This chapter introduces the formal second quantization method for fermions in quantum field theory, and the connection to second quantization of bosons is shown. The picture of fermions as rotations between two states is presented, which helps the reader to see where the Pauli exclusion rule comes from. Finally, Dirac’s original derivation of his equation for relativistic motion of fermions is given.

All examples seen in the preceding chapters have dealt with a single particle. In this chapter, the theory is expanded to systems with several identical particles. Here, ‘many’ in practice means two. However, this does allow the introduction of several central aspects. Perhaps the most important one is spin, which is the topic of the first part. Central elements in this context are the Stern–Gerlach experiment and the Pauli matrices. The characteristics of these matrices are studied in some detail as they play crucial roles in the remainder of the book. The concept of entanglement in quantum physics is introduced – exemplified using both the two-particle spin wave function and the combined spin–space wave function for a single particle. Due to the Pauli principle, the importance of spin and exchange symmetry in a many-body context is hard to underestimate. The fact that identical particles are indistinguishable has implications for the symmetry of the wave function. This, in turn, has significant consequences for the structure of the system – including its ground state. This is investigated by performing calculations of energy estimates. Most of these apply the variational principle, but also the notion of self-consistent field and the Hartree–Fock method are introduced.

Political technology' is not a term much used in the West. But spin and political consulting are outdated labels. Spin doctors and political consultants do more than spin or consult; they also meet the definition of engineers of the political system. Particularly in the United States, where a different type of political universe has been constructed: Political Action Committees, dark money and astroturfing.

A good knowledge of special relativity and quantum mechanics is essential for studying particle physics. Even if the reader is assumed to be already familiar with these two theories, a brief review of special relativity is given in this chapter with emphasis on the covariant and contravariant notations, which may be less well mastered but are very useful in particle physics. Important aspects of quantum mechanics for particle physics, such as the angular momentum, are also addressed.

This chapter discusses modification of the fluid dynamic point forces due to proximity to the wall and due to neighboring particles, where the latter focuses on the Richardson–Zaki exponent. In addition, particle collision with other particles and with walls is discussed for normal and tangential restitution. This includes effects of viscoelasticity, spin, plasticity, fluid viscosity, and adhesion for solid particles as well as deformation and wetting for fluid particles.

All news is fake news, because all reports are to some extent ‘made up’ by the time they are received by a mediated consumer distanced from the original source. ‘Fake’, from the Latin facere (to make, to do), is a member of the family of making words that includes fact, factory, fashion, artificial, and face. It is ironic that the standard test for whether news is ‘fake’ is to subject it to ‘fact-checking’. Facts themselves are things – artefacts – that we make through artificial processes of Creation and Production. Any ‘fact’ deserving of the name is something established by some process involving human skill and judgment. What matters is not whether news or facts are made up – they always are – but how they are made up and what relation there is between the thing at source and the thing as made up for public reception. Public reception also plays its part in the broadcast of fake news. We therefore need to think in terms of ‘receiver responsibility’, from the case of the journalist who receives the factual grain of a promising story to the editor who publishes journalists’ copy to the online user who re-tweets a tweet.

Through the Stern-Gerlach experiment, we demonstrate several key concepts about quantum mechanics: quantum mechanics is probabilistic; spin measurements are quantized; quantum measurements disturb the system. We show how to describe the state of a quantum mechanical system mathematically using a ket, which represents all the information we can know about that state.

For a proper quantum mechanical description of multiple-particle systems, we must account for the indistinguishability of fundamental particles. The symmetrization postulate requires that the quantum state vector of a system of identical particles be either symmetric or antisymmetric with respect to exchange of any pair of identical particles within the system. Nature dictates that integer spin particles – bosons – have symmetric states, while half-integer spin particles – fermions – have antisymmetric states. The best-known manifestation of this is the Pauli exclusion principle, which limits the number of electrons in given atomic levels and leads to the structure of the periodic table.

We learn how to use perturbation theory to solve more realistic problems that do not admit exact solutions. We learn degenerate and nondegenerate perturbation theory and apply them to a variety of problems, including spin magnetic moments in magnetic fields and the Stark effect in hydrogen.

We give a mini-review of representations of the Poincaré group, emphasizing the role of irreducible unitary representations in identifying ‘elementary particles’. Young tableaux are used to identify irreducible representations of the little group and thus the particle content of the excitation spectrum of a string.

Wigner’s idea, that to each elementary particle is associated an irreducible representation of the Poincare group gives fundamental importance to these representations. They are non-trivial mathematical objects. We strive to give a mathematically sound and complete description of the physically relevant representations, and the multiple ways they can be presented, while avoiding the pitfall of relying on advanced representation theory. The representations corresponding to massive particles depend, besides the mass, on a single non-negative integer which corresponds to the spin of the particle. The representations which correspond to massless particles depend on an integer, the helicity, which is a property somewhat similar to the spin. We investigate that action of parity and the operation of taking a “mirror image” of a particle. Finally we provide a brief account of Dirac’s equation.

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.

Our modern understanding of atoms, molecules, solids, atomic nuclei, and elementary particles is largely based on quantum mechanics. Quantum mechanics grew in the mid-1920s out of two independent developments: the matrix mechanics of Werner and the wave mechanics of Erwin Schrödinger. For the most part this chapter follows the path of wave mechanics, which is more convenient for all but the simplest calculations. The general principles of the wave mechanical formulation of quantum mechanics are laid out and provide a basis for the discussion of spin, identical particles. and scattering processes. The general principles are supplemented with the canonical formalism to work out the Schrödinger equation for charged particles in a general electromagnetic field. The chapter ends with the unification of the approaches of wave and matrix mechanics by Paul Dirac, and a modern approach, known as Hilbert space, is briefly described.

Many different phases of matter can be characterized by the symmetries that they break.The Ising model for interacting spins illustrates this idea.In the absence of a magnetic field, there is a critical temperature, below which there is ferromagnetic ordering, and above which there is not.The magnetization is the order parameter for this transition: it is non-zero only when there is ferromagnetic ordering.The ferromagnetic phase transition in the Ising model is explored using the approximate method of mean field theory.Exact solutions are known for the Ising model in one and two dimensions and are discussed, along with numerical solutions using Monte Carlo simulations.Finally, the ideas of broken symmetry and their relationship to phase transitions are placed in the general framework of Landau theory and compared to results from mean field theory.

The concepts of angular momentum, spin and magnetic moment are worked out using standard quantum mechanical formalism. The concepts of intrinsic spin of a pointlike particle is contrasted with the intrinsic angular momentum of composite particles. The Larmor frequency and the magnetic resonance of non-interacting spins are introduced. The quantum statistics of a system of spins is overviewed, before introducing the thermodynamics of a spin system in a static frame of reference. Nuclear magnetic phase transitions are briefly reviewed.