Hamiltonian anomalies of bound states in QED

The Bound State in QED is described in systematic way by means of nonlocal irreducible representations of the nonhomogeneous Poincare group and Dirac’s method of quantization. As an example of application of this method we calculate triangle diagram Para-Positronium → γγ. We show that the Hamiltonian approach to Bound State in QED leads to anomaly-type contribution to creation of pair of parapositronium by two photon.     

Real Hamiltonian forms of Hamiltonian systems

We introduce the notion of a real form of a Hamiltonian dynamical system in analogy with the notion of real forms for simple Lie algebras. This is done by restricting the complexified initial dynamical system to the fixed point set of a given involution. The resulting subspace is isomorphic (but not symplectomorphic) to the initial phase space. Thus to each real Hamiltonian system we are able to associate another nonequivalent (real) ones. A crucial role in this construction is played by the assumed analyticity and the invariance of the Hamiltonian under the involution. We show that if the initial system is Liouville integrable, then its complexification and its real forms will be integrable again and this provides a method of finding new integrable systems starting from known ones. We demonstrate our construction by finding real forms of dynamics for the Toda chain and a family of Calogero-Moser models. For these models we also show that the involution of the complexified phase space induces a Cartan-like involution of their Lax representations.Received: 8 October 2003, Published online: 8 June 2004PACS: 02.30.Ik Integrable systems - 45.20.Jj Lagrangian and Hamiltonian mechanics     

Hamiltonian cosmology of bigravity

This article is written as a review of the Hamiltonian formalism for the bigravity with de Rham–Gabadadze–Tolley (dRGT) potential, and also of applications of this formalism to the derivation of the background cosmological equations. It is demonstrated that the cosmological scenarios are close to the standard ΛCDM model, but they also uncover the dynamical behavior of the cosmological term. This term arises in bigravity regardless on the choice of the dRGT potential parameters, and its scale is given by the graviton mass. Various matter couplings are considered.     

Relativistic Hamiltonian dynamics

A manifestly covariant relativistic hamiltonian dynamics is presented for a closed system of N particles in mutual interaction. The “no-interaction theorem” is overcome by use of relativistic center-of-mass variables instead of individual particle variables. The theory permits canonical quantization.     

Synthetic Hamiltonian mechanics

This paper deals with some infinitesimal aspects of Hamiltonian mechanics from the standpoint of synthetic differential geometry. Fundamental results concerning Hamiltonian vector fields, Poisson brackets, and momentum mappings are discussed. The significance of the Lie derivative in the synthetic context is also consistently stressed. In particular, the notion of an infinitesimally Euclidean space is introduced, and the Jacobi identity of vector fields with respect to Lie brackets is established naturally for microlinear, infinitesimally Euclidean spaces by using Lie derivatives instead of a highly combinatorial device such as P. Hall's 42-letter identity.     

Lorentz-Covariant Hamiltonian Formalism

The dynamics of a classical system can be expressed by means of Poissonbrackets. In this paper we generalize the relation between the usual noncovariantHamiltonian and the Poisson brackets to a covariant Hamiltonian and new bracketsin the frame of Minkowski space. These brackets can be related to those usedby Feynman in his derivation Maxwell's equations. The case of curved space isalso considered with the introduction of Christoffel symbols, covariant derivatives,and curvature tensors.     

Conformal Hamiltonian systems

Vector fields whose flow preserves a symplectic form up to a constant, such as simple mechanical systems with friction, are called “conformal”. We develop a reduction theory for symmetric conformal Hamiltonian systems, analogous to symplectic reduction theory. This entire theory extends naturally to Poisson systems: given a symmetric conformal Poisson vector field, we show that it induces two reduced conformal Poisson vector fields, again analogous to the dual pair construction for symplectic manifolds. Conformal Poisson systems form an interesting infinite-dimensional Lie algebra of foliate vector fields. Manifolds supporting such conformal vector fields include cotangent bundles, Lie–Poisson manifolds, and their natural quotients.     

Monte Carlo Hamiltonian

We construct an effective Hamiltonian via Monte Carlo from a given action. This Hamiltonian describes physics in the low energy regime. We test it by computing spectrum, wave functions and thermodynamical observables (average energy and specific heat) for the free system and the harmonic oscillator. The method is shown to work also for other local potentials.     

Renormalization of Hamiltonian QCD

We study to one-loop order the renormalization of QCD in the Coulomb gauge using the Hamiltonian formalism. Divergences occur which might require counter-terms outside the Hamiltonian formalism, but they can be cancelled by a redefinition of the Yang–Mills electric field.     

Vibration-rotation-inversion Hamiltonian of formaldehyde

An expression for the Hamiltonian of a vibrating-rotating-inverting formaldehyde molecule is derived. In this derivation, we have used one curvilinear coordinate corresponding to the angle between the CO bond and the bisector of the angle H?H, and five rectilinear coordinates (linearized valence coordinates). Making use of the zeroth-order Hamiltonian, we have fitted to least-squares (i) the three observed $\text{Δ}\text{G(v}{}_4\text{+}\text{1}\text{2}\text{)}$ values for inversion of H₂CO and (ii) five of D₂CO, both belonging to the $\text{A}\text{?}{}^1\text{A}{}_2$ excited electronic states, in two separate calculations. For this, we have employed two model potential functions: one consisting of a sum of quadratic and Gaussian and the other a sum of quadratic and Lorentzian terms. In each case, the refined parameters, when transferred to the isotopic molecules (D₂CO and HDCO in the one case; H₂CO and HDCO in the other), could not account for their observed $\text{Δ}\text{G(v}{}_4\text{+}\text{1}\text{2}\text{)}$ values to the expected degree. We attribute the discrepancies to the inadequacy of the model chosen for the formaldehyde molecule which takes into account only one large amplitude bending motion and which neglects vibration-inversion interactions.We have also obtained a number of quadratic squared sum relations among the Coriolis coupling constants $\text{ζ}{}_{\mathrm{kl}}{}^{\mathrm{\alpha }}$ and the inertial constants $\text{a}{}_{\mathrm{k}}{}^{\mathrm{\alpha \beta }}$. These are applicable to any molecule undergoing a large amplitude bending motion provided the reference configuration is chosen as described in the text.     

Hamiltonian BRST-anti-BRST theory

The hamiltonian BRST-anti-BRST theory is developed in the general case of arbitrary reducible first class systems. This is done by extending the methods of homological perturbation theory, originally based on the use of a single resolution, to the case of a biresolution. The BRST and the anti-BRST generators are shown to exist. The respective links with the ordinary BRST formulation and with thesp(2)-covariant formalism are also established.Maître de recherches au Fonds National de la Recherche Scientifique.     

Global gravitational anomalies from perturbative chiral anomalies

The relation between (4k ? 2)-dimensional global gravitational anomalies and perturbative chiral anomalies in 4k dimensions is clarified using an open space generalization of the Atiyah-Patodi-Singer index theorem. These anomalies are then reduced to a chiral anomaly in a two-dimensional Schwinger model. It is argued that “all” anomalies can be similarly reduced.     

Auxiliary-field anomalies

In general, quantum corrections to matter-supergravity couplings uniquely determine what are acceptable auxiliary fields for N = 1 supergravity, and partially determine those for N = 2. This is because one-loop corrections produce anomalies in not only the local superscale transformations, but also in the local (Poincaré) supersymmetry transformations themselves, except for special cases: in particular, for N = 1 the $\text{n =}\text{1}\text{3}$ minimimal set of auxiliary fields is uniquely chosen.     

Hamiltonian formalism of fractional systems

In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional equations of motion are derived using the Hamiltonian formalism. The approach is illustrated with a simple-fractional oscillator in a free motion and under an external force. Besides the behavior of the coupled fractional oscillators is analyzed. The natural extension of this approach to continuous systems is stated. The interpretation of the mechanics is discussed.