Abstract
The author considers the Gel'fand-Yaglom equations for free fields of arbitrary spin, in the particular case when the field transforms according to a direct sum of inequivalent irreducible finite representations of the proper Lorentz group. Under the assumption that the theory carries neither physical states of zero charge or energy density and that the mass-spin states are non-degenerate, he obtains the precise forms of the minimal and characteristic polynomials of the s blocks of the L0 matrix, which are then used to obtain new necessary and sufficient conditions that the theory be quantizable. The representation according to which the field transforms can be depicted graphically in a simple way and he takes advantage of this to use some simple ideas of graph theory to obtain results. The conclusion is that it will probably be necessary to allow repeated irreducible representations of the proper Lorentz group for theories of spin greater than eight to be quantizable.
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