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This chapter begins with a normal-mode analysis of the classical electromagnetic field in an ideal cavity. The resulting expression for the electromagnetic energy has the same form as the energy of a collection of harmonic oscillators, called radiation oscillators. This analogy is the basis for a quantization conjecture in which the classical mode amplitudes are replaced by photon creation and annihilation operators, obeying a version of the canonical commutation relations of quantum mechanics. Fock space is constructed by repeated application of creation operators to the vacuum state. Pure and mixed quantum states of light are described, respectively, by Fock-space vectors satisfying the Schrödinger equation, and density operators satisfying the quantum Liouville equation. The notions of normal ordering and vacuum fluctuations are introduced, and the latter is used to explain the Casimir effect.

The cavity-mode quantization conjecture of Chapter 2 is replaced by local commutation relations — which are independent of the size and shape of the cavity — between field-operator components. This step eliminates the previous dependence on the classical boundary conditions at the ideal cavity wall. The cavity annihilation and creation operators are respectively replaced by the positive- and negative-frequency parts of the vector potential. A simple ad hoc model provides similar results for quantized fields in a passive, linear dielectric. It is shown that the total electromagnetic angular momentum cannot, in general, be expressed as the sum of well defined orbital- and spin-parts. The chapter ends with a discussion of localizability for photons, in which it is shown that there is no photon position operator, no position-space photon wave function, and no local photon number operator.

Angular momentum is one of the more abstract topics in quantum mechanics. This chapter begins with a review of classical angular momentum. Quantum-mechanical angular momentum is then introduced by obtaining operators from their classical counterparts. The commutation relations amongst those operators and the corresponding uncertainty relations then follow. Eigenstates and eigenvalues of the angular momentum operators are then introduced (but not derived). From these follow the raising and lowering operators. Physical meaning is imparted to quantum angular momentum by considering predictions for sequential measurements, and by comparing the conceptual pictures of angular momentum in classical and quantum mechanics. Finally, quantum spin and orbital angular momenta, and the deep differences between them, are discussed.

This chapter examines China’s case-guiding system as a recent reform measure adopted in 2010 to standardize judicial sentencing, and discusses its potential impact on China’s death penalty practice in the future.

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.

The chapter develops a correctness theory of truth (CTT) for semantic information to explain how well-formed and meaningful data may become truthful. After the introduction, Section 8.2 defends the possibility of translating semantic information propositionally (i). In Section 8.3, i is polarized into a query (Q) and a result (R). In Section 8.4, [Q+R] is transformed into a Boolean question and its relative yes/no answer [Q+A]. In Sections 8.5 and 8.6, it is argued that (1) A is the correct answer to Q if and only if (2) A correctly saturates Q by verifying and validating it; that (2) if and only if (3) [Q+A] generates an adequate model (m) of the relevant system (s) identified by Q; that (3) if and only if (4) m is a proxy of s and (5) proximal access to m commutes with the distal access to s; and that (5) if and only if (6) reading/writing m enables one to read/write s. Section 8.7 discusses CTT in connection with the semantic paradoxes. The final section draws a general conclusion about CTT.

Basic notions of hyperbolic geometry are presented: geodesics, light rays, tangent bundles etc. These are systematically derived from properties of Minkowski space: the boundary of hyperbolic space is given by the light cone, geodesics by planes intersecting the light cone in two rays, and horospheres by affine hyperplanes parallel to a light ray and intersecting the hyperbolic space. Intrinsic formulae related to the hyperbolic distance are obtained by using only elementary linear algebra within R^1,d . Harmonic conjugation is also discussed in this framework. Poincaré coordinates are extended to the boundary ∂H ^d . The geodesic and horocyclic flows are defined by the right action of A ^d on frames. The classical ball and upper-half-space models are presented, and the latter is related to Poincaré coordinates. A commutation relation in PSO(1, d) is established. Stable leaves and the Busemann function are introduced, and some physical interpretations are given.

The chapter begins by defining the fundamental distinction between competence and performance and introduces various methodological approaches used in the linguistics of humor, with a focus on the principle of commutation. The problem of identifying humor is then addressed, including a review of traditional “markers’ of humor (i.e., laughter). A triangulation method incorporating several markers and semantic analysis is proposed to replace it. In particular the inadequacy of using laughter is investigated by reviewing non-mirthful laughter. Finally the idea of keying, from ethnomethodology is used to explain how a situation may be seen as humorous or not, depending on its keying.

This chapter reconstructs the environment in which the verb to be will be understood in a new way. It identifies three radical changes, conceptually distinct but not completely unrelated, that come into play when looking at language from the structuralist perspective. First, if what matters are the relationships, elements that have the same relationships to other elements of the system must be considered equivalent and, in principle, they can be exchanged (the commutation principle). Second, given an incompletely filled grid that combines certain properties, if the grid is correct, it will lead to the discovery of new objects that could fill the empty grid spaces (the retrieval principle). Third, not all possible combinations of primitive elements are used in every language (the redundancy principle).

This chapter shows how the manorial system was influenced by the economic boom of the later eighteenth century. The older school saw the benefits of inflation as largely ‘creamed off’ by the nobles. While revisionists have moderated this view, and evidence of peasant wealth has become more plentiful, the deeply-ingrained idea that subjection was irreconcilable with peasant well-being still prevails. In fact, peasant wealth was not ‘exceptional’, for most peasants held secure tenure and were protected from rent increases, in this period a significant wealth transfer from nobles to peasants. Although incentives to invest were poor, many peasants commuted labour dues to maximize their profits. A significant minority chose to use some of the rising value of their surpluses in more ostentatious consumption, part of the European ‘Industrious Revolution’ posited by Jan de Vries.

This chapter examines how conscription became a debatable issue between Union soldiers and Northern civilians during the Civil War. A large number of Union soldiers entered the service under the terms of the Enrollment Act of 1863. Conscription has always caused controversy in American military history, particularly in its inaugural use in the Civil War. Sharp differences of opinion arose between those who favored conscription as a means of creating the armies necessary to win the war and those who were not comfortable with the expansion of government power. The most significant debate sparked by conscription was across the experience divide between soldiers in the field and the civilians at home. Union soldiers were dismayed to find that many Northerners were not in favor of the draft. They even found conscription opponents within their own communities and even their own families. This chapter also discusses the controversies surrounding commutation, substitution, and the bounty system in the military.

Following Aquinas, John Finnis treats the domain of justice as the domain of duties owed to others, or (equivalently) actions to the performance of which others have a right. This chapter challenges the Aquinas–Finnis position from a more classically Aristotelian direction. It argues that the domain of justice is the domain of the allocative. There are cases in which duties owed to others are not allocative, and cases in which allocative considerations are not duties owed to others. The examination of these cases leads into a discussion of the forms of justice. Finnis initially gets these forms wrong, and is eventually led to abandon the search for forms altogether, by his Thomist mischaracterisation of the domain of justice. The chapter nevertheless finds much common ground with Finnis on justice, particularly on the place of justice in law.

This chapter recounts the last days of the Massie–Fortescue murder trial, the sentencing of the group, and the subsequent commutation of their sentences by Governor Lawrence McCully Judd in the spring of 1932. Local audiences in the islands and malihini haoles saw the outcome of the case much differently from those on the “Mainland.” While the trial’s unorthodox legal ending provided closure to the Massie case for most continental Americans, it left an open wound for local audiences. The trial’s outcome illustrated and maintained a boundary between “locals” and “mainlanders” that was crucial to the formation of local identity in Hawaii.

This chapter explains how executive clemency—the state governor’s power to reduce a death penalty to a lesser sentence—helped make capital punishment function in North Carolina. With mandatory death sentences for four serious crimes (murder, rape, burglary, and arson), judges had no choice but to pronounce a death penalty upon conviction. This left it up to the governor to decide whether or not the convicted person would be executed. Although this power was reserved for the governor, it was soon transferred to a parole board, which reviewed each death sentence and invited community comment. This process most benefitted teenagers and women. Perversely, more African Americans received commutations than whites because of the high rate of error in their trials.

The notion of free quantum field is thoroughly discussed in the linearised setting associated with the choice of a detector. The discussion requires attention to certain details that are often overlooked in the standard literature. Explicit expressions for generic fields, Dirac fields, gauge fields and ghost fields are laid down, as well the ensuing free-field expressions of important functionals. The relations between super-commutators of free fields and propagators, and the canonical super-commutation rules, follow from the above results.

This chapter sets out the second-quantisation treatment for systems of identical bosons or fermions. Key features such as the symmetrisation principle and super-selection rules are presented. Orthogonality and completeness requirements for mode functions are set out, and Fock states are defined in terms of mode occupancy, leading to the definitions of mode annihilation and creation operators and the determination of their commutation (bosons) or anticommutation (fermions) rules. Important two-mode states such as binomial states and relative-phase eigenstates are considered. Field creation and annihilation operators are defined and related to multi-particle position eigenstates. The Hamiltonian is expressed in terms of both mode operators and field operators, state dynamics being treated via Liouville–von Neumann, master or Matsubara equations for the density operator. Normal ordering is introduced and applied to expressions for the vacuum state projector. Multi-particle position probabilities are considered and related to normally ordered quantum correlation functions.

This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the associative and distributive rules are stated, highlighting the anticommuting multiplication property of g-numbers and the consequences that all powers of g-numbers are zero and they do not have inverses. Complex conjugation rules are also stated, highlighting a second major difference between c- and g-numbers. The concepts of monomials and their order are introduced for Grassmann variables. Grassmann functions are defined, their division into even and odd functions noted, and commutation properties of even and odd Grassmann functions are obtained. Important results for exponential functions involving Grassmann variables and functions are derived, including the Baker–Hausdorff theorem. Linear transformations of Grassmann variables are defined.

Focusing on systems of many identical particles, Chapter 2 introduces appropriate operators to describe their properties in terms of Schwinger’s “measurement symbols.” The latter are then factorized into “creation” and “annihilation” operators, whose fundamental properties and commutation/anticommutation relations are derived in conjunction with the Pauli exclusion principle. This leads to “second quantization” with the Hamiltonian, number, linear and angular momentum operators expressed in terms of the annihilation and creation operators, as well as the occupation number representation. Finally, the concept of coherent states, as eigenstates of the annihilation operator, having minimum uncertainty, is introduced and discussed in detail.