Towards an algebraic quantum chromodynamics

Abstract

We outline a quantum theory of quarks and gluons based on fields with values taken from a noncommutative Jordan algebra. These fields automatically satisfy a triality rule: Quark-antiquark and three-quark states are color singlets. If the elements of the algebra are position dependent, the theory leads to a minimal gauge-invariant coupling between quarks and gluons. The quantization of such a theory is outlined; we find that only color-singlet clusters of quarks and antiquarks have particle properties. The color-nonsinglet fields do not support a representation of the Lorentz group; in fact, their phases are undefined. We conjecture that this can be remedied by a coupling between space-time and flavor symmetries as suggested by Hawking and Pope. Such a coupling naturally leads to one-third-integer values of the quark charges.