Our fundamental theories, that is, the quantum theory and general relativity, are invariant under time reversal. Only when we treat systems from the point of view of thermodynamics, that is, averaging between many subsystem components, an arrow of time emerges. The relation between thermodynamic and the quantum theory has been fertile, deeply explored and still a source of new investigations. The relation between the quantum theory and gravity, while it has not yet brought an established theory of quantum gravity, has certainly sparked in-depth analysis and tentative new theories. On the other hand, the connection between gravity and thermodynamics is less investigated and more puzzling. I review a selection of results in covariant thermodynamics, such as the construction of a covariant notion of thermal equilibrium by considering tripartite systems. I discuss how such construction requires a relational take on thermodynamics, similar to what happens in the quantum theory and in gravity.

According to the standard account of time reversal, namely the account found in physics books, a time-reversal transformation involves a temporal operator 𝑇 that, when acting on a sequence of states, inverts the order with which states happen, and suitably changes the properties of the entities in the state so as to make the theory time-reversal invariant. This ‘symmetry first’ approach imposes symmetries on the theory: the changes in the states are a consequence of requiring the theory to be time-reversal invariant. Some (Albert, Callender) find this view unjustified: we discover a theory has a given symmetry, on the basis of the theory’s ontology, not the other way around. So, they propose a ‘metaphysics first’ approach, sometimes dubbed ‘pancake account’ of time reversal: 𝑇 inverts the order of the states but does nothing else. Consequently, since there are no obvious independent reasons for the state to change as 𝑇 prescribes to preserve time-reversal symmetry, then the theory is not time-reversal invariant. In this chapter I wish to further motivate the pancake account of time reversal by arguing the standard account is far more problematical than has been suggested. Moreover, I defend the pancake account from recent objections raised by Roberts. Finally, since I value symmetries, I propose an alternative account, which aims at retaining the best of both approaches: the 𝑇 operator changes the order of the states, it leaves the state unaffected (like the pancake account), but also makes the theory time-reversal invariant (like the standard account).

In 1925, as matrix mechanics was taking shape, Lucy Mensing (1901−1995), who earned her PhD with Lenz and Pauli in Hamburg, came to Göttingen as a postdoc. She was the first to apply matrix mechanics to diatomic molecules, using the new rules for the quantization of angular momentum. As a byproduct, she showed that orbital angular momentum can only take integer values. Impressed by this contribution, Pauli invited her to collaborate on the susceptibility of gases. She then went to Tübingen, where many of the spectroscopic data were obtained that drove the transition from the old to the new quantum theory. It is hard to imagine better places to be in those years for young quantum physicists trying to make a name for themselves. This chapter describes these promising early stages of Mensing’s career and asks why she gave it up three years in. We argue that it was not getting married and having children that forced Lucy Mensing, now Lucy Schütz, out of physics, but the other way around. Frustration about her own research in Tübingen and about the prevailing male-dominated climate in physics led her to choose family over career.

In recent years, Grete Hermann (1901–1984) has been rediscovered as a principal figure in the history and philosophy of quantum physics. In particular, her criticism of Johann von Neumann’s so-called “no hidden variables” proof is a focal point of interest. Did she really find a mistake in this proof? We argue that the whole debate is misleading. It fits too well with the image of a forgotten woman who disproved a result of a mathematical genius, but it is neither historically nor systematically justified. Despite Hermann’s challenging thoughts on quantum physics, her impressive and important achievements were in ethics and politics. We offer a new and broader reading of Hermann’s interpretation of quantum physics and try to build a bridge between her works on quantum physics and ethics. In doing so, we focus on her interpretation of Heisenberg’s cut as a methaphorical device to argue against Leonard Nelson’s theory of free will and for freedom and responsibility as cornerstones of any democratic society.

I introduce quaternions by recounting the story of how Hamilton discovered them, but in far more detail than other authors give. This detail is necessary for the reader to understand why Hamilton wrote his quaternion equations in the way that he did. I describe the role of quaternions in rotation, show how to convert between them and matrices, and discuss their role in modern computer graphics. I describe a modern problem in detail whereby Hamilton’s original definition has been ‘hijacked’ in a way that has now produced much confusion. I end by describing how quaternions play a role in topology and quantum mechanics.

Chapter 2 serves as an introduction to the fundamental principles of quantum mechanics, focusing on closed systems. It begins with the historic Stern–Gerlach experiment, highlighting the discovery of quantum spin. The narrative then shifts to the mathematical framework of quantum mechanics, covering inner product spaces, Hilbert spaces, and linear operators. These concepts are crucial for understanding the behavior and manipulation of quantum states, the core of quantum information theory.

The chapter further explores the encoding of information in quantum states, emphasizing qubits, and discusses quantum measurements, revealing the probabilistic nature of quantum mechanics. Additionally, it addresses hidden variable models, offering insights into the deterministic versus probabilistic interpretations of quantum phenomena.

Unitary evolution and the Schrödinger equation are introduced as mechanisms for the time evolution of quantum states, showcasing the deterministic evolution in the absence of measurements. This section underscores the dynamic aspect of quantum systems, pivotal for advancements in quantum information theory.

Did Werner Heisenberg and Carl Friedrich von Weizsäcker compromise with the Nazis? The story begins with Albert Einstein, who became a target for conservative physicists like Philipp Lenard and Johannes Stark who could not follow Einstein’s physics, and the early Nazi Party that rejected Einstein as a Jew as well as his pacifism and internationalism. When Hitler came to power, Lenard and Stark gained great influence. Stark in particular tried to accumulate power but steadily lost influence through conflicts with other Nazis. When Stark’s nemesis, the theoretical physicist Arnold Sommerfeld, was going to retire and be succeeded by Werner Heisenberg, Stark launched a vicious attack on Heisenberg in the SS newspaper. Heisenberg appealed to SS Leader Heinrich Himmler and thanks to support from the aeronautical engineer Ludwig Prandtl was eventually rehabilitated by the SS. The established physics community then launched a counterattack against the “Aryan Physics” of Lenard and Stark, which included writing Einstein out of the history of relativity theory. This was arguably Heisenberg’s greatest compromise with Nazism.

This monograph is about the change in meaning and scope of human rights rules, principles, ideas and concepts, and the interrelationships and related actors, on moving from the physical domain into the online domain. The transposition into the digital reality can alter the meaning of well-established offline human rights to a wider or narrower extent; it can turn positivity into negativity and vice versa. The digital human rights realm has different layers of complexity in comparison with the offline realm.

This project uses the procedure of root-finding to resolve the eigenvalue problem of a rectangular quantum well. The procedure is applied to determine the first two to three energy levels of a simplified model of a hydrogen atom, represented by the rectangular quantum well. The project also explores the eigenvalue problem within various physics fields. While the project involves simple mathematical operations, it is rooted in complex physical concepts like quantum mechanics, often unfamiliar to first-year students. The fundamentals of quantum mechanics are introduced, providing enough understanding for successful project execution. The project initially focuses on the central object, the quantum state, and its probabilistic nature in quantum mechanics. The Schrödinger equation, an eigenvalue problem, is used to find state functions. This project explores eigenenergies and eigenfunctions within a rectangular finite quantum well, treating the well as a simplistic 1D model of the hydrogen atom.

Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.

Basic concepts of quantum mechanics: Schroedinger equation; Dirac notation; the energy representation; expectation value; Hermite operators; coherent superposition of states and motion in the quantum world; perturbation Hamiltonian. Time-dependent perturbation theory: harmonic perturbation. Transition rate: Fermi’s golden rule. The density matrix; pure and mixed states. Temporal dependence of the density operator: von Neuman equation. Randomizing Hamiltonian. Longitudinal and transverse relaxation times. Density matrix and entropy.

In this chapter, we review basic concepts from quantum mechanics that will be required for the study of superconducting quantum circuits. We review the fundamental idea of energy quantization and how this can be formalized, using Dirac's ideas, to develop a quantum mechanical description that is consistent with the classical theory for a comparable object. We review the notions of quantum state, observable and projective and generalized measurements, particularizing some of these ideas to the simple case of a two-dimensional object or qubit.

Here we discuss the possible relation ofour generalconjecture on global attractors ofnonlinear Hamiltonian PDEs todynamicaltreatment of Bohr's postulates and of wave--particle duality, which are fundamental postulates of quantum mechanics, in the context of couplednonlinear Maxwell--SchrödingerandMaxwell--Dirac equations. The problem of adynamicaltreatment was the main inspiration for our theoryof global attractors ofnonlinear Hamiltonian PDEs.