Quantization of Fields of Arbitrary Spin

Abstract

The search for relativistic field equations for elementary particles of arbitrary spin is a long standing problem of Quantum Field Theory. Although much work has been done on relativistic field theories describing fields with various mass-spin spectra, little has been done on picking out those theories which are quantizable, that is, those for which a particle interpretation exists which is consistent with the basic postulates of quantum theory. In this thesis we examine theories based on the relativistic field equation (Lμϑμ + X)Ψ = 0. The condition for ecahi tata on is expressed in terms of certain positive definiteness requirements on the eigenvectors corresponding to the non-zero eigenvalues of Lo. By restricting the. discussion to a specific type of theory, in which spin states are not repeated, these positive definiteness requirements are expressed in terms of certain simple trace conditions. A systematic procedure for finding representations of lo for quantizable theories is given, and illustrated in the case of spin 0,1,2 fields. In this procedure use is made of graphs depicting the representations of the Lorentz Group according to which % transforms ina relativistic field theory. Some simple results of graph theory are applied to develop the theory and also to discuss the properties of the Lo matrix. Further possibilities of this graphical approach are briefly discussed.