Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of to include any Hamiltonian for which a certain non-resonance condition holds. Received: 5 October 1999 / Accepted: 2 February 2000     

Interacting Fermi Liquid in Two Dimensions¶at Finite Temperature.¶Part I: Convergent Attributions

Using the method of a continuous renormalization group around the Fermi surface, we prove that a two-dimensional interacting system of Fermions at low temperature T is a Fermi liquid in the domain , where K is some numerical constant. According to [S1], this means that it is analytic in the coupling constant λ, and that the first and second derivatives of the self energy obey uniform bounds in that range. This is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermion systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof. Received: 27 July 1999 / Accepted: 31 May 2000     

Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity

A Wheeler-DeWitt quantum constraint operator for four-dimensional, non-perturbative Lorentzian vacuum quantum gravity is defined in the continuum. The regulated Wheeler-DeWitt constraint operator is finite, does not require any renormalization and the final operator is anomaly-free and at least symmetric.

The technique introduced here can also be used to produce a couple of completely well-defined regulated operators including but not exhausting (i) the Euclidean Wheeler-DeWitt operator, (ii) the generator of the Wick rotation transform that maps solutions to the Euclidean Hamiltonian constraint to solutions to the Lorentzian Hamiltonian constraint, (iii) length operators, (iv) Hamiltonian operators of the matter sector and (v) the generators of the asymptotic Poincaré group including the quantum ADM energy.     

Alien calculus and non perturbative effects in Quantum Field Theory

In many domains of physics, methods for dealing with non-perturbative aspects are required. Here, I want to argue that a good approach for this is to work on the Borel transforms of the quantities of interest, the singularities of which give non-perturbative contributions. These singularities in many cases can be largely determined by using the alien calculus developed by Jean Écalle. My main example will be the two point function of a massless theory given as a solution of a renormalization group equation.     

Wakimoto Realizations of Current Algebras: An Explicit Construction

A generalized Wakimoto realization of can be associated with each parabolic subalgebra of a simple Lie algebra according to an earlier proposal by Feigin and Frenkel. In this paper the proposal is made explicit by developing the construction of Wakimoto realizations from a simple but unconventional viewpoint. An explicit formula is derived for the Wakimoto current first at the Poisson bracket level by Hamiltonian symmetry reduction of the WZNW model. The quantization is then performed by normal ordering the classical formula and determining the required quantum correction for it to generate by means of commutators. The affine-Sugawara stress-energy tensor is verified to have the expected quadratic form in the constituents, which are symplectic bosons belonging to and a current belonging to . The quantization requires a choice of special polynomial coordinates on the big cell of the flag manifold . The effect of this choice is investigated in detail by constructing quantum coordinate transformations. Finally, the explicit form of the screening charges for each generalized Wakimoto realization is determined, and some applications are briefly discussed. Received: 19 December 1996 / Accepted: 21 March 1997     

A Markov chain approach to renormalization group transformations

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov–Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear important information about all six of the critical exponents. This illustrates the universality property of the renormalization group map in this case.     

An exact,finite, gauge-invariant,non-perturbative approach to QCD renormalization

A particular choice of renormalization, within the simplifications provided by the non-perturbative property of Effective Locality, leads to a completely finite, non-perturbative approach to renormalized QCD, in which all correlation functions can, in principle, be defined and calculated. In this Model of renormalization, only the Bundle chain-Graphs of the cluster expansion are non-zero. All Bundle graphs connecting to closed quark loops of whatever complexity, and attached to a single quark line, provided no ‘self-energy’ to that quark line, and hence no effective renormalization. However, the exchange of momentum between one quark line and another, involves only the cluster-expansion’s chain graphs, and yields a set of contributions which can be summed and provide a finite color-charge renormalization that can be incorporated into all other QCD processes. An application to High Energy elastic pp scattering is now underway.     

Nonperturbative Calculations in Truncated Fock Space in LFD

A non-perturbative approach based on the Fock decomposition of the state vector and its truncation is discussed. In order the non-perturbative renormalization procedure after truncation could eliminate infinities, it should be the sector dependent. We clarify the meaning of this procedure in a toy model. Then we demonstrate stability, relative to the increasing cutoff, of the anomalous magnetic moment found using the sector dependent renormalization scheme in Yukawa model.     

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

Fractal dimension of Siegel disc boundaries

Using renormalization techniques, we provide rigorous computer-assisted bounds on the Hausdorff dimension of the boundary of Siegel discs. Specifically, for Siegel discs with golden mean rotation number and quadratic critical points we show that the Hausdorff dimension is less than 1.08523. This is done by exploiting a previously found renormalization fixed point and expressing the Siegel disc boundary as the attractor of an associated Iterated Function System. Received: 26 January 1998 / Received in final form: 5 June 1998 / Accepted: 11 June 1998     

Systematic Renormalization Scheme in Non-Perturbative Calculations Within Covariant Light-Front Dynamics

We advocate the use of the recently developed Taylor-Lagrange renormalization scheme for non-perturbative calculations of the properties of bound state systems. As an example, we show how this scheme can be applied to the calculation of non-perturbative corrections to the mass of a fermion within the covariant formulation of light-front dynamics.     

D = 5, N = 2 Geometric Higher Curvature Supergravity in the Second-Order Canonical Theory

The supersymmetric extension of the five-dimensional Chern–Simons gravity is studied from the Hamiltonian point of view. This model containing the Gauss–Bonnet term quadratic in the Riemann curvature is the gauge theory of the supergroup SU(2,2/1). In the first order, the theory has a polynomial structure, but the second-order leads to a nonpolynomial structure for both the Hamiltonian and the supersymmetry transformation rules of the fields. The second-order theory has the advantage that the apparent gauge degrees of freedom are unambiguously removed leaving only the physical ones. This important feature is analyzed by constructing the second-order Hamiltonian theory. The gauge invariances of the model and the generator of time evolution are found.     

Renormalization group invariant average momentum of non-singlet parton densities

We compute, within the Schrödinger functional scheme, a renormalization group invariant renormalization constant for the first moment of the non-singlet parton distribution function. The matching of the results of our non-perturbative calculation with the ones from hadronic matrix elements allows us to obtain eventually a renormalization group invariant average momentum of non-singlet parton densities, which can be translated into a preferred scheme at a specific scale.     

Field theoretical approach to inhomogeneous ionic systems: thermodynamic consistency with the contact theorem,Gibbs adsorption and surface tension

Using a statistical field approach we investigate the structure of an electrolyte solution in contact with a neutral impenetrable wall. The Hamiltonian contains the Coulomb interaction and the ideal entropy. At the level of the quadratic approximation, the Hamiltonian yields the Debye-Hückel theory in the bulk. Analytic expressions of the charge-charge and potential-potential inhomogeneous correlation functions are obtained. Exact asymptotic results for point ion charge correlation functions are obtained and the profile for the fluctuation of the electric potential is calculated. We also consider the term beyond the quadratic expansion of the ideal entropy in the Hamiltonian. With this term a higher order coupling between charge density and number density produces a non-trivial profile for the total ion density. This density profile is consistent with the contact theorem and the related surface tension calculated from the Gibbs adsorption isotherm.     

Occupation numbers from functional integral

Occupation numbers for non-relativistic interacting particles are discussed within a functional integral formulation. For bosons at zero temperature the Bogoliubov theory breaks down for strong couplings as well as for low-dimensional models. We find that the leading behavior of the occupation numbers for small momentum is governed by a quadratic time derivative in the inverse propagator that is not contained in the Bogoliubov theory. We propose to use a functional renormalization group equation for the occupation numbers in order to implement systematic non-perturbative extensions beyond the Bogoliubov theory. We also discuss interacting fermions, in particular the issue of pseudogaps.     

Renormalization in Coulomb gauge QCD

In the Coulomb gauge of QCD, the Hamiltonian contains a non-linear Christ–Lee term, which may alternatively be derived from a careful treatment of ambiguous Feynman integrals at 2-loop order. We investigate how and if UV divergences from higher order graphs can be consistently absorbed by renormalization of the Christ–Lee term. We find that they cannot.     

Non-perturbative QEG corrections to the Yang–Mills beta function

We discuss the non-perturbative renormalization group evolution of the gauge coupling constant by using a truncated form of the functional flow equation for the effective average action of the Yang–Mills-gravity system. Our result is consistent with the conjecture that quantum Einstein gravity (QEG) is asymptotically safe and has a vanishing gauge coupling constant at the non-trivial fixed point.     

Large N_ c in chiral perturbation theory

The construction of the effective Lagrangian relevant for the mesonic sector of QCD in the large limit meets with a few rather subtle problems. We thoroughly examine these and show that, if the variables of the effective theory are chosen suitably, the known large counting rules of QCD can unambiguously be translated into corresponding counting rules for the effective coupling constants. As an application, we demonstrate that the Kaplan–Manohar transformation is in conflict with these rules and is suppressed to all orders in . The anomalous dimension of the axial singlet current generates an additional complication: The corresponding external field undergoes nonmultiplicative renormalization. As a consequence, the Wess–Zumino–Witten term, which accounts for the U(3)U(3) anomalies in the framework of the effective theory, contains pieces that depend on the running scale of QCD. The effect only shows up at nonleading order in , but requires specific unnatural parity contributions in the effective Lagrangian that restore renormalization group invariance. Received: 14 July 2000 / Published online: 27 October 2000