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.     

Non-commutative standard model: model building

A non-commutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: (1) although we can have non-commutative U(n) (which we denote by _U* (n)) gauge theory we cannot have non-commutative SU(n) and (2) the charges in non-commutative QED are quantized to just . We show how the latter problem with charge quantization, as well as with the gauge group, can be resolved by taking the gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the non-commutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the non-commutative standard model (NCSM). In addition, among several peculiar features of our model, we address the inherent CP violation and new neutrino interactions. Received: 23 January 2003, Published online: 18 June 2003     

Lipshitz continuity of gap boundaries for Hofstadter-like spectra

We consider an effective HamiltonianH representing the motion of a single-band-two-dimensional electron in a uniform magnetic field. ThenH belongs to the rotation algebra, namely the algebra of continuous functions over a non-commutative 2-torus. We define a non-commutative analog of smooth functions by mean of elements of classC ^l,n , wherel andn characterize respectively the degree of differentiability with respect to the magnetic field and the torus variables. We show that ifH is of classC ^1,3+ , the gap boundaries of the spectrum ofH are Lipshitz continuous functions of the magnetic field at each point for which the gap is open.URA 505, CNRS     

The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.     

Deformation Quantization of the n-tuple Point

Contrary to the classical methods of quantum mechanics, the deformation quantization can be carried out on phase spaces which are not even topological manifolds. In particular, the Moyal star product gives rise to a canonical functor F from the category of affine analytic spaces to the category of associative (in general, non-commutative) ℂ-algebras. Curiously, if X is the n-tuple point, x ⁿ =0, then F(X) is the algebra of n×n matrices. Received: 4 November 1998 / Accepted: 3 March 1999     

Riesz Transforms on Deformed Fock Spaces

The Fock Von Neumann algebra , equipped with its canonical trace τ, is spanned by n hermitian operators acting on a Hilbert Fock space some commutation relations between and are defined by the n×n hermitian matrix A. We define a Riesz transform , where is the number operator, ∇^′ is aninner derivation (unbounded in general) and . Let 1<p<∞. We prove that is equivalent to for every with null trace, with constants which do not depend on n. Received: 24 November 1998 / Accepted: 2 March 1999     

q-Gaussian Processes: Non-commutative and Classical Aspects

We examine, for −1<q<1, q-Gaussian processes, i.e. families of operators (non-commutative random variables) – where the a _t fulfill the q-commutation relations for some covariance function – equipped with the vacuum expectation state. We show that there is a q-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of q-Gaussian processes possesses a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret [FB]. Received: 25 July 1996 / Accepted: 17 September 1996     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.     

Non-Commutative Martingale Inequalities

We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L ^p -martingale via its integrand, and then extend the Ito-Clifford integral theory in L ², developed by Barnett, Streater and Wilde, to L ^p for all 1<p<∞. We include an appendix on the non-commutative analogue of the classical Fefferman duality between $H ¹ and BMO. Received: 20 March 1997 / Accepted: 21 March 1997     

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.     

Open Wilson lines as closed strings in non-commutative field theories

Open Wilson line operators and the generalized star product have been studied extensively in non-commutative gauge theories. We show that they also show up in non-commutative scalar field theories as universal structures. We first point out that the dipole picture of non-commutative geometry provides an intuitive argument for robustness of the open Wilson lines and generalized star products therein. We calculate the one-loop effective action of the non-commutative scalar field theory with cubic self-interaction and show explicitly that the generalized star products arise in the non-planar part. It is shown that, in the low-energy, large non-commutativity limit, the non-planar part is expressible solely in terms of the scalar open Wilson line operator and descendants, the latter being interpreted as composite operators representing a closed string. Received: 11 September 2001 / Revised version: 24 October 2001 / Published online: 14 December 2001     

Trace Dynamics and a non-commutative special relativity

Trace Dynamics is a classical dynamical theory of non-commuting matrices in which cyclic permutation inside a trace is used to define the derivative with respect to an operator. We use the methods of Trace Dynamics to construct a non-commutative special relativity. We define a line-element using the Trace over space–time coordinates which are assumed to be operators. The line-element is shown to be invariant under standard Lorentz transformations, and is used to construct a non-commutative relativistic dynamics. The eventual motivation for constructing such a non-commutative relativity is to relate the statistical thermodynamics of this classical theory to quantum mechanics.     

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the “replica trick” whereby the entropy is obtained as the derivative with respect to n of the trace of the nth power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the nth power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.     

Quantum Logic and Non-Commutative Geometry

We propose a general scheme for the “logic” of elementary propositions of physical systems, encompassing both classical and quantum cases, in the framework given by Non-Commutative Geometry. It involves Baire^*-algebras, the non-commutative version of measurable functions, arising as envelope of the C ^*-algebras identifying the topology of the (non-commutative) phase space. We outline some consequences of this proposal in different physical systems. This approach in particular avoids some problematic features appearing in the definition of physical states in the standard (W ^*-)algebraic approach to classical mechanics.     

Strong and electroweak interactions and their unification with non-commutative space-time

Quantum field theories based on non-commutative space-time (NCQFT) have been extensively studied recently. However no NCQFT model which can uniquely describe the strong and electroweak interactions has been constructed. This prevents one to make a consistent and systematic study of non-commutative space-time. In this work we construct a NCQFT model based on the trinification gauge group . A unique feature of this model, that all matter fields (fermions and Higgs bosons) are assigned to (anti-) fundamental representations of the factor SU(3) groups, allows us to construct a NCQFT model for strong and electroweak interactions and their unification without ambiguities. This model provides an example which allows one to make a consistent and systematic study of non-commutative space-time phenomenology. We also comment on some related issues regarding extensions to E₆ and models. Received: 18 September 2002 / Revised version: 21 February 2003 / Published online: 5 May 2003     

Duality in noncommutative topologically massive gauge field theory revisited

We introduce a master action in non-commutative space, out of which we obtain the action of the non-commutative Maxwell-Chern-Simons theory. Then, we look for the corresponding dual theory at both first and second order in the non-commutative parameter. At the first order, the dual theory happens to be, precisely, the action obtained from the usual commutative self-dual model by generalizing the Chern-Simons term to its non-commutative version, including a cubic term. Since this resulting theory is also equivalent to the non-commutative massive Thirring model in the large fermion mass limit, we remove, as a byproduct, the obstacles arising in the generalization to non-commutative space, and to the first non-trivial order in the non-commutative parameter, of the bosonization in three dimensions. Then, performing calculations at the second order in the non-commutative parameter, we explicitly compute a new dual theory which differs from the non-commutative self-dual model and, further, differs also from other previous results and involves a very simple expression in terms of ordinary fields. In addition, a remarkable feature of our results is that the dual theory is local, unlike what happens in the non-Abelian, but commutative case. We also conclude that the generalization to non-commutative space of bosonization in three dimensions is possible only when considering the first non-trivial corrections over ordinary space.Received: 12 November 2003, Published online: 23 March 2004M. Botta Cantcheff: mbotta_c@ictp.trieste.itP. Minces: Permanent address Centro Brasileiro de Pesquisas Físicas (CBPF), Departamento de Teoria de Campos e Partículas (DCP), Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil     

QuantumK-systems

We generalize the classical notion of aK-system to a non-commutative dynamical system by requiring that an invariantly defined memory loss be 100%. We give some examples of quantumK-systems and show that they cannot contain any quasi-periodic subsystem.     

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.     

Asymptotic behaviour and general properties of harmonic generation susceptibilities

We use the Kubo response function formalism to derive the asymptotic behaviour of the harmonic generation susceptibilities to all orders n. The results show a stringent correspondence with the ones previously obtained from the classical anharmonic oscillator model. They are characterized by a dependence and a coefficient proportional to the trace of the (n+1)th derivative of the potential energy on the equilibrium density matrix. Using the above results we derive new Kramers-Kr?nig relations and sum rules for all orders of harmonics susceptibilities. Received 17 April 2000     

Matrix De Rham Complex and Quantum A-infinity algebras

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A _∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin–Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006–48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein–Leites simple associative algebras with odd trace q(N), and gl(N|N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin–Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.