Newtonian mechanics was the first great synthesis of modern physics. It provided the main theory about the workings of the physical world from the date of its first presentation (1687) until the beginnings of the twentieth century.

The study of composite systems typically requires their analysis into simpler systems. In classical physics, the simplest systems are particles, that is, pointlike bodies that move in space. A particle is traditionally described by three position coordinates and three momenta, so the associated state space is ${ℝ}^6$. In Chapter 14, we will see that this description is incomplete: Particles also have an additional degree of freedom called spin.

The fundamental kinematical symmetry is the invariance under transformations between inertial reference frames. In the regime of small velocities, this symmetry corresponds to the Galilei group. However, the Galilei symmetry is only approximate. The exact symmetry, in the absence of gravity, is defined by the Poincaré group, which we analyse here.

In Chapter 11, we saw that by Wigner’s theorem, symmetry transformations in quantum theory are represented by unitary or antiunitary operators. Then, we focused exclusively in the symmetry of space rotations. Since our focus was so narrow, we did not have to introduce the most appropriate language for the description of symmetries, namely group theory.

In Chapter 9, we presented the quantum rule of combination of subsystems through the tensor product. In this chapter, we will discuss a key elaboration of this rule that applies to composite systems with a specific symmetry, namely, invariance under exchange of identical particles.

Spin was introduced as part of the effort to understand the structure of atoms prior to the development of mature quantum theory. Variations of Bohr’s model described atoms in terms of three quantum numbers, roughly similar to $n,𝓁$, and $m$ that appear when solving the Schrödinger equation in central potentials. In this context, Pauli proposed that, in each atom, there exists at most one electron for each triplet of quantum numbers. This proposal is Pauli’s famous “exclusion principle,” which we will analyze in Chapter 15.

The first step towards quantum theory was a response to a problem that could not be addressed by the concepts and methods of classical physics: the radiation from black bodies.

One of the most important concepts of classical mechanics is that of a closed system. A closed system is loosely defined as a system whose components interact only with each other, and it is characterized by phase space volume conservation and energy conservation – see Section 1.2.

We saw in Chapter 2 that Born’s statistical interpretation of the wave function was one of the building blocks of quantum theory. According to Born’s interpretation, the wave function of a particle at a given moment of time defined a probability density $|\psi (x){|}^2$ with respect to the position $x$. This result is generalized to state vectors of an Hilbert space and to general observables through the following procedure.

In Chapter 1, we presented the fundamental principles of classical physics, and then we motivated and presented the fundamental principles of quantum physics. The two sets of principles are summarized and compared in Table 10.1.

This chapter addresses a basic integer encoding problem whose impact on the total memory footprint and speed performance of the underlying application is too easily underestimated or neglected. The problem consists of squeezing the space (in bits) required to store an increasing sequence of integers, and then supporting efficient query operations such as decompressing the sequence from the beginning or from some other position, checking whether an integer occurs in the sequence, or finding the smallest integer larger than the queried one. This problem occurs in several common applications, such as in the storage of the posting lists of search engines, or of the adjacency lists of trees and graphs, or of the encoding of sequences of offsets (pointers). The integer coders here discussed, analyzed, and illustrated with many running examples are Elias’ γ- and δ-codes, Rice’s code, PForDelta code, variable-byte code, (s, c)-dense codes, interpolative code, and, finally, the very elegant and powerful Elias–Fano code.

This chapter addresses a problem related to lists, the basic data structure underlying the design of many algorithms that manage interconnected items. It starts with an easy-to-state but I/O-inefficient solution derived from the optimal one designed for the classic RAM model; it then discusses increasingly sophisticated solutions that are elegant and efficient in the two-level memory model, and are still simple enough to be implemented with a few lines of code. The treatment of this problem will also allow us to highlight a subtle relation between parallel computation and I/O-efficient computation, which can be deployed to derive efficient disk-aware algorithms from efficient parallel algorithms.