Renormalization and quantum field theory

The aim of this paper is to describe how to use regularization and renormalization to construct a perturbative quantum field theory from a Lagrangian. We first define renormalizations and Feynman measures, and show that although there need not exist a canonical Feynman measure, there is a canonical orbit of Feynman measures under renormalization. We then construct a perturbative quantum field theory from a Lagrangian and a Feynman measure, and show that it satisfies perturbative analogues of the Wightman axioms, extended to allow time-ordered composite operators over curved spacetimes.Comment: 30 pages Revised version fixes a gap in the definition of Feynman measure, and has other minor change

Baikov-Lee Representations Of Cut Feynman Integrals

We develop a general framework for the evaluation of $d$-dimensional cut Feynman integrals based on the Baikov-Lee representation of purely-virtual Feynman integrals. We implement the generalized Cutkosky cutting rule using Cauchy's residue theorem and identify a set of constraints which determine the integration domain. The method applies equally well to Feynman integrals with a unitarity cut in a single kinematic channel and to maximally-cut Feynman integrals. Our cut Baikov-Lee representation reproduces the expected relation between cuts and discontinuities in a given kinematic channel and furthermore makes the dependence on the kinematic variables manifest from the beginning. By combining the Baikov-Lee representation of maximally-cut Feynman integrals and the properties of periods of algebraic curves, we are able to obtain complete solution sets for the homogeneous differential equations satisfied by Feynman integrals which go beyond multiple polylogarithms. We apply our formalism to the direct evaluation of a number of interesting cut Feynman integrals.Comment: 37 pages; v2 is the published version of this work with references added relative to v

Feynman formulae and phase space Feynman path integrals for tau-quantization of some L\'evy-Khintchine type Hamilton functions

This note is devoted to representation of some evolution semigroups. The semigroups are generated by pseudo-differential operators, which are obtained by different (parametrized by a number $\tau$) procedures of quantization from a certain class of functions (or symbols) defined on the phase space. This class contains functions which are second order polynomials with respect to the momentum variable and also some other functions. The considered semigroups are represented as limits of $n$-fold iterated integrals when $n$ tends to infinity (such representations are called Feynman formulae). Some of these representations are constructed with the help of another pseudo-differential operators, obtained by the same procedure of quantization (such representations are called Hamiltonian Feynman formulae). Some representations are based on integral operators with elementary kernels (these ones are called Lagrangian Feynman formulae and are suitable for computations). A family of phase space Feynman pseudomeasures corresponding to different procedures of quantization is introduced. The considered evolution semigroups are represented also as phase space Feynman path integrals with respect to these Feynman pseudomeasures. The obtained Lagrangian Feynman formulae allow to calculate these phase space Feynman path integrals and to connect them with some functional integrals with respect to probability measures

Loop lessons from Wilson loops in N=4 supersymmetric Yang-Mills theory

N=4 supersymmetric Yang-Mills theory exhibits a rather surprising duality of Wilson-loop vacuum expectation values and scattering amplitudes. In this paper, we investigate this correspondence at the diagram level. We find that one-loop triangles, one-loop boxes, and two-loop diagonal boxes can be cast as simple one- and two- parametric integrals over a single propagator in configuration space. We observe that the two-loop Wilson-loop "hard-diagram" corresponds to a four-loop hexagon Feynman diagram. Guided by the diagrammatic correspondence of the configuration-space propagator and loop Feynman diagrams, we derive Feynman parameterizations of complicated planar and non-planar Feynman diagrams which simplify their evaluation. For illustration, we compute numerically a four-loop hexagon scalar Feynman diagram.Comment: 20 pages, many figures. Two references added. Published versio