We find dispersion laws for the photon propagating in the presence of mutually orthogonal constant external electric and magnetic fields in the context of the $theta $-expanded noncommutative QED. We show that there is no birefringence to the first o
rder in the noncommutativity parameter $% theta .$ By analyzing the group velocities of the photon eigenmodes we show that there occurs superluminal propagation for any direction. This phenomenon depends on the mutual orientation of the external electromagnetic fields and the noncommutativity vector. We argue that the propagation of signals with superluminal group velocity violates causality in spite of the fact that the noncommutative theory is not Lorentz-invariant and speculate about possible workarounds.
We develop a general technique for finding self-adjoint extensions of a symmetric operator that respect a given set of its symmetries. Problems of this type naturally arise when considering two- and three-dimensional Schrodinger operators with singul
ar potentials. The approach is based on constructing a unitary transformation diagonalizing the symmetries and reducing the initial operator to the direct integral of a suitable family of partial operators. We prove that symmetry preserving self-adjoint extensions of the initial operator are in a one-to-one correspondence with measurable families of self-adjoint extensions of partial operators obtained by reduction. The general construction is applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the electron in the magnetic field of an infinitely thin solenoid.
We analyze renormalizability properties of noncommutative (NC) theories with a bifermionic NC parameter. We introduce a new 4-dimensional scalar field model which is renormalizable at all orders of the loop expansion. We show that this model has an i
nfrared stable fixed point (at the one-loop level). We check that the NC QED (which is one-loop renormalizable with usual NC parameter) remains renormalizable when the NC parameter is bifermionic, at least to the extent of one-loop diagrams with external photon legs. Our general conclusion is that bifermionic noncommutativity improves renormalizablility properties of NC theories.
