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H. Grosse

Publications and source records attributed to H. Grosse.

At least 19 recordsLinked to original sources
arXiv2006

A nontrivial solvable noncommutative ϕ^3 model in 4 dimensions

We study the quantization of the noncommutative selfdual ϕ^3 model in 4 dimensions, by mapping it to a Kontsevich model. The model is shown to be renormalizable, provided one additional counterterm is included compared to the 2-dimensional case which can be interpreted as divergent shift of the field ϕ. The known results for the Kontsevich model allow to obtain the genus expansion of the free energy and of any n-point function, which is finite for each genus after renormalization. No coupling constant or wavefunction renormalization is required. A critical coupling is determined, beyond which the model is unstable. This provides a nontrivial interacting NC field theory in 4 dimensions.

hep-th↗
arXiv2005

Renormalization of the noncommutative phi^3 model through the Kontsevich model

We point out that the noncommutative selfdual phi^3 model can be mapped to the Kontsevich model, for a suitable choice of the eigenvalues in the latter. This allows to apply known results for the Kontsevich model to the quantization of the field theory, in particular the KdV flows and Virasoro constraints. The 2-dimensional case is worked out explicitly. We obtain nonperturbative expressions for the genus expansion of the free energy and some n-point functions. The full renormalization for finite coupling is found, which is determined by the genus 0 sector only. All contributions in a genus expansion of any n-point function are finite after renormalization. A critical coupling is determined beyond which the model is unstable. The model is free of UV/IR diseases.

hep-th↗
arXiv2003

Space/time noncommutative field theories and causality

As argued previously, amplitudes of quantum field theories on noncommutative space and time cannot be computed using naive path integral Feynman rules. One of the proposals is to use the Gell-Mann--Low formula with time-ordering applied before performing the integrations. We point out that the previously given prescription should rather be regarded as an interaction point time-ordering. Causality is explicitly violated inside the region of interaction. It is nevertheless a consistent procedure, which seems to be related to the interaction picture of quantum mechanics. In this framework we compute the one-loop self-energy for a space/time noncommutative ϕ^4 theory. Although in all intermediate steps only three-momenta play a role, the final result is manifestly Lorentz covariant and agrees with the naive calculation. Deriving the Feynman rules for general graphs, we show, however, that such a picture holds for tadpole lines only.

hep-th↗
arXiv2003

Non-commutative U(1) Super-Yang-Mills Theory: Perturbative Self-Energy Corrections

The quantization of the non-commutative N=1, U(1) super-Yang-Mills action is performed in the superfield formalism. We calculate the one-loop corrections to the self-energy of the vector superfield. Although the power-counting theorem predicts quadratic ultraviolet and infrared divergences, there are actually only logarithmic UV and IR divergences, which is a crucial feature of non-commutative supersymmetric field theories.

hep-th↗
arXiv2002

The Landau-Problem on the θ-Deformed Two-Torus

We study the Landau-problem on the $θ$-deformed two-torus and use well-known projective modules to obtain perturbed spectra. For a strong magnetic field B the problem can be restricted to one particular Landau-level. First we represent generators of the algebra of the non-commutative torus as finite dimensional matrices. A second approach leads to a reducible representation with a $θ$-dependent center. For a simple periodic potential, the rational part of the Hofstadter-butterfly spectrum is obtained.

quant-ph↗
arXiv2002

IR-Singularities in Noncommutative Perturbative Dynamics?

We analyse the IR-singularities that appear in a noncommutative scalar quantum field theory on $\mathcal{E}_4$. We demonstrate with the help of the quadratic one-loop effective action and an appropriate field redefinition that no IR-singularities exist. No new degrees of freedom are needed to describe the UV/IR-mixing.

hep-th↗
arXiv2001

Noncommutative spin-1/2 representations

In this letter we apply the methods of our previous paper hep-th/0108045 to noncommutative fermions. We show that the fermions form a spin-1/2 representation of the Lorentz algebra. The covariant splitting of the conformal transformations into a field-dependent part and a θ-part implies the Seiberg-Witten differential equations for the fermions.

hep-th↗
arXiv2001

Noncommutative Lorentz Symmetry and the Origin of the Seiberg-Witten Map

We show that the noncommutative Yang-Mills field forms an irreducible representation of the (undeformed) Lie algebra of rigid translations, rotations and dilatations. The noncommutative Yang-Mills action is invariant under combined conformal transformations of the Yang-Mills field and of the noncommutativity parameter θ. The Seiberg-Witten differential equation results from a covariant splitting of the combined conformal transformations and can be computed as the missing piece to complete a covariant conformal transformation to an invariance of the action.

hep-th↗
arXiv2001

Field Theory on the q-deformed Fuzzy Sphere II: Quantization

We study the second quantization of field theory on the q-deformed fuzzy sphere for real q. This is performed using a path-integral over the modes, which generate a quasiassociative algebra. The resulting models have a manifest U_q(su(2)) symmetry with a smooth limit q -> 1, and satisfy positivity and twisted bosonic symmetry properties. A systematic way to calculate n-point correlators in perturbation theory is given. As examples, the 4-point correlator for a free scalar field theory and the planar contribution to the tadpole diagram in ϕ^4 theory are computed. The case of gauge fields is also discussed, as well as an operator formulation of scalar field theory in 2_q + 1 dimensions. An alternative, essentially equivalent approach using associative techniques only is also presented. The proposed framework is not restricted to 2 dimensions.

hep-th↗
arXiv2000

The Superfield Formalism Applied to the Noncommutative Wess-Zumino Model

We introduce the notion of superoperators on noncommutative R^4 and re-investigate in the framework of superfields the noncommutative Wess-Zumino model as a quantum field theory. In a highly efficient manner we are able to confirm the result that this model is renormalizable to all orders.

hep-th↗
arXiv1999

Strong Connections and Chern-Connes Pairing in the Hopf-Galois Theory

We reformulate the concept of connection on a Hopf-Galois extension $B\subseteq P$ in order to apply it in computing the Chern-Connes pairing between the cyclic cohomology $HC^{2n} (B)$ and $K_0 (B)$. This reformulation allows us to show that a Hopf-Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz-Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern-Connes pairing for the super line bundles associated to super Hopf fibration.

math.QA↗
arXiv1999

Some properties of the one-dimensional generalized point interactions (a torso)

This text is a part of an unfinished project which deals with the generalized point interaction (GPI) in one dimension. We employ two natural parametrizations, which are known but have not attracted much attention, to express the resolvent of the GPI Hamiltonian as well as its spectral and scattering properties. It is also shown that the GPI yields one of the simplest models in which a non-trivial Berry phase is exhibited. Furthermore, the generalized Kronig-Penney model corresponding to the GPI is discussed. We show that there are three different types of the high-energy behaviour for the corresponding band spectrum.

math-ph↗
arXiv1999

Generalized functions for quantum fields obeying quadratic exchange relations

The axiomatic formulation of quantum field theory (QFT) of the 1950's in terms of fields defined as operator valued Schwartz distributions is re-examined in the light of subsequent developments. These include, on the physical side, the construction of a wealth of (2-dimensional) soluble QFT models with quadratic exchange relations, and, on the mathematical side, the introduction of the Colombeau algebras of generalized functions. Exploiting the fact that energy positivity gives rise to a natural regularization of Wightman distributions as analytic functions in a tube domain, we argue that the flexible notions of Colombeau theory which can exploit particular regularizations is better suited (than Schwartz distributions) for a mathematical formulation of QFT.

math-ph↗
arXiv1998

A treatment of the Schwinger Model within Noncommutative Geometry

A free spinor field on a noncommutative sphere is described starting from a canonical realization of the enveloping algebra U(u(2|1)). The gauge extension of the model - the Schwinger model on a noncommutative sphere is defined and the model is quantized. The model contains only finite number degrees of freedom and is nonperturbatively UV-regular. The chiral anomaly and the effective actions are calculated. In the nomcommutative limit standard formulas are recovered.

hep-th↗
arXiv1996

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.

hep-th↗
arXiv1996

On Finite 4D Quantum Field Theory in Non-Commutative Geometry

The truncated 4-dimensional sphere $S^4$ and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping the SO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.

hep-th↗