The fermion-fermion effective potential in the Maxwell-Chern-Simons theory

Abstract

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

Comment on ‘‘Attractive forces between electrons in (2+1)-dimensional QED’’

“…It is to be emphasized that such a procedure is fully consistent with the analysis of Refs. [1] and [2] which claim validity for e 2 /m <3C 1. In fact the vertex corrections which are shown there to be of order e 2 /m in Ref.…”

“…[l] and [2] and the numerical results presented there cannot be expected to be relevant to the high-T c problem.…”

“…[l] in the 0 in -* -<*> limit except for the omission in the latter of the repulsive (g 2 /2nr) 2 term. It is to be emphasized that the present approach fully realizes the ideal of Ref.…”

“…It has recently been claimed by Girotti et al [1,2] that electron-electron bound states are possible in QED3. They have furthermore used their model to perform extensive numerical calculations in order to calculate transition temperatures in high-r c superconductors.…”

Relativistic quantum scattering on a cone

“…Also analyzing the fermion-fermion effective potential for a Maxwell-ChernSimons theory [5] and a massive Thirring model [6] in a (2+1)-dimensional space-time, a two-dimensional delta-function potential emerges in both models under specific circumstances, and also a systematic approach to provide a self-adjoint extension of the respective Hamiltonians operators were performed. The inclusion in our system of the non-minimal coupling between the scalar field and scalar curvature of an idealized cosmic string space-time, as far as we know, is the first example where an effective two-dimensional delta function interaction naturally occurs in a relativistic quantum mechanical problem.…”