The diagrammatic method is one of the most powerful tools of theoretical physics. It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory. The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams ( graphs ). Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs . There exist several kinds of diagrams. We will try to present them in a systematic way. In Sect. 20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure. This formalism is used extensively in classical statistical physics. It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables. We use the term “Gaussian integration” in a rather broad sense. Beside commuting “bosonic” variables, we also consider anti-commuting “fermionic” variables, where we use the Berezin integral with respect to a Gaussian weight. Even in the case of commuting variables, the “Gaussian integral” is not necessarily meant in the sense of measure theory. It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight. However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.
The diagrammatic method is one of the most powerful tools of theoretical physics. It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory. The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams ( graphs ). Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs . There exist several kinds of diagrams. We will try to present them in a systematic way. In Sect. 20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure. This formalism is used extensively in classical statistical physics. It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables. We use the term “Gaussian integration” in a rather broad sense. Beside commuting “bosonic” variables, we also consider anti-commuting “fermionic” variables, where we use the Berezin integral with respect to a Gaussian weight. Even in the case of commuting variables, the “Gaussian integral” is not necessarily meant in the sense of measure theory. It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight. However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.
DiagrammaticsThe diagrammatic method is one of the most powerful tools of theoretical physics.It allows us to efficiently organize perturbative computations in statistical physics, quantum many-body theory and quantum field theory.The main feature of this method is a representation of individual terms of a perturbative expansion as diagrams (graphs).Diagrams consist of vertices representing terms in the perturbation, lines representing pairings between vertices and, possibly, external legs.There exist several kinds of diagrams.We will try to present them in a systematic way.In Sect.20.1 we present a diagrammatic formalism whose goal is to organize integration of polynomials with respect to a Gaussian measure.This formalism is used extensively in classical statistical physics.It also plays an important role in quantum physics, especially in the Euclidean approach, since many quantum quantities can be expressed in terms of Gaussian integrals over classical variables.We use the term "Gaussian integration" in a rather broad sense.Beside commuting "bosonic" variables, we also consider anti-commuting "fermionic" variables, where we use the Berezin integral with respect to a Gaussian weight.Even in the case of commuting variables, the "Gaussian integral" is not necessarily meant in the sense of measure theory.It denotes an algebraic operation performed on polynomials (or formal power series), which in the case of a positive definite covariance coincides with the usual integral with a Gaussian weight.However, we allow the covariance to be complex, or even negative definite, and do not insist that the operation have a measure theoretic meaning.We distinguish two kinds of spaces on which we perform the integrals: real and complex.As in many other places in our work, we treat these two cases in parallel.Of course, the difference between the real (i.e.neutral), and the complex (i.e.charged) formalism is mainly that of a different notation.In particular, charged lines need to be equipped with an arrow, whereas neutral lines need not.The terminology that we use is inspired by quantum field theory.Therefore, the variables that enter the integral are associated with "particles"; they are divided into "bosons" and "fermions", each subdivided into "neutral" and "charged" particles.In the main part of the chapter we describe the diagram formalism used in quantum many-body physics and quantum field theory.As a preparation, we include a brief Sect.20.2 devoted to the basic terminology of perturbation theory
At present, the only general method available for gaining knowledge from the fundamental principles about the dynamics of a system is the perturbative study. This consists in dividing the Hamiltonian into one part representing a simpler well-understood problem and a nontrivial part, the effect of which is studied order by order. The expressions resulting from perturbation theory quickly become unwieldy. This chapter deals with propagators. It introduces the retarded and advanced propagator and describes their perturbation theoretic structure in a potential in terms of diagrams. A convenient method of representing perturbative expressions by diagrams was invented by Feynman. Besides the appealing aspect of representing perturbative expressions by drawings, the diagrammatic method can also be used directly for reasoning and problem solving. The easily recognizable topology of diagrams makes the diagrammatic method a powerful tool for constructing approximation schemes as well as exact equations that may hold true beyond perturbation theory.
Abstract In this work, the perturbative aspects of quantum mechanics (QM) and quantum field theory (QFT), to a large extent, are studied with functional (path or field) integrals and functional techniques. This physics textbook thus begins with a discussion of algebraic properties of Gaussian measures, and Gaussian expectation values for a finite number of variables. The important role of Gaussian measures is not unrelated to the central limit theorem of probabilities, although the interesting physics is generally hidden in essential deviations from Gaussian distributions. A few algebraic identities about Gaussian expectation values, in particular Wick's theorem are recalled. Integrals over some type of formally complex conjugate variables, directly relevant for boson systems are defined. Fermion systems require the introduction of Grassmann or exterior algebras, and the corresponding generalization of the notions of differentiation and integration. Both for complex and Grassmann integrals, Gaussian integrals, and Gaussian expectation values are calculated, and generalized Wick's theorems proven. The concepts of generating functions and Legendre transformation are recalled.
Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the `bold Feynman diagrammatic expansion' involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In Part I of this two-part series, we illustrate how the combinatorial properties of Feynman diagrams in MBPT can be studied in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature) and provide a self-contained explanation of Feynman diagrams in this setting. We prove the combinatorial validity of the bold diagrammatic expansion, with methods generalizable to several types of field theories and interactions. Our treatment simplifies the presentation and numerical study of known techniques in MBPT such as the self-consistent Hartree-Fock approximation (HF), the second-order Green's function approximation (GF2), and the GW approximation. The bold diagrams are closely related to the Luttinger-Ward (LW) formalism, which was proposed in 1960 but whose analytic properties have not been rigorously established. The analytical study of the LW formalism in the setting of Gibbs measures will be the topic of Part II.
Abstract Chapter 2 is rather descriptive and introduces the notion of functional (path and field) integrals, for boson (the holomorphic formalism) as well as fermion systems (this necessitates the introduction of Grassmann integration) as they are used in physics. Prior to the second half of the twentieth century, the technical tools of theoretical physics were mainly differential or partial differential equations. However, when systematic investigations of large scale systems with quantum of statistical fluctuations began, new tools were required. This led to the development of functional integration. In this chapter, the role of Gaussian measures and Gaussian expectation values is emphasized, leading to Wick’s theorem, a tool for perturbative calculations. Functional integrals provide remarkable tools to study the classical limit and barrier penetration. They provide a simple bridge between non–relativistic quantum mechanics and quantum field theory.
Applying Feynman diagrammatics to non-fermionic strongly correlated models with local constraints might seem generically impossible for two separate reasons: (i) the necessity to have a Gaussian (non-interacting) limit on top of which the perturbative diagrammatic expansion is generated by Wick’s theorem, and (ii) Dyson’s collapse argument implying that the expansion in powers of coupling constant is divergent. We show that for arbitrary classical lattice models both problems can be solved/circumvented by reformulating the high-temperature expansion (more generally, any discrete representation of the model) in terms of Grassmann integrals. Discrete variables residing on either links, plaquettes, or sites of the lattice are associated with the Grassmann variables in such a way that the partition function (as well as all correlation functions) of the original system and its Grassmann-field counterpart are identical. The expansion of the latter around its Gaussian point generates Feynman diagrams. Our work paves the way for studying lattice gauge theories by treating bosonic and fermionic degrees of freedom on equal footing.