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Raimar Wulkenhaar

Publications and source records attributed to Raimar Wulkenhaar.

At least 55 records · Page 3Linked to original sources
arXiv2012

A numerical approach to harmonic non-commutative spectral field theory

We present a first numerical investigation of a non-commutative gauge theory defined via the spectral action for Moyal space with harmonic propagation. This action is approximated by finite matrices. Using Monte Carlo simulation we study various quantities such as the energy density, the specific heat density and some order parameters, varying the matrix size and the independent parameters of the model. We find a peak structure in the specific heat which might indicate possible phase transitions. However, there are mathematical arguments which show that the limit of infinite matrices is very different from the original spectral model.

math-ph↗
arXiv2011

8D-spectral triple on 4D-Moyal space and the vacuum of noncommutative gauge theory

Observing that the Hamiltonian of the renormalisable scalar field theory on 4-dimensional Moyal space A is the square of a Dirac operator D of spectral dimension 8, we complete (A,D) to a compact 8-dimensional spectral triple. We add another Connes-Lott copy and compute the spectral action of the corresponding U(1)-Yang-Mills-Higgs model. We find that in the Higgs potential the square ϕ^2 of the Higgs field is shifted to ϕ* ϕ+ const X_μ* X^μ, where X_μis the covariant coordinate. The classical field equations of our model imply that the vacuum is no longer given by a constant Higgs field, but both the Higgs and gauge fields receive non-constant vacuum expectation values.

hep-th↗
arXiv2011

Spectral geometry of the Moyal plane with harmonic propagation

We construct a `non-unital spectral triple of finite volume' out of the Moyal product and a differential square root of the harmonic oscillator Hamiltonian. We find that the spectral dimension of this triple is d but the KO-dimension is 2d. We add another Connes-Lott copy and compute the spectral action of the corresponding U(1)-Yang-Mills-Higgs model. We find that the `covariant coordinate' involving the gauge field combines with the Higgs field to a unified potential, yielding a deep unification of discrete and continuous parts of the geometry.

math.OA↗
arXiv2009

Progress in solving a noncommutative quantum field theory in four dimensions

We study the noncommutative ϕ^4_4-quantum field theory at the self-duality point. This model is renormalisable to all orders as shown in earlier work of us and does not have a Landau ghost problem. Using the Ward identity of Disertori, Gurau, Magnen and Rivasseau, we obtain from the Schwinger-Dyson equation a non-linear integral equation for the renormalised two-point function alone. The non-trivial renormalised four-point function fulfils a linear integral equation with the inhomogeneity determined by the two-point function. These integral equations are the starting point for a perturbative solution. In this way, the renormalised correlation functions are directly obtained, without Feynman graph computation and further renormalisation steps

hep-th↗
arXiv2009

Non-compact spectral triples with finite volume

In order to extend the spectral action principle to non-compact spaces, we propose a framework for spectral triples where the algebra may be non-unital but the resolvent of the Dirac operator remains compact. We show that an example is given by the supersymmetric harmonic oscillator which, interestingly, provides two different Dirac operators. This leads to two different representations of the volume form in the Hilbert space, and only their product is the grading operator. The index of the even-to-odd part of each of these Dirac operators is 1. We also compute the spectral action for the corresponding Connes-Lott two-point model. There is an additional harmonic oscillator potential for the Higgs field, whereas the Yang-Mills part is unchanged. The total Higgs potential shows a two-phase structure with smooth transition between them: In the spontaneously broken phase below a critical radius, all fields are massive, with the Higgs field mass slightly smaller than the NCG prediction. In the unbroken phase above the critical radius, gauge fields and fermions are massless, whereas the Higgs field remains massive.

hep-th↗
arXiv2008

On the vacuum states for noncommutative gauge theory

Candidates for renormalisable gauge theory models on Moyal spaces constructed recently have non trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in the position space for these vacuum configurations in two and four dimensions.

hep-th↗
arXiv2007

Noncommutative Induced Gauge Theory

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the one-loop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, 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 oscillator term, which for the noncommutative $ϕ^4$-theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

hep-th↗
arXiv2004

Power-counting theorem for non-local matrix models and renormalisation

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.

hep-th↗
arXiv2004

Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base

We prove that the real four-dimensional Euclidean noncommutative ϕ^4-model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R^4. The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

hep-th↗
arXiv2004

Renormalisation of ϕ^4-theory on noncommutative R^4 to all orders

We present the main ideas and techniques of the proof that the duality-covariant four-dimensional noncommutative ϕ^4-model is renormalisable to all orders. This includes the reformulation as a dynamical matrix model, the solution of the free theory by orthogonal polynomials as well as the renormalisation by flow equations involving power-counting theorems for ribbon graphs drawn on Riemann surfaces.

hep-th↗
arXiv2004

The β-function in duality-covariant noncommutative ϕ^4-theory

We compute the one-loop β-functions describing the renormalisation of the coupling constant λand the frequency parameter Ωfor the real four-dimensional duality-covariant noncommutative ϕ^4-model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the β_λ-function remains non-negative. Both β_λand β_Ωvanish at the one-loop level for the duality-invariant model characterised by Ω=1. Moreover, β_Ωalso vanishes in the limit Ω\to 0, which defines the standard noncommutative ϕ^4-quantum field theory. Thus, the limit Ω\to 0 exists at least at the one-loop level.

hep-th↗
arXiv2003

Renormalisation of ϕ^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 ϕ^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.

hep-th↗
arXiv2002

Quantum field theories on noncommutative R^4 versus theta-expanded quantum field theories

I recall the main motivation to study quantum field theories on noncommutative spaces and comment on the most-studied example, the noncommutative R^4. That algebra is given by the *-product which can be written in (at least) two ways: in an integral form or an exponential form. These two forms of the *-product are adapted to different classes of functions, which, when using them to formulate field theory, lead to two versions of quantum field theories on noncommutative R^4. The integral form requires functions of rapid decay and a (preferably smooth) cut-off in the path integral, which therefore should be evaluated by exact renormalisation group methods. The exponential form is adapted to analytic functions with arbitrary behaviour at infinity, so that Feynman graphs can be used to compute the path integral (without cut-off) perturbatively.

hep-th↗
arXiv2002

Quantisation of θ-expanded non-commutative QED

We analyse two new versions of θ-expanded non-commutative quantum electrodynamics up to first order in θand first loop order. In the first version we expand the bosonic sector using the Seiberg-Witten map, leaving the fermions unexpanded. In the second version we leave both bosons and fermions unexpanded. The analysis shows that the Seiberg-Witten map is a field redefinition at first order in θ. However, at higher order in θthe Seiberg-Witten map cannot be regarded as a field redefinition. We find that the initial action of any θ-expanded massless non-commutative QED must include one extra term proportional to θwhich we identify by loop calculations.

hep-th↗
arXiv2002

Non-renormalizability of θ-expanded noncommutative QED

Computing all divergent one-loop Green's functions of θ-expanded noncommutative quantum electrodynamics up to first order in θ, we show that this model is not renormalizable. The reason is a divergence in the electron four-point function which cannot be removed by field redefinitions. Ignoring this problem, we find however clear hints for new symmetries in massless θ-expanded noncommutative QED: Four additional divergences which would be compatible with gauge and Lorentz symmetries and which are not reachable by field redefinitions are absent.

hep-th↗
arXiv2001

Renormalization of the noncommutative photon self-energy to all orders via Seiberg-Witten map

We show that the photon self-energy in quantum electrodynamics on noncommutative $\mathbb{R}^4$ is renormalizable to all orders (both in $θ$ and $\hbar$) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the $θ$-deformed classical action which are necessary to compensate the divergences coming from loop integrations.

hep-th↗
arXiv2000

Renormalization of noncommutative Yang-Mills theories: A simple example

We prove by explicit calculation that Feynman graphs in noncommutative Yang-Mills theory made of repeated insertions into itself of arbitrarily many one-loop ghost propagator corrections are renormalizable by local counterterms. This provides a strong support for the renormalizability conjecture of that model.

hep-th↗