Self-organized criticality (SOC) is a theoretical concept that describes the statistics of nonlinear processes. It is a fundamental principle common to many nonlinear dissipative systems in the universe. Due to its ubiquity, SOC is a law of nature, for which we derive a theoretical framework and specific macroscopic physical models. Introduced by Bak, Tang, and Wiesenfeld in 1987, the SOC concept has been applied to laboratory experiments of sandpiles, to human activities such as population growth, language, economy, traffic jams, or wars, to biophysics, geophysics, magnetospheric physics, solar physics, stellar physics, and to galactic physics and cosmology. From an observational point of view, the hallmark of SOC behavior is the power law shape of occurrence frequency distributions of spatial, temporal, and energy scales, implying scale-free nonlinear processes. Power laws are neither a necessary nor a sufficient condition for SOC behavior, because intermittent turbulence produces power law-like size distributions also. A novel trend that is ongoing in current SOC research is a paradigm shift from “microscopic” scales toward “macroscopic” modeling based on physical scaling laws.

The starting point for string theory is the idea that the elementary constituents of the theory, which in quantum field theory are assumed to be point-like, are in fact one-dimensional objects, namely strings. As time evolves, a string sweeps out a Riemann surface whose topology governs the interactions that result from joining and splitting strings. The Feynman–Polyakov prescription for quantum mechanical string amplitudes amounts to summing over all topologies of the Riemann surface, for each topology integrating over the moduli of the Riemann surface, and for each value of the moduli solving a conformal field theory. Modular invariance plays a key role in the reduction of the integral over moduli to an integral over a single copy of moduli space and, in particular, is responsible for rendering string amplitudes well behaved at short distances. In this chapter, we present a highly condensed introduction to key ingredients of string theory and string amplitudes, relegating the important aspects of toroidal compactification and T-duality to Chapter 13 and a discussion of S-duality in Type IIB string theory to Chapter 14.

Derivation of the Born approximation and criteria for its validity; applications of the Born approximation to scattering in Coulomb and Yukawa potentials; derivation of the optical theorem; perturbative expansion of the scattering wave function and scattering amplitude; scattering in a hard-sphere potential at low and high energy; scattering in potential well and resonances; partial wave expansion of the integral equation for scattering in a central potential; scattering in a spin-dependent potential; phase shifts in Born approximation; effective range theory; phase shifts at high energy and the eikonal approximation for the scattering amplitude.

The trip first takes in Japan, then Hong Kong, and Singapore. In Singapore, he loses a case containing all of his notes, and learns to lecture without them. He visits some Indian institutes and then flies to Israel. After tourist stopovers in Istanbul and Athens, they arrive finally in London as guests of Abdus Salam. He begins work on a research project on bound states in strong forces, which leads nowhere. He gives invited talks throughout the UK. They spend some time in Italy before attending the Rochester Conference of 1962, in Geneva. Weinberg receives news that he has been promoted to associate professor. They return to New York via Lisbon, after 280 days away.

A Riemann surface is a connected complex manifold of two real dimensions or equivalently a connected complex manifold of one complex dimension, also referred to as a complex curve. In this appendix, we shall review the topology of Riemann surfaces, their homotopy groups, homology groups, uniformization, construction in terms of Fuchsian groups, as well as their emergence from two-dimensional orientable Riemannian manifolds. All these ingredients provide crucial mathematical background for two-dimensional conformal field theory on higher genus Riemann surfaces and its application to string theory.

In this chapter, we shall draw together a number of different strands of inquiry addressed in Chapters 5, 12, and 13. We shall study the interplay between superstring amplitudes, their low-energy effective interactions, Type IIB supergravity, and the S-duality symmetry of Type IIB superstring theory. We begin with a brief review of Type IIB supergravity which, in particular, provides the massless sector of Type IIB superstring theory. We then discuss how the SL(2,R) symmetry of Type IIB supergravity is reduced to the SL(2,Z) symmetry of Type IIB superstring theory via an anomaly mechanism. We conclude with a discussion of how the low-energy effective interactions induced by string theory on supergravity may be organized in terms of modular functions and modular graph forms under this SL(2,Z) symmetry, and match the predictions provided by perturbative calculations of Chapter 12.

The orbital angular momentum operator is defined and its commutation relations with the position and momentum operators, and generally with vector operators, are obtained; the relationship between the square of the momentum operator and the square of the orbital angular momentum is derived; the spectrum of the square and z-component of the orbital angular momentum is obtained by solving the Schröedinger equation near the origin; the radial equation is derived and the spherical harmonics are obtained as solutions of the associated Legendre equation.

In this chapter, we shall discuss modular forms for the congruence subgroups introduced in Chapter 6. We shall obtain the dimension formulas for the corresponding rings of modular forms and cusp forms, describe the fields of modular functions on the modular curves introduced in Chapter 6, and construct the associated Eisenstein series. Throughout the chapter, we shall make use of the correspondence between modular forms and differential forms, viewed as sections of holomorphic line bundles on the compact Riemann surface of the modular curve. We shall provide concrete examples of modular forms for the standard congruence subgroups and apply the results to the theorems of Lagrange and Jacobi on counting the number of representations of an integer as a sum of squares.

In this appendix, we collect some basic results in number theory, including the Chinese remainder theorem, its application to solving polynomial equations, the Legendre and Jacobi quadratic residue symbols, quadratic reciprocity, its application to solving quadratic equations modulo N, and a brief introduction to Dirichlet characters and Dirichlet L-functions.

We close with a brief introduction to Galois theory and illustrate the application of these mathematical ideas in physics through examples from conformal field theory.

In this chapter, we construct differential equations in the modular parameter and find solutions to these equations in simple cases. The solutions can generically be assembled into vector-valued modular forms, which have proven fruitful in recent works in mathematics and physics. We will establish that, in general, each component of a vector-valued modular form is a modular form for a congruence subgroup.

Examples of addition of two angular momenta; derivation of the addition formula for two spherical harmonics; implications of conservation of angular momentum and parity for two-body decays; addition of three angular momenta.

Closely related variants of modular forms, including quasi-modular forms, almost-holomorphic modular forms, Maass forms, non-holomorphic Eisenstein series, mock modular forms, and quantum modular forms, are introduced and their properties are analyzed.

The author describes his parents’ upbringing and move to New York around the time of the Great Depression. The young Weinberg is encouraged to read widely and later takes inspiration from Norse myths from the Poetic Edda.

In this penultimate chapter, we shall discuss dualities in Yang–Mills theories with extended supersymmetry in four-dimensional Minkowski space-time. We briefly review supersymmetry multiplets of states and fields and the construction of supersymmetric Lagrangian theories with N = 1, 2, and 4 Poincaré supersymmetries. We then discuss the SL(2,Z) Montonen–Olive duality properties of the maximally supersymmetric N = 4 theory and the low-energy effective Lagrangians for N = 2 theories via the Seiberg–Witten solution. We shall close this chapter with a discussion of dualities of N = 2 superconformal gauge theories, which possess interesting spaces of marginal gauge couplings. In some cases, these spaces of couplings can be identified with the moduli spaces for Riemann surfaces of various genera.

The lowering, raising, and number operators are defined and their properties are studied; the eigenfunctions of the harmonic oscillator Hamiltonian are derived; coherent or quasi-classical states are obtained and their properties examined. Several applications are discussed, including, among others, the isotropic harmonic oscillator in N dimensions and a model for a one-dimensional crystal.