Chern–Simons soliton and a critical study of the energy momentum tensor in noncommutative field theory

Abstract

We have shown unambiguously the existence of solitons in the non-commutative (NC) extension of Chern–Simons–Higgs model. The analysis is done at the classical level (since solitons are essentially classical objects) and in the first non-trivial order in θ, the only spatial noncommutativity parameter. At the same time, we have exposed an inadequacy in the conventional definitions of the energy momentum tensor (EMT) in the present context but this pathology appears to be generic to NC field theories. This is reflected in the fact that the BPS soliton equations (obtained from the EMT) are not compatible with the full variational equations of motion, requiring further imposition a restriction on the form of the Higgs field, contrary to the commutative spacetime case. Both in the Lagrangian and Hamiltonian formulations of the problem, we concentrate on the canonical and symmetric forms of the energy–momentum tensor. In the Hamiltonian scheme, constraint analysis and the induced Dirac brackets are derived. In fact the EMT behaves properly as the spacetime translation generators and their actions on the fields are discussed in detail. The effects of noncommutativity on the soliton solutions have been analyzed carefully and we have come up with some interesting results. Comparing the relative strengths of the noncommutative effects, we have shown that there is a universal character in the noncommutative correction to the magnetic field—it depends only on θ. On the other hand, in the cases of all other observables of physical interest, such as the potential profile, soliton mass or the electric field, the parameters θ as well as τ (the latter comprising solely of commutative Chern–Simons–Higgs model parameters) appear with similar weightage.