Properties of Current Algebra at Infinite Momentum

Abstract

The sum rules of current algebra at infinite momentum can be considered as a system of coupled matrix equations, which we call the algebra of form factors. In addition to the equations following from sum rules, the form factors must also satisfy a complicated kinematic relation known as the angular condition. Presumably, the set of all hadron states (including the continuum) provides the basis for a representation of the algebra of form factors in which the angular condition is satisfied. It was conjectured by Dashen and Gell-Mann that a much smaller set of states containing only stable and resonant hadrons might also provide a representation. If this is true, the algebra of form factors could be used to predict many properties of hadrons. In the following paper, we attack the problem of finding a representation in which the angular condition is satisfied and in which all states have the same isospin. We obtain a large class of solutions which we suspect, but have not been able to prove, actually includes all solutions. None of our solutions has a physically acceptable mass spectrum. One purpose of the present paper is to discuss, in the proper physical context, the implications of the above-mentioned result. We discuss the algebra of form factors and the angular condition in detail, stressing those features which are general and not restricted to particular solutions. It is shown, for example, how one can incorporate additional equations following from the commutators of time components of currents with space components. We then consider the special problem of finding representations where all states have the same isospin. The relevance of this problem in the program of Dashen and Gell-Mann is discussed in detail.