Abstract
The Dirac Hamiltonian formalism is applied to a system in $(2+1)$-dimensions consisting of a Dirac field $\psi$ minimally coupled to Chern-Simons $U(1)$ and $SO(2,1)$ connections, $A$ and $\omega$, respectively. This theory is connected to a supersymmetric Chern-Simons form in which the gravitino has been projected out (unconventional supersymmetry) and, in the case of a flat background, corresponds to the low energy limit of graphene. The separation between first-class and second-class constraints is performed explicitly, and both the field equations and gauge symmetries of the Lagrangian formalism are fully recovered. The degrees of freedom of the theory in generic sectors shows that the propagating states correspond to fermionic modes in the background determined by the geometry of the graphene sheet and the nondynamical electromagnetic field. This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector $\psi=0$, which is of interest for perturbation theory. For the latter the analysis is adapted to the case of manifolds with boundary, and the corresponding Dirac brackets together with the centrally extended charge algebra are found. Finally, the $SU(2)$ generalization of the gauge group is briefly treated, yielding analogous conclusions for the degrees of freedom.
Highlights
That we have been able to explore, but it is supposedly restored at a sufficiently high energies
This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector ψ = 0, which is of interest for perturbation theory
We have carried out the Dirac analysis for constrained Hamiltonian systems for the action composed of a spin-1/2 Dirac field minimally coupled to an electromagnetic potential and to the Lorentz connection in (2 + 1)-dimensions
Summary
Hi = (∂iAj − ∂j Ai) πj − Ai∂j πj + ∂iωja − ∂j ωia πaj − ωia∂j πaj + ∂ieaj − ∂jeai pja − eai ∂jpja + ∂iψχ + χ∂iψ , which can be readily seen to generate spatial diffeomorphisms on phase space functions F as {F, ξiHi} = LξF. It can be directly checked that Ja, Kand Υ generate SO(2, 1)×U(1)×W eyl transformations over all the fields and momenta They satisfy the (weakly vanishing) Poisson relations (B.4). The absence of spatial derivatives in Υ implies that such symmetry is generated by a purely local constraint with no associated asymptotic charges, as explained in detail below (recent examples of this fact can be found in [15] and references therein, see [16] for a thorough discussion). The corresponding symmetry breaking, leads to physical consequences as we will discuss
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