Wilson loops in lattice gauge theories
Jensén, Johanna (2022)
Jensén, Johanna
2022
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi-fe2022043031665
https://urn.fi/URN:NBN:fi-fe2022043031665
Tiivistelmä
This thesis is based on the article "Wilson loops in finite Abelian lattice gauge theories" by M. Forsström, J. Lenells and F. Viklund. In the article, lattice gauge theories on Z^4 for a finite Abelian structure group are considered and the expectation value for the Wilson loop observable at weak coupling is computed. The purpose of this thesis is to explain this article in more detail and to give the theory necessary to understand the article.
In this thesis, we consider the lattice Z^4, the structure group Z_n and a faithful and one-dimensional representation. Basic theory for groups, representations and lattices is discussed. To state the main result, several definitions, e.g. the Wilson loop observable and Wilson action, are given. The main result is given as a theorem, where we have an inequality for the limit of the expectation value of the Wilson loop observable.
The theory necessary to prove the main result is given in this thesis. Theory for discrete exterior calculus is given in the third chapter. This includes theory for k-cells and k-forms as well as definitions and applications for both the exterior derivative and the co-derivative. Furthermore, two versions of the Poincaré lemma are given and applied to problems, e.g. for writing the given measure as a measure on plaquette configurations instead of spin configurations. The Hodge dual of the lattice Z^r is defined and both examples and lemmas, which are important for later proofs, are given.
In the fourth chapter, vortices and oriented surfaces are defined using the theory from the previous chapter. It is important to note that these definitions might differ from other sources. Various lemmas are stated and proved. The most important result in this chapter is a proposition, in which a probability is computed, that is applied several times in the proof of the main result.
Since the limit of the expectation of the Wilson loop observable is computed, both its existence and translation invariance must be proved. A more general theorem, which proves the existence and translation invariance for a real-valued function, is given and proved with Ginibre's inequality. This theorem is then applied to the Wilson loop observable.
The last part of this thesis is the proof of the main theorem. To prove this theorem, the theory and results given in the earlier chapters are applied. The proof is divided into two parts, which are then combined to achieve the desired result.
In this thesis, we consider the lattice Z^4, the structure group Z_n and a faithful and one-dimensional representation. Basic theory for groups, representations and lattices is discussed. To state the main result, several definitions, e.g. the Wilson loop observable and Wilson action, are given. The main result is given as a theorem, where we have an inequality for the limit of the expectation value of the Wilson loop observable.
The theory necessary to prove the main result is given in this thesis. Theory for discrete exterior calculus is given in the third chapter. This includes theory for k-cells and k-forms as well as definitions and applications for both the exterior derivative and the co-derivative. Furthermore, two versions of the Poincaré lemma are given and applied to problems, e.g. for writing the given measure as a measure on plaquette configurations instead of spin configurations. The Hodge dual of the lattice Z^r is defined and both examples and lemmas, which are important for later proofs, are given.
In the fourth chapter, vortices and oriented surfaces are defined using the theory from the previous chapter. It is important to note that these definitions might differ from other sources. Various lemmas are stated and proved. The most important result in this chapter is a proposition, in which a probability is computed, that is applied several times in the proof of the main result.
Since the limit of the expectation of the Wilson loop observable is computed, both its existence and translation invariance must be proved. A more general theorem, which proves the existence and translation invariance for a real-valued function, is given and proved with Ginibre's inequality. This theorem is then applied to the Wilson loop observable.
The last part of this thesis is the proof of the main theorem. To prove this theorem, the theory and results given in the earlier chapters are applied. The proof is divided into two parts, which are then combined to achieve the desired result.
Kokoelmat
- 111 Matematiikka [38]