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.

In this chapter we consider two examples of the situation when the classicalobservables should be described by a noncommutative (quantum-like)probability space. A possible experimental approach to find quantum-like correlationsfor classical disordered systems is discussed. The interpretation ofnoncommutative probability in experiments with classical systems as a resultof context (complex of experimental physical conditions) dependence ofprobability is considered.

This chapter is devoted to the Bohr complementarity principle.This is one of the basic quantum principles. We dissolve it into separate subprincipleson contextuality, incompatibility, complementary-completeness,and individuality. We emphasize the role of the contexuatlity component. It is not highlighted in the foundational discussions. ByBohr, the outputs of measurements are resulted from the complexinteraction between a system and measurement context, the values ofquantum observables cannot be treated as objective properties of systems.Such Bohr contextuality is more general than joint measurement contextuality(JMC) considered in the discussions on the Bellinequality. JMC is a very special form of the Bohr contextuality. The incompatibility component is always emphasized and often referred as the wave-particle duality. The principle ofinformation complementary-completeness represents Bohr’s claim on completenessof quantum theory. The individuality principle is basic for thenotion of phenomenon used by Bohr to emphasize the individuality and discreteness of outputs of measurements. Individuality plays the crucial role in distinguishing quantum and classical optics entanglements.

We show that for two classical Brownian particles there exists an analog ofcontinuous-variable quantum entanglement: The common probability distributionof the two coordinates and the corresponding coarse-grained velocitiescannot be prepared via mixing of any factorized distributions referring tothe two particles in separate. This is possible for particles which interactedin the past, but do not interact in the present. Three factors are crucial forthe effect: (1) separation of time-scales of coordinate and momentum whichmotivates the definition of coarse-grained velocities; (2) the resulting uncertaintyrelations between the coordinate of the Brownian particle and thechange of its coarse-grained velocity; (3) the fact that the coarse-grained velocity,though pertaining to a single Brownian particle, is defined on a commoncontext of two particles. The Brownian entanglement is a consequenceof a coarse-grained description and disappears for a finer resolution of theBrownian motion. We discuss possibilities of its experimental realizations inexamples of macroscopic Brownian motion.

This chapter presents the basics of the mathematical formalism and methodologyof the prequantum classical statistical field theory (PCSFT). In theBild-conception framework, PCSFT gives an example of acausal theoretical model (CTM) beyond QM, considered as observationalmodel (OM). Generally CTM-OM correspondence is not as straightforwardas in Bell’s model with hidden variables. In PCSFT hidden variables are randomfields fluctuating at spatial and temporal scales which are essentiallyfiner than those approached by the present measurement technology. Thekey element of the PCSFT-QM correspondence is mapping of the complexcovariance operator of a subquantum random field to the density operator.For compound systems, the situation is more complicated. Here PCSFT providestwo descriptions of compound systems with random fields valued intensor vs. Cartesian product of the Hilbert spaces of subsystems. The lattermodel matches representation of compound systems in classical statisticalmechanics. Both approaches are used for measure-theoretic representationof the correlations violating the Bell inequalities.

The aim of this chapter is to attract attention of experimenters to the originalBell (OB) inequality which was shadowed by the common considerationof the CHSH inequality. Since this chapter is directed to experimenters, herewe present the standard viewpoint on the violation of the Bell inequality andthe EPR argument. There are two reasonsto test the OB inequality and not the CHSH inequality. First, theOB inequality is a straightforward consequence of the EPR argumentation.And only this inequality is related to the EPR–Bohr debate.The second distinguishing feature of the OB inequality was emphasizedby Pitowsky. He pointed out that the OB inequality provides a higherdegree of violations of classicality than the CHSH inequality. Thus, by violating the OBinequality it is possible to approach a higher degree of deviation from classicality.The main problem is that the OB inequality is derived under theassumption of perfect (anti-)correlations. However, the last few years have been characterizedby the amazing development of quantum technologies. Nowadays,there exist sources producing with very high probability the pairs of photonsin the singlet state. Moreover, the efficiency of photon detectors wasimproved tremendously. In any event one can start by proceeding with thefair sampling assumption.

The aim of this chapter is to highlight the possibility of applying the mathematicalformalism and methodology of quantum theory to model behaviourof complex biosystems, from genomes and proteins to animals, humans, ecologicaland social systems. Such models are known as quantum-like and theyshould be distinguished from genuine quantum physical modeling of biologicalphenomena. One of the distinguishing features of quantum-like models istheir applicability to macroscopic biosystems, or to be more precise, to informationprocessing in them. Quantum-like modeling has the base in quantuminformation theory and it can be considered as one of the fruits of the quantuminformation revolution. Since any isolated biosystem is dead, modelingof biological as well as mental processes should be based on theory of opensystems in its most general form – theory of open quantum systems. In thischapter we advertise its applications to biology and cognition, especiallytheory of quantum instruments and quantum master equation. We mentionthe possible interpretations of the basic entities of quantum-like models withspecial interest to QBism as maybe the most useful interpretation.

In this chapter we develop the contextual approach to quantum mechanics.This approach matches with the views ofBohr who emphasized that the quantum description represents complexes ofexperimental physical conditions, in the modern terminology – experimentalcontexts. In this chapter we formalize the contextual approach on the basisof contextual probability theory which is closely connected with generalizedprobability theory (but interpretationally not identical with it). The contextualprobability theory serves as the basis of the contextual measurementmodel (CMM). The latter covers measurements in classical, quantum, andquasi-classical physics.

This chapter is a step towards understanding why quantum nonlocalityis a misleading concept. Metaphorically speaking, quantum nonlocality isJanus faced. One face is an apparent nonlocality of the state update basedon the Luders projection postulate. It can be referred as intrinsic quantumnonlocality. And the other face is subquantumnonlocality: by introducing a special model with hidden variables onederives the Bell inequality and claims that its violation implies the existenceof mysterious instantaneous influences between distant physical systems(Bell nonlocality). According to the Luders projection postulate, aquantum measurement performed on one of the two distant entangled physicalsystems, say on S1, modifies instantaneously the state of S2. Therefore, ifthe quantum state is considered to be an attribute of the individual physicalsystem (Copenhagen interpretation) and if one assumes thatexperimental outcomes are produced in a random way one arrives at the contradiction. It is a primary source of speculation about aspooky action at the distance. But Einstein had already pointed out that the quantum paradoxes disappear, ifone adopts the statistical interpretation.