The effective nonrelativistic potential VT describing the fermion-fermion interaction in the Maxwell-Chern-Simons theory is derived to the lowest order in perturbation theory. As expected, VT is not invariant under parity and time-reversal transformations. The quantum dynamics generated by Vr becomes exactly solvable at the limits where either the Maxwell or the ChernSimons terms disappear; in neither case electron-electron bound states show up. However, numerical calculations indicate that fermion-fermion bound states do exist in the general case.Motivated by recent discussions [1] about the consistency of the nonrelativistic limit of certain relativistically invariant quantum field theories, we consider in this letter the problem of determining the effective electron-electron low energy potential arising from the Maxwell-Chern-Simons (MCS) theory. We shall also investigate whether this potential defines a physically sensible and nontrivial quantum dynamics. In particular, the existence of electronelectron bound states is one of our main concerns in this work.As known [ 2 ], the MCS theory is a (2 + l )-dimensional model describing the coupling of charged fertalons (~, ~) of mass m and electric charge e to the gauge field potential A~ via the lagrangian density -2--22 (OuAU) (O.A ") + li(r/uOuV --½i(Ouq)y~v-e(z7,A~¢-rn~¢ ,where F,, = O~A,-GA,, 0 >/0 is the topological mass, 2 is a gauge parameter and a is a dimensionless real Supported in part by Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico (CNPq), Brazil.parameter (0 ~< a ~< 1 ) enabling us to modulate the intensity of the Maxwell term. Throughout this paper we use natural units (c--h=l). Our metric is goo=-gl~ = -g22 =l, while for the y matrices we adopt the representation 7° = o3, 71 = ia~, 72 = itr2; ai, i= 1, 2, 3, are the Pauli spin matrices. Neither parity nor time-reversal are, separately, symmetries of the model.The contribution of order e 2 to the elastic scattering amplitude e-+ e-~ e-+ e-(M611er scattering ) is given bywhere _e 2Here, pl, P2 (P~, P~ ) are the on-shell two-momenta of the initial (final) electrons, k-p't -Pl =P2 --P'2 is the transferred momentum and Du~(k) designates the free photon propagator. One can easily check that 0370-2693/92/$ 05.00
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.