This chapter focuses on parts of the basic underlying quantum-mechanical structure which are critical to realizing the relativity and locality constraints discussed in the subsequent chapters. It begins with a review of the basic operator formalism underlying standard quantum mechanics, paying particular attention to dynamics (time evolution) and symmetries. The chapter then turns to the reformulation of quantum dynamics as a sum over histories (the ‘path-integral’ approach) due to Feynman and Dirac, which has enormous conceptual and technical utility in quantum field theory. Finally, the chapter considers those aspects of quantum scattering theory which are central in teasing out the intricate physical content of field theory.
This chapter focuses on parts of the basic underlying quantum-mechanical structure which are critical to realizing the relativity and locality constraints discussed in the subsequent chapters. It begins with a review of the basic operator formalism underlying standard quantum mechanics, paying particular attention to dynamics (time evolution) and symmetries. The chapter then turns to the reformulation of quantum dynamics as a sum over histories (the ‘path-integral’ approach) due to Feynman and Dirac, which has enormous conceptual and technical utility in quantum field theory. Finally, the chapter considers those aspects of quantum scattering theory which are central in teasing out the intricate physical content of field theory.
This chapter, based on the background information of the previous chapter, will review the formalism of quantum mechanics – its fundamental mathematical structure (the basic contents of Hilbert space representation and operator algebra representation) and physical assumptions (kinematics, dynamics, and measurement theory). These are the premise and foundation for discussing the quantum histories interpretation. Then, in combination with the viewpoints of some literature in recent years, we will explore what is the interpretation of quantum mechanics, analyze the relationship between the theory of quantum mechanics and its interpretation, and declare the author’s basic stance on these two issues. These two are the primary issues that all interpretive work in the philosophy of physics will face (not limited to the interpretation of quantum mechanics), and the necessary clarification here is also conducive to the subsequent discussion. Three common terms – theory, formalism, and interpretation – are explicated and differentiated by offering the necessary definitions of their meanings, based on the works of Acuña and Ruetsche. The clarification will help to make the subsequent related discussions more precise.
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.
This chapter provides a self-contained introduction to the basic aspects of Quantum Mechanics, focusing on what is must for Quantum Field Theory. The notions of state space, unitary operators, self-adjoint operators, and projective representation are covered as well as Heisenberg’s uncertainty principle. A complete proof of Stone’s theorem is given, but the spectral theory is covered only at the heuristic level. We provide an introduction to Dirac’s formalism, which is almost universally used in physics literature. The time-evolution is described in both the Schrödinger and the Heisenberg picture. A complete treatment of the harmonic oscillator, providing an introduction to the fundamental idea of creation and annihilation operators concludes the chapter.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
In this chapter, the physical content of previous chapters is summarized, and the reader is suided towards some key areas of modern physics: the Feynman formulation, the unificaton of quantum theory with special relativity, and dualities in quantum mechanics. This very short chapter is intended to be a bridge aimed at motivating readers to attend first course in quantum field theory. Towards an overall view At this stage, the student or the general reader might want to know what picture of the physical world emerges from the material covered so far and from the areas related to it but not covered in our book, so as to be introduced to what is in sight not only for the beginner but also for the scientific community. For this purpose, we summarize very briefly the previous pictures and we arrive at three key concepts: the Lagrangian in quantum mechanics via the Feynman sum over histories; the unification of quantum mechanics and special relativity; and new duality symmetries in quantum mechanics. From Schrödinger to Feynman Quantum mechanics provides a probabilistic description of the world on atomic or sub-atomic scale. It tells us that, on such scales, the world can be described by a Hilbert space structure, or suitable generalizations. Even in the relatively simple case of the hydrogen atom, the appropriate Hilbert space is infinite-dimensional, but finite-dimensional Hilbert spaces play a role as well.
Abstract The standard von Neumann-Dirac formulation of quantum mechanics is presented as a set of five basic rules. We discuss each rule is discussed in turn paying particular attention to the conceptual history of the theory. Of central importance is the standard interpretation of states (the eigenvalue-eigenstate link) and the dynamical laws of the theory (the random collapse dynamics and the deterministic linear dynamics) and how the interpretation and dynamics work together to predict and explain the results of basic quantum experiments. While the focus is on the behavior of electrons, we also briefly consider how the theory uses the same mathematical formalism to treat other phenomena like the behavior of neutral K mesons and qbits in a quantum computer.