Canonical field anticommutators in the extended gauged Rarita-Schwinger theory

Abstract

We reexamine canonical quantization of the gauged Rarita-Schwinger theory using the extended theory, incorporating a dimension 1 2 auxiliary spin-1 2 field Λ, in which there is an exact off-shell gauge invariance. In Λ ¼ 0 gauge, which reduces to the original unextended theory, our results agree with those found by Johnson and Sudarshan, and later verified by Velo and Zwanziger, which give a canonical Rarita-Schwinger field Dirac bracket that is singular for small gauge fields. In gauge covariant radiation gauge, the Dirac bracket of the Rarita-Schwinger fields is nonsingular, but does not correspond to a positive semidefinite anticommutator, and the Dirac bracket of the auxiliary fields has a singularity of the same form as found in the unextended theory. These results indicate that gauged Rarita-Schwinger theory is somewhat pathological, and cannot be canonically quantized within a conventional positive semidefinite metric Hilbert space. We leave open the questions of whether consistent quantizations can be achieved by using an indefinite metric Hilbert space, by path integral methods, or by appropriate couplings to conventional dimension

Analysis of a gauged model with a spin-12field directly coupled to a Rarita-Schwinger spin-32field

“…(1) The coupled model eliminates the discontinuity in the number of degrees of freedom when an external field is present, relative to the zero field case, discussed in detail in [8]. (2) The nonperturbative behavior of the Dirac bracket, already clear from the zero mass limit of [4,5], and discussed in detail in [6][7][8] is eliminated.…”

“…(2) The nonperturbative behavior of the Dirac bracket, already clear from the zero mass limit of [4,5], and discussed in detail in [6][7][8] is eliminated. The coupled model admits a perturbative expansion in powers of the gauge coupling g, as well as permitting calculation of the perturbative chiral anomaly.…”

“…(1) and (2) already break the free Rarita-Schwinger field fermionic gauge symmetry ψ μ → ψ μ þ ∂ μ ϵ, so there is no change in the degree of freedom counting when the model is gauged by replacing the derivative ∂ μ by the gauge-covariant derivative D μ ¼ ∂ μ þ gA μ , with A μ an anti-Hermitian gauge field. This avoids the vexing issue of a discontinuity in the degree of freedom counting when a Rarita-Schwinger field is gauged, which is analyzed in detail in [8]. The SUð8Þ gauge model with field content set out in Table I has a number of interesting properties, which make it special, and possibly unique.…”

“…This work showed that in the massless case all modes are luminal, and that by adding an auxiliary field, the gauged theory could be extended to have a full off-shell fermionic gauge invariance. A detailed investigation of the extended theory by the author, Henneaux, and Pais [8] showed that the extended model still has problems. When covariant gauge fixing is correctly implemented, the extended model has nonpositive anticommutators, and more seriously from the point of view of quantization, the Dirac brackets in the constrained theory have a weak field singularity in the massless case that is shifted from the spin-3 2 bracket to the auxiliary field bracket but is not removed.…”

“…II-IX, introduces the coupled model and reprises in this context the analysis of Refs. [6][7][8]. In Sec.…”