This chapter is devoted to several attempts at showing the necessity of quantum mechanics. It first examines whether necessity arguments can be extracted from the history of quantum mechanics, both on the matrix-mechanics side and on the wave-mechanics side. It then recounts the stunning mathematical discovery (1979) that the only one-parameter deformation of the Lie algebra of classical Hamiltonian mechanics is the Moyal algebra, which is isomorphic to the algebra of infinitesimal evolutions of quantum mechanics. The chapter goes on with an account of “quantum logic” as an attempt to justify the matrix-density representation of quantum states and the unitary evolution of these states by a natural logic of Yes–No experimental questions. The last section is a discussion of Lucien Hardy’s (2001) and more recent proposals of “natural axioms” for statistical measurement correlations, leading again to the matrix-density formulation of quantum mechanics.
This chapter is devoted to several attempts at showing the necessity of quantum mechanics. It first examines whether necessity arguments can be extracted from the history of quantum mechanics, both on the matrix-mechanics side and on the wave-mechanics side. It then recounts the stunning mathematical discovery (1979) that the only one-parameter deformation of the Lie algebra of classical Hamiltonian mechanics is the Moyal algebra, which is isomorphic to the algebra of infinitesimal evolutions of quantum mechanics. The chapter goes on with an account of “quantum logic” as an attempt to justify the matrix-density representation of quantum states and the unitary evolution of these states by a natural logic of Yes–No experimental questions. The last section is a discussion of Lucien Hardy’s (2001) and more recent proposals of “natural axioms” for statistical measurement correlations, leading again to the matrix-density formulation of quantum mechanics.
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.
The chief aim of this chapter is to give brief accounts of topics in nonrelativistic quantum mechanics that are not always treated in elementary texts. We begin with the Hilbert space formulation of quantum mechanics as set down by von Neumann in his book Mathematical Foundations of Quantum Mechanics (von Neumann, 1955), which will henceforth be referred to as von Neumann's book. Much of our concern will be with continuous spectra, which cannot be discussed adequately in the Dirac formalism. The density matrix, which will play a key role in Chapters 8 and 10, will be treated in some detail. The section on formalism is followed by one on the probability interpretation; the latter is included because Sewell's theory of measurement suggests a subtle reformulation of a part of it. These are followed by sections on superselection rules and the Galilei group, which is the relativity group of quantum mechanics. They are based on the pioneering works of Wigner and Bargmann. The last section is devoted to the fundamental theorems of von Neumann and Stone, and to Reeh's observation on the physical significance of the failure of the Stone–von Neumann uniqueness theorem at the Lie algebra level. The formalism of quantum mechanics By quantum mechanics we shall mean the nonrelativistic quantum theory of a system with a finite number, N, of particles. The number N is assumed fixed. It will be convenient, for later recall, to divide the material into subsections.
This chapter introduces the quantum mechanics. It explores the formulation in terms of the operator calculus. The chapter shows how nonrelativistic quantum kinematics emerges from the general theory. It discusses how physical quantities are related to symmetries through unitary transformations, and consider translations. The chapter also introduces the statistical operator and the density matrix. It introduces the consistent history interpretation of quantum mechanics. The true core of quantum mechanics is the rule for assigning the amplitude to an event that can be effected in indistinguishable ways. The chapter examines the presentation of Feynman's space-time approach since it offers the most intuitive formulation of quantum mechanics. A physical quantity is said to be conserved if it for any state of the system stays constant in time. In quantum mechanics this therefore means that its probability distribution in any state is independent of time.
Abstract In addition to recounting some contemporary scientific history, Chapter 6 describes the hypothesis that matter, like light, can display wavelike properties, and the creation of the various formulations of quantum mechanics. That matter could have a wavelength was proposed in 1924 by Louis de Broglie, who presented a specific formula for calculating it, one that was verified experimentally in 1927. However, de Broglie’s hypothesis was overshadowed by the creation of three versions of quantum mechanics in 1925/26. The first, denoted matrix mechanics, was proposed by Werner Heisenberg. It was quickly and successfully applied by Wolfgang Pauli to the hydrogen atom. Paul Dirac introduced the next version, which was followed by that of Erwin Schrödinger via a wave equation whose solutions, denoted wave functions, were soon interpreted byMax Born to be related to the probability that certain outcomes or events will occur: classical-physics determinism was thereby removed from quantum mechanics.
Our purpose in this chapter is to present the key concepts of quantum mechanics in the language of Hilbert spaces. The reader who has not previously met the physical ideas motivating quantum mechanics, and some of the more elementary applications of Schrödinger's equation, is encouraged to read any of a number of excellent texts on the subject such as [1–4]. Otherwise, the statements given here must to a large extent be taken on trust – not an altogether easy thing to do, since the basic assertions of quantum theory are frequently counterintuitive to anyone steeped in the classical view of physics. Quantum mechanics is frequently presented in the form of several postulates, as though it were an axiomatic system such as Euclidean geometry. As often presented, these postulates may not meet the standards of mathematical rigour required for a strictly logical set of axioms, so that little is gained by such an approach. We will do things a little more informally here. For those only interested in the mathematical aspects of quantum mechanics and the role of Hilbert space see [5–8]. Many of the standard applications, such as the hydrogen atom, will be omitted here as they can be found in all standard textbooks, and we leave aside the enormous topic of measurement theory and interpretations of quantum mechanics. This is not to say that we need be totally comfortable with quantum theory as it stands. Undoubtedly, there are some philosophically disquieting features in the theory, often expressed in the form of socalled paradoxes.
The purpose of this dissertation is to review the salient features of non-Hermitian quantum mechanics. An introduction to Hermitian quantum mechanics is included to make this review as accessible as possible. Attempts at formulating a consistent physical theory are introduced, before examining non-Hermitian theories’ uses as convenient computational frameworks. Particular emphasis is placed on recent developments in open quantum systems that utilise non-Hermitian Hamiltonians. Chapter four introduces a logic that maps a non-Hermitian Hamiltonian onto a non-Hamiltonian algebra that has a Hermitian Hamiltonian. This was put forward by Sergi, who then goes on to show its application to a two level system. The time evolution is then derived in terms of the density matrix model. This system can then be used to analyse different types of decay such as coherence and population difference. This serves to illustrate the usefulness of the approach.