Projecting to sets A and B are the elementary operations used by the RRR algorithm to find solutions in their intersection. This chapter covers all the projections that arise in this book.

The periodic table is one of the most iconic images in science. All elements are classified in groups, ranging from metals on the left that go bang when you drop them in water through to gases on the right that don’t do very much at all. The purpose of this chapter is to start to look at the periodic table from first principles, to understand the structure and patterns that lie there.

Many of the most interesting things in fluid mechanics occur because simple flows are unstable. If they get knocked a little bit, the fluid curls up into interesting shapes, or dissolves into some messy turbulent flow. In this chapter, we start to understand how these processes can happen.

Any education in theoretical physics begins with the laws of classical mechanics. The basics of the subject were laid down long ago by Galileo and Newton and are enshrined in the famous equation $F=ma$ that we all learn in school. But there is much more to the subject and, in the intervening centuries, the laws of classical mechanics were reformulated to emphasise deeper concepts such as energy, symmetry, and action. This textbook describes these different approaches to classical mechanics, starting with Newton’s laws before turning to subsequent developments such as the Lagrangian and Hamiltonian approaches. The book emphasises Noether’s profound insights into symmetries and conservation laws, as well as Einstein’s vision of spacetime, encapsulated in the theory of special relativity. Classical mechanics is not the last word on theoretical physics. But it is the foundation for all that follows. The purpose of this book is to provide this foundation.

Much of classical mechanics treats particles as infinitesimally small. But most of our world is not like this. Planets and cats and tennis balls are not infinitesimally small, but have an extended size and this can be important for many applications. The purpose of this chapter is to understand how to describe the complicated motion of extended objects as they tumble and turn.

Jane Dewey (1900−1976) was the only woman in a group that John Slater described as the lucky generation of US physicists: those born near the beginning of the twentieth century and who spent time in Europe, learning with the leading quantum physicists of the era. After completing a PhD at the Massachusetts Institute of Technology in 1925, Dewey went to Niels Bohr’s Institute for Theoretical Physics in Copenhagen. She worked on the Stark effect in helium, a key test of the recently formulated quantum mechanics. Bohr praised her skills in a fellowship application, and Karl Compton later supported her (unsuccessful) efforts to land a permanent job. Although Dewey did pioneering work in the field of quantum optics, the conditions she encountered made it difficult for her to continue on this research path. Her promising abilities did not translate into a successful academic career as they did for many of the men of the lucky generation. Perhaps she was not lucky enough. Or was luck conditional on being a man? This chapter argues that subtle – yet, structural – gender discriminatory practices contributed to her gradual exclusion from physics research, and ultimately from academia.

The purpose of this chapter is to understand how quantum particles react to magnetic fields. There are a number of reasons to do be interested in this. First, quantum particles do extraordinary things when subjected to magnetic fields, including forming exotic states of matter known as quantum Hall fluids. But, in addition, magnetic fields bring a number of new conceptual ideas to the table. Among other things, this is where we first start to see the richness that comes from combining quantum mechanics with the gauge fields of electromagnetism.

For many systems, the full information of an underlying Markovian decription is not accessible due to limited spatial or temporal resolution. We first show that such an often inevitable coarse-graining implies that, rather than the full entropy production, only a lower bound can be retrieved from coarse-grained data. As a technical tool, it is derived that the Kullback–Leibler divergence decreases under coarse-graining. For a discrete time-series obtained from an underlying time-continuous Markov dynamics, it is shown how the analysis of n-tuples leads to a better estimate with increasing length of the tuples. Finally, state-lumping as one strategy for coarse-graining an underlying Markov model is shown explicitly to yield a lower bound for the entropy production. However, in general, it does not yield a consistent interpretation of the first law along coarse-grained trajectories as exemplified with a simple model.

The difference between quantum and classical mechanics does not involve just a small tweak. Instead it is a root and branch overhaul of the entire framework. In this chapter we introduce the key concept that underlies this new framework: the quantum state, as manifested in the wavefunction.