Duality in scalar field theory on noncommutative phase spaces

Abstract

We describe a novel duality symmetry of Φ 4 2n -theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of noninteracting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regularization we show, to all orders of perturbation theory, that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories.

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

“…As discussed in [12], in the case of complex Φ 4 -theory deformed as described above, a finite matrix dimension N amounts to having both an ultraviolet and infrared cutoff at the same time, as there exist duality transformations which exchange infrared and ultraviolet divergences, while the action and all regularized Green's functions of the model are duality invariant. In this paper we will carry this approach one step further and describe precisely how to remove the regularization, i.e.…”

“…It was shown in [12] that there is a natural regularization of these models which allows, in a simple manner, to make precise mathematical sense of the functional integrals defining the quantum field theory. Rather than expanding the boson fields in plane waves, it is more natural to expand them in the eigenfunctions of the Landau Hamiltonian P 2 i and thereby diagonalize the free part of the action.…”

Relativistic corrections to the Moyal-Weyl spacetime

“…And it is precisely around the Planck-length scale where noncommutative spacetimes are supposed to be non-negligible, [LS02,Sza04].…”

Non-commutative Renormalization

“…Moyal interactions were noticed to obey a certain Langmann-Szabo duality [41], which exchanges space and momentum variables. Grosse and Wulkenhaar realized that the propagator should be modified to also respect this symmetry [40].…”