Date of Award
Doctor of Philosophy (PhD)
The concept of an infinite set of conserved charges in 1+1 dimensional quantum field theories has received considerable attention. Such sets have been found both for continuum theories such as the sine-Gordon equation1 and lattice theories such as the XYZ spin chain.2 The presence of an infinite set of conserved charges it closely related to the solution of the theory by the inverse scattering method.3 Knowledge of the form of the conserved charges can be of value in determining how to implement an inverse scattering solution to a given theory. It has also been shown, in many cases, that the existence of an infinite set of local conserved charges implies the lack of particle production in scattering.4 For interacting field theories such as those mentioned above (the XYZ spin chain model has been shown to have the massive Thirring model as its continuum limit5) this implies a much more restricted form of the S-matrix than would be expected for such theories. Clearly therefore, as has been the case with so many other symmetries and conservation laws, infinite sets of conserved charges provide a great deal of useful information about a theory. Of course, in the physics of the real world one is not interested in theories in only two dimensions but rather in four dimensions. In four dimensions the property of possessing an infinite set of conserved charges appears at first to be too strong to be useful. It has been shown in 3+1 dimensions that theories which possess this property along with several other reasonable requirements are necessarily free theories, provided the charges are constructed from local ·currents.28 Such theories are therefore uninteresting. The possibility is left open, however, for the case of non-local charges. It has been speculated that the Yang-Mills equations in loop space, which are similar to the equations of the two-dimensional chiral models, many possess charges of this sort.6,7 Polyakov has shown how one might construct conserved charges for the loop Yang Mills theory in 2+1 dimensions in analogy with the conserved charges of the chiral model.7
Grady, Michael Patrick, "Infinite Sets of Conserved Charges and Duality in Quantum Field Theory" (1982). Student Theses and Dissertations. 455.