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Chapter 11 is the first of four chapters that discuss various issues connected with the Standard Model of fundamental interactions at the microscopic scale. It discusses the important notion of gauge invariance, first Abelian and then non–Abelian, the basic geometric structure that generates interactions. It relates it to the concept of parallel transport. Due to gauge invariance, not all components of the gauge field are dynamical and gauge fixing is required (with the problem of Gribov copies in non–Abelian theories). The quantization of non–Abelian gauge theories is briefly discussed, with the introduction of Faddeev–Popov ghost fields and the appearance of BRST symmetry.

This chapter focuses on abelian gauge theory, whose physical realization is Quantum Electrodynamics (QED). The chapter is organized as follows. It begins with elementary considerations about the massive vector field in perturbation theory. It shows that coupling to matter field leads to field theories that are renormalizable in four dimensions only if the vector field is coupled to a conserved current. In the latter case the massless vector limit can be defined. The corresponding field theories are gauge invariant. It then discusses the specific properties of gauge invariant theories and mentions the IR problem of physical observables. It quantizes gauge theories starting directly from first principles. The formal equivalence between different gauges is established. Regularization methods are presented which allow overcoming the new diffculties one encounters in gauge theories. The abelian gauge symmetry, broken by gauge fixing terms, then leads to a set of WT identities which are used to prove the renormalizability of the theory. The gauge dependence of correlation functions in a set of covariant gauges is determined. Renormalization group equations follow and the RG β-function is calculated at leading order. As an introduction to the next chapter, the abelian Higgs mechanism is analyzed. Finally, the chapter comments about stochastic quantization of gauge theories.

A geometrical derivation of Abelian and non- Abelian gauge theories. The Faddeev–Popov quantisation. BRST invariance and ghost fields. General discussion of BRST symmetry. Application to Yang–Mills theories and general relativity. A brief history of gauge theories.

In a remarkable lecture Julian Schwinger delivered at the University of Nottingham on July 14, 1993, on the occasion of his receiving an honorary degree, entitled ‘The Greening of quantum field theory: George and I’, he summarised the central role Green's function played throughout his career. Schwinger then went on to recount his experience at the Massachusetts Institute of Technology's Radiation Laboratory during World War II and traced the influences of George Green on his own works. This chapter chronicles Schwinger's research in relation to Green's function, his first trip to Europe, and his work on the gauge invariance and vacuum polarization, the quantum action principle, electrodynamic displacements of energy levels, quantum field theory, and condensed matter physics.

This chapter derives the kinetic equations for the two-component distribution function in a gauge-invariant form. The collision integrals for interaction of excitations with impurities, phonons, and with each other are written down. The gauge-invariant expressions for electron density, electric current, heat current, and order parameter are obtained. Kinetic equations for dirty superconductors are derived. Heat conduction in superconducting state is considered.

A brief introduction to particle physics and quantum field theory (QFT) is presented in the main thread of chapter 5. The impasse of unification in particle physics is historically reviewed, showing that the dynamical paradigm pervades the development of particle physics and QFT. Thus, as with the conundrums of general relativity and quantum mechanics, dynamical explanation in the mechanical universe is responsible for the impasse regarding unification in particle physics as per QFT. It is shown that RBW’s adynamical approach provides an entirely new view of unification and particle physics. Philosophy of Physics for Chapter 5 uses RBW to resolve the interpretational issues of gauge invariance, gauge fixing, the Aharonov–Bohm effect, regularization, and renormalization, and largely discharges the problems of Poincaré invariance in a graphical approach, inequivalent representations, and Haag’s theorem. Foundational Physics for Chapter 5 shows how classical field theory is related to QFT and introduces gauge fields per QFT.

This chapter expresses the fields E(r, t) and B(r, t) in terms of the electromagnetic potentials Ф(r, t) and A(r, t), and shows that these potentials are defined only up to a gauge transformation. This leads the reader naturally to the Coulomb and Lorenz gauges which are usually encountered in textbooks. The inhomogeneous wave equations whose solutions are the retarded electromagnetic potentials are also considered, as well as the Lienard–Wiechert potentials for an arbitrarily moving point charge. A few questions are included in which the Lagrangian and Hamiltonian of a point charge are expressed in terms of Ф and A. The chapter concludes by deriving a multipole expansion for the dynamic vector potential which will provide the starting point in our treatment of electromagnetic radiation later on in Chapter 11

Arguably the most important achievement of quantum field theory is quantum electrodynamics (QED), which describes the interaction between electrons and photons, and this theory is presented in outline in this chapter.

This chapter studies the structure of Maxwell’s equations in a vacuum and the action from which they are derived, while emphasizing the consequences of their gauge invariance. Gauge invariance, on the one hand, allows one of the components of the magnetic potential to be chosen freely. Here, the chapter shows how the gauge-invariant version of the Maxwell equations in the vacuum can also be derived directly by extremizing. On the other hand, the chapter argues that gauge invariance imposes a constraint on the initial conditions such that in the end the general solution has only two ‘degrees of freedom’. Finally, the chapter develops the Hamiltonian formalisms in the Maxwell theory and compares them to the formalisms using non-gauge-invariant or massive vector fields.

This chapter presents Maxwell equations determining the electromagnetic field created by an ensemble of charges. It also derives these equations from the variational principle. The chapter studies the equation’s invariances: gauge invariance and invariance under Poincaré transformations. These allow us to derive the conservation laws for the total charge of the system and also for the system energy, momentum, and angular momentum. To begin, the chapter introduces the first group of Maxwell equations: Gauss’s law of magnetism, and Faraday’s law of induction. It then discusses current and charge conservation, a second set of Maxwell equations, and finally the field–energy momentum tensor.

These lecture notes deal with Ising models with interactions containing products of more than two spins. After a number of examples have been given, some general statements are presented. Of particular interest is a gauge-invariant Ising model in four dimensions. This has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson loop, yielding an interaction that increases proportionally to the distance and thus corresponds to quark confinement. The other phase yields a perimeter law allowing for a quark–gluon plasma.

This chapter describes theories that combine the ideas of gauge symmetry and spontaneous symmetry breaking. It explains that this combination gives rise to massive spin-1 bosons. This construction is used to propose fundamental equations for the weak interaction. The predictions of these equations for high-energy neutrino scattering are worked out and compared to experiment.

So far, general relativity has been viewed from the four-dimensional Lagrangian perspective. This chapter introduces the (3+1)-dimensional Hamiltonian formalism, starting with the ADM form of the metric and extrinsic curvature. The Hamiltonian form of the action is served, and the nature of the constraints—and, more generally, of constraints and gauge invariance in Hamiltonian systems—is discussed. The formalism is used to count the physical degrees of freedom of the gravitational field. The chapter ends with a discussion of boundary terms and the ADM energy.

Using a gauge-invariant formulation of Green’s function, the electric-field dependent kinetic equations are derived in Born and RPA (dynamically screened) approximation. The feedback and Debye–Onsager relaxation effects are discussed and explicitly calculated for two- and three-dimensional systems. It is found that only the asymmetrically screened result in accordance with the asymmetric cummulant expansion of chapter 11 can describe the correct relaxation effect. The conductivity with electron-electron interaction is presented and the adiabatic as well as isothermal approximations introduced. All expressions are calculated for an example of a quasi two-dimensional electron gas.