M. Asorey

Some Remarks on High Derivative Quantum Gravity

We analyze the perturbative implications of the most general high derivative approach to quantum gravity based on a diffeomorphism-invariant local action. In particular, we consider the superrenormalizable case with a large number of metric derivatives in the action. The structure of ultraviolet divergences is analyzed in some detail. We show that they are independent of the gauge-fixing condition and the choice of field reparametrization. The cosmological counterterm is shown to vanish under certain parameter conditions. We elaborate on the unitarity problem of high derivative approaches and the distribution of masses of unphysical ghosts. We also discuss the properties of the low energy regime and explore the possibility of having a multiscale gravity with different scaling regimes compatible with Einstein gravity at low energies. Finally, we show that the ultraviolet scaling of matter theories is not affected by the quantum corrections of high derivative gravity. As a consequence, asymptotic freedom is stable under those quantum gravity corrections.

GLOBAL THEORY OF QUANTUM BOUNDARY CONDITIONS AND TOPOLOGY CHANGE

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ=∂M. The space ℳ of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M. In this sense, the change of topology of M is connected with the nontrivial structure of ℳ. The space ℳ itself can be identified with the unitary group of the Hilbert space of boundary data . This description, is shown to be equivalent to the classical von Neumanns description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold 𝒞_. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space ℳ is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self-adjoint boundary conditions, the space 𝒞_ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold 𝒞_ is dual of the Maslov class of ℳ. The phenomena are illustrated with some simple low dimensional examples.

Consistency of the regularization of gauge theories by high covariant derivatives

We show that regularization of gauge theories by higher covariant derivatives and gauge-invariant Pauli-Villars regulators is a consistent method if the Pauli-Villars vector fields are considered in a covariant {alpha} gauge with {alpha}{ne}0 and a given auxiliary preregularization is introduced in order to uniquely define the regularization. The limit {alpha}{r_arrow}0 in the regulating Pauli-Villars fields is pathological and the original Slavnov proposal in the covariant Landau gauge is not correct because of the appearance of massless modes in the regulators which do not decouple when the ultraviolet regulator is removed. In such a case the method does not correspond to the regularization of a pure gauge theory but that of a gauge theory in interaction with massless ghost fields. However, a minor modification of the Slavnov method provides a consistent regularization even for such a case. The regularization that we introduce also solves the problem of overlapping divergences in a way similar to geometric regularization and yields the standard values of the {beta} and {gamma} functions of the renormalization group equations. {copyright} {ital 1996 The American Physical Society.}

Quantum evolution as a parallel transport

We point out that the evolution of a quantum system can be considered as a parallel transport of unitary operators in Hilbert spaces along the time with respect to a generalized connection. The different quantum representations of the system are shown to correspond to the choices of cross sections in the principal fiber bundle where the generalized connection is defined. This interpre‐ tation of time evolution allows us to solve the problem of the formulation of the evolution of a quantum particle in a four‐dimensional gauge field.

Attractive and repulsive Casimir vacuum energy with general boundary conditions

Abstract The infrared behaviour of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth–Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials.

Chern-Simons theory and BCS superconductivity

Abstract We study the relationship between the holomorphic unitary connection of Chern–Simons theory with temporal Wilson lines and the Richardsons exact solution of the reduced BCS Hamiltonian. We derive the integrals of motion of the BCS model, their eigenvalues and eigenvectors as a limiting case of the Chern–Simons theory.

Chern-Simons theory and geometric regularization

Abstract We analyze a regularization of Chern-Simons theory based on a geometrical interpretation of the functional integral in the covariant formalism. Perturbative calculations show the existence of two independent finite renormalization of the gauge field and the Chern-Simons coupling constant. Both renormalizations do not depend on the space-time volume and the gauge condition. Coupling constant renormalization, unlike gauge field renormalization, is also independent of the regulating parameters. The method opens the possibility of a non-perturbative approach in the covariant formalism.

Isoperiodic classical systems and their quantum counterparts

Abstract One-dimensional isoperiodic classical systems have been first analyzed by Abel. Abel’s characterization can be extended for singular potentials and potentials which are not defined on the whole real line. The standard shear equivalence of isoperiodic potentials can also be extended by using reflection and inversion transformations. We provide a full characterization of isoperiodic rational potentials showing that they are connected by translations, reflections or Joukowski transformations. Upon quantization many of these isoperiodic systems fail to exhibit identical quantum energy spectra. This anomaly occurs at order O( ℏ 2 ) because semiclassical corrections of energy levels of order O( ℏ ) are identical for all isoperiodic systems. We analyze families of systems where this quantum anomaly occurs and some special systems where the spectral identity is preserved by quantization. Conversely, we point out the existence of isospectral quantum systems which do not correspond to isoperiodic classical systems.

Nodes, monopoles and confinement in (2+1)-dimensional gauge theories

Abstract In the presence of Chern-Simons interactions the wave functionals of physical states in 2 + 1-dimensional gauge theories vanish at a number of nodal points. We show that those nodes are located at some classical configurations which carry a non-trivial magnetic charge. In abelian gauge theories this fact explains why magnetic monopoles are suppressed by Chern-Simons interactions. In non-abelian theories it suggests a relevant role for nodal gauge field configurations in the confinement mechanism of Yang-Mills theories. We show that the vacuum nodes correspond to the chiral gauge orbits of reducible gauge fields with non-trivial magnetic monopole components.

Generalized canonical transformations for time‐dependent systems

We introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space. We prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable. The theory of generating functions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.