Feynman amplitudes with confinement included

Amplitudes for any multipoint Feynman diagram are written taking into account vacuum background confining field. Higher order gluon exchanges are treated within background perturbation theory. For amplitudes with hadrons in initial or final states vertices are shown to be expressed by the corresponding wave function with the renormalized z factors. Examples of two-point functions, three-point functions (form factors), and decay amplitudes are explicitly considered. The text was submitted by the author in English.     

A points-splitting and dimensional renormalization

We consider Feynman amplitudes which are doubly regularized by means of complete points splitting of vertices and continuation in the dimension of space-time. We show how to construct a subtraction operator which leads to polynomial counterterms and to a renormalized amplitude which is finite as the regularizations are removed in either order, and corresponds to the dimensionally renormalized result in the limit of no points splitting.     

Renormalization of Feynman amplitudes and parametric integral representation

A new substraction formula is presented to renormalize Feynman amplitudes written in Schwinger's integral representation. The substractions are generated by an operator acting on the integrand, which only depends on the total number of internal lines but is completely independent of the structure of the graph. This formulation is also valid for non-renormalizable theories and is shown to reduce to Zimmermann'sR-operation for scalar theories. It satisfies in any case Bogoliubov's recursive formula and yields an explicit tool for actual computations of renormalized Feynman amplitudes with a minimal number of substractions.     

Diagrammatic analysis of correlations in polymer fluids: Cluster diagrams via Edwards’ field theory

Edwards’ functional integral approach to the statistical mechanics of polymer liquids is amenable to a diagrammatic analysis in which free energies and correlation functions are expanded as infinite sums of Feynman diagrams. This analysis is shown to lead naturally to a perturbative cluster expansion that is closely related to the Mayer cluster expansion developed for molecular liquids by Chandler and co-workers. Expansion of the functional integral representation of the grand-canonical partition function yields a perturbation theory in which all quantities of interest are expressed as functionals of a monomer-monomer pair potential, as functionals of intramolecular correlation functions of non-interacting molecules, and as functions of molecular activities. In different variants of the theory, the pair potential may be either a bare or a screened potential. A series of topological reductions yields a renormalized diagrammatic expansion in which collective correlation functions are instead expressed diagrammatically as functionals of the true single-molecule correlation functions in the interacting fluid, and as functions of molecular number density. Similar renormalized expansions are also obtained for a collective Ornstein-Zernicke direct correlation function, and for intramolecular correlation functions. A concise discussion is given of the corresponding Mayer cluster expansion, and of the relationship between the Mayer and perturbative cluster expansions for liquids of flexible molecules. The application of the perturbative cluster expansion to coarse-grained models of dense multi-component polymer liquids is discussed, and a justification is given for the use of a loop expansion. As an example, the formalism is used to derive a new expression for the wave-number dependent direct correlation function and recover known expressions for the intramolecular two-point correlation function to first-order in a renormalized loop expansion for coarse-grained models of binary homopolymer blends and diblock copolymer melts.     

Renormalization in the complete Mellin representation of Feynman amplitudes

The Feynman amplitudes are renormalized in the formalism of the CM representation. This Mellin-Barnes type integral representation, previously introduced for the study of asymptotic behaviours, is shown to have the following interesting property: in contrast with the usual subtraction procedures, the renormalization leaves the CM integrand unchanged, and only results into translations of the integration path. The explicit CM representation of the renormalized amplitudes is given. In addition, the dimensional regularization and the extension to spinor amplitudes are sketched.     

Renormalization of position space amplitudes in a massless QFT

Ultraviolet renormalization of position space massless Feynman amplitudes has been shown to yield associate homogeneous distributions. Their degree is determined by the degree of divergence while their order—the highest power of logarithm in the dilation anomaly—is given by the number of (sub)divergences. In the present paper we review these results and observe that (convergent) integration over internal vertices does not alter the total degree of (superficial) ultraviolet divergence. For a conformally invariant theory internal integration is also proven to preserve the order of associate homogeneity. The renormalized 4-point amplitudes in the φ⁴ theory (in four space-time dimensions) are written as (non-analytic) translation invariant functions of four complex variables with calculable conformal anomaly.Our conclusion concerning the (off-shell) infrared finiteness of the ultraviolet renormalized massless φ⁴ theory agrees with the old result of Lowenstein and Zimmermann [23].     

Integral representation for the dimensionally renormalized Feynman amplitude

A compact convergent integral representation for dimensionally renormalized Feynman amplitudes is explicitly constructed. The subtracted integrand is expressed as a distribution in the Schwinger -parametric space, and is obtained by applying upon the bare integrand a new subtraction operatorR' which respects Zimmermann's forest structure.     

The convergence of BPH renormalization

The convergence of the integrals defining BPH renormalized Feynman amplitudes is derived from the known additive structure of analytic renormalization.     

Circuit based graphs for renormalized perturbation theory

A generalization of graph theory is introduced and used to obtain Feynman parametric formulas relevant to renormalized amplitudes. The generalization of graph theory is based upon circuit coefficients instead of the usual incidence matrix. The parametric formulas presented are valid for amplitudes which have been renormalized, as in the Zimmermann formulation, by subtracting Taylor terms in momentum space.The research reported in this paper was supported in part by the Max-Planck-Institut für Physik, München, by the German and American Fulbright Commission in Bonn, and by the Department of Mathematics of the University of Virginia, Charlottesville.     

Hamiltonian Feynman-Kac and Feynman Formulae for Dynamics of Particles with Position-Dependent Mass

A Feynman formula is a representation of the semigroup, generated by an initial-boundary value problem for some evolutionary equation, by a limit of integrals over Cartesian powers of some space E, the integrands being some elementary functions. The multiple integrals in Feynman formulae approximate integrals with respect to some measures or pseudomeasures on sets of functions which take values in E and are defined on a real interval. Hence Feynman formulae can be used both to calculate explicitly solutions for such problems, to get some representations for these solutions by integrals over functions taking values in E (such representations are called Feynman-Kac formulae), to get approximations for transition probability of some diffusion processes and transition amplitudes for quantum dynamics and to get computer simulations for some stochastic and quantum dynamics. The Feynman formula is called a Hamiltonian Feynman formula if the space, Cartesian products of which are used, is the phase space of a classical Hamiltonian system; the corresponding Feynman-Kac formula is called a Hamiltonian Feynman-Kac formula. In the latter formula one integrates over functions taking values in the same phase space. In a similar way one can define Lagrangian Feynman formulae and Lagrangian Feynman-Kac formulae substituting the phase space by the configuration space.     

Minimal renormalization without ɛ-expansion: Amplitude functions in three dimensions

The minimal subtraction scheme and the Borel resummation method are used to calculate the amplitudes of renormalized correlation functions aboveT _c for then-vector model in three dimensions. Accurate representations are given for the effective amplitudes of the renormalized expressions of the correlation length, of the susceptibility and of the specific heat forn=1, 2, 3. The resummed higher-order contributions turn out to yield only small corrections to the low-order approximations. These results complement a recent calculation of renormalization-group functions and provide the basis for accurate analyses of the critical behavior in three dimensions including the amplitude functions.     

The Temperature Zero Limit

We prove that, for a broad class of many-fermion models, the amplitudes of renormalized Feynman diagrams converge to their temperature zero values in the limit as the temperature tends to zero.     

Renormalization of QED with planar binary trees

The Dyson relations between renormalized and bare photon and electron propagators and are expanded over planar binary trees. This yields explicit recursive relations for the terms of the expansions. When all the trees corresponding to a given power of the electron charge are summed, recursive relations are obtained for the finite coefficients of the renormalized photon and electron propagators. These relations significantly decrease the number of integrals to carry out, as compared to the standard Feynman diagram technique. In the case of massless quantum electrodynamics (QED), the relation between renormalized and bare coefficients of the perturbative expansion is given in terms of a Hopf algebra structure. Received: 23 March 2000 / Revised version: 24 November 2000 / Published online: 6 April 2001     

On renormalization and complex space-time dimensions

The method of using the dimension of space-time as a complex parameter introduced recently to regularize Feynman amplitudes is extended to an arbitrary Feynman graph. The method has promise of being particularly well-suited to gauge theories. It is shown how the renormalized amplitude, together with the Lagrangian counter-terms, may be extracted directly, following the method of analytic renormalization.     

The borel transform in Euclidean ϕ v 4 local existence for Rev < 4

We consider the ⁴ theory in Euclidean space of complex dimensionv and prove that, for Rev < 4 the renormalized Feynman amplitudes grow at worst exponentially in the number of vertices in the graph. This implies that the Borel transform of any Schwinger function may be defined in a neighborhood of the origin in the Borel plane.Research partially supported by National Science Foundation, Grant Number Phy 77-02277 A 01     

Regularized and renormalized Bethe-Salpeter equations: Some aspects of irreducibility and asymptotic completeness in renormalizable theories

Results on the links between 2-particle irreducibility and asymptotic completeness are presented in the framework of a renormalized Bethe-Salpeter formalism, introduced recently by J. Bros from an axiomatic viewpoint, for the most simple class of renormalizable theories. These results, which involve therenormalized 2-particle irreducible kernelG (i.e. from the perturbative viewpoint the sum of renormalized Feynman amplitudes of 2-particle irreducible graphs in the channel considered), complement the general quasi-equivalence previously established by Bros forregularized (non-renormalized) Bethe-Salpeter kernels. On the one hand, a formal derivation of (2-particle) asymptotic completeness from the irreducibility ofG is given. On the other hand, the links between regularized and renormalized kernels are investigated. This analysis provides in particular a converse derivation (up to some assumptions) of the 2-particle irreducibility ofG from asymptotic completeness. As a byproduct, it also provides a more explicit justification of previous heuristic derivations by K. Symanzik of integral equations betweenF and various differences of values ofG, and a simple alternative derivation of the recently proposed renormalized Bethe-Salpeter equation.     

Quantum-statistical kinetic equations

Considering a homogeneous normal quantum fluid consisting of identical interacting fermions or bosons, we derive an exact quantum-statistical generalized kinetic equation with a collision operator given as explicit cluster series where exchange effects are included through renormalized Liouville operators. This new result is obtained by applying a recently developed superoperator formalism (Liouville operators, cluster expansions, symmetrized projectors,P _q rule, etc.) to nonequilibrium systems described by a density operator(t) which obeys the von Neumann equation. By means of this formalism a factorization theorem is proven (being essential for obtaining closed equations), and partial resummations (leading to renormalized quantities) are performed. As an illustrative application, the quantum-statistical versions (including exchange effects due to Fermi-Dirac or Bose-Einstein statistics) of the homogeneous Boltzmann (binary collisions) and Choh-Uhlenbeck (triple collisions) equations are derived.     

Superconducting State in the Lagrangian Formalism of the Generalized Hubbard Model

The nonperturbative large-N expansion applied to the generalized Hubbard model describing N-fold-degenerate correlated bands is considered. Our previous results, obtained in the framework of the Lagrangian formalism for the normal-state case, are extended to the superconducting state. The standard Feynman diagrammatics is obtained and the renormalized physical quantities are computed and analyzed. Our purpose is to obtain the 1/N corrections to the renormalized boson and fermion propagators when a state with Cooper-pair condensation (i.e., the superconducting state) is considered.