Abstract Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.
Abstract Chapter 7 covers the main reasons for adopting the methods of quantum field theory (QFT) to study the critical phenomena. It presents both the canonical quantization and the path integral formulation of the field theories as well as the analysis of the perturbation theory. The chapter also covers transfer matrix formalism and the Euclidean aspects of QFT, the field theory of the Ising model, Feynman diagrams, correlation functions in coordinate space, the Minkowski space and the Legendre transformation and vertex functions. Everything in this chapter will be needed sooner or later, since it highlights most of the relevant aspects of quantum field theory.
In this chapter we review the main ideas of the canonical quantization in nonrelativistic quantum mechanics and optics, which we suppose the reader learned in previous courses. We first consider the least action principle and the action functional for classical many-body systems and also for the Schrödinger wavefunction, interpreted as the complex field of a classical field theory. Then, after having analyzed the Legendre transformation from the Lagrangian to the Hamiltonian formulation, we illustrate the canonical quantization of particles, and finally we extend the approach to the canonical quantization of bosonic and fermionic fields, considering also the role of temperature.
Abstract Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.
Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.
Abstract Chapter 8 introduces the key ideas of the renormalization group, including how they provide a theoretical scheme and a proper language to face critical phenomena. It covers the scaling transformations of a system and their implementations in the space of the coupling constants and reducing the degrees of freedom. From this analysis, the reader is led to the important notion of relevant, irrelevant and marginal operators and then to the universality of the critical phenomena. Furthermore, the chapter also covers (as regards the RG) transformation laws, effective Hamiltonians, the Gaussian model, the Ising model, operators of quantum field theory, universal ratios, critical exponents and β-functions.
Abstract Quantum field theory with boundaries is a very rich subject both from purely theoretical point of view or applicative aims. Chapter 21 presents the basis results relative to critical systems and massive integrable field theories. It covers stress-energy tensor in boundary conformal field theories and integrable deformations of these theories (including the single Virasoro algebra), reflection amplitudes, conformal boundary operators and states (including the Hilbert space and Ishibashi state), non-renormalizable states of the Hilbert space, operator product expansion involving a boundary operator, massive integrable boundary field theory, the massive boundary Ising model, boundary magnetic fields and correlation functions.
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.