The study of the quantum–classical correspondence has been focused on the quantum measurement problem. However, most of the discussion in the preceding chapters is motivated by a broader question: Why do we perceive our quantum Universe as classical? Therefore, emergence of the classical phase space and Newtonian dynamics from the quantum Hilbert space must be addressed. Chapter 6 starts by re-deriving decoherence rate for non-local superpositions using the Wigner representation of quantum states. We then discuss the circumstances that, in some situations, make classical points a useful idealization of the quantum states of many-body systems. This classical structure of phase space emerges along with the (at least approximately reversible) Newtonian equations of motion. Approximate reversibility is a non-trivial desideratum given that the quantum evolution of the corresponding open system is typically irreversible. We show when such approximately reversible evolution is possible. We also discuss quantum counterparts of classically chaotic systems and show that, as a consequence of decoherence, their evolution tends to be fundamentally irreversible: They produce entropy at the rate determined by the Lyapunov exponents that characterize classical chaos. Thus, quantum decoherence provides a rigorous rationale for the approximations that led to Boltzmann’s H-theorem.

Chapter 5 explores the consequences of decoherence. We live in a Universe that is fundamentally quantum. Yet, our everyday world appears to be resolutely classical. The aim of Chapter 5 is to discuss how preferred classical states, and, more generally, classical physics, arise, as an excellent approximation, on a macroscopic level of a quantum Universe. We show why quantum theory results in the familiar “classical reality” in open quantum systems, that is, systems interacting with their environments. We shall see how and why, and to what extent, quantum theory accounts for our classical perceptions. We shall not complete this task here—a more detailed analysis of how the information is acquired by observers is needed for that, and this task will be taken up in Part III of the book. Moreover, Chapter 5 shows that not just Newtonian physics but also equilibrium thermodynamics follows from the same symmetries of entanglement that led to Born’s rule (in Chapter 3).

Chapter 3 describes how quantum entanglement leads to probabilities based on a symmetry, but—in contrast to subjective equal likelihood based solely on ignorance—it is an objective symmetry of known quantum states. Entanglement-assisted invariance (or envariance for short) relies on quantum correlations: One can know the quantum state of the whole and use this to quantify the resulting ignorance of the states of parts. Thus, quantum probability is, in effect, an objective consequence of the Heisenberg-like indeterminacy between global and local observables. This derivation of Born’s rule is based on the consistent subset of quantum postulates. It justifies statistical interpretation of reduced density matrices, an indispensable tool of decoherence theory. Hence, it gives one the mandate to explore—in Part II of this book—the fundamental implications of decoherence and its consequences using reduced density matrices and other customary tools.

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.

The famously controversial 1935 paper by Einstein, Podolsky, and Rosen (EPR) took aim at the heart of quantum mechanics. The paper provoked responses from leading theoretical physicists of the day, and brought entanglement and nonlocality to the forefront of discussion. This book looks back at when the EPR paper was published and explores those intense. conversations in print and in private correspondence. These offer significant insight into the minds of pioneering quantum physicists, including Bohr, Schrödinger and Einstein himself. Offering the most complete collection of sources to date – many published or translated here for the first time – this text brings a rich new context to this pivotal moment in physics history.

Schrödinger’s reaction to the EPR paper is less widely known than, say, Bohr’s, and yet our analysis shows that it fits rather nicely with contemporary concerns in foundations of quantum mechanics. Taking the lead both from the EPR paper and from Pauli’s remarks in their correspondence, Schrödinger shows that EPR’s locality considerations lead to the assignment of values to all quantum mechanical observables, but that under apparently mild assumptions this then leads to contradictions of the von Neumann type. This dilemma (as he explicitly calls it) is thus similar to more recent debates between nonlocality on the one hand and no-go results on the other (whether through violation of the Bell inequalities, the Kochen–Specker theorem, or what you will). We shall first look at Schrödinger’s fundamental worries in the years leading up to 1935. The chapter then discusses in detail the direct reaction by Schrödinger to EPR. It will, however, not exhaust our discussion of Schrödinger, who is a recurring character in the book, having poked and prodded his peers on EPR during the whole summer and autumn of 1935.

This is a revision of John Trimmer’s English translation of Schrödinger’s famous ‘cat paper’, originally published in three parts in Naturwissenschaften in 1935.

This chapter details not only the prehistory of EPR but also examines the structure and logic of the EPR paper – including Einstein’s own preferred version of the argument for incompleteness. We here attempt a seamless interweaving of the excellent extant literature with additional details that have emerged from our work and the recent work of others. Some examples of new aspects in this prehistory of EPR include evidence of a ‘proto’ photon-box thought experiment Einstein had developed in connection with his ill-starred collaboration with Emil Rupp in 1926. We also describe the potential importance to this prehistory of Einstein’s paper with Tolman and Podolsky and of Einstein’s seminar and discussions with Schrödinger in Berlin in the early 1930s.

This is a reprinting of Schrödinger’s famous pair of papers delivered at the Cambridge Philosophical Society in late 1935 and 1936, wherein he first coins the term ‘entanglement’ to describe interacting quantum systems. The first paper (1935) is given here in full; section 4 of the second paper (1936) is reprinted as an appendix.

As Stephen Dedalus walks upon Sandymount Strand in Ulysses, he thinks, “the land a maze of dark cunning nets … Ringsend: wigwams of brown steersman and master mariners. Human shells” (3.154–57). This thought evokes Ireland’s complicated position as an island nation and its entanglements with fellow colonized peoples. For Ireland’s cultural mariners of the twentieth century, navigating such currents requires a knowledge not only of sea but also of sky. In the “Wandering Rocks” episode of Ulysses, a chapter where the city of Dublin is the prominent star, the sections are separated by a series of three asterisks also known as a dinkus. As a writer for the Paris Review explains, a dinkus is “used as a section break in a text. It’s the flatlining of an asterism (⁂), which in literature is a pyramid of three asterisks and in astronomy is a cluster of stars.” Asterisms serve as a striking intervention into the textual groundswells of Joyce’s Ulysses that ultimately connect to Derek Walcott’s own navigations in Omeros as a means of paternal inheritance and transatlantic affiliation.

The sphinx is a good test case illustrating the complexities of studying Greek hybrids. The pronounced sexuality of modern sphinxes (notably those of Moreau and Ingres) sets them apart from Greek examples, which themselves are very different from the sphinxes of Egypt and the Ancient Near East. Common to all is the blurring of human/animal boundaries, a phenomenon going back to the Palaeolithic. Modern comparisons from New Guinea and Africa confirm that there is an animal dimension at the heart of being human. Hybrids, born of this mixing, are polymorphous, polysemic and polyvalent. Around the hybrid there lurks a host of questions: what bits have been mixed, how exactly are the parts combined, and is the mixture taxonomically fitting or anomalous? Each of these questions shapes our response to a hybrid, affirming the power of hybridity to challenge (or affirm) categories and taxonomies. And since taxonomies are the proof of our comprehending the world by classifying phenomena, hybridity represents a culture’s uneasiness with the limits of its epistemology. If such things exist, even if only in our stories and imagination, how certain is certainty?

Problems involving calculations of various properties associated with the density operator and entropies and their relations to more general situations in physics are included.

Born in the year of the liberation Korea from Japanese colonisation, Younghi Pagh-Paan (*1945) grew up during the Korean war and the subsequent division of her homeland. Although she trained in Seoul, her career as a composer properly started with her move to Freiburg in Germany in 1974. The result was a culture shock, and, throughout much of her career, Pagh-Paan struggled with her displacement and endeavoured to reconcile her gender and cultural identity as an Asian woman with Western modernism; vowing, in her own words, ‘[n]ot [to] write music that distances me from what […] I perceive inside me as the root of our culture’. This chapter discusses Pagh-Paan’s career and her aesthetic beliefs, such as her commitment to the student movement and democratic opposition in her country and her syncretistic religiosity that embraces the different spiritual traditions of her country, such as Shamanism and Taoism, as well as her fervent Catholicism. Analysing the reflection of these ideas in her music I conclude that, transcending notions of cultural contrast or ‘East-meets-West fusion’, Pagh-Paan’s work is a response to more than a century of intimate entanglements between Western and Korean culture.

This chapter presents the surprising mathematical result that classical systems can indeed have entanglement. However, the degree to which they can be entangled is strictly limited, while quantum systems have no limit to their amount of entanglement.

This chapter presents some basic calculations that show counterintuitive or unexpected results. First, it is shown that the Planck spectrum of light, which played an important role in the history of quantum mechanics, doesn’t say anything about the existence of indivisible particles. Second, a brief discussion of “chaos theory” shows that jumpy and unpredictable behavior can occur in classical systems. Last, the concept of “entanglement” is introduced as a basic property of quantum systems.