The gluon SU(3) gauge field is studied, with “quarks” only as static sources. Wedescribe confinement by referring to the Wegner–Wilson loop and its strong-couplingexpansion 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 theLuescher term as a Casimir effect. The Fredenhagen–Marcu operator provides a soundconfinement criterion. In the confined phase we discuss the glueball spectrum, thePolyakov 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).

A commonly used method in combinatorics for bounding the number of certain objects from above involves a smart application of Shannon entropy. Notably the precision of this application can be increased by three methods: marginal bound, pairwise bound (Shearer’s lemma and generalization, see Theorem 1.8), and the chain rule (exact calculation).

In Chapter 7, we give three applications using the above three methods, respectively, in order of increasing difficulty:

1. (1) enumerating binary vectors of a given average weight;

2. (2) counting triangles and other subgraphs; and

3. (3) Brégman’s theorem.

We sketch the highlights of Part I, which introduces the concepts of quantum fieldtheory.

Chapter 26 evaluates the rate-distortion function for Gaussian and Hamming sources. We also discuss the important foundational implication that an optimal (lossy) compressor paired with an optimal error-correcting code together form an optimal end-to-end communication scheme (known as the joint source–channel coding separation principle). This principle explains why “bits” are the natural currency of the digital age.

In Chapter 31 we study three commonly used techniques for proving minimax lower bounds, namely, Le Cam’s method, Assouad’s lemma, and Fano’s method. Compared to the results in Chapter 29, which are geared toward large-sample asymptotics in smooth parametric models, the approach here is more generic, less tied to mean-squared error, and applicable in non-asymptotic settings such as nonparametric or high-dimensional problems. The common rationale of all three methods is reducing statistical estimation to hypothesis testing.

In Chapter 2 we introduced the Kullback–Leibler (KL) divergence that measures the dissimilarity between two distributions. This turns out to be a special case of the family of f-divergences between probability distributions, introduced by Csiszár. Like KL divergence, f-divergences satisfy a number of useful properties: operational significance, invariance to bijective transformations, data-processing inequality, variational representations (à la Donsker–Varadhan), and local behavior.

The purpose of Chapter 7 is to establish these properties and prepare the ground for applications in subsequent chapters. The important highlight is a joint-range theorem of Harremoës and Vajda, which gives the sharpest possible comparison inequality between arbitrary f-divergences (and puts an end to a long sequence of results starting from Pinsker’s inequality – Theorem 7.10).

Chapter 3 defines perhaps the most famous concept in the entire field of information theory, mutual information. It was originally defined by Shannon, although the name was coined later by Robert Fano. It has two equivalent expressions (as a Kullback–Leibler divergence and as difference of entropies), both having their merits. In this chapter, we collect some basic properties of mutual information (non-negativity, chain rule, and the data-processing inequality). While defining conditional information, we also introduce the language of directed graphical models, and connect the equality case in the data-processing inequality with Fisher’s concept of sufficient statistics. The connection between information and estimation is furthered in Section 3.7*, in which we relate mutual information and minimum mean-squared error in Gaussian noise (I-MMSE relation). From the latter we also derive the entropy-power inequality, which plays a central role in high-dimensional probability and concentration of measure.

In Chapter 30 we describe a strategy for proving the statistical lower bound we call the mutual information method (MIM), which entails comparing the amount of information data provides with the minimum amount of information needed to achieve a certain estimation accuracy. Similar to Section 29.2, the main information-theoretical ingredient is the data-processing inequality, this time for mutual information as opposed to f-divergences.

There is a criminal trial in progress and the prosecutor tells the jury that the defendant’s DNA was found at the crime scene. This, says the prosecutor, is enough to prove that the defendant did commit the crime. In fact, someone’s DNA was indeed found at the crime scene, and it could be the defendant’s, according to a DNA expert who testifies that there is only a 1 in a million chance that the DNA could be from someone else. Even so, the prosecutor insists, that means there is a 99.99999% chance that the DNA is from the defendant, which is so overwhelmingly conclusive that any reasonable person would agree the DNA must be from the defendant. The prosecutor presses the jury for a conviction.

We stay in the framework of the low-energy effective theory of QCD in terms ofNambu–Goldstone bosons fields and consider effects due to their topology. We distinguishthe cases of Nf = 2 or Nf >= 3 light quark flavors and discuss in both cases how thegauge anomaly cancelation is manifest in the effective theory, the role of G-parity, andthe neutral pion decay into two photons, which does not explicitly depend on the number ofcolors, Nc . For Nf >= 3 we introduce the Wess–Zumino–Novikov–Witten term in a 5thdimension, we discuss the intrinsic parity of light meson fields and their electromagneticinteractions. In this context, we clarify the question whether there is low-energyevidence for Nc = 3, and we address again therole of technicolor.

What interest could rational agents have in acting lawfully if not for the order, stability, and other collective goods that law brings to society? Why should it otherwise matter to them that their actions are lawful? It would matter to them, of course, if acting unlawfully made them liable to punishment. But in that case their interest in acting lawfully would not come from seeing it as a good thing. It would come, rather, from seeing it as the surest way to avoid a bad thing, something they have an interest in escaping. Yet the challenge to an ethics like Kant’s that represents lawfulness as the essence of moral action is to explain what could interest rational agents in acting lawfully regardless of how the law is enforced, regardless, that is, of whether it is enforced by threats of punishment or incentives to obey. The question, then, that confronts a defender of Kant’s ethics is why a rational agent should regard an action’s being lawful as a condition of its being reasonable to do. If he cannot give an answer to this question, the charge of excessive formalism will stick.

The Higgs mechanism is introduced, first for scalar QED and then with the Higgs doublet,which takes us to the gauge bosons in the electroweak sector of the Standard Model. Nextwe discuss variants of “spontaneous symmetry breaking” patterns, which deviate from theStandard Model, in the continuum and on the lattice. Finally we consider a “smallunification” of the electroweak gauge couplings, as a toy model for the concept of GrandUnified Theories (to be address in Chapter 26).

The topological charge of smooth Yang–Mills gauge fields is discussed, describing inparticular the SU(2) instanton. This leads to the Adler–Bell–Jackiw anomaly and to θ-vacuum states, which are similar to energy bands in a crystal. Wefinally discuss the Atiyah–Singer index theorem in the continuum and more explicitly onthe lattice.

So far our discussion of channel coding was mostly following the same lines as the M-ary hypothesis testing (HT) in statistics. In Chapter 18 we introduce a key departure from this: The principal and most interesting goal in information theory is the design of the encoder mapping an input message to the channel input. Once the codebook is chosen, the problem indeed becomes that of M-ary HT and can be tackled by standard statistical tools. However, the task of choosing the encoder has no exact analogs in statistical theory (the closest being design of experiments). It turns out that the problem of choosing a good encoder will be much simplified if we adopt a suboptimal way of testing M-ary HT, based on thresholding information density.