The free electromagnetic field is quantized canonically and with the functional integral. We emphasize the roles of the Gauss law, helicity, and gauge fixing in the continuum. We also derive Planck’s formula for black-body radiation and apply it to the cosmic microwave background.

The company culture at the sports equipment manufacturer, unsurprisingly, is that most employees regularly participate in outside-of-work sports activities.

So far our discussion on information-theoretic methods has been mostly focused on statistical lower bounds (impossibility results), with matching upper bounds obtained on a case-by-case basis. In Chapter 32 we will discuss three information-theoretic upper bounds for statistical estimation under KL divergence (Yang–Barron), Hellinger (Le Cam–Birgé), and total variation (Yatracos) loss metrics. These three results apply to different loss functions and are obtained using completely different means. However, they take on exactly the same form, involving the appropriate metric entropy of the model. In particular, we will see that these methods achieve minimax optimal rates for the classical problem of density estimation under smoothness constraints.

In 1900, sponge divers near the Greek island of Antikythera came upon a Roman shipwreck bestrewn with ancient statues, jewelry, and other treasure, and an odd looking, very encrusted gear mechanism. Retrieved and then studied for decades, it took until 1974 for scientists to unravel the mystery of the mechanism’s purpose. It was a computer, designed and built in Egypt around 100 BCE, that could predict the positions of the sun, moon, and planets, and the timings of solar eclipses and Olympic Games events.

This enthusiastic introduction to the fundamentals of information theory builds from classical Shannon theory through to modern applications in statistical learning, equipping students with a uniquely well-rounded and rigorous foundation for further study. The book introduces core topics such as data compression, channel coding, and rate-distortion theory using a unique finite blocklength approach. With over 210 end-of-part exercises and numerous examples, students are introduced to contemporary applications in statistics, machine learning, and modern communication theory. This textbook presents information-theoretic methods with applications in statistical learning and computer science, such as f-divergences, PAC-Bayes and variational principle, Kolmogorov’s metric entropy, strong data-processing inequalities, and entropic upper bounds for statistical estimation. Accompanied by additional stand-alone chapters on more specialized topics in information theory, this is the ideal introductory textbook for senior undergraduate and graduate students in electrical engineering, statistics, and computer science.

Chapter 29 gives an exposition of the classical large-sample asymptotics for smooth parametric models in fixed dimensions, highlighting the role of Fisher information introduced in Chapter 2. Notably, we discuss how to deduce classical lower bounds (Hammersley–Chapman–Robbins, Cramér–Rao, van Trees) from the variational characterization and the data-processing inequality (DPI) of χ2-divergence in Chapter 7.

In high-energy scattering processes, hadrons can be described as a set of partons. This picture is compatible with QCD, where the partons are identified as quarks, anti-quarks, and gluons. In this picture, we consider electron–positron annihilation, which can lead to hadrons or a muon–anti-muon pair. The R-ratio of the cross sections for these scenarios allows us to identify the number of colors, Nc = 3, experimentally. Next we discuss deep inelastic electron–nucleon scattering, which leads to the concepts of the Bjorken variable, structure functions, the parton distribution function, Bjorken scaling, the Callan–Gross relation, and the DGLAP evolution equation. The hadronic tensor takes us to the scaling functions, where high-energy neutrino–nucleon scattering provides further insight, in particular a set of constraints which are expressed as sum rules.

Starting from 2-flavor QCD, isospin symmetry is employed in order to explain the multiplets of light baryons and mesons, from a constituent quark perspective. Next we involve the strange quark and arrive at meson mixing as well as the Gell-Mann–Okubo formula for the baryon multiplet splitting. Regarding QCD from first principles, we comment on lattice simulation results for the hadron masses. At last we discuss the hadron spectrum in a hypothetical world with Nc=5 colors.

Ethics, like other branches of philosophy, springs from seemingly simple questions. What makes honest actions right and dishonest ones wrong? Why is death a bad thing for the person who dies? Is there anything more to happiness than pleasure and freedom from pain? These are questions that naturally occur in the course of our lives, just as they naturally occurred in the lives of people who lived before us and in societies with different cultures and technologies from ours. They seem simple, yet they are ultimately perplexing. Every sensible answer one tries proves unsatisfactory upon reflection. This reflection is the beginning of philosophy. It turns seemingly simple questions into philosophical problems. And with further reflection, we plumb the depths of these problems.

This enthusiastic introduction to the fundamentals of information theory builds from classical Shannon theory through to modern applications in statistical learning, equipping students with a uniquely well-rounded and rigorous foundation for further study. The book introduces core topics such as data compression, channel coding, and rate-distortion theory using a unique finite blocklength approach. With over 210 end-of-part exercises and numerous examples, students are introduced to contemporary applications in statistics, machine learning, and modern communication theory. This textbook presents information-theoretic methods with applications in statistical learning and computer science, such as f-divergences, PAC-Bayes and variational principle, Kolmogorov’s metric entropy, strong data-processing inequalities, and entropic upper bounds for statistical estimation. Accompanied by additional stand-alone chapters on more specialized topics in information theory, this is the ideal introductory textbook for senior undergraduate and graduate students in electrical engineering, statistics, and computer science.

The fermion content of the Standard Model is extended to 3 generations. For the lepton we discuss universality, and for the quarks the GIM mechanism, the CKM quark mixing matrix, and its CP violating parameter. Similarly, for the leptons we construct the PMNS mixing matrix, and we describe how neutrino oscillation was observed. Since the Standard Model is complete now, we provide an overview over its parameters and take another look from an unconventional low-energy perspective.

This chapter first focuses on the QCD vacuum θ and the related strong CP problem. We review the manifestation of theta in the QCD action or alternatively in the mass matrix, and in chiral perturbation theory. Beyond the Standard Model, the Peccei–Quinn formalism turns θ into an axion field. We discuss that approach and its implications in astrophysics and cosmology. Finally we consider the corresponding parameters for the SU(2)L gauge field and for QED. The former can be absorbed by field-redefinitions, but the latter leads to a linear combination of these two vacuum angles, which persists as a parameter of the Standard Model, but which is often ignored.

The Standard Model is reduced to Quantum Chromodynamics (QCD) only by a limitation to moderate energies. We review crucial properties like asymptotic free and the role of chiral symmetry. The latter is analyzed both in the continuum and on the lattice. In particular, the Ginsparg–Wilson relation for lattice Dirac operators allow us to properly address the hierarchy problem which appears fermion masses at the non-perturbative level.

The gluon SU(3) gauge field is studied, with “quarks” only as static sources. We describe confinement by referring to the Wegner–Wilson loop and its strong-coupling expansion on the lattice. The way back to the continuum is related to asymptotic freedom. We discuss the strength of the strong interaction, its low-energy string picture, and the Luescher term as a Casimir effect. The Fredenhagen–Marcu operator provides a sound confinement criterion. In the confined phase we discuss the glueball spectrum, the Polyakov loop, and center symmetry. We also consider deconfinement at high temperature, and finally the case of a G(2) gauge group instead of SU(3).