In this article an energy correction is calculated in the time independent perturbation setup using a regularised ultraviolet finite Hamiltonian on the noncommutative Minkowski space. The correction to the energy is invariant under rotation and translation but is not Lorentz covariant and this leads to a distortion of the dispersion relation. In the limit where the noncommutativity vanishes the common quantum field theory on the commutative Minkowski space is reobtained.
The Heisenberg spin 1/2 chain is revisited in the perturbative RG approach with special focus on the transition of the critical exponents. We give a compact review that first order RG in the couplings is sufficient to derive the exact transition from ν = 1 to ν = 2/3, if the boson radius obtained in the bosonization procedure is replaced by the exact radius obtained in the Bethe approach. We explain the fact, that from the bosonization procedure alone, the critical exponent can not be derived correctly in the isotropic limit Jz → J. We further state that this fact is important if we consider to bosonize the antiferromagnetic super spin chain for the quantum Hall effect.
We derive a composite fermion model for the fractional quantum Hall effect from relativistic quantum electrodynamics. With simple arguments it is shown that a special Chern Simons transformation of the Dirac electrons in four spacetime dimensions leads in the low energy limit to a single particle Hamiltonian for composite fermions in three dimensions with correction terms such as Rashba-or Dresselhaus-spin-orbit coupling and zitterbewegung. Furthermore we provide a mechanism to quantum-mechanically project the quantum fields defined in the four dimensional Minkowski space to three dimensions. This leads to a relativistic field theory and especially a composite fermion field theory in three dimension. This projection map can be combined with the projection onto a Landau level or composite fermion Landau level respectively. This results in a quasi relativistic quantum field theory on a noncommutative plane. The phenomenological models resulting from this approach are discussed and allow a systematical exploration of the effects of the spin and the condensation in a Landau level. We expect from the relativistic approach corrections in terms of spin-orbit coupling effects. From the projection onto Landau levels we expect a modification of the dispersion relation and a modified composite fermion mass. Furthermore, the BRST quantization for Chern Simons theories with compact gauge group is reviewed and the phenomenological consequences within a composite fermion model with spin are discussed. The connection to Wess Zumino Witten theories is recalled and a possible link between the corresponding central charge of the related affine Lie algebra and the composite fermion filling factor is pointed out.
We derive a composite fermion model for the fractional quantum Hall effect from relativistic quantum electrodynamics. With simple arguments it is shown that a special Chern Simons transformation of the Dirac electrons in four spacetime dimensions leads in the low energy limit to a single particle Hamiltonian for composite fermions in three dimensions with correction terms such as Rashba‐ or Dresselhaus‐spin‐orbit coupling and zitterbewegung. Furthermore we provide a mechanism to quantum‐mechanically project the quantum fields defined in the four dimensional Minkowski space to three dimensions. This leads to a relativistic field theory and especially a composite fermion field theory in three dimension. This projection map can be combined with the projection onto a Landau level or composite fermion Landau level respectively. This results in a quasi relativistic quantum field theory on a noncommutative plane. The phenomenological models resulting from this approach are discussed and allow a systematical exploration of the effects of the spin and the condensation in a Landau level. We expect from the relativistic approach corrections in terms of spin‐orbit coupling effects. From the projection onto Landau levels we expect a modification of the dispersion relation and a modified composite fermion mass. Furthermore, the BRST quantization for Chern Simons theories with compact gauge group is reviewed and the phenomenological consequences within a composite fermion model with spin are discussed. The connection to Wess Zumino Witten theories is recalled and a possible link between the corresponding central charge of the related affine Lie algebra and the composite fermion filling factor is pointed out.
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