The size distribution of solar energetic particle (SEP) events, which represent a more energetic subset than flare events, is mostly found to follow power law distribution functions, rather than Poissonian random distribution functions. However, the numerical value of the power law slope is generally flatter than the slopes of the flare size distributions in hard X-rays, soft X-rays, and EUV (Hudson 1978), which can be explained in at least four different ways: (i) normal flares and proton flares are produced by two fundamentally different acceleration mechanisms; (ii) proton flares behave differently than normal flares; (iii) the fractal dimensionality of SEP events is different from normal flares; (iv) proton flares are subject to a selection bias toward the most energetic events and thus are not a representative sample of large flares. Nevertheless, the standard fractal-diffusive SOC model can explain the observed slopes of SEP size distributions, but observations reveal deviations from straight power law functions, or broken power law slopes, and thus are not unique and need to be modeled in more detail.

We focus on the statistics of SOC-related solar flare parameters in soft X-ray wavelengths, including their size and waiting time distributions. An early SOC model assumed a linear increase of the energy storage, but this pioneering model is not consistent with the expected correlation between the waiting time interval and the subsequently dissipated energy. The Neupert effect in solar flares implies a correlation between the hard X-ray fluence and the soft X-ray flux, which predicts identical size distributions for these two parameters. Quantifying of thermal flare energies in soft X-ray emitting plasma needs also to include radiative and conductive losses. The intermittency and bursty variability of the solar dynamo implies a nonstationary SOC driver, which yields a universal value for the power law slope of fluxes, but the power law slopes of waiting times vary with the flare rate. While our focus encompasses primarily SOC models, alternative models in terms of MHD turbulence can explain some characteristics of SOC features also, such as size distribution functions, Fourier spectra, and structure functions.

The Schroedinger equation for a particle in a potential is introduced and the general properties of its solutions are discussed; the uncertainity relations are derived; the Gram--Schimdt procedure for orthonormalizing a set of independent wave functions is introduced; the time evolution of the expectation values of the position and momentum operatorsfor a particle in a potential and in an electromagnetic field are derived.

For physicists who study elementary particles and quantum field theory, the 1970s was a golden age. It saw the experimental confirmation of the electroweak theory, and the extension of that thinking would lead us to a successful theory of strong interactions as well. All the fundamental forces of nature, except for gravity, would be unified in what became known as the “Standard Model.” By the end of 1973, there was some experimental verification of the electroweak theory. Weinberg agrees to write The First Three Minutes, which was published in 1977. Louise visits Stanford Law School, accompanied by Weinberg, who finds his host department cold. In 1977, he collaborates with Ben Lee of Fermilab, who tragically died in a car accident later that year. Louise is invited to teach at University of Texas Law School, in the summer of 1979, after which she was offered a full professorship. The Weinbergs taught in their respective universities and met in Cambridge in the holidays. Weinberg’s Nobel Prize, shared with Salam and Glashow, is announced in October 1979, ahead of the ceremony that December.

A natural set of mutually commuting linear operators acting on the space of modular forms are the Hecke operators. They map holomorphic functions to holomorphic functions, weight-k modular forms to weight-k modular forms, and weight-k cusp forms to weight-k cusp forms. For the full modular group SL(2,Z), the Hecke operators map the space of holomorphic modular forms into itself and map the subspace of cusp forms into itself. For congruence subgroups, the Hecke operators map weight-k modular forms of one congruence subgroup into those of another congruence subgroup. Hecke operators commute with the Laplace–Beltrami operator on the upper half plane so that Maass forms and cusp forms are simultaneous eigenfunctions of all Hecke operators. Finally, given a modular form with positive integer Fourier coefficients, the Hecke transforms also have positive integer Fourier coefficients. For this reason, Hecke operators are relevant in a number of physical problems, such as two-dimensional conformal field theory, that we shall discuss.

In Chapter 3, we introduced SL(2,Z) as the automorphism group of a two-dimensional lattice with an arbitrary modulus. For every value of the modulus, the lattice also possesses a ring of endomorphisms which multiply the lattice by a nonvanishing integer to produce a sublattice of the original lattice. Multiplying the lattice by an arbitrary complex number gives a lattice that will generally not be a sublattice of the original lattice. However, for special values of the modulus, referred to as singular moduli, and associated special values of the complex-valued multiplying factor, the lattice obtained by multiplication will be a sublattice of the original lattice and the ring of endomorphisms will be enlarged. This phenomenon is referred to as complex multiplication. From a mathematics standpoint, various modular forms take on special values at singular moduli, as illustrated by the fact that the j-function is an algebraic integer. From a physics standpoint, the enlargement of the endomorphism ring has arithmetic consequences in conformal field theory, as illustrated by the fact that conformal field theories corresponding to toroidal compactifications at singular moduli are rational conformal field theories as will be discussed in Chapter 13.

Weinberg takes up a National Science Foundation predoctoral fellowship to study at the Niels Bohr Institute in Copenhagen. He is encouraged to take up research on nuclear alpha decay. His advisor, Gunnar Källén, tasks him with studying the Lee model. He plans to obtain his PhD from Princeton.