In this chapter, we discuss the modular properties of quantum field theories of scalar fields that take values in a d-dimensional torus with a flat metric and a constant anti-symmetric tensor. The problem is of great interest in quantum field theory and string theory in view of the fact that such toroidal compactifications admit solutions using free-field theory methods on the worldsheet, preserve Poincaré supersymmetries and may be used to relate different perturbative string theories via T-duality. Toroidal compactifications produce large duality groups, which we shall derive and which generalize the full modular group SL(2,Z). The quantum field theories of toroidal compactification on a worldsheet torus for a singular modulus is shown to be a rational conformal field theory.

Exchange and permutation operators; symmetrizer and antisymmetrizer; two bosons or two fermions in a central potential; scattering amplitudes of two identical particles in a central potential; the Fermi gas; theory of white dwarf stars; the Thomas-Fermi approximation for many-electron atoms.

Weinberg takes a summer job in the Atomic Beam Group at Princeton, calculating the trajectories of beam particles through the experimental equipment. He describes the culture of close relations of graduate students in physics with the younger faculty and its emphasis on research rather than course work. He details the various courses he took, along with the personalities of the Princeton professors at the time. Sam Treiman agrees to be his PhD advisor for a thesis on strong interactions in decay processes.

The size distribution of waiting times are found to have an exponential distribution in the case of a stationary Poissonian process. In reality, however, the waiting time distributions reveal power law-like distribution functions, which can be modeled in terms of non-stationary Poisson processes by a superposition of Poissonian distribution functions with time-varying event rates. We model the time evolution of such waiting time distributions by polynomial, sinusoidal, and Gaussian functions, which have exact analytical solutions in terms of the incomplete Gamma function, as well as in terms of the Pareto type-II approximation, which has a power law slope of , where represents the linear time evolution, or with representing nonlinear growth rates, which have a power law slope of . Our mathematical modeling confirms the existence of significant deviations from ideal power law size distributions (of waiting times), but no correlation or significant interval–size relationship exists, as would be expected for a simple (linear) energy storage-dissipation model.

Weinberg returns to the “The Cosmological Constant Problem” and suggests an anthropic principle solution. Anthropic reasoning could make it possible for us to calculate the effective vacuum energy. Observations of dark energy in 1998 show that the expansion of the universe is accelerating. This observational result is not inconsistent with the notion of a possible multiverse – the issue has not been settled.

Wave-particle duality and the Davisson-Germer experiment are briefly discussed; wave packets are defined and their features, including phase velocity, group velocity, and spreading, are examined; the stationary phase method is presented; free-particle wave functions are introduced, and the equivalence between coordinate- and momentum-space representations of these wave functions is emphasized.

The occurrence frequency distributions (size distributions) are the most important diagnostics for self-organized criticality systems. There are at least three formats for size distributions: (i) the differential size distribution function, (ii) the cumulative size distribution function, and (iii) the rank-order plot. Each of the three formats (or methods) has at least three ranges of event sizes: (i) a range with statistically incomplete sampling; (ii) an inertial range or power law fitting range with statistically complete sampling; and (iii) a range bordering finite system sizes. Only the intermediate range with power law behavior should be used to determine the power law slope from fitting the observed size distributions. The establishment of power law functions in a given observed size distribution depends crucially on the choice of the fitting range, which should have a logarithmic range of at least 2–3 decades. Often the fitted distribution functions exhibit significant deviations from an ideal power law and can be fitted better with alternative functions, such as log-normal distributions, Pareto type-II distributions, and Weibull distributions.

Among stellar systems, we find many with applications of SOC, such as stellar flares or pulsar glitches. Stellar flares occur mostly in the wavelength ranges of ultraviolet, soft X-rays and UV, and in visible light. A breakthrough in new stellar data was accomplished with the Kepler spacecraft, which allowed unprecedented detections of exoplanets, while the same light curves could be searched for large stellar flares. Exploiting these promising new datasets, one finds that most stellar flare datasets exhibit dominant size distributions that converges to a power law slope of , regardless of the star type. The size distributions of pulsar glitches are mostly found outside of the valid range of the Standard FD-SOC model and thus require a different model. Power law fits are not always superior to fits with the log-normal function or Weibull function. This discrepancy between observed and modeled power law slopes in stellar SOC systems is mostly due to small-number statistics of the samples, incomplete sampling near the lower threshold, and due to ill-defined power law fitting ranges, which can cause significant deviations from ideal power laws.

From the statistics of solar radio bursts, we learn that we can discriminate between three diagnostic regimes: (i) the incoherent regime where the radio burst flux is essentially proportional to the flare volume (with a power law slope of ), as it occurs for gyroemission, gyroresonance emission, gyrosynchrotron emission; (ii) the coherent regime that implies a nonlinear scaling between the radio flux and the flare volume ; as it occurs for the electron beam instability, the loss-cone instability, or maser emission; and (iii) the exponential regime that does not display a power law function, but rather an exponential cutoff as expected for random noise distributions. Thus, the power law slopes offer a useful diagnostic to verify the flux–volume scaling law and to discriminate between coherent and incoherent radio emission processes, as well as to distinguish between SOC processes and non-SOC processes. An additional diagnostic comes from the inertial range of power law fits: SOC-related power law size distributions should extend over multiple decades, while power law ranges of less than one decade are most likely not related to SOC processes.

Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.

Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.

Construction ofunitary operators inducing space and time translations, and rotations; the anti-unitary operator inducing time reversal; consequences of invariance under a symmetry transformation; periodic potentials and Bloch waves; the Kronig-Penney model; the ammonia molecule and broken parity symmetry; consequences of time reversal invariance on the scattering amplitude of spinless particles; Kramers degeneracy.

Can we claim that the dynamics of the solar wind is consistent with a SOC system? Observationally we find that magnetic field and kinetic energy fluctuations measured in the solar wind exhibit power law distributions, which is consistent with a SOC system. What about the driver, instability, and avalanches expected in a SOC system? The driver mechanism is the acceleration of the solar wind in the solar corona itself, a process that basically follows the hydrodynamic model of Parker (1958), and may be additionally complicated by the presence of nonlinear wave–particle interactions, such as ion-cyclotron resonance. Then, the instability threshold, triggering extreme bursts of magnetic field fluctuations, the avalanches of solar wind SOC events, can be caused by dissipation of Alfven waves, onset of turbulence, or by the ion-cyclotron instability. Thus, in principle the generalized SOC concept can be applied to the solar wind, if there is a system-wide threshold for an instability that causes extreme magnetic field fluctuations.

Weinberg collaborates with Ed Witten. He becomes the youngest member of the Saturday Club of Boston. Weinberg signs up to write The Discovery of Subatomic Particles. After their continued separation due to teaching, Weinberg grows to like Austin more and more, with its social scene that crossed from academia into the public sphere. He negotiates with the Universioty of Texas for a position in Austin as the Josey Regental Chair in Science beginning in 1982. He joins the Headliners Club in Austin. Weinberg helps found the Jerusalem Winter School in Theoretical Physics. He begins exploring physical theories in higher dimensions. He attends the Shelter Island Conference in 1983. He is elected to the Philosophical Society of Texas and joined the Town and Gown Club in Austin, but quits the latter over its male-only stance, to help form a rival, the Tuesday Club (of Austin). In mid-1980s, he becomes seriously interested in string theory.

Congruence subgroups form a countable infinite class of discrete non-Abelian subgroups of SL(2,Z) and play a particularly prominent role in deriving the arithmetic properties of modular forms. In this chapter, we study various aspects of congruence subgroups, including their elliptic points, cusps, and topological properties of the associated modular curve. Jacobi theta-functions, theta-constants, and the Dedekind eta-function are used as examples of modular forms under congruence subgroups that are not modular forms under the full modular group SL(2,Z).