The Luttinger-Schwinger Model

We study the Luttinger-Schwinger model, i.e. the (1+1) dimensional model of massless Dirac fermions with a non-local 4-point interaction coupled to a U(1)-gauge field. The complete solution of the model is found using the boson-fermion correspondence, and the formalism for calculating all gauge invariant Green functions is provided. We discuss the role of anomalies and show how the existence of large gauge transformations implies a fermion condensate in all physical states. The meaning of regularization and renormalization in our well-defined Hilbert space setting is discussed. We illustrate the latter by performing the limit to the Thirring-Schwinger model where the interaction becomes local.Comment: 19 pages, Latex, to appear in Annals of Physics, download problems fixe

Two and Three Loops Beta Function of Non Commutative Φ44\Phi^4_4 Theory

The simplest non commutative renormalizable field theory, the $\phi_4^4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe at one loop, as shown by H. Grosse and R. Wulkenhaar. We extend this result up to three loops. If this remains true at any loop, it should allow a full non perturbative construction of this model.Comment: 24 pages, 7 figure

Induced Gauge Theory on a Noncommutative Space

We consider a scalar $\phi^4$ theory on canonically deformed Euclidean space in 4 dimensions with an additional oscillator potential. This model is known to be renormalisable. An exterior gauge field is coupled in a gauge invariant manner to the scalar field. We extract the dynamics for the gauge field from the divergent terms of the 1-loop effective action using a matrix basis and propose an action for the noncommutative gauge theory, which is a candidate for a renormalisable model.Comment: Typos corrected, one reference added; eqn. (49) corrected, one equation number added; 30 page

Renormalization of Non-Commutative Phi^4_4 Field Theory in x Space

In this paper we provide a new proof that the Grosse-Wulkenhaar non-commutative scalar Phi^4_4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler and could be more adapted to the future study of these theories (in particular at the non-perturbative or constructive level).Comment: 32 pages, v2: correction of lemmas 3.1 and 3.2 with no consequence on the main resul

Renormalisation of \phi^4-theory on noncommutative R^2 in the matrix base

As a first application of our renormalisation group approach to non-local matrix models [hep-th/0305066], we prove (super-)renormalisability of Euclidean two-dimensional noncommutative \phi^4-theory. It is widely believed that this model is renormalisable in momentum space arguing that there would be logarithmic UV/IR-divergences only. Although momentum space Feynman graphs can indeed be computed to any loop order, the logarithmic UV/IR-divergence appears in the renormalised two-point function -- a hint that the renormalisation is not completed. In particular, it is impossible to define the squared mass as the value of the two-point function at vanishing momentum. In contrast, in our matrix approach the renormalised N-point functions are bounded everywhere and nevertheless rely on adjusting the mass only. We achieve this by introducing into the cut-off model a translation-invariance breaking regulator which is scaled to zero with the removal of the cut-off. The naive treatment without regulator would not lead to a renormalised theory.Comment: 26 pages, 44 figures, LaTe

Noncommutative Induced Gauge Theories on Moyal Spaces

Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4-D of the one-loop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative $\phi^4$-theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.Comment: 24 pages, 6 figures. Talk given at the "International Conference on Noncommutative Geometry and Physics", April 2007, Orsay (France). References updated. To appear in J. Phys. Conf. Se

Geometry of the Grosse-Wulkenhaar Model

We define a two-dimensional noncommutative space as a limit of finite-matrix spaces which have space-time dimension three. We show that on such space the Grosse-Wulkenhaar (renormalizable) action has natural interpretation as the action for the scalar field coupled to the curvature. We also discuss a natural generalization to four dimensions.Comment: 16 pages, version accepted in JHE

Generalized local interactions in 1D: solutions of quantum many-body systems describing distinguishable particles

As is well-known, there exists a four parameter family of local interactions in 1D. We interpret these parameters as coupling constants of delta-type interactions which include different kinds of momentum dependent terms, and we determine all cases leading to many-body systems of distinguishable particles which are exactly solvable by the coordinate Bethe Ansatz. We find two such families of systems, one with two independent coupling constants deforming the well-known delta interaction model to non-identical particles, and the other with a particular one-parameter combination of the delta- and (so-called) delta-prime interaction. We also find that the model of non-identical particles gives rise to a somewhat unusual solution of the Yang-Baxter relations. For the other model we write down explicit formulas for all eigenfunctions.Comment: 23 pages v2: references adde

Two Color Entanglement

We report on the generation of entangled states of light between the wavelengths 810 and 1550 nm in the continuous variable regime. The fields were produced by type I optical parametric oscillation in a standing-wave cavity build around a periodically poled potassium titanyl phosphate crystal, operated above threshold. Balanced homodyne detection was used to detect the non-classical noise properties, while filter cavities provided the local oscillators by separating carrier fields from the entangled sidebands. We were able to obtain an inseparability of I=0.82, corresponding to about -0.86 dB of non-classical quadrature correlation.Comment: 4 pages, 2 figure

UV/IR duality in noncommutative quantum field theory

We review the construction of renormalizable noncommutative euclidean phi(4)-theories based on the UV/IR duality covariant modification of the standard field theory, and how the formalism can be extended to scalar field theories defined on noncommutative Minkowski space.Comment: 12 pages; v2: minor corrections, note and references added; Contribution to proceedings of the 2nd School on "Quantum Gravity and Quantum Geometry" session of the 9th Hellenic School on Elementary Particle Physics and Gravity, Corfu, Greece, September 13-20 2009. To be published in General Relativity and Gravitatio