This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincare sphere on polarization optics.

The moral inquiry into the nature of justice is often marginalized, as contemporary discussions focus primarily on political considerations. Yet, the need to examine the moral framework of justice is fundamental. What is the point of justice, after all, if not to establish a moral framework for political interactions? The question resonates especially when recognizing that no monocultures exist over time and space. In examining Plato's thoughts on individual and social morality (dikaiosune) and Aristotle's focus on individual virtue (arete) and the pursuit of well-being (eudaimonia), the author proffers foundational resources for today's discussions of justice. Moreover, he brings this nexus of thought into conversation with theories that have emerged over the centuries, such as Kant's concept of due concern and respect, individual rights and responsibilities as in Hobbes, Locke and J.S. Mill. Thus, he engages contemporary disputes of justice including distributive vs. relational schemes, choice vs. chance considerations, group rights theories, value pluralism, the right vs. the good, as well as international and future justice. His inquiry concludes with the provision of a deontological framework set against a liberal political context, justice as right actions, for further examination of questions of justice.

Justice as Right Actions presents an original theory of justice anchored in the analytical philosophical tradition. In contrast to many contemporary approaches, the theory provides normative guidance, rather than focusing solely on political structures and institutions, as the question of justice is seen to comprise both a moral inquiry concerned with questions of good and bad, right and wrong, and a political inquiry, concerned with the nature of the polity and how individuals relate to it. Presenting a relational account of justice, rather than a distributive account - the latter, so much more prevalent in current studies - communications are seen as the key to the theory, both in the substantive sense as a discursive method of resolving disputes, as well as instrumentally, in the transmission of concepts, especially values through time. Rule-oriented in approach, justice as right actions attempts to be value-neutral, acknowledging, however, an underlying thin theory of the good, including concepts of rationality, autonomous moral agency, equal concern and respect for others, as well as plurality of values. Its political context is liberalism, with components of negative liberty and equality of concern and respect, while underscoring as well, the concepts of tolerance and social diversity. In this study, the original theory of Justice as Right Actions is also contrasted with and situated among contemporary accounts of justice, including the most important theoretical works on the topic in the past half-century. Thus, the study also serves as a valuable review and critique of such major contemporary accounts of justice.

The concept of phase space plays a decisive role in the study of the transition from classical to quantum physics. This is particularly the case in areas such as nonlinear dynamics and chaos, geometric quantization and the study of the various semi-classical theories, which are the setting of the present volume. Much of the content is devoted to the study of the Wigner distribution. This volume gives the first complete survey of the progress made by both mathematicians and physicists. It will serve as an excellent reference for further research.

This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincar sphere on polarization optics.