We describe a novel duality symmetry of Φ 4 2n -theory defined on noncommutative Euclidean space and with noncommuting momentum coordinates. This duality acts on the fields by Fourier transformation and scaling. It is an extension, to interactions defined with a star-product, of that which arises in quantum field theories of noninteracting scalar particles coupled to a constant background electromagnetic field. The dual models are in general of the same original form but with transformed coupling parameters, while in certain special cases all parameters are essentially unchanged. Using a particular regularization we show, to all orders of perturbation theory, that this duality also persists at the quantum level. We also point out various other properties of this class of noncommutative field theories.

We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the Groenewold-Moyal star-product. Explicit results are presented for all Green's functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed non-perturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinite-dimensional noncommutative algebra.

A detailed derivation of a two dimensional (2D) low energy effective model for spinless fermions on a square lattice with local interactions is given. This derivation utilizes a particular continuum limit that is justified by physical arguments. It is shown that the effective model thus obtained can be treated by exact bosonization methods. It is also discussed how this effective model can be used to obtain physical information about the corresponding lattice fermion system.

We study the time evolution of a 1D interacting fermion system described by the Luttinger model starting from a non-equilibrium state defined by a smooth temperature profile T (x). As a specific example we consider the case when T (x) is equal to T L (T R ) far to the left (right). Using a series expansion in = 2(T R − T L ) (T L + T R ), we compute the energy density, the heat current density, and the fermion two-point correlation function for all times t ≥ 0. For local (delta-function) interaction, the first two are computed to all orders giving simple exact expressions involving the Schwarzian derivative of the integral of T (x). For non-local interaction, breaking scale invariance, we compute the non-equilibrium steady state (NESS) to all orders and the evolution to first order in . The heat current in the NESS is universal even when conformal invariance is broken by the interaction, and its dependence on T L,R agrees with numerical results for the XXZ spin chain. Moreover, our analytical formulas predict peaks at short times in the transition region between different temperatures and show dispersion effects that, even if non-universal, are qualitatively similar to ones observed in numerical simulations for related models, such as spin chains and interacting lattice fermions.