Moyal Space Research Articles

Photon scattering by an electric field in noncommutative spacetime

As is known, the existence of a small noncommutativity between coordinates would generate nonlocal self-interactions in the electromagnetic theory. To explore some consequences of this effect on the propagation of photons we consider Moyal space half-filled with a static and homogeneous electric field and analyze electromagnetic fluctuations on top of this step-like background. Both the localization of photons and the possibility of photon production by strong electric fields are addressed. Several aspects of the Klein paradox in this setup are discussed as well.

On the UV/IR mixing of Lie algebra-type noncommutatitive ϕ4-theories

We show that a UV divergence of the propagator integral implies the divergences of the UV/IR mixing in the two-point function at one-loop for a ϕ4-theory on a generic Lie algebra-type noncommutative space-time. The UV/IR mixing is defined as a UV divergence of the planar contribution and an IR singularity of the non-planar contribution, the latter being due to the former UV divergence, and the UV finiteness of the non-planar contribution. Some properties of this general treatment are discussed. The UV finiteness of the non-planar contribution and the renormalizability of the theory are not treated but commented. Applications are performed for the Moyal space, having a UV/IR mixing, and the κ-Minkowski space for which the two-point function at one-loop is finite.

Exact Solutions v.s. Perturbative Calculations of Finite Φ3-Φ4 Hybrid-Matrix-Model

There is a matrix model corresponding to a scalar field theory on noncommutative spaces called Grosse-Wulkenhaar model (Φ4 matrix model), which is renormalizable by adding a harmonic oscillator potential to scalar Φ4 theory on Moyal spaces. There are more unknowns in Φ4 matrix model than in Φ3 matrix model, for example, in terms of integrability. We then construct a one-matrix model (Φ3-Φ4 Hybrid-Matrix-Model) with multiple potentials, which is a combination of a 3-point interaction and a 4-point interaction, where the 3-point interaction of Φ3 is multiplied by some positive definite diagonal matrix M. This model is solvable due to the effect of this M. In particular, the connected ∑i=1BNi-point function G|aN11⋯aN11|⋯|a1B⋯aNBB| of Φ3-Φ4 Hybrid-Matrix-Model is studied in detail. This ∑i=1BNi-point function can be interpreted geometrically and corresponds to the sum over all Feynman diagrams (ribbon graphs) drawn on Riemann surfaces with B boundaries (punctures). Each |a1i⋯aNii| represents Ni external lines coming from the i-th boundary (puncture) in each Feynman diagram. First, we construct Feynman rules for Φ3-Φ4 Hybrid-Matrix-Model and calculate perturbative expansions of some multipoint functions in ordinary methods. Second, we calculate the path integral of the partition function Z[J] and use the result to compute exact solutions for 1-point function G|a| with 1-boundary, 2-point function G|ab| with 1-boundary, 2-point function G|a|b| with 2-boundaries, and n-point function G|a1|a2|⋯|an| with n-boundaries. They include contributions from Feynman diagrams corresponding to nonplanar Feynman diagrams or higher genus surfaces.

Revisiting NCQED and scattering amplitudes

Research progress on the noncommutative gauge theories on the Moyal space is discussed in this minireview. We first present a brief overview on the development of gauge theories on Moyal space, with an emphasis on the role of Seiberg–Witten maps. Two important relations induced by reversible Seiberg–Witten maps, namely the formal equivalence of the on-shell DeWitt background field effective action in general and the explicit identical relation between tree-level scattering amplitudes in noncommutative quantum electrodynamics (NCQED), are described in some detail. We then proceed to the properties of the tree-level two-by-two scattering amplitudes in NCQED, including a forward scattering singularity in NCQED Compton scattering. After covering some phenomenological perspectives of noncommutative Yang–Mills (NCYM)-based models, outlooks for the future are given at the end.

Solution of the self-dual \u03a64 QFT-model on four-dimensional Moyal space

Previously the exact solution of the planar sector of the self-dual Φ4-model on 4-dimensional Moyal space was established up to the solution of a Fredholm integral equation. This paper solves, for any coupling constant λ > − frac{1}{uppi} , the Fredholm equation in terms of a hypergeometric function and thus completes the construction of the planar sector of the model. We prove that the interacting model has spectral dimension 4 − 2 frac{arcsin left(uplambda uppi right)}{uppi} for |λ| < frac{1}{uppi} . It is this dimension drop which for λ > 0 avoids the triviality problem of the matricial {varPhi}_4^4 -model. We also establish the power series approximation of the Fredholm solution to all orders in λ. The appearing functions are hyperlogarithms defined by iterated integrals, here of alternating letters 0 and −1. We identify the renormalisation parameter which gives the same normalisation as the ribbon graph expansion.

How Prof. Zeidler Supported Our Research on Exact Solution of Quantum Field Theory Toy Models

Over many years, we developed the construction of the ϕ4-model on four-dimensional Moyal space. The solution of the related matrix model mathcal {Z}[E,J]=int d{Phi } exp (text {tr}(J{Phi }- E{Phi }^{2} -frac {lambda }{4} {Phi }^{4})) is given in terms of the solution of a non-linear equation for the 2-point function and the eigenvalues of E. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. Locality is fulfilled. The Schwinger 2-point function is reflection positive in special cases.

Integrability and positivity in quantum field theory on noncommutative geometry

We review a sequence of papers in which we construct the λϕ4⋆4-model and the λϕ2,4,6⋆3-models on noncommutative Moyal space by a common method. Thereby we show that not only the Kontsevich model λΦ3 but also the λΦ44-model is integrable in a certain scaling limit which corresponds to infinitely large Moyal deformation parameter. Surprisingly, this limit gives rise to Schwinger functions on commutative Euclidean space. Our explicit formulae permit us to discuss reflection positivity of these Schwinger functions.

Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model

We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm Tr}\ (-\Delta M^2-\frac{\lambda}{4}M^4)$. The integral is over the space of Hermitian $(\Lambda+1)\times(\Lambda+1)$ matrices, the matrix $\Delta$, which is not a multiple of the identity matrix, encodes the dynamics and $\lambda>0$ is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series, hence is a well defined analytic function of the coupling constant in certain analytic domain of $\lambda$, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual $\phi^4$ theory on the 2 dimensional Moyal space, also called the 2 dimensional Grosse-Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.

The Gribov problem in noncommutative gauge theory

After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.

The [formula omitted] and [formula omitted] matricial QFT models have reflection positive two-point function

We extend our previous work (on D=2) to give an exact solution of the ΦD3 large-N matrix model (or renormalised Kontsevich model) in D=4 and D=6 dimensions. Induction proofs and the difficult combinatorics are unchanged compared with D=2, but the renormalisation – performed according to Zimmermann – is much more involved. As main result we prove that the Schwinger 2-point function resulting from the ΦD3-QFT model on Moyal space satisfies, for real coupling constant, reflection positivity in D=4 and D=6 dimensions. The Källén–Lehmann mass spectrum of the associated Wightman 2-point function describes a scattering part |p|2≥2μ2 and an isolated broadened mass shell around |p|2=μ2.

Revisiting quantum mechanics on non-commutative space–time

We construct an effective commutative Schrödinger equation in Moyal space–time in (1+1)-dimension where both t and x are operator-valued and satisfy tˆ,xˆ=iθ. Beginning with a time-reparametrised invariant form of an action we identify the actions of various space–time coordinates and their conjugate momenta on quantum states, represented by Hilbert–Schmidt operators. Since time is also regarded as a configuration space variable, we show how an ‘induced’ inner product can be extracted, so that an appropriate probabilistic interpretation is obtained. We then discuss several other applications of the formalism developed so far.

Photon self-interaction on deformed spacetime

A novel nonlocal four-photon interaction on the deformed spacetime is derived and studied in the three selected models (I, II, III). The first two models (I, II) are obtained via two distinct second-order $\theta$-exact Seiberg-Witten maps of the noncommutative U(1) gauge field strength on Moyal space. The third one (III), inspired by the manifestly gauge invariant structures emerging in the first two, due to the model generality has been constructed with a different set of freedom parameters. The physical relevancy of all models is analyzed by evaluating the four-photon-tadpole diagram, which, when combined with the bubble graph, enables us to fully consider all contributions to the one-loop photon polarization tensor. For an arbitrary noncommutative matrix theta^\mu\nu, the full quadratic IR divergence cancellation in the one-loop photon two-point function is obtained with particular combinations of Seiberg-Witten map/gauge-symmetry freedom parameters in models I and II. Finally, our model III enables complete elimination of all pathological terms in the photon polarization tensor at one-loop order if a special value for the theta matrix is chosen.

On the Fixed Point Equation of a Solvable 4D QFT Model

The regularisation of the \({\lambda {\upphi }^{4}_{4}}\)-model on noncommutative Moyal space gives rise to a solvable QFT model in which all correlation functions are expressed in terms of the solution of a fixed point problem. We prove that the non-linear operator for the logarithm of the original problem satisfies the assumptions of the Schauder fixed point theorem, thereby completing the solution of the QFT model.

θ-exact Seiberg-Witten maps at ordere3

We study two distinct $\ensuremath{\theta}$-exact Seiberg-Witten (SW) map expansions, (I) and (II), respectively, up to order ${e}^{3}$ for the gauge parameter, gauge field, and gauge field strengths of the noncommutative ${\mathrm{U}}_{\ensuremath{\star}}(1)$ gauge theory on the Moyal space. We derive explicitly the closed-form expression for the SW map ambiguity between the two and observe the emergence of several new totally commutative generalized star products. We also identify the additional gauge freedoms within each of the ${e}^{3}$-order field-strength expansions and define corresponding sets of deformation/ratio/weight parameters, $(\ensuremath{\kappa},{\ensuremath{\kappa}}_{i})$ and $(\ensuremath{\kappa},{\ensuremath{\kappa}}_{i}^{\ensuremath{'}})$, for these two SW maps, respectively.

On the energy-momentum tensor in Moyal space

We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the latter two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice-versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line.

Pair production of Dirac particles in a $$d+1$$ d + 1 -dimensional noncommutative space–time

This work addresses the computation of the probability of fermionic particle pair production in \((d+1)\)-dimensional noncommutative Moyal space. Using Seiberg–Witten maps, which establish relations between noncommutative and commutative field variables, up to the first order in the noncommutative parameter \(\theta \), we derive the probability density of vacuum–vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent, and space-dependent electric fields are considered and discussed.

Wong's Equations and Charged Relativistic Particles in Non-Commutative Space

In analogy to Wong's equations describing the motion of a charged relativistic point particle in the presence of an external Yang-Mills field, we discuss the motion of such a particle in non-commutative space subject to an external $U_\star(1)$ gauge field. We conclude that the latter equations are only consistent in the case of a constant field strength. This formulation, which is based on an action written in Moyal space, provides a coarser level of description than full QED on non-commutative space. The results are compared with those obtained from the different Hamiltonian approaches. Furthermore, a continuum version for Wong's equations and for the motion of a particle in non-commutative space is derived.

Noncommutative quantum field theory

AbstractWe summarize our recent construction of the ϕ4‐model on four‐dimensional Moyal space. This is achieved by solving the quartic matrix model for a general external matrix in terms of the solution of a non‐linear equation for the 2‐point function and the eigenvalues of that matrix. The β‐function vanishes identically. For the Moyal model, the theory of Carleman type singular integral equations reduces the construction to a fixed point problem. The resulting Schwinger functions in position space are symmetric and invariant under the full Euclidean group. The Schwinger 2‐point function is reflection positive iff the diagonal matrix 2‐point function is a Stieltjes function.

Gauge fields on non‐commutative spaces and renormalization

AbstractConstructing renormalizable models on non‐commutative spaces constitutes a big challenge. Only few examples of renormalizable theories are known, such as the scalar Grosse‐Wulkenhaar model. Gauge fields are even more difficult, since new renormalization techniques are required which are compatible with the inherently non‐local setting on the one hand, and also allow to properly treat the gauge symmetry on the other hand. In this proceeding (which is based on my talk given at the “Workshop on Noncommutative Field Theory and Gravity” in Corfu/Greece, September 8–15, 2013), I focus on this last point and present new extensions to existing renormalization schemes (which are known to work for gauge field theories in commutative space) adapted to non‐commutative Moyal space.

Self-Dual Noncommutative $${\phi^4}$$ ϕ 4 -Theory in Four Dimensions is a Non-Perturbatively Solvable and Non-Trivial Quantum Field Theory

We study quartic matrix models with partition function Z[E,J]=\int dM \exp(trace(JM-EM^2-(\lambda/4)M^4)). The integral is over the space of Hermitean NxN-matrices, the external matrix E encodes the dynamics, \lambda>0 is a scalar coupling constant and the matrix J is used to generate correlation functions. For E not a multiple of the identity matrix, we prove a universal algebraic recursion formula which gives all higher correlation functions in terms of the 2-point function and the distinct eigenvalues of E. The 2-point function itself satisfies a closed non-linear equation which must be solved case by case for given E. These results imply that if the 2-point function of a quartic matrix model is renormalisable by mass and wavefunction renormalisation, then the entire model is renormalisable and has vanishing \beta-function. As main application we prove that Euclidean \phi^4-quantum field theory on four-dimensional Moyal space with harmonic propagation, taken at its self-duality point and in the infinite volume limit, is exactly solvable and non-trivial. This model is a quartic matrix model, where E has for N->\infty the same spectrum as the Laplace operator in 4 dimensions. Using the theory of singular integral equations of Carleman type we compute (for N->\infty and after renormalisation of E,\lambda) the free energy density (1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linear integral equation. Existence of a solution is proved via the Schauder fixed point theorem. The derivation of the non-linear integral equation relies on an assumption which we verified numerically for coupling constants 0<\lambda\leq (1/\pi).