Superluminal Noncommutative Photons

With the help of the Seiberg-Witten map, one can obtain an effective action of a deformed QED from a noncommutative QED. Starting from the deformed QED, we investigate the propagation of photons in the background of electromagnetic field, up to the leading order of the noncommutativity parameter. In our setting (both the electric and magnetic fields are parallel to the coordinate axis x and the nonvanishing component of the noncommutativity parameter is θ), we find that the electric field has no any effect on the propagation of photons, but the velocity of photons can be larger than the speed of light (c = 1) when the propagating direction of photons is perpendicular to the direction of background magnetic field, while the light-cone condition does not change when the propagating direction is parallel to the background magnetic field. The causality associated with the superluminal photons is discussed briefly. ∗email address: [email protected] 1 Recently much attention has been focused on the noncommutativity of spacetime and its consequences in various aspects. Many interesting phenomena have been found. For example, there exists an intriguing UV/IR mixing [1] in the perturbative dynamics of noncommutative field theories (field theories in a noncommutative spacetime). The degrees of freedom are reduced drastically in the non-planar sector of noncommutative field theories [2–4]. The energy levels of the hydrogen atom and the Lamb shift in the noncommutative quantum electrodynamics (QED) have been found to deviate from those in the usual QED both on the classical and quantum levels, from which a constraint on the noncommutativity parameter of spacetime can be obtained by comparing with experimental data [5]. The noncommutativity of spacetime has also significant affections on the cosmology of the early universe. For instance, it has been shown that the noncommutativity in spatial coordinates can generate magnetic field in the early universe on a horizon scale, and due to its dependence of the number density of massive charged particles, one can trace back the temperature dependence of the noncommutativity scale from the bounds on the primordial magnetic field coming from nucleosynthesis [6]. Further, it was argued that due to the noncommutativity of spacetime, the inflation induced fluctuations become non-Gaussian and anisotropic, and the short distance dispersion relations are modified [7]. By definition, the coordinates in a noncommutative spacetime satisfy [x, x ] = iθ , (1) where θ is an antisymmetric constant matrix and is of dimension of (length). The action of a noncommutative field theory can be constructed by replacing the ordinary products between field operators by the ⋆-products in the action of the corresponding ordinary field theory. The ⋆-product is defined as follows, f ⋆ g(x) = e 1 2 θμν∂xμ∂yν f(x)g(y)|y=x. (2) The Lagrangian of the photon sector of the noncommutative QED is [8] L1 = − 1 4 F̂μν ⋆ F̂ μν , (3)