4-6.4
- Rajaraman, (above), 3.2, 3.4
- From the math side, if you want to read up on homotopy equivalence, homotopy groups, their computation, or more:
  - Nakahara evidently has a good chapter for familiarity
  - Allen Hatcher's Algebraic Topology. He has an online copy here. This text is useful for formalism and as a guide for computation of homotopy groups. And it's just beautiful. For example: the big theorem in Coleman's 3.7 is a trivial consequence of a short exact sequence involving fiber bundles in Hatcher's 4.2. His online copy on vector bundles is also rather cool and worth it for the beauty.

Andrew Lytle
Abstract - I will discuss instanton effects in the 1+1d abelian higgs model. I will develop a formalism for dilute instanton gas sums, beginning in the conceptually simpler realm of a particle in a potential. I will extend this formalism to study how instantons affect the vacuum structure of the abelian higgs model.
Refs -
- Coleman, (above), 7.2-7.4
- Rajaraman, (above), 10

Joaquin Drut
Abstract - I will describe Polyakov's proof of charge confinement, and if time permits comment briefly on Son & Kovchegov's calculation of the critical temperature for the deconfinement transition.
Refs -
- Polyakov, Nucl. Phys. B120 (1977) 429 (and references therein)
- Polyakov, "Gauge Fields and Strings", Harwood Academic Publisher, 1987
- Son and Kovchegov, JHEP 0301:050,2003 (hep-th/0212230)
- Rubakov, "Classical Theory of gauge fields", Princeton Univ. Pr. (2002)

Ethan Thompson
Abstract - The existence of theta vacua for Yang-Mills theories in 4 dimensions will be shown to arise from a careful canonical quantization of the theory. Along the way, some confusions about where the homotopy group arises, what is continuously deformable to what, and how infinite energies come into play will hopefully be explained. Interactions between instantons will be explained, as will the concepts of 'large' and 'small' instantons and the different physics that follows from the two. Following Joe's work last week, the effects of adding one or two massless quarks will be investigated.
Refs -
- See refs in part I
- Callen, Dashen and Gross, Phys. Rev. D17 (1978), 2717, sections 2, 3 and 5.