Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Gravity coupled with matter and the foundation of non-commutative geometry

We first exhibit in the commutative case the simple algebraic relations between the algebra of functions on a manifold and its infinitesimal length elementds. Its unitary representations correspond to Riemannian metrics and Spin structure whileds is the Dirac propagatords=x−x=D ⁻¹, whereD is the Dirac operator. We extend these simple relations to the non-commutative case using Tomita's involutionJ. We then write a spectral action, the trace of a function of the length element, which when applied to the non-commutative geometry of the Standard Model will be shown ([CC]) to give the SM Lagrangian coupled to gravity. The internal fluctuations of the non-commutative geometry are trivial in the commutative case but yield the full bosonic sector of SM with all correct quantum numbers in this slightly non-commutative case. The group of local gauge transformations appears spontaneously as a normal subgroup of the diffeomorphism group.     

Differential algebras in non-commutative geometry

We discuss the differential algebras used in Connes' approach to Yang-Mills theories with spontaneous symmetry breaking. These differential algebras generated by algebras of the form functions ⊗ matrix are shown to be skew tensor products of differential algebras with a specific matrix algebra. For that we derive a general formula for differential algebras based on tensor products of algebras. The result is used to characterize differential algebras which appear in models with one symmetry breaking scale.     

Ward identities for non-commutative geometry

We interpret the cocycle condition in quantum field theory as a set of integrated Ward identities for non-commutative geometry.Dedicated to Res Jost and Arthur WightmanSupported in part by the National Science Foundation under Grants DMS/PHY 88-16214 and INT 87-22044     

The action functional in non-commutative geometry

We establish the equality between the restriction of the Adler-Manin-Wodzicki residue or non-commutative residue to pseudodifferential operators of order –n on ann-dimensional compact manifoldM, with the trace which J. Dixmier constructed on the Macaev ideal. We then use the latter trace to recover the Yang Mills interaction in the context of non-commutative differential geometry.     

CMB constraints on non-commutative geometry during inflation

We investigate the primordial power spectrum of the density perturbations based on the assumption that space is non-commutative in the early stage of inflation, and constrain the contribution from non-commutative geometry using CMB data. Due to the non-commutative geometry, the primordial power spectrum can lose rotational invariance. Using the k-inflation model and slow-roll approximation, we show that the deviation from rotational invariance of the primordial power spectrum depends on the size of non-commutative length scale L _s but not on sound speed. We constrain the contributions from the non-commutative geometry to the covariance matrix of the harmonic coefficients of the CMB anisotropies using five-year WMAP CMB maps. We find that the upper bound for L _s depends on the product of sound speed and slow-roll parameter. Estimating this product using cosmological parameters from the five-year WMAP results, the upper bound for L _s is estimated to be less than 10^?27 cm at 99.7% confidence level.     

On finite 4D quantum field theory in non-commutative geometry

The truncated 4-dimensional sphereS ⁴ and the action of the self-interacting scalar field on it are constructed. The path integral quantization is performed while simultaneously keeping theSO(5) symmetry and the finite number of degrees of freedom. The usual field theory UV-divergences are manifestly absent.Participating in Project No. P8916-PHY of the Fonds zur Förderung der wissenschaftlichen Forschung in ÖsterreichPartially supported by the grant GAR 210/96/0310     

Muong — 2 measurements and non-commutative geometry of quantum beams

We discuss a completely quantum mechanical treatment of the measurement of the anomalous magnetic moment of the muon. A beam of muons move in a strong uniform magnetic field and a weak focusing electrostatic field. Errors in the classical beam analysis are exposed. In the Dirac quantum beam analysis, an important role is played by non-commutative muon beam coordinates leading to a discrepancy between the classical and quantum theories. We obtain a quantum limit to the accuracy achievable in BNL type experiments. Some implications of the quantum corrected data analysis for supersymmetry are briefly mentioned.     

Conformally symmetric wormhole solutions supported by non-commutative geometry in f(Q,T ) gravity

This paper investigates wormhole solutions within the framework of extended symmetric teleparallel gravity, incorporating non-commutative geometry, and conformal symmetries. To achieve this, we examine the linear wormhole model with anisotropic fluid under Gaussian and Lorentzian distributions. The primary objective is to derive wormhole solutions while considering the influence of the shape function on model parameters under Gaussian and Lorentzian distributions. The resulting shape function sa...     

Supersymmetry in the non-commutative plane

The supersymmetric extension of a model introduced by Lukierski, Stichel, and Zakrewski in the non-commutative plane is studied. The Noether charges associated to the symmetries are determined. Their Poisson algebra is investigated in the Ostrogradski-Dirac formalism for constrained Hamiltonian systems. It is shown to provide a supersymmetric generalization of the Galilei algebra with a two-dimensional central extension.     

Graded non-commutative geometries

We present a short exposition of graded finite non-commutative geometries. The theory that serves as an example is based on the algebra of matrices M_n . This non-commutative algebra replaces the algebra of functions on a manifold. Consequently, vector fields (differentiations), forms and connections are constructed. The gauge theory can be introduced without the notion of internal manifold. We discuss some physical application, the similarities with the standard model, and the graded version of this geometry.     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space.The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity.First,new Poisson brackets have been defined in non-commutative phase space.They contain corrections due to the non-commutativity of coordinates and momenta.On the basis of this new Poisson brackets,a new modified second law of Newton has been obtained.For two cases,the free particle and the harmonic oscillator,the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys.Rev.D,2005,72:025010).The consistency between both methods is demonstrated.It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space.but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

Non-hermitian quantum mechanics in non-commutative space

A recent investigation of the possibility of having a -symmetric periodic potential in an optical lattice stimulated the urge to generalize non-hermitian quantum mechanics beyond the case of commutative space. We thus study non-hermitian quantum systems in non-commutative space as well as a -symmetric deformation of this space. Specifically, a -symmetric harmonic oscillator together with an iC(x ₁+x ₂) interaction are discussed in this space, and solutions are obtained. We show that in the deformed non-commutative space the Hamiltonian may or may not possess real eigenvalues, depending on the choice of the non-commutative parameters. However, it is shown that in standard non-commutative space, the iC(x ₁+x ₂) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not -symmetric. A complex interacting anisotropic oscillator system also is discussed.     

Some results in non-commutative ergodic theory

We study some properties of invariant states on aC*-algebraA with a groupG of automorphisms. Using the concept ofG-factorial state, which is a non-commutative generalization of the concept of ergodic measure, in general wider in scope thanG-ergodic state, we show that under a certain abelianity condition on (A,G), which in particular holds for the quasi-local algebras used in statistical mechanics, two differentG-ergodic states are disjoint. We also define the concept ofG-factorial linear functional, and show that under the same abelianity condition such a functional is proportional to aG-ergodic state. This generalizes an earlier result for complex ergodic measures.     

A non-commutative binomial formula

We prove a binomial formula for variables x and y satisfying a quadratic relation xy = ax² + qyx + by². Such relations are important in quantum group theory and non-commutative geometry.     

On non-commutative geodesic motion

In this work we study the geodesic motion on a noncommutative space–time. As a result we find a non-commutative geodesic equation and then we derive corrections of the deviation angle per revolution in terms of the non-commutative parameter when we specify the problem of Mercury’s perihelion. In this way, we estimate the noncommutative parameter based in experimental data.     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity. First, new Poisson brackets have been defined in non-commutative phase space. They contain corrections due to the non-commutativity of coordinates and momenta. On the basis of this new Poisson brackets, a new modified second law of Newton has been obtained. For two cases, the free particle and the harmonic oscillator, the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys. Rev. D, 2005, 72: 025010). The consistency between both methods is demonstrated. It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space, but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

Chiral decomposition in the non-commutative Landau problem

The decomposition of the non-commutative Landau (NCL) system into two uncoupled one-dimensional chiral components, advocated by Alvarez et al. (2008) [1], is generalized to nonvanishing electric fields. This allows us to discuss the main properties of the NCL problem including its exotic Newton–Hooke symmetry and its relation to the Hall effect. The “phase transition” when the magnetic field crosses a critical value determined by the non-commutative parameter is studied in detail.     

Quantized equations of motion in non-commutative theories

Quantum field theories based on interactions which contain the Moyal star product suffer, in the general case when time does not commute with space, from several diseases: quantum equation of motions contain unusual terms, conserved currents cannot be defined and the residual spacetime symmetry is not maintained. All these problems have the same origin: time ordering does not commute with taking the star product. Here we show that these difficulties can be circumvented by a new definition of time ordering: namely with respect to a light-cone variable. In particular the original spacetime symmetries and translation invariance turn out to be respected. Unitarity is guaranteed as well.Received: 12 January 2005, Revised: 16 March 2005, Published online: 18 May 2005