`Understanding non-abelian Yang-Mills theory is one of the major challenges of theoretical physics. It is helpful to find simpler analogous problems whose solution might shed some light on this problem.'The two dimensional system is exactly solvable, and hopefully will provide some insight into the quantization in higher dimensions. The problem has strong topological considerations which are potentially very important in the full theory.
The amount of theoretical work one has to cover before being able to solve problems of real practical value is rather large, but this circumstance is an inevitable consequence of the fundamental part played by transformation theory and is likely to become more pronounced in the theoretical physics of the future.- P.A.M. Dirac.
Since the development of quantum theory considerable progress has been made in developing encompassing principles which provide physical theories. One of the most successful principles has been that of \emph{gauge invariance}, which served as a guide to the development of quantum electrodynamics. Q.E.D. has undergone rigorous experimental testing, and lays claim to some of the most accurate predictions known (footnote).
The success of Q.E.D. encouraged Chen Ning Yang and Robert Mills to attempt to develop a theory of the strong interaction by examining non-abelian gauge invariance. Q.E.D. only displayed abelian gauge invariance. The work of Yang and Mills 1954 did not succeed in developing a theory of the strong interaction, but it did lay the groundwork for later theories such as quantum chromodynamics and the so called electroweak theory, which united the electromagnetic and weak interactions as a single theory. Yang-Mills theory has been a fruitful area of research and continues to be a useful tool of exploration.
Gauge invariance seems to be a basic requirement of physical theories, but there are accompanying complications. As a consequence of gauge invariance, the canonical variables in the Hamiltonian formulation of the theory are not all independent as a result of the presence of constraints (footnote). Constraints add mathematical complication to the quantization. Gauge theories can also display confinement and asymptotic freedom, which means the coupling strength of the interaction increases with increasing separation and almost disappears with small separation. Unfortunately most solutions to physical gauge theories are perturbative, so in cases where asymptotic freedom is present, they only work for short distances(footnote). This means that the perturbative solutions break down for longer distance, low energy interactions. Considerable work is being conducted in an attempt to develop an an approach other than perturbation theory to address this problem.
This dissertation addresses a quantization of pure Yang-Mills theory in two dimensions; one time and one spatial. Because the quantization is conducted in two dimensions there is an exact solution. However as a further consequence of the two dimensional restriction there are no propagating modes in the theory, which means there are no `particles'. Despite this, a careful quantization of the theory has non-trivial solutions because of topological considerations. The theory is an important illustration of the need to be careful in quantizing an apparently trivial classical theory. This message is of relevance to areas other than high energy physics, such as condensed matter physics, where both lower dimensional systems and non-trivial topological effects are present. Examples include the Bohm-Aharonov effect and flux quantization.
Another reason why two dimensional Yang-Mills theory is an interesting problem is that it is a constrained system (as is four dimensional Yang-Mills theory). In the simplified setting of two dimensions, it is possible to see more easily how the constraints are important in the quantum theory.
The topological aspects of gauge theory are of particular interest as they are a current field of study, not only in High Energy Physics but in Mathematical Physics. Publications such as (cite) and (cite) all address the consequences of topology on gauge theories. Several major surveys (footnote), such as (cite) and (cite), examine topological results relating to two dimensional gauge theories and how this relation to string theories.
The results of this thesis are not new. In particular S.G.Rajeev (cite) addressed the quantization of pure Yang-Mills theory on a cylinder in 1988. This dissertation examines the same problem, and gives the same results but attains them by a different method. Where Rajeev reduces the problem to a question in group theory, this dissertation retains a ``physical'' formalism through out. The physical formalism clearly enunciates the constraints and gauge invariance in pure Yang-Mills theory and identifies the physical degrees of freedom, aspects of the theory which are difficult to identify in Rajeev's paper.
Chapter 1 discusses the basics of gauge invariance, with particular reference to quantum electrodynamics. Generalizing quantum electrodynamics to a non-trivial group structure provides a logical approach to the Yang-Mills action. Chapter 2 deals with reformulating Yang-Mills theory in Hamiltonian form, and discusses the existence and effect of constraints. These form a necessary prelude to quantization, which proceeds along the methods outlined by P.A.M. Dirac (cite). Chapter 3 follows a canonical quantization of the Hamiltonian theory, and translates the classical results of Chapter 2 into an operator form. The final chapter, Chapter 4 derives a general solution to the quantized theory, and examines some specific examples.
Yang-Mills theory provides a valuable introduction to mathematical physics. The quantization given in this paper is almost a closed case, although further work in this field, for instance coupling to point particles, such as in the article (cite), is possible.
In a path integral formulation gauge invariance also complicates the quantization of the theory, although the constraints do not make a direct appearance.
Or equivalently, high energy interactions.
The survey (cite) was released in November 1994 to HEP-TH, and gives a good summary of 2D Yang-Mills theory.
K. S. Gupta and R.J. Henderson and S. G. Rajeev and O.T. Turgut, Yang-Mills Theory on a Cylinder Coupled to Point Particles", hep-th, 1993.
J. Harvey and J. Polchinski, Recent Directions in Particle Theory, World Scientific,1992.
R. Jackiw, Constrained Quantization Without Tears, hep-th, 9306075, 1993.
M. Blau and G. Thompson, Quantum Yang-Mills Theory on Arbitary Surfaces, International Journal of Modern Physics, 7(16):3781-806, 1992.
S.G. Rajeev, Yang-Mills Theory on a Cylinder, Physics Letters B, 212(2):203-5, 1988.
E. Witten, Two Dimensional Dauge Theories Revisited, Journal of Geometry and Physics, 9:303, 1992.
S. Cordes and G. Moore and S. Ramgoolam}, Lectures on 2D Yang-Mills Theory, Equivalent Chomology and Topological Field Theories, Yale Pre-Print YCTP, 1994. hep-th 9411210.
W.E. Burcham and M. Jobes, Nuclear and Particle Physics, Longman,
1995.