Gauge-invariant spinor theories

Abstract

It is demonstrated that in spinor theories with nonlinear quadratic self-interaction, gauge symmetries connected withx-dependent internal symmetry transformations can be established without the introduction of additional vector fields if the spinor field operator has the noncanonical length dimension −1/2. In this case the theory is scale invariant at small distances and hence formally renormalizable. Operator products can be defined according to prescriptions given by Zimmermann and Wilson. The usual role of the gauge fields in these spinor theories is taken over by the formally constructed vector and axial vector «currents» of the noncanonical spinor fields which have the correct length dimension of boson operators. Physical fermion fields are related to deverivatives of these spinor fields or to 3-products of these fields. The noncanonical spinor field hence may be regarded as a «spinor potential» in the sense that its relation to a physical spinor field is similar to the relation of the vector potential to the physical electromagnetic field. The unobservable «spinor potential» acts in a state space with indefinite metric. Examples forSU n ⊗SU n gauge-invariant theories are given. The Heisenberg nonlinear spinor theory is shown to beSU 2 gauge-invariant, and hence the parity symmetry version of Durr to beSU 2⊗SU 2 gauge-invariant. This formal invariance, however, does not necessarily imply that the summetry holds in Hilbert space.