Abstract
We formulate the worldline quantization (a.k.a. deformation quantization) of a massive fermion model coupled to external higher spin sources. We use the relations obtained in this way to show that its regularized effective action is endowed with an L∞ symmetry. The same result holds also for a massive scalar model.
Highlights
We focused on massive scalar and Dirac fermion models, but, no doubt, the same method can be applied to other elementary fields
The worldline quantization of field theory is based on the Weyl quantization of a particle in quantum mechanics, where the coordinates in the phase space are replaced by position and momentum operator and observables are endowed with a suitable operator ordering
In this paper we have carried out the worldline quantization of a Dirac fermion field coupled to external sources
Summary
2.1 Fermion linearly coupled to higher spin fields Let us consider a free fermion theory. We see that the symmetric tensor field hμμ1...μn is linearly coupled to the HS (higher spin) current. In terms of symbols, δεhμ(x, p) = ∂xμε(x, p) − i[hμ(x, p) ∗, ε(x, p)] ≡ Dx∗με(x, p) This will be referred to hereafter as HS transformation, and the corresponding symmetry HS symmetry. The transformations of ψ are somewhat different They can be expressed as Moyal product of symbols δεψ(x, p) = iε(x, p) ∗ ψ(x, p). We want to understand the conservation law ensuing from the HS symmetry of the interacting classical action (2.5) We evaluate this expression on the classical solution, in which case the first two terms vanish (remember that h is the background field). We see that the HS ε-transform is of the Lie algebra type
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