The standard model on non-commutative space-time

We consider the standard model on a non-commutative space and expand the action in the non-commutativity parameter . No new particles are introduced; the structure group is . We derive the leading order action. At zeroth order the action coincides with the ordinary standard model. At leading order in we find new vertices which are absent in the standard model on commutative space-time. The most striking features are couplings between quarks, gluons and electroweak bosons and many new vertices in the charged and neutral currents. We find that parity is violated in non-commutative QCD. The Higgs mechanism can be applied. QED is not deformed in the minimal version of the NCSM to the order considered. Received: 29 November 2001 / Published online: 25 January 2002     

Spherically symmetric non-commutative space: d=4

In order to find a non-commutative analog of Schwarzschild or Schwarzschild–de Sitter black hole we investigate spherically symmetric spaces generated by four non-commutative coordinates in the frame formalism. We present two solutions which, however, do not possess the prescribed commutative limit. Our analysis indicates that the appropriate non-commutative space might be found as a subspace of a higher-dimensional space.     

Some phenomenological consequences of the time-ordered perturbation theory of QED on non-commutative spacetime

A framework was recently proposed for doing perturbation theory on non-commutative (NC) spacetime. It preserves the unitarity of the S matrix and differs from the naive, popular approach already at the lowest order in perturbation when time does not commute with space. In this work, we investigate its phenomenological implications at linear colliders, especially the TESLA at DESY, through the processes of . We find that some NC effects computed previously are now modified and that there are new processes which now exhibit NC effects. Indeed, the first two processes get corrected at tree level as opposed to the null result in the naive approach, while the third one coincides with the naive result only in the low energy limit. The impact of the earth's rotation is incorporated. The NC signals are generally significant when the NC scale is comparable to the collider energy. If this is not the case, the non-trivial azimuthal angle distribution and day-night asymmetry of events due to Lorentz violation and the earth's rotation will be useful in identifying signals. We also comment briefly on the high energy behavior of the cross section that grows linearly in the center of mass energy squared and argue that it does not necessarily contradict some statements, e.g., the Froissart-Martin bound, achieved in ordinary theory. Received: 29 November 2002 / Revised version: 11 March 2003 / Published online: 5 May 2003 RID="a" ID="a" e-mail: liaoy@itp.uni-leipzig.de RID="b" ID="b" e-mail: dehne@itp.uni-leipzig.de     

Supersymmetric quantum mechanics on non-commutative space

We construct a supersymmetric quantum mechanics in terms of two real supercharges on non-commutative space in arbitrary dimensions. We obtain the exact eigenspectra of the two- and three-dimensional non-commutative superoscillators. We further show that a reduction in the phase space occurs for a critical surface in the space of parameters. At this critical surface, the energy spectrum of the bosonic sector is infinitely degenerate, while the degeneracy in the spectrum of the fermionic sector gets enhanced by a factor of two for each pair of reduced canonical coordinates. For the two-dimensional non-commutative “inverted superoscillator”, we find exact eigenspectra with a well-defined ground state for certain regions in the parameter space, which have no smooth limit to the ordinary commutative space.Received: 24 February 2005, Revised: 21 April 2005, Published online: 22 June 2005PACS: 03.65.-w, 03.65.Fd, 11.30.Pb, 11.10.Nx     

Non-commutative superstring worldsheet

In this paper we consider the worldsheet of the superstring as a non-commutative space. Some additional terms can be added to the superstring action, such that for an ordinary worldsheet they are zero. The expansion of this extended action up to the first order of the non-commutativity parameter leads to the new supersymmetric action for the string. For the closed superstring, we obtain the boundary state that describes a brane. From the open string point of view, the new boundary conditions on the worldsheet bosons generalize the non-commutativity of spacetime. Finally, we suggest some definitions for the non-commutativity parameter of the superstring worldsheet. Received: 19 April 2002 / Published online: 18 October 2002 RID="a" ID="a" e-mail: kamani@theory.ipm.ac.ir     

Non-singular Brans–Dicke collapse in deformed phase space

We study the collapse process of a homogeneous perfect fluid (in FLRW background) with a barotropic equation of state in Brans–Dicke (BD) theory in the presence of phase space deformation effects. Such a deformation is introduced as a particular type of non-commutativity between phase space coordinates. For the commutative case, it has been shown in the literature (Scheel, 1995), that the dust collapse in BD theory leads to the formation of a spacetime singularity which is covered by an event horizon. In comparison to general relativity (GR), the authors concluded that the final state of black holes in BD theory is identical to the GR case but differs from GR during the dynamical evolution of the collapse process. However, the presence of non-commutative effects influences the dynamics of the collapse scenario and consequently a non-singular evolution is developed in the sense that a bounce emerges at a minimum radius, after which an expanding phase begins. Such a behavior is observed for positive values of the BD coupling parameter. For large positive values of the BD coupling parameter, when non-commutative effects are present, the dynamics of collapse process differs from the GR case. Finally, we show that for negative values of the BD coupling parameter, the singularity is replaced by an oscillatory bounce occurring at a finite time, with the frequency of oscillation and amplitude being damped at late times.     

Born–Infeld–Chern–Simons theory: Hamiltonian embedding,duality and bosonization

In this paper we study in detail the equivalence of the recently introduced Born–Infeld self-dual model to the Abelian Born–Infeld–Chern–Simons model in 2+1 dimensions. We first apply the improved Batalin, Fradkin and Tyutin scheme, to embed the Born–Infeld self-dual model to a gauge system and show that the embedded model is equivalent to Abelian Born–Infeld–Chern–Simons theory. Next, using Buscher's duality procedure, we demonstrate this equivalence in a covariant Lagrangian formulation and also derive the mapping between the n-point correlators of the (dual) field strength in Born–Infeld–Chern–Simons theory and of basic field in Born–Infeld self-dual model. Using this equivalence, the bosonization of a massive Dirac theory with a non-polynomial Thirring type current–current coupling, to leading order in (inverse) fermion mass is also discussed. We also rederive it using a master Lagrangian. Finally, the operator equivalence between the fermionic current and (dual) field strength of Born–Infeld–Chern–Simons theory is deduced at the level of correlators and using this the current–current commutators are obtained.     

非对易空间的三模谐振子能谱

在非对易空间中,用不变本征算符方法(IEO),对非耦合、坐标耦合、动量耦合三种三模谐振子系统能谱进行求解,并将求解结果与一般对易空间的能谱进行比较分析.通过比较发现,当非对易参数为零时,所求能级差还原到了与普通空间相对应的一般量子系统哈密顿量能级差,验证了推导结果的正确性;同时讨论了耦合系数对非对易空间能谱的影响. 关键词： 不变本征算符 非对易空间 三模谐振子能谱 能级差     

Noncommutative two-dimensional gauge theories

We elaborate on the dynamics of noncommutative two-dimensional gauge field theories. We consider U(N) gauge theories with fermions in either the fundamental or the adjoint representation. Noncommutativity leads to a rather non-trivial dependence on theta (the noncommutativity parameter) and to a rich dynamics. In particular the mass spectrum of the noncommutative U(1) theory with adjoint matter is similar to that of ordinary (commutative) two-dimensional large-NSU(N) gauge theory with adjoint matter. The noncommutative version of the ?t Hooft model receives a non-trivial contribution to the vacuum polarization starting from three-loops order. As a result the mass spectrum of the noncommutative theory is expected to be different from that of the commutative theory.     

Generalized Spiked Harmonic Oscillator in Non-commutative Space

In this paper non-commutative Schrodinger equation is considered for generalized Spiked harmonic oscillator potential. The energy shift due to non-commutativeity is obtained via the perturbation theory. Furthermore we show that the degeneracy of the initial spectral line is broken in transition from commutative space to non-commutative space.     

Bounded Boolean Powers and Free Product of GMV-Algebras

GMV-algebras are a non-commutative generalization of MV-algebras and they describe a non-commutative many-valued Łukasiewicz logic. We introduce a bounded Boolean power of GMV-algebras. Using the bounded Boolean power of MV-algebras, we show that the free product of an MV-algebra and a Boolean algebra is again a commutative MV-algebra, however the free product of MV-algebras in the variety of GMV-algebras is in general non-commutative (Dvurečenskij and Holland, Algebra Univers., (2010), doi:). In addition, we analyze also a topological version of the Boolean power and we show that these constructions are equivalent.     

Lorentz Invariant Generalization of the Linear Self-Dual Constrained Theory

We propose a manifestly Lorentz invariant action with a modified linear self-dual constraint, which contains a self-dual field or a free scalar field theory according to the parameter α introduced. We obtain the Floreanini-Jackiw's formulation for any values of α by imposing the linear self-dual constraint in phase space.     

The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.     

Gravitomagnetism and Non-commutative Geometry

Similarity between the gravitoelectromagnetism and the electromagnetism is discussed. We show that the gravitomagnetic field (similar to the magnetic field) can be equivalent to the non-commutative effect of the momentum sector of the phase space when one maintains only the first order of the non-commutative parameters. This is performed through two approaches. In one approach, by employing the Feynman proof, the existence of a Lorentz-like force in the gravitoelectromagnetism is indicated. The appearance of such a force is subjected to the slow motion and the weak field approximations for stationary fields. The analogy between this Lorentz-like force and the motion equation of a test particle in a non-commutative space leads to the mentioned equivalency. In fact, this equivalency is achieved by the comparison of the two motion equations. In the other and quietly independent approach, we demonstrate that a gravitomagnetic background can be treated as a Dirac constraint. That is, the gravitoelectromagnetic field can be regarded as a constrained system from the sense of the Dirac theory. Indeed, the application of the Dirac formalism for the gravitoelectromagnetic field reveals that the phase space coordinates have non-commutative structure from the view of the Dirac bracket. Particularly, the gravitomagnetic field as a weak field induces the non-trivial Dirac bracket of the momentum sector which displays the non-commutativity.     

Characterizations of Categories of Commutative C*-Subalgebras

We aim to characterize the category of injective *-homomorphisms between commutative C*-subalgebras of a given C*-algebra A. We reduce this problem to finding a weakly terminal commutative subalgebra of A, and solve the latter for various C*-algebras, including all commutative ones and all type I von Neumann algebras. This addresses a natural generalization of the Mackey–Piron programme: which lattices are those of closed subspaces of Hilbert space? We also discuss the way this categorified generalization differs from the original question.     

Relation Between Relativistic and Non-Relativistic Quantum Mechanics as Integral Transformation

A formulation of quantum mechanics (QM) in the relativistic configurational space (RCS) is considered. A transformation connecting the non-relativistic QM and relativistic QM (RQM) has been found in an explicit form. This transformation is a direct generalization of the Kontorovich–Lebedev transformation. It is shown also that RCS gives an example of non-commutative geometry over the commutative algebra of functions.     

An integrable noncommutative version of the sine-Gordon system

Using the bicomplex approach we discuss an integrable noncommutative system in two-dimensional Euclidean space. It is described by an equation of motion which reduces to the ordinary sine-Gordon equation when the noncommutation parameter is removed, plus a constraint equation which is nontrivial only in the noncommutative case. The implications of this constraint, which is required by integrability but seems to reduce the space of classical solutions, remain to be understood. We show that the system has an infinite number of conserved currents and we give the general recursive relation for constructing them. For the particular cases of lower spin nontrivial currents we work out the explicit expressions and perform a direct check of their conservation. These currents reduce to the usual sine-Gordon currents in the commutative limit. We find classical “localized” solutions to first order in the noncommutativity parameter and describe the Backlund transformations for our system. Finally, we comment on the relation of our noncommutative system to the commutative sine-Gordon system.     

Vortex condensation in the dual Chern–Simons–Higgs model

The contribution of nontrivial vacuum (topological) excitations, more specifically vortex configurations of the self-dual Chern–Simons–Higgs model, to the functional partition function is considered. By using a duality transformation, we arrive at a representation of the partition function in terms of which explicit vortex degrees of freedom are coupled to a dual gauge field. By matching the obtained action to a field theory for the vortices, the physical properties of the model in the presence of vortex excitations are then studied. In terms of this field theory for vortices in the self-dual Chern–Simons–Higgs model, we determine the location of the critical value for the Chern–Simons parameter below which vortex condensation can happen in the system. The effects of self-energy quantum corrections to the vortex field are also considered.     

Quantisation of $\theta$-expanded non-commutative QED

We analyse two new versions of -expanded non-commutative quantum electrodynamics up to first order in and 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 . However, at higher order in the Seiberg-Witten map cannot be regarded as a field redefinition. We find that the initial action of any -expanded massless non-commutative QED must include one extra term proportional to which we identify by loop calculations. Received: 3 July 2002 / Published online: 7 October 2002 RID="a" ID="a" e-mail: jesper@hep.itp.tuwien.ac.at Work supported by The Danish Research Agency. RID="b" ID="b" e-mail: raimar.wulkenhaar@mis.mpg.de Schloe?mann fellow