Quantum mechanical degrees of freedom come in many kinds and varieties and are variables that can take values in many different kinds of spaces (manifolds). Compact and noncompact degrees of freedom are two important types. In essence, a compact degree of freedom takes values in a finite range whereas the noncompact take values in an infinite range. The simplest compact degree of freedom is the Ising spin variable μ that takes two values, namely μ = ±1, and which was studied in Chapters 8 and 9. In Chapters 11 to 16, the noncompact degree of freedom x – takingvaluesonthe real line, that is x ∈[−∞, +∞] – was studied for various models. Compact degrees of freedom have many specific properties not present for the noncompact case, and the focus of this chapter is on these properties. The degrees of freedom that take values in compact spaces for two different cases are discussed, with each case having its own specific features, as follows: • A degree of freedom taking values on a circle S 1 . This case has a nontrivial topological structure that occurs in a wide range of problems, with the simplest case being of a quantum mechanical particle moving on a circle S 1 . • A degree of freedom taking values on a two-dimensional sphere S 2 , a compact space. This degree of freedom can represent a quantum particle moving on a sphere and is the simplest case of a quantum particle moving on a curved manifold.
Quantum mechanical degrees of freedom come in many kinds and varieties and are variables that can take values in many different kinds of spaces (manifolds). Compact and noncompact degrees of freedom are two important types. In essence, a compact degree of freedom takes values in a finite range whereas the noncompact take values in an infinite range. The simplest compact degree of freedom is the Ising spin variable μ that takes two values, namely μ = ±1, and which was studied in Chapters 8 and 9. In Chapters 11 to 16, the noncompact degree of freedom x – takingvaluesonthe real line, that is x ∈[−∞, +∞] – was studied for various models. Compact degrees of freedom have many specific properties not present for the noncompact case, and the focus of this chapter is on these properties. The degrees of freedom that take values in compact spaces for two different cases are discussed, with each case having its own specific features, as follows: • A degree of freedom taking values on a circle S 1 . This case has a nontrivial topological structure that occurs in a wide range of problems, with the simplest case being of a quantum mechanical particle moving on a circle S 1 . • A degree of freedom taking values on a two-dimensional sphere S 2 , a compact space. This degree of freedom can represent a quantum particle moving on a sphere and is the simplest case of a quantum particle moving on a curved manifold.
The simplest possible quantum degree of freedom has only two discrete possible values and is called an Ising spin. This could be a spin system having only two possible values, namely “up” or “down,” or this could be a quantum particle that can occupy only two possible positions. To simplify the problem to its bare essentials, a two-state system is studied in zero-dimension space and in Euclidean time; the two-state system occupies a single point and, as it evolves in time, it can point either up or down. Furthermore, time evolution is simplified by discretizing time into a finite time lattice. The two-state degree of freedom in zero-dimension space propagating on a time lattice is called the Ising model. The one-dimensional Ising model is a toy model that is studied to develop the essential ideas of quantum mechanics. The concepts of state space, Hamiltonian, evolution kernel, and path integration have a discrete and simple representation in the Ising model. The Ising model can be viewed as a statistical mechanical system in thermal equilibrium, with the Ising spins occupying a one-dimensional space lattice, and hence is called the one-dimensional Ising model. The Ising model can be defined on higher dimensional lattices and forms one of the bedrocks of theoretical statistical mechanics. In Sections 8.1 to 8.5 the degree of freedom, state space, Hamiltonian, path integral and correlator for the Ising model are discussed. In Section 8.6, the technique of spin decimation is illustrated using an Ising model with coupling depending on the lattice site.
This Festschrift in honour of J. A. de Azcarraga gives an introduction to the concept of duality, i.e., to the relativity of the notion of a quantum, in the context of the quantum mechanics of a finite number of degrees of freedom. Although the concept of duality arises in string and M--theory, Vafa has argued that it should also have a counterpart in quantum mechanics, before moving on to second quantisation, fields, strings and branes. We illustrate our analysis with the case when classical phase space is complex projective space, but our conclusions can be generalised to other complex, symplectic phase spaces, both compact and noncompact.
An examination of the postulates of quantum mechanics reveals a number of fundamental mathematical constructs that form its theoretical underpinnings. Many of the results that are summarized in this Chapter will only become clear after reading the rest of the book and a re-reading may be in order. The dynamical variables of classical mechanics are superseded by the quantum degree of freedom. An exhaustive and complete description of the indeterminate degree of freedom is given by its state function, which is an element of a state space that, in general, is an infinite-dimensional linear vector space. The properties of the indeterminate degree of freedom are extracted from its state vector by the linear action of operators representing experimentally observable quantities. Repeated applications of the operators on the state function yield the average value of the operator for the state [Baaquie (2013e)]. The conceptual framework of quantum mechanics is discussed in Section 2.1. The concepts of degree of freedom, state space and operators are briefly reviewed in Sections 2.3–2.5. Three distinct formulations of quantum mechanics emerge from the superstructure of quantum mechanics and these are briefly summarized in Sections 2.7–2.9. The Copenhagen quantum postulate The Copenhagen interpretation of quantum mechanics, pioneered by Niels Bohr and Werner Heisenberg, provides a conceptual framework for the interpretation of the mathematical constructs of quantum mechanics and is the standard interpretation that is followed by the majority of practicing physicists [Stapp (1963), Dirac (1999)].
Continuous degrees of freedom The quantum systems we have discussed so far have been described by finite-dimensional Hilbert spaces. Basic measurements on such systems have a finite number of possible outcomes, and a quantum state predicts a discrete probability distribution over these. Now we wish to extend our theory to handle systems with one or more continuous degrees of freedom, such as the position of a particle that can move in one dimension. This will require an extension of our theory to Hilbert spaces of infinite dimension, and to systems with continuous observables. There is a philosophical issue here. How do we know that there really are infinitely many distinct locations for a particle? The short answer is, we don't. It might be that space itself is both discrete (at the tiniest scales) and bounded (at the largest), so that the number of possible locations of a particle is some very large but finite number. If this is the case, then the continuum model for space is nothing more than a convenient approximation. Infinity is just a simplified way of describing a quantity that is immense, but still finite. In this section, we will adopt this view of infinity. We will imagine that any continuous variable is really an approximation of a “true” discrete variable. This idea will motivate the continuous quantities and operations that we need.
This work analyzes the fundamental distinction between finite-dimensional quantum systems and infinite-dimensional wavefunction systems. It introduces three mathematically independent notions of dimension—linear dimension, parameter dimension, and degree-of-freedom dimension—and shows that finite-dimensional systems are “triple-finite,” while wavefunction systems such as L^2(R) are “triple-infinite.” The paper distinguishes geometric continuity on finite-dimensional projective manifolds (e.g., the Bloch sphere) from structural continuity arising from the uncountable degrees of freedom of continuous configuration spaces. It further explains why infinite-dimensional systems cannot be reduced to finite-dimensional ones, drawing on functional analysis, spectral theory, and the properties of unbounded operators. Intermediate systems such as L^2([0,L]) and the extension to quantum field theory are also discussed, illustrating how spatial topology shapes spectral and dynamical behavior. The framework clarifies the categorical difference between finite- and infinite-dimensional quantum state spaces and highlights their complementary roles in quantum theory. 本文系统分析了有限维量子系统与无限维波函数系统之间的根本差异。文章提出三种彼此独立的“维度”概念——线性维度、参数维度与自由度维度,并指出有限维系统是“三重有限”,而波函数系统(如 L^2(R))是“三重无限”。有限维系统的连续性来自射影流形的几何结构(如 Bloch 球),而无限维系统的连续性来自连续构型空间所带来的不可数自由度。文章进一步利用泛函分析与谱理论论证了无限维系统对有限维系统的不可约化性,并讨论了中间系统 L^2([0,L]) 以及量子场论中的更高层次结构连续性。该框架揭示了有限维与无限维量子态空间在范畴上的本质差异,并说明它们在量子理论中所扮演的互补角色。
Quantum mechanics is one of the pillars of modern physics, however rather difficult to teach at the introductory level due to the conceptual difficulties and the required advanced mathematics. Nevertheless, attempts to identify relevant features of quantum mechanics and to put forward concepts of how to teach it have been proposed.1–8 Here we present an approach to quantum physics based on the simplest quantum mechanical system—the quantum bit (qubit).1 Like its classical counterpart—the bit—a qubit corresponds to a two-level system, i.e., some system with a physical property that can admit two possible values. While typically a physical system has more than just one property or the property can admit more than just two values, in many situations most degrees of freedom can be considered to be fixed or frozen. Hence a variety of systems can be effectively described as a qubit. For instance, one may consider the spin of an electron or atom, with spin up and spin down as two possible values, and where other properties of the particle such as its mass or its position are fixed. Further examples include the polarization degree of freedom of a photon (horizontal and vertical polarization), two electronic degrees of freedom (i.e., two energy levels) of an atom, or the position of an atom in a double well potential (atom in left or right well). In all cases, only two states are relevant to describe the system.