Quantum Groups, Correlation Functions and Infrared Divergences

We show in two simple examples that for one-dimensional quantum chains with quantum group symmetries, the correlation functions of local operators are, in general, infrared divergent. If one considers, however, correlation functions invariant under the quantum group, the divergences cancel out.Comment: 7 pages, BONN-HE-93-0

The formal path integral and quantum mechanics

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.Comment: 33 pages, many TikZ diagrams, submitted to _Journal of Mathematical Physics

Entropy on Spin Factors

Recently it has been demonstrated that the Shannon entropy or the von Neuman entropy are the only entropy functions that generate a local Bregman divergences as long as the state space has rank 3 or higher. In this paper we will study the properties of Bregman divergences for convex bodies of rank 2. The two most important convex bodies of rank 2 can be identified with the bit and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman divergence that satisfies sufficiency then the convex body is spectral and if the Bregman divergence is monotone then the convex body has the shape of a ball. A ball can be represented as the state space of a spin factor, which is the most simple type of Jordan algebra. We also study the existence of recovery maps for Bregman divergences on spin factors. In general the convex bodies of rank 2 appear as faces of state spaces of higher rank. Therefore our results give strong restrictions on which convex bodies could be the state space of a physical system with a well-behaved entropy function.Comment: 30 pages, 6 figure

A simple and efficient numerical scheme to integrate non-local potentials

As nuclear wave functions have to obey the Pauli principle, potentials issued from reaction theory or Hartree-Fock formalism using finite-range interactions contain a non-local part. Written in coordinate space representation, the Schrodinger equation becomes integro-differential, which is difficult to solve, contrary to the case of local potentials, where it is an ordinary differential equation. A simple and powerful method has been proposed several years ago, with the trivially equivalent potential method, where non-local potential is replaced by an equivalent local potential, which is state-dependent and has to be determined iteratively. Its main disadvantage, however, is the appearance of divergences in potentials if the wave functions have nodes, which is generally the case. We will show that divergences can be removed by a slight modification of the trivially equivalent potential method, leading to a very simple, stable and precise numerical technique to deal with non-local potentials. Examples will be provided with the calculation of the Hartree-Fock potential and associated wave functions of 16O using the finite-range N3LO realistic interaction.Comment: 8 pages, 2 figures, submitted to Eur. Phys. J.

Not all adiabatic vacua are physical states

Adiabatic vacua are known to be Hadamard states. We show, however that the energy-momentum tensor of a linear Klein-Gordon field on Robertson-Walker spaces developes a generic singularity on the initial hypersurface if the adiabatic vacuum is of order less than four. Therefore, adiabatic vacua are physically reasonable only if their order is at least four. A certain non-local large momentum expansion of the mode functions has recently been suggested to yield the subtraction terms needed to remove the ultraviolet divergences in the energy-momentum tensor. We find that this scheme fails to reproduce the trace anomaly and therefore is not equivalent to adiabatic regularisation.Comment: 13 pages, LaTex2

Quantum Fluctuations for de Sitter Branes in Bulk AdS(5)

The vacuum expectation value of the square of the field fluctuations of a scalar field on a background consisting of {\it two} de Sitter branes embedded in an anti-de Sitter bulk are considered. We apply a dimensional reduction to obtain an effective lower dimensional de Sitter space equation of motion with associated Kaluza-Klein masses and canonical commutation relations. The case of a scalar field obeying a restricted class of mass and curvature couplings, including massless, conformal coupling as a special case, is considered. We find that the local behaviour of the quantum fluctuations suffers from surface divergences as we approach the brane, however, if the field is {\it constrained} to its value on the brane from the beginning then surface divergences disappear. The ratio of $$ between the Kaluza-Klein spectrum and the lowest eigenvalue mode is found to vanish in the limit that one of the branes goes to infinity.Comment: 14 pages, no figures, to appear in Prog. Theor. Phy

Effective Action, Conformal Anomaly and the Issue of Quadratic Divergences

For massless $\phi^4$ theory, we explicitly compute the lowest order non-local contributions to the one-loop effective action required for the determination of the trace anomaly. Imposing exact conformal invariance of the local part of the effective action, we argue that the issue of quadratic divergences does not arise in a theory where exact conformal symmetry is only broken by quantum effects. Conformal symmetry can thus replace low energy supersymmetry as a possible guide towards stabilizing the weak scale and solving the hierarchy problem, if (i) there are no intermediate scales between the weak scale and the Planck scale, and (ii) the running couplings exhibit neither Landau poles nor instabilities over this whole range of energies.Comment: 17 page

Massless Rotating Fermions Inside a Cylinder

We study rotating thermal states of a massless quantum fermion field inside a cylinder in Minkowski space-time. Two possible boundary conditions for the fermion field on the cylinder are considered: the spectral and MIT bag boundary conditions. If the radius of the cylinder is sufficiently small, rotating thermal expectation values are finite everywhere inside the cylinder. We also study the Casimir divergences on the boundary. The rotating thermal expectation values and the Casimir divergences have different properties depending on the boundary conditions applied at the cylinder. This is due to the local nature of the MIT bag boundary condition, while the spectral boundary condition is nonlocal

Convective damping of buoyancy anomalies and its effect on lapse rates in the tropical lower troposphere

International audienceIn actively convecting regions of the tropics, lapse rates in the lower troposphere (2.0 km to 5.2 km) vary with height in a way which is inconsistent with both reversible moist adiabatic and pseudoadiabatic assumptions. It is argued that this anomalous behavior arises from the tendency for the divergence of a convective buoyancy anomaly to be primarily offset by the collective divergence of all other updrafts and downdrafts within one Rossby radius of deformation. (Ordinarily, convective divergences are at least partially offset by an induced radiative divergence in the background atmosphere.) If convective divergences are balanced purely by other convective divergences, it would force the vertical clear sky radiative mass flux to be independent of altitude. This is consistent with what is observed at several radiosonde locations in the Western Tropical Pacific between 2.0 and 5.2 km. It is conjectured, that at tropical locations where SST's exceed 27°C over a region whose horizontal extent exceeds the local Rossby radius, this condition on the clear sky radiative mass flux serves to partially constrain the range of physically allowed mean temperature and moisture profiles in the lower troposphere