Construction of convergent simplicial approximations of quantum fields on Riemannian manifolds

We construct simplicial approximations of random fields on Riemannian manifolds of dimensiond. We prove convergence of the fields to the continuum limit, for arbitraryd in the Gaussian case and ford=2 in the non-Gaussian case. In particular we obtain convergence of the simplicial approximation to the continuum limit for quantum fields on Riemannian manifolds with exponential interaction.Dedicated to Res Jost and Arthur WightmanBiBoS Research Centre     

Maximally causal quantum mechanics

We present a new causal quantum mechanics in one and two dimensions developed recently at TIFR by this author and V Singh. In this theory both position and momentum for a system point have Hamiltonian evolution in such a way that the ensemble of system points leads to position and momentum probability densities agreeing exactly with ordinary quantum mechanics     

Two-dimensional global manifolds of vector fields

We describe an efficient algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. Larger and larger pieces of a manifold are grown until a sufficiently long piece is obtained. This allows one to study manifolds geometrically and obtain important features of dynamical behavior. For illustration, we compute the stable manifold of the origin spiralling into the Lorenz attractor, and an unstable manifold in zeta(3)-model converging to an attracting limit cycle. (c) 1999 American Institute of Physics.     

Simple example of quantum causal structure

The axioms of quantum causal structure and the definition of AlexandrovT-structure are presented in an improved form and an illustrative example is shown.     

Dirac-Hestenes spinor fields on Riemann-Cartan manifolds

In this paper we study Dirac-Hestenes spinor fields (DHSF) on a four-dimensional Riemann-Cartan spacetime (RCST). We prove that these fields must be defined as certain equivalence classes of even sections of the Clifford bundle (over the RCST), thereby being certain particular sections of a new bundle named the spin-Clifford bundle (SCB). The conditions for the existence of the SCB are studied and are shown to be equivalent to Geroch's theorem concerning the existence of spinor structures in a Lorentzian spacetime. We introduce also the covariant and algebraic Dirac spinor fields and compare these with DHSF, showing that all three kinds of spinor fields contain the same mathematical and physical information. We clarify also the notion of (Crumeyrolle's) amorphous spinors (Dirac-Kähler spinor fields are of this type), showing that they cannot be used to describe fermionic fields. We develop a rigorous theory for the covariant derivatives of Clifford fields (sections of the Clifford bundle, CB) and of Dirac-Hestenes spinor fields. We show how to generalize the original Dirac-Hestenes equation in Minkowski spacetime for the case of RCST. Our results are obtained from a variational principle formulated through the multiform derivative approach to Lagrangian field theory in the Clifford bundle.     

Riemannian structure on manifolds of quantum states

A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state vector (generalized coherent states manifold hereafter noted G.C.S.M.) is studied in details; the geometrical properties of some wellknown G.C.S.M. are reviewed and an explicit expression for the scalar Riemannian curvature is given in the general case. The physical meaning of such Riemannian structures (which have been recently introduced to describe collective manifolds in nuclear physics) is discussed. It is shown on examples that the distance between nearby states is related to quantum fluctuations; in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.Equipe de Recherche Associée au C.N.R.S.     

Atomic focusing by quantum fields: Entanglement properties

The coherent manipulation of the atomic matter waves is of great interest both in science and technology. In order to study how an atom optic device alters the coherence of an atomic beam, we consider the quantum lens proposed by Averbukh et al. [1] to show the discrete nature of the electromagnetic field. We extend the analysis of this quantum lens to the study of another essentially quantum property present in the focusing process, i.e., the atom–field entanglement, and show how the initial atomic coherence and purity are affected by the entanglement. The dynamics of this process is obtained in closed form. We calculate the beam quality factor and the trace of the square of the reduced density matrix as a function of the average photon number in order to analyze the coherence and purity of the atomic beam during the focusing process.     

Jacobi fields on statistical manifolds of negative curvature

Two entropic dynamical models are considered. The geometric structure of the statistical manifolds underlying these models is studied. It is found that in both cases, the resulting metric manifolds are negatively curved. Moreover, the geodesics on each manifold are described by hyperbolic trajectories. A detailed analysis based on the Jacobi equation for geodesic spread is used to show that the hyperbolicity of the manifolds leads to chaotic exponential instability. A comparison between the two models leads to a relation among statistical curvature, stability of geodesics and relative entropy-like quantities. Finally, the Jacobi vector field intensity and the entropy-like quantity are suggested as possible indicators of chaoticity in the ED models due to their similarity to the conventional chaos indicators based on the Riemannian geometric approach and the Zurek-Paz criterion of linear entropy growth, respectively.     

Geometric phase in the causal quantum theories

The projective-geometric formulation of geometric phase for any ensemble in the causal quantum theories is given. This formulation generalizes the standard formulation of geometric phase to any causal ensemble including the cases of a single causal trajectory, the experimental geometric phase and the classical geometric phase.     

A causal quantum theory in phase space

The causal theory for the coherent state representation of quantum mechanics is derived. The general conditions for the classical limit are given and it is shown that phase space classical mechanics can be obtained as a limit even for stationary states, in contrast to the de Broglie-Bohm quantum theory of motion.     

On fermi quantum fields constructed from bose quantum fields and their applications

In former papers a representation of the quantum Fermi and para-Fermi fields was proposed. This representation is such that the only basic quantum entities are Bose quantum fields. In this paper we show several possibilities of application: (i) to lower the number of elementary particles; (ii) to describe as separate states of a fundamental particle other particles that presently are considered as different, and to induce an ordering among them; (iii) to obtain relations among the quantum numbers of those particles; (iv) to obtain a physical picture of some unstable particles. This article is concerned with the physical interpretation of the formalism, and some of the statements that are contained here have a conjectural character.     

Cosmology and the causal interpretation of quantum mechanics

We answer the objections raised recently by Tipler to the trajectory interpretation of quantum mechanics by analyzing in detail the scattering of a particle from a semi-transparent surface. We conclude that the notion of the “wavefunction of the universe” poses no particular problems for this interpretation.     

Renormalisation of out-of-equilibrium quantum fields

We consider the initial value problem and its renormalisation in the framework of the two-particle-irreducible (2PI) effective action. We argue that in the case of appropriately chosen self-consistent initial conditions, the counterterms needed to renormalise the system in equilibrium are also sufficient to renormalise its time evolution. In this way we improve on Gaussian initial conditions which have the disadvantage of generically not showing a continuum limit. For a more detailed discussion see [Sz. Borsányi and U. Reinosa, arXiv:0809.0496].     

Supersymmetry transformation of quantum fields

If supersymmetry is realized non-linearly as is the case when auxiliary fields are eliminated and/or one works in the Wess–Zumino gauge it is usually incorporated in terms of BRS transformations and Slavnov–Taylor identities. On the vertex functional susy transformations act even non-locally. Furthermore, the gauge fixing term breaks supersymmetry. In the present paper we clarify in which sense supersymmetry is still a symmetry of the system and how it is realized on the level of quantum fields. We treat the Wess–Zumino model as an example for chiral models, SQED and massless SYM as prototypes of gauge theories.     

Finding finite-time invariant manifolds in two-dimensional velocity fields

For two-dimensional velocity fields defined on finite time intervals, we derive an analytic condition that can be used to determine numerically the location of uniformly hyperbolic trajectories. The conditions of our main theorem will be satisfied for typical velocity fields in fluid dynamics where the deformation rate of coherent structures is slower than individual particle speeds. We also propose and test a simple numerical algorithm that isolates uniformly finite-time hyperbolic sets in such velocity fields. Uniformly hyperbolic sets serve as the key building blocks of Lagrangian mixing geometry in applications. (c) 2000 American Institute of Physics.     

A class of quantum fields constructed from Markoff fields

A class of fields with the mass parameter is constructed from Gaussian fields. The procedure of linearization of the Hamiltonian is shown to be equivalent to some linear transformation on the space indexing the Gaussian process.