The fascinating new world inside the nucleon, of quarks, gluons and colour, the nuclear strong force. How quantum chromodynamics (QCD) was discovered: probing the nucleons with scattering experiments and with increasing energy e+e− colliders, where quarks and gluons appear as hadronic jets.

The colour charges are three. Being the gauge of QCD non-Abelian, the gluons, not only the quarks, are ‘coloured’. How colour charges bind three quarks or a quark–antiquark pair forming hadrons that have zero overall colour charges.

The QCD coupling constant runs as the fine-structure constant, but with increasing momentum transfer, it decreases, instead of growing. Quarks become ‘free’, when they are very close to each other. Only a very small fraction of the proton mass is due to the quark masses, 99% being the energy of the colour field. The QCD vacuum, the status of minimum energy, a very active medium indeed, beautiful to study.

When matter first appeared in the universe, in the first microsecond after the Big Bang, quarks and gluons moved freely in a hot ‘soup’, the quark–gluon plasma. It is created in the laboratory in the ultra-relativistic heavy ion colliders and theoretically analysed with lattice QCD

In classical electrodynamics (CED), the most important quantities are the electric and magnetic fields, which directly determine the forces. In quantum electrodynamics (QED), the potentials are the most important quantities; they determine the energy and momentum exchange between the EM field and matter. Gauge invariance, which in CED is just a mathematical curiosity, becomes fundamental in QED, ruling the gauge symmetry that determines the interaction itself.

The Lamb experiment that opened the way is discussed in detail.

Feynman diagrams are graphic representations of mathematical expressions of scattering or decay amplitudes. Without going into the mathematics, we use them to visually suggest the underlying physics. We show how the propagator describes virtual particles, and how uncertainty and relativity principles, joined, imply the existence of antimatter.

The fine-structure constant, which is the dimensionless expression of the electromagnetic charge, depends on the momentum transfer between the probe and the target charge in the scattering experiment we are performing. The ‘running’ of the coupling constants is a property of all the interactions.

The highest precision measurements and theoretical predictions of the magnetic moments of the electron and of the muon. The precision frontier to search for new physics.

This chapter introduces how we can use the quantum fields introduced in the previous chapter to access amplitudes and, thus, measurable quantities, such as the cross sections and the particle lifetime. More specifically, an educational tour of quantum electrodynamics (QED), which describes the interaction of electrons (or any charged particles) with photons, is proposed. Although this chapter uses concepts from quantum field theory, it is not a course on that topic. Rather, the aim here is to expose the concepts and prepare the reader to be able to do simple calculations of processes at the lowest order. The notions of gauge invariance and the S-matrix are, however, explained. Many examples of Feynman diagrams and the calculation of the corresponding amplitudes are detailed. Summation and spin averaging techniques are also presented. Finally, the delicate concept of renormalisation is explained, leading to the notion of the running coupling constant.

This chapter includes a broad survey of numerical approaches for multiphase flow using classifications based on particle reference frame, relative velocity magnitude, and relative particle size to the grid size. These approaches are then considered for Brownian motion and turbulent diffusion. Throughout this chapter, comments are made on selecting the numerical approach for a given flow based on the approach’’s ability to capture physics and computational requirements (best tools for the job and costs).

The Lagrangian for a charge in electric and magnetic fields is presented. The acceleration of charges in particle accelerators, in laser-produced plasmas and in the production of cosmic rays is described. Emission from charges in magnetic fields is treated in some detail. Synchrotron radiation, undulators, and free electron laser radiation output is examined.

Here, we discuss concepts of duality for convex optimization problems, and algorithms that make use of these concepts. We define the Lagrangian function and its augmented Lagrangian counterpart. We use the Lagrangian to derive optimality conditions for constrained optimization problems in which the constraints are expressed as linear algebraic conditions. We introduce the dual problem, and discuss the concepts of weak and strong duality, and show the existence of positive duality gaps in certain settings. Next, we discuss the dual subgradient method, the augmented Lagrangian method, and the alternating direction method of multipliers (ADMM), which are useful for several types of data science problems.

We provide the briefest introduction to Lagrangian and Hamiltonian Mechanics, and we explore several routes by which the physicists argue that the massive scalar field is a quantization of a natural classical field, the Klein-Gordon field.

Chapter 5 described quantum mechanics in the context of particles moving in a potential. This application of quantum mechanics led to great advances in the 1920s and 1930s in our understanding of atoms, molecules, and much else. But, starting around 1930 and increasingly since then, theoretical physicists have become aware of a deeper description of matter, in terms of fields. Just as Einstein and others had much earlier recognized that the energy and momentum of the electromagnetic field is packaged in bundles, the particles later called photons, so also there is an electron field whose energy and momentum is packaged in particles, observed as electrons, and likewise for every other sort of elementary particle. Indeed, in practice this is what we now mean by an elementary particle: it is the quantum of some field that appears as an ingredient in whatever seem to be the fundamental equations of physics at any stage in our progress.

A notable advance in mechanics took place nearly a century after Newton in the work of the French mathematician and physicist Joseph-Louis Lagrange (1736--1813). We already introduced variational principles in Chapter 3, and showed that they can give the equations of motion of nonrelativistic particles subject to arbitrary conservative forces. Although useful, that is not enough: now we need to see if other kinds of forces can be included in the variational approach. In this chapter we will find that forces of constraint can be included as well, which provides us with deep insights into mechanics and also enormous simplifications in problem solving. We will introduce Hamilton’s principle and the Lagrangian, concepts that are so elegant that we are encouraged to place them at the very heart of classical mechanics. We are further encouraged to do so in the following chapter, the capstone chapter to Part I of the book, where we show how they naturally emerge as we take the classical limit of the vastly more comprehensive theory of quantum mechanics.

We will next move to discuss the Standard Model (SM) of electroweak and strong interactions. We will present the basic SM Lagrangian, andwe will discuss various properties of this simplest theory, but we will not do any phenomenology at this stage. There are many excellent books on the dynamics of the SM, and we will in most cases skip derivations of well-known formulae to save space and in particular energy for the later steps that are not covered in detail in textbooks. The most important result of this step will be collection of the Feynman rules for those interactions present in the SM that we will need for explicit calculations as we proceed. For readers' convenience we will collect these rules in an Appendix.

We will begin by presenting the general structure of gauge theories, stressing their symmetries and their breakdown. The structure of the Lagrangians in this chapter will be fundamental for the full book. We will encounter Lagrangians involvingspin-0, spin-1/2, and spin-1 fields. We will also collect literature in which further details can be found. But the information provided in this chapter should be sufficient for following the book.

This chapter discusses the basic concepts in many-body dynamics.From the Lagrangian, Hamiltonian, canonical transformations and time transformations to Hamilton-Jacobi equations. This content can be found in most classical dynamics textbooks

Lagrangian discrete models are the most ancient and still most effective models used in mechanics to predict the behavior of complex systems. Their structure is presented here, in a very classical way, which follows the classical presentation given by Levi-Civita. Their flexibility is highlighted, together with their capacity to be used in very efficient numerical codes. They must be regarded as a preferred tool also in the formulation of problems in the theory of metamaterials.