Heisenberg Evolution in a Quantum Theory of Noncommutative Fields

Abstract

A quantum theory of noncommutative fields was recently proposed by Carmona, Cortez, Gamboa and Mendez (hep-th/0301248). The implications of the noncommutativity of the fields, intended as the requirements $[\phi,\phi^{+}]=\theta\delta^{3}(x-x^{\prime}), [\pi,\pi^{+}]=B\delta^{3}(x-x^{\prime}) $, were analyzed on the basis of an analogy with previous results on the so-called ``noncommutative harmonic oscillator construction". Some departures from Lorentz symmetry turned out to play a key role in the emerging framework. We first consider the same hamiltonian proposed in hep-th/0301248, and we show that the theory can be analyzed straightforwardly within the framework of Heisenberg evolution equation without any need of making reference to the ``noncommutative harmonic oscillator construction". We then consider a rather general class of alternative hamiltonians, and we observe that violations of Lorentz invariance are inevitably encountered. These violations must therefore be viewed as intrinsically associated with the proposed type of noncommutativity of fields, rather than as a consequence of a specific choice of Hamiltonian.