Discrete nonlocal waves

Abstract

We study generic waves without rotational symmetry in (2+1)dimensional noncommutative scalar field theory. In the representation chosen, the radial coordinate is naturally rendered discrete. Nonlocality along this coordinate, induced by noncommutativity, accounts for the angular dependence of the fields. The exact form of standing and propagating waves on such a discrete space is found in terms of finite series. A precise correspondence is established between the degree of nonlocality and the angular momentum of a field configuration. At small distance no classical singularities appear, even at the location of the sources. At large radius one recovers the usual commutative behaviour.

Natural discretization in noncommutative field theory

“…The generator of rotations in the NC plane, turns out [4] to beN =â †â . The variation of the field φ under an infinitesimal rotation α is then δ φ iα[N, φ ].…”

“…Using its poles-displaying expansion [12] to monitor the multiplicative cancellation between zeroes and poles as σ → −m we obtain a second solution [4] …”

“…Here we report on the first non-radially symmetric exact solution [4], with discreteness and nonlocality explicit at the level of the fundamental degrees of freedom. Its continuum limit -required in order to use our methodology for usual field theories -is well defined.…”

“…We report progress [4] on a different approach [5], in which discretization is obtained assuming that the field space coordinates form a noncommutative algebra. One chooses a discrete representation of this algebra [6], obtaining a denumerable number of points.…”

“…A straightforward adaptation to the discrete case of the method of variation of constants (or of the method of Green functions) suggests [4,6] to search for…”