Tree weights and renormalization in QFT

In this chapter we define specific tree weights which appear natural when considering a certain approach to non-perturbative renormalization in QFT, namely the constructive renormalization. Several examples of such tree weights are explicitly given in Appendix A. A fundamental step in QFT is to compute the logarithm of functional integrals used to define the partition function of a given model This comes from a fundamental theorem of enumerative combinatorics, stating the logarithm counts the connected objects. The main advantage of the perturbative expansion of a QFT into a sum of Feynman amplitudes is to perform this computation explicitly: the logarithm of the functional integral is the sum of Feynman amplitudes restricted to connected graphs. The main disadvantage is that the perturbative series indexed by Feynman graphs typically diverges.

Approximate evaluation of the functional integrals generated by the Dirac equation with pseudospin symmetry

In this paper, the matrix-valued functional integrals generated by the Dirac equation with relativistic Hamiltonian are considered. The Dirac Hamiltonian contains scalar and vector potentials. The sum of the scalar and vector potentials is equal to zero, i.e., the case of pseudospin symmetry is investigated. In this case, a Schrödinger-type equation for the eigenvalues and eigenfunctions of the relativistic Hamiltonian generating the functional integral is constructed. The eigenvalues and eigenfunctions of the Schrödinger-type operator are found using the Sturm sequence method and the reverse iteration method. A method for the evaluation of matrix-valued functional integrals is proposed. This method is based on the relation between the functional integral and the kernel of the evolution operator with the relativistic Hamiltonian and the expansion of the kernel of the evolution operator in terms of the found eigenfunctions of the relativistic Hamiltonian.

Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.

Quantum gauge theories

This chapter, which is the last chapter in Part I, is devoted to an extensive discussion of quantum gauge theories, which is based on functional integrals and Lagrangian quantization. After introducing the notion of a Yang-Mills gauge theory, the Faddeev-Popov method (also known as the DeWitt-Faddeev-Popov procedure) is explained. Starting from this point, the BRST symmetry is formulated, and the corresponding Ward identities (called Slavnov-Taylor identities in some cases) established. More specialized subjects, such as the gauge dependence of effective action and the background field method, are dealt with in detail. In addition, Yang-Mills theory is analyzed as a primary example of general theorems concerning the renormalization of gauge theories.

Quantum fields in curved spacetime: renormalization

In this chapter, which forms the central chapter of Part II, the effective action of quantum matter fields in curved spacetime is formulated in terms of functional integrals. A qualitative, albeit incompletely conclusive, analysis of divergences and renormalization in curved space is given. Both non-covariant and covariant methods of calculations are discussed. Normal coordinates and local momentum representation are used to derive the effective potential. The basic elements of the Schwinger-DeWitt technique are elaborated in detail, resulting in the general formulas for one-loop divergences. The heat-kernel technique is also discussed.

Functional integrals

This chapter presents an alternative approach to the quantization of fields, an approach that will be critically important for the development of quantum field theory in curved space, which is the subject of the second part of the book. It starts by providing a description of a functional integral in quantum mechanics, concentrating on the representation of an evolution operator. It then considers the functional representation of the Green functions and the generating functional in quantum field theory, including for fermionic theories. After that, perturbative calculations of the generating functionals and their general properties are formulated. The chapter ends with a brief description of ζ‎-regularization as a technique for defining functional determinants.

Semiclassical approximation of functional integrals

In this paper, we consider a semiclassical approximation of special functional integrals with respect to the conditional Wiener measure. In this apptoximation we use the expansion of the action with respect to the classical trajectory. In so doing, the first three terms of expansion are taken into account. Semiclassical approximation may be interpreted as an expansion in powers of the Planck constant. The novelty of this work is the numerical analysis of the accuracy of the semiclassical approximation of functional integrals. A comparison of the results is used for numerical analysis. Some results are obtained by means of semiclassical approximation, while the other by means of the functional integrals calculation method based on the expansion in eigenfunctions of the Hamiltonian generating a functional integral.