Noncommutative U(1) gauge theory from a worldline perspective

Abstract

We study pure noncommutative U(1) gauge theory representing its one-loop effective action in terms of a phase space worldline path integral. We write the quadratic action using the background field method to keep explicit gauge invariance, and then employ the worldline formalism to write the one-loop effective action, singling out UV-divergent parts and finite (planar and non-planar) parts, and study renormalization properties of the theory. This amounts to employ worldline Feynman rules for the phase space path integral, that nicely incorporate the Fadeev-Popov ghost contribution and efficiently separate planar and non-planar contributions. We also show that the effective action calculation is independent of the choice of the worldline Green's function, that corresponds to a particular way of factoring out a particle zero-mode. This allows to employ homogeneous string-inspired Feynman rules that greatly simplify the computation.

Non-commutativity in polar coordinates

“…A common starting point in such a setting is the introduction of the fundamental commutator between spatial coordinates x i ,x j = iθ i j (1) for constant non-commutativity parameters assembled into a real, skew-symmetric matrix θ i j . Much progress has been made in understanding various aspects of these noncommutative spaces [5] and their application to quantum a e-mail: jedwards@ifm.umich.mx field theory [6][7][8][9], where the effects of UV / IR mixing have played an important rôle. There is also interest in generalisations of (1), perhaps most famously to describe the "fuzzy sphere" [10,11].…”

U(N) Yang-Mills in non-commutative space time

“…is given by a transition amplitude x|e −T δ 2 S ghost |x which can be computed in terms of a phase-space {x(t), p(t)} path integral of a particle on the circle [10,12] with Hamiltonian δ 2 S ghost . The symbol tr in (2.27) denotes the trace over the color indices in δ 2 S ghost .…”

“…Indeed, it has been shown that non-commutative field theories are linked to phase-space worldline path integrals. This was developped in [10,11] for (complex-) scalar field theories and in [12,13] for a U(1) gauge theory.…”

Worldline colour fields and non-Abelian quantum field theory

“…Its origins can be traced to Feynman [6], with a revival of interest following Bern and Kosower's Master Formulae for field theory amplitudes derived from string theory [7,8]. It is recently finding application in a wide range of physical problems, including multi-loop effective actions and scattering amplitudes [3,9], constant field QED and the Schwinger effect [10,11], gravitational interactions [12,13], higher spin fields [14,15], quantum fields in non-commutative space-time [16,17] and the form factor decomposition of the four gluon vertex reported by Schubert elsewhere in these proceedings.…”