Several experimental facts cannot be explained by classical physics (Newtonian mechanics and Maxwell’s equations): the observed black-body radiation spectrum, the stability of atoms and associated spectral lines, the heat capacities of solids, and several others. The problems posed in this chapter are meant to illustrate and analyze the failure of classical physics in explaining these phenomena and how this failure points to the need for a radically new treatment.

The goal of this book is to exhibit the profound and myriad interrelations between the mathematics of modular forms and the physics of string theory. Our presentation is intended to be informal but mathematically precise, logically complete, and reasonably self-contained. The exposition is kept as simple as possible so as to be accessible to adventurous undergraduates, motivated graduate students, and dedicated professionals interested in the interface between theoretical physics and pure mathematics. Assuming little more than a knowledge of complex function theory, we introduce elliptic functions and elliptic curves as a lead-in to modular forms and their various deep generalizations. Following an economical introduction to string theory, its perturbative expansion, toroidal compactification, and supergravity limit are used to illustrate the power of modular invariance in physics. Dualities and their realization via modular forms in Yang–Mills theories with extended supersymmetry are studied both via the Seiberg–Witten solution and via their superconformal phase. Appendices are included to review foundational topics, and 75 exercises with detailed solutions give the reader ample opportunity for practice.

Back in Berkeley, Weinberg reconsiders how we understand some of the theories of physics, that is, why they actually are true. He begins teaching a general relativity course starting from physical principles, rather than the usual geometric approach. These course notes later became the basis for his book Gravitation and Cosmology. Around this time, Louise was pregnant, so Weinberg avoided opportunities to travel to spend more time at home. He begins working on functional analysis, but discovers the Russian Faddeev has already done foundational work in this area. Weinberg then reexamines what he knows about quantum field theory, and jettisons the Heisenberg–Pauli canonical formalism, taking particles as his starting point. This led him to a clearer understaning of antimatter. He embarks on a series of papers about massless particles. in 1964, he is promoted to full professor. Louise applied to Harvard Law School, prompting a move to Cambridge, Mass.

In the 1980s, two groups of physicists in Europe and America began to lay plans for a high energy proton accelerator that could settle the question of electroweak symmetry breaking. In 1984, Weinberg is appointed to the SSC Board of Overseers, and this work would occupy his time for much of the next decade. Weinberg testifies before Congress in favor of the SSC project and starts to appreciate the role of pork-barrel politics in the siting decisions. The Texas site of Waxahachie, near Dallas, is approved in 1988. The project’s construction funding is approved but faces ongoing challenges from other competing areas of science. Changes in the specifications, required by the science goals, lead to increases in the costs, resulting in bad press. In 1993, the funding was cut, and the SSC was killed off. Weinberg writes a successful trade book, Dreams of a Final Theory.

Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.

General derivation of the eigenvalues and eigenstates of the square and z-component of the angular momentum; relationship between the angular momentum and the harmonic oscillator in two dimensions; transformation of states and operators under rotations; algebraic derivation of the hydrogen atom spectrum.

Bound states in one-dimensional finite and infinite wells and delta-function potentials, and combinations of these, are obtained; the WKB method for bound states is introduced; the consequences of a parity-invariant potential for the eigenfunctions are derived.

In Chapter 2, the dependence of elliptic functions on the points in the torus was studied for a fixed lattice. In this chapter, it is the dependence on the lattice that will be investigated. The modular group SL(2,Z) is introduced as the group of automorphisms of the lattice, and its generators, elliptic points, and cusps are identified. The hyperbolic geometry of the Poincaré upper half plane is reviewed, and the fundamental domain for SL(2,Z) is constructed. Modular forms and cusp forms are defined and shown to form a polynomial ring. They are related to holomorphic Eisenstein series, the discriminant function, the Dedekind eta-function, and the j-function and are expressed in terms of Jacobi theta-functions. The Fourier and Poincaré series representations of Eisenstein series are analyzed as well.

Shortly after moving to Berkeley, Weinberg slips a disc and is bedbound. He reads Chandrasekhar’s stellar physics book, which helped spark his interest in astrophysics. They decide to stay on the San Francisco side of the bay. At that time, Berkeley was the world’s leading center of experimental research on elementary particles and the newly commissioned Bevatron was the latest particle accelerator. Weinberg resolves to do some work that will be useful to Berkeley experimenters and sets about studying muon physics. In Spring 1960, he is offered and accepts a tenure-track position as an assistant professor. He is invited to join JASON, the group of defense consultants. He begins teaching and learns that he loves it. He decides to take a year abroad via an Alfred Sloan Fellowship and he and Louise buy a round-the-world ticket.

In the summer of 1991, Weinberg receives the National Medal of Science from President George H. W. Bush. He describes various visits and internationl trips. Through the early 1990s at the University of Texas at Austin, he taught a course on the quantum theory of fields, which was published as a two-volume treatise on “The Quantum Theory of Fields.” (A third volume on supersymmetry would follow in 2000.) Around this time, he begins publishing popular science in The New York Review of Books. He gives the dedicatory address at the opening of the university’s Hobby–Eberly Telescope.

Research in “complex physics” or “nonlinear physics” is rapidly expanding across various science disciplines, for example, in mathematics, astrophysics, geophysics, magnetospheric physics, plasma physics, biophysics, and sociophysics. What is common among these science disciplines is the concept of “self-organized criticality systems,” which is presented here in detail for observed astrophysical phenomena, such as solar flares, coronal mass ejections, solar energetic particles, solar wind, stellar flares, magnetospheric events, planetary systems, and galactic and black-hole systems. This book explains fundamental questions: Why do power laws, as hallmarks of self-organized criticality, exist? What power law index is predicted for each astrophysical phenomenon? Which size distributions have universality? What can waiting time distributions tell us about random processes? This book is the first monograph that tests comprehensively astrophysical observations of self-organized criticality systems. The highlight of this book is a paradigm shift from microscopic concepts (such as the traditional cellular automaton algorithms) to macroscopic concepts (formulated in terms of physical scaling laws).

Disheartened by the cancelation of the Superconducting Super Collider, Weinberg turns his attention to the cosmological constant. It must behave like a vacuum energy density, and can be adjusted to cancel the energy in fluctuating fields. Today the effective vacuum energy density, including the cosmological constant, has come to be called “dark energy.”

The generalized fractal-diffusive SOC model predicts the probability distribution functions for each parameter as a function of the dimensionality, diffusive spreading exponent, fractal dimension, and type of (coherent/incoherent) radiation process. The waiting time distributions are predicted by the FD-SOC model to follow a power law with a slope of during active and contiguously flaring episodes, while an exponential cutoff is predicted for the time intervals of quiescent periods. This dual regime of the waiting time distribution predict both persistence and memory during the active periods, and stochasticity during the quiescent periods. These predictions provide useful constraints of the physical parameters and underlying scaling laws. Significant deviations from the size distributions predicted by the FD-SOC model could indicate problems with the measurements or data analysis. The generic FD-SOC model is considered to have universal validity and explains the statistics and scaling between SOC parameters but does not reveal the detailed physical mechanism that governs the instabilities and energy dissipation in a particular SOC process.

The fractal nature in avalanching systems with SOC is investigated here for phenomena in the solar photosphere and transition region. In the standard SOC model, the fractal Hausdorff dimension is expected to cover the range of [1, 2], with a mean of for 2-D observations projected in the plane-of-sky, and the range of [2, 3], with a mean of for real-world 3-D structures. Observations of magnetograms and with IRIS reveal four groups: (i) photospheric granulation with a low fractal dimension of ; (ii) transition region plages with a low fractal dimension of ; (iii) sunspots at transition region heights with an average fractal dimension of ; and (iv) active regions at photospheric heights with an average fractal dimension of . Phenomena with a low fractal dimension indicate sparse curvilinear flows, while high fractal dimensions indicate near space-filling flows. Investigating the SOC parameters, we find a good agreement for the event areas and mean radiated fluxes in events in transition region plages.