Spectral families of quantum stochastic integrals

We develop a theory of spectral integration for quantum stochastic integrals of certain families of processes driven by creation, conservation and annihilation processes in Fock space. These give a non-commutative generalisation of classical stochastic integrals driven by Poisson random measures. A stochastic calculus for these processes is developed and used to obtain unitary operator valued solutions of stochastic differential equations. As an application we construct stochastic flows on operator algebras driven by Lévy processes with finite Lévy measure.     

Essay: Cosmic Censorship: The Role of Quantum Gravity

The cosmic censorship hypothesis introduced by Penrose thirty years ago is still one of the most important open questions in classical general relativity. In this essay we put forward the idea that cosmic censorhip is intrinsically a quantum gravity phenomena. To that end, we construct a gedanken experiment in which cosmic censorship is violated within the purely classical framework of general relativity. We prove, however, that quantum effects restore the validity of the conjecture. This suggests that classical general relativity is inconsistent and that cosmic censorship might be enforced only by a quantum theory of gravity.     

Hamilton formalism in non-cummutative geometry

We study the Hamilton formalism for Connes-Lott models, i.e. for Yang-Mills theory in non-commutative geometry. The starting point is an associative *-algebra A which is of the form A = C (I, A_s), where A_s is itself an associative *-algebra. With appropriate choice of a K-cycle over A it is possible to identify the time-like part of the generalized differential algebra constructed out of A. We define the non-commutative analogue of integration on space-like surfaces via the Dixmier trace restricted to the representation of the space-like part A_s of the algebra. Due to this restriction it is possible to define the Lagrange function resp. Hamilton function also for Minkowskian space-time. We identify the phase-space and give a definition of the Poisson bracket for Yang-Mills theory in non-commutative geometry. This general formalism is applied to a model on a two-point space and to a model on Minkowski space-time x two-point space.     

Conformal Symmetry and Quantum Relativity

The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincare symmetry of field theory can be extended to the larger conformal symmetry. We use these symmetries to define quantum observables associated with positions in space-time, in the spirit of Einstein theory of relativity. This conception of localization may be applied to massive as well as massless fields. Localization observables are defined as to obey Lorentz covariant commutation relations and in particular include a time observable conjugated to energy. While position components do not commute in the presence of a nonvanishing spin, they still satisfy quantum relations which generalize the differential laws of classical relativity. We also give of these observables a representation in terms of canonical spatial positions, canonical spin components, and a proper time operator conjugated to mass. These results plead for a new representation not only of space-time localization but also of motion.     

The principle of equivalence and theories of gravity

We construct classical theories of gravity on the basis of special relativity and the Einstein-Infeld accelerating-elevator thought experiment. The resulting theories share most of the main features of general relativity, namely the nonlinear character of the theory, the metrical significance of the gravitational potentials and the geodesic equation of particle motion. They differ from general relativity in at most nonlinear terms in the gravitational constant G in their equations of particle motion and field equations.     

Area preserving transformationsin non-commutative space and NCCS theory

We propose a heuristic rule for the area transformation on the non-commutative plane. The non-commutative area preserving transformations are quantum deformations of the classical symplectic diffeomorphisms. The area preservation condition is formulated as a field equation in the non-commutative Chern-Simons gauge theory. A higher-dimensional generalization is suggested and the corresponding algebraic structure - the infinite-dimensional sin-Lie algebra - is extracted. As an illustrative example the second-quantized formulation for electrons in the lowest Landau level is considered.Received: 13 June 2003, Revised: 11 September 2003, Published online: 7 November 2003     

Gravitational and electroweak unification by replacing diffeomorphisms with larger group

The covariance group for general relativity, the diffeomorphisms, is replaced by a group of coordinate transformations which contains the diffeomorphisms as a proper subgroup. The larger group is defined by the assumption that all observers will agree whether any given quantity is conserved. Alternatively, and equivalently, it is defined by the assumption that all observers will agree that the general relativistic wave equation describes the propagation of light. Thus, the group replacement is analogous to the replacement of the Lorentz group by the diffeomorphisms that led Einstein from special relativity to general relativity, and is also consistent with the assumption of constant light velocity that led him to special relativity. The enlarged covariance group leads to a non-commutative geometry based not on a manifold, but on a nonlocal space in which paths, rather than points, are the most primitive invariant entities. This yields a theory which unifies the gravitational and electroweak interactions. The theory contains no adjustable parameters, such as those that are chosen arbitrarily in the standard model.     

Gravity in non-commutative geometry

We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.Dedicated to H. ArakiSupported in part by the Swiss National Foundation (SNF)     

Unitary evolutions and horizontal lifts in quantum stochastic calculus

Unitarity is proved for a class of solutions of quantum stochastic differential equations with unbounded coefficients. The resulting processes are then used to construct algebraic quantum diffusions. Applications include an existence proof for a class of diffusions on the non-commutative two-torus and a geometric interpretation for diffusions driven by the classical Poisson process.     

Chiral dynamics and meson with non-commutative dipole field in gauge/gravity dual

Apply the T-duality and smeared twist to the D3-brane solution one can construct the supergravity backgrounds which may dual to supersymmetric or non-supersymmetric non-commutative dipole field theory. We introduce D7-brane probe into the dual supergravity background to study the chiral dynamics and meson spectrum therein. We first find that the non-commutative dipole field does not induce the chiral symmetry breaking even if the supersymmetry was completely broken, contrast to the conventional believing that the chiral symmetry will be broken in the non-supersymmetric theory. Next, we find that the dipole field does not modify the meson spectrum in the supersymmetric theory while it will reduce the meson bound-state energy in the non-supersymmetric theory. We also evaluate the static quark–anti-quark potential and see that the dipole field has an effect to produce attractive force between the quark and anti-quark.     

Making classical and quantum canonical general relativity computable through a power series expansion in the inverse cosmological constant

We consider general relativity with a cosmological constant as a perturbative expansion around a completely solvable diffeomorphism invariant field theory. This theory is the lambda --> infinity limit of general relativity. This allows an explicit perturbative computational setup in which the quantum states of the theory and the classical observables can be explicitly computed. An unexpected relationship arises at a quantum level between the discrete spectrum of the volume operator and the allowed values of the cosmological constant.     

Relativity theory: What is reality?

In classical Newtonian physics there was a clear understanding of “what reality is.? Indeed in this classical view, reality at a certain time is the collection of all what is actual at this time, and this is contained in “the present.? Often it is stated that three-dimensional space and one-dimensional time hare been substituted by four-dimensional space-time in relativity theory, and as a consequence the classical concept of reality, as that which is “present,? cannot be retained. Is reality then the four-dimensional manifold of relativity theory? And if so, what is then the meaning of “change in time?? This problem confronts a geometric view (as the Einsteinian interpretation of relativity theory) with a process view (where reality changes constantly in time). In this paper we investigate this problem, taking into account our insight into the nature of reality as it came by analyzing the problems of quantum mechanics. We show that with an Einsteinian interpretation of relativity theory, reality is indeed four-dimensional, but there is no contradiction with the process view, where this reality changes in time.     

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     

Recovering MOND from extended metric theories of gravity

We show that the Modified Newtonian Dynamics (MOND) regime can be fully recovered as the weak-field limit of a particular theory of gravity formulated in the metric approach. This is possible when Milgrom’s acceleration constant is taken as a fundamental quantity which couples to the theory in a very consistent manner. As a consequence, the scale invariance of the gravitational interaction is naturally broken. In this sense, Newtonian gravity is the weak-field limit of general relativity and MOND is the weak-field limit of that particular extended theory of gravity. We also prove that a Noether’s symmetry approach to the problem yields a conserved quantity coherent with this relativistic MONDian extension.     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Special Relativity Cannot Be Derived from Galilean Mechanics Alone

A recent paper suggested that if Galilean covariance was extended to signals and interactions, the resulting theory would contain such anomalies as would have impelled physicists towards special relativity even without empirical prompts. I analyze this claim. Some so-called anomalies turn out to be errors. Others have classical analogs, which suggests that classical physicists would not have viewed them as anomalous. Still others, finally, remain intact in special relativity, so that they serve as no impetus towards this theory. I conclude that Galilean covariance is insufficient to derive special relativity.