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We show that it is in principle possible to construct dualities between commutative and non-commutative theories in a systematic way. This construction exploits a generalization of the exact renormalization group equation (ERG). We apply this to the simple case of the Landau problem and then generalize it to the free and interacting non-canonical scalar field theory. This constructive approach offers the advantage of tracking the implementation of the Lorentz symmetry in the non-commutative dual theory. In principle, it allows for the construction of completely consistent non-commutative and non-local theories where the Lorentz symmetry and unitarity are still respected, but may be implemented in a highly non-trivial and non-local manner.
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The effect of non-commutativity on electromagnetic waves violates Lorentz invariance: in the presence of a background magnetic induction field b, the velocity for propagation transverse to b differs from c, while propagation along b is unchanged. In principle, this allows a test by the Michelson-Morley interference method. We also study non-commutativity in another context, by constructing the theory describing a charged fluid in a strong magnetic field, which forces the fluid particles into their lowest Landau level and renders the fluid dynamics non-commutative, with a Moyal product determined by the background magnetic field.
We discuss a non--commutative integration calculus arising in the mathematical description of anomalies in fermion--Yang--Mills systems. We consider the differential complex of forms $u_0ccr{eps}{u_1}cdotsccr{eps}{u_n}$ with $eps$ a grading operator on a Hilbert space $cH$ and $u_i$ bounded operators on $cH$ which naturally contains the compactly supported de Rham forms on $R^d$ (i.e. $eps$ is the sign of the free Dirac operator on $R^d$ and $cH$ a $L^2$--space on $R^d$). We present an elementary proof that the integral of $d$--forms $int_{R^d}trac{X_0dd X_1cdots dd X_d}$ for $X_iinMap(R^d;gl_N)$, is equal, up to a constant, to the conditional Hilbert space trace of $Gamma X_0ccr{eps}{X_1}cdotsccr{eps}{X_d}$ where $Gamma=1$ for $d$ odd and $Gamma=gamma_{d+1}$ (`$gamma_5$--matrix) a spin matrix anticommuting with $eps$ for $d$ even. This result provides a natural generalization of integration of de Rham forms to the setting of Connes non--commutative geometry which involves the ordinary Hilbert space trace rather than the Dixmier trace.
General non-commutative supersymmetric quantum mechanics models in two and three dimensions are constructed and some two and three dimensional examples are explicitly studied. The structure of the theory studied suggest other possible applications in physical systems with potentials involving spin and non-local interactions.
In this paper we study non-commutative massive unquenched Chern-Simons matter theory using its gravity dual. We construct this novel background by applying a TsT-transformation on the known parent commutative solution. We discuss several aspects of this solution to the Type IIA supergravity equations of motion and, amongst others, check that it preserves ${cal N}=1$ supersymmetry. We then turn our attention to applications and investigate how dynamical flavor degrees of freedom affect numerous observables of interest. Our framework can be regarded as a key step towards the construction of holographic quantum Hall states on a non-commutative plane.
We compute the spectrum of cosmological perturbations in a scenario in which inflation is driven by radiation in a non-commutative space-time. In this scenario, the non-commutativity of space and time leads to a modified dispersion relation for radiation with two branches, which allows for inflation. The initial conditions for the cosmological fluctuations are thermal. This is to be contrasted with the situation in models of inflation in which the accelerated expansion of space is driven by the potential energy of a scalar field, and in which the fluctuations are of quantum vacuum type. We find that, in the limit that the expansion of space is almost exponential, the spectrum of fluctuations is scale-invariant with a slight red tilt. The magnitude of the tilt is different from what is obtained in a usual inflationary model with the same expansion rate during the period of inflation. The amplitude also differs, and can easily be adjusted to agree with observations.
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