A chiral spin theory in the framework of an invariant evolution parameter formalism

B. Sarel, L. P. Horwitz

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We present a formulation for the construction of first-order equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with off-shell particles with spin. Our basic requirement is that the second-order equation in the theory is of the Schrödinger-Stueckelberg type, which exhibits features of both the Klein-Gordon and Schrödinger equations. This requirement restricts the structure of the first-order equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the first-order system, we are able to obtain second-order forms containing the transverse and longitudinal momentum relative to a timelike vector tμtμ = -1 used to maintain covariance of the theory. This timelike vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space-time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac's equation and its solutions is obtained in the highly constrained limit in which pμ is proportional to tμ on mass shell. The chiral nature of the theory is apparent. We define the discrete symmetries of the theory, and find that they are represented by states which are pure left or right handed.

Original languageEnglish
Pages (from-to)2963-2995
Number of pages33
JournalJournal of Mathematical Physics
Issue number6
StatePublished - Jun 1997


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