A simple harmonic oscillator, or oscillator in short, is one of the most important systems in physics as well as in quantum mechanics. The path integral and Hamiltonian of a simple harmonic oscillator are leading exemplars for the description of a wide class of physical systems. An oscillator is a theoretical model of great utility, primarily due to its simplicity that allows for the exact derivation of many results.

The simple harmonic oscillator can be generalized to the case of infinitely many oscillators, and is also the starting point for analysis of many nonlinear quantum systems.

Path integrals for the simple harmonic oscillator are all based on Gaussian integration, which has been discussed in Section 7.2. The harmonic oscillator is described by Gaussian path integrals, and is a bedrock of the general theory of path integration. Moreover, all perturbation expansions of nonlinear path integrals about the oscillator path integral, which includes the semi-classical expansion, are based on results of Gaussian integration.

The properties of the harmonic oscillator are analyzed in the coordinate space representation, and its path integral as well as its correlation functions are analyzed. All formulas and derivations are for Euclidean time.

In Sections 11.1 and 11.2 the Hamiltonian and state space of the oscillator are studied and the correlator is derived using state space methods. In Section 11.2 the infinite time oscillator is introduced and in Sections 11.3–11.5 the path integral for the oscillator is evaluated.

Quantum mechanics is undoubtedly one of the most accurate and important scientific theories in the history of science. The theoretical foundations of quantum mechanics have been discussed in depth in Baaquie (2013e), where the main focus is on the interpretation of the mathematical symbols of quantum mechanics and on its enigmatic superstructure. In contrast, the main focus of this book is on the mathematics of path integral quantum mechanics.

The traditional approach to quantum mechanics has been to study the Schrödinger equation, one of the cornerstones of quantum mechanics, and which is a special case of partial differential equations. Needless to say, the study of the Schrödinger equation continues to be a central task of quantum mechanics, yielding a steady stream of new and valuable results.

Interestingly enough, there are two other formulations of quantum mechanics, namely the operator approach of Heisenberg and the path integral approach of Dirac–Feynman, that provide a mathematical framework which is independent of the Schrödinger equation. In this book, the Schrödinger equation is never directly solved; instead the Hamiltonian operator is analyzed and path integrals for different quantum and classical random systems are studied to gain an understanding of quantum mathematics.

I became aware of path integrals when I was a graduate student, and what intrigued me most was the novelty, flexibility and versatility of their theoretical and mathematical framework. I have spent most of my research years in exploring and employing this framework.

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:

1. • A degree of freedom taking values on a circle S1. 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 S1.

2. • A degree of freedom taking values on a two-dimensional sphere S2, 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.

Operators represent physically observable quantities, as discussed in Section 2.5. The structure and property of operators depend on the nature of the degree of freedom; operators act on the state space and in particular on the state vector of a given degree of freedom. The significance of operators in the interpretation of quantum mechanics has been discussed in Baaquie (2013e).

The operators discussed in this chapter are mostly based on the continuous degree of freedom, which is analyzed in Section 3.1. Hermitian operators represent physically observable properties of a degree of freedom and their mathematical properties are defined in Section 3.3. The coordinate and momentum operators are the leading exemplar of a pair of noncommuting Hermitian operators and these are studied in some detail in Section 3.4. The Weyl operators yield, as in Section 3.5, a finite-dimensional example of the shift and scaling operators; Section 3.8 provides a unitary representation of the coordinate and momentum operators.

The term self-adjoint operator is used for Hermitian operators when there is a need to emphasize the importance of the domain of the Hilbert space on which the operators act – a topic not usually discussed in most books on quantum mechanics. Sections 3.10 and 3.11 discuss the concept of self-adjoint operators, in particular the crucial role played by the domain for realizing the property of self-adjointness. It is shown in Section 3.12 how the requirement of self-adjointness yields a non-trivial extension of Hamiltonians that include singular interactions.

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 book studies the mathematical aspect of path integrals and Hamiltonians – which emerge from the formulation of quantum mechanics. The theoretical framework of quantum mechanics provides the mathematical tools for studying both quantum indeterminacy and classical randomness. Many problems arising in quantum mechanics as well as in vastly different fields such as finance and economics can be addressed by the mathematics of quantum mechanics, or quantum mathematics in short. All the topics and subjects in the various chapters have been specifically chosen to illustrate the structure of quantum mathematics, and are not tied to any specific discipline, be it quantum mechanics or stochastic systems.

The book is divided into the following six parts, in accordance with the Chapter dependency flowchart given below.

• Part one addresses the Fundamental principles of path integrals and (Hamiltonian) operators and consists of five chapters. Chapter 2 is on the Mathematical structure of quantum mechanics and introduces the mathematical framework that emerges from the quantum principle. Chapters 3 to 6 discuss the mathematical pillars of quantum mathematics, starting from the Feynman path integral, summarizing Hamiltonian mechanics and introducing path integral quantization.

• Part two is on Stochastic processes. Stochastic systems are dissipative and are shown to be effectively modeled by the path integral. Chapter 7 is focused on the application of quantum mathematics to classical random systems and to stochastic processes.

Path integrals are by and large defined directly in terms of the configuration space representation of the quantum entity's degrees of freedom and employ the Lagrangian description of the quantum entity; most of the path integrals in this book follow this approach.

The Hamiltonian provides another independent approach for defining path integrals and is discussed in this chapter. Two important path integrals, which are directly based on the Hamiltonian, are the following: (a) one that is defined on the degree of freedom's phase space, defined as the tensor product of the degree of freedom space and its canonical conjugate momentum space; and (b) path integrals using the coherent state basis instead of the coordinate basis. Path integrals defined on phase space, or for coherent states, are both based on the Hamiltonian.

To put in the foreground the role of the Hamiltonian in quantum mechanics, the canonical equations connecting the Lagrangian to the Hamiltonian are discussed. A brief review of Hamiltonian mechanics, also called the canonical equations, is given in Section 5.1 and the connection of symmetries with conservation laws is discussed in Section 5.2. The Hamiltonian is derived from the Lagrangian in Section 5.3, for both Minkowski and Euclidean time. Phase space path integrals are defined in Section 5.4. Canonical quantization based on the Poisson brackets is discussed in Section 5.5, and Dirac brackets required for quantizing constrained systems are derived in Section 5.7. Coherent states and their path integrals are discussed in Sections 5.10 to 5.14.

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)].

Introduction to part five

The most widely used actions in quantum mechanics have a kinetic term that is the velocity squared of the degree of freedom and a potential term that depends on the degree of freedom. The kinetic term fixes the equal time commutation equation of the degree of freedom, as shown in Section 6.7. Furthermore, the velocity term in the Lagrangian entails that all the possible indeterminate paths obey two boundary conditions, which in turn yields a state space that depends on the degree of freedom only. In Section 11.13, this property of the indeterminate paths was used for deriving the state function of the harmonic oscillator.

The action with acceleration has a kinetic term that is given by the acceleration squared of the degree of freedom, in addition to the usual velocity and potential terms. It is an example of higher derivative Lagrangians, discussed in Simon (1990). The higher derivative quantum systems have many remarkable properties not present for the usual cases studied so far.

The action with acceleration arises in many diverse fields and has been widely studied; it describes the behaviour of “stiff” polymers, of cell walls, of the formation of microemulsions, the properties of chromoelectric flux lines in quantum chromodynamics, as well as the Big Bang singularity in cosmology.

There is no conservation of probability in the Euclidean formulation of the acceleration Lagrangian. A narrow definition of a Hamiltonian, and the one that holds in quantum mechanics but is not suitable for our case, is that the Hamiltonian defines a probability conserving unitary evolution. A broader definition of the Hamiltonian is adopted that is consistent with both statistical mechanics and quantum mechanics. Namely, the Hamiltonian – which is equivalent to the transfer matrix in statistical mechanics – is the generator of infinitesimal translations in time for both the Minkowski and Euclidean cases.

The Hamiltonian derived for the acceleration Lagrangian in Chapter 13 is studied in this chapter using the concepts of the state space and operators. The Hamiltonian for unequal and real frequencies ω1,ω2 is pseudo-Hermitian and is explicitly mapped to a Hermitian Hamiltonian using a nontrivial differential operator. An explicit expression is obtained for the matrix element of the differential operator. The mapping fails for a critical value of the coupling constants.

The Hamiltonian and state space are shown to have real and complex branches that are separated by the critical point. At the critical point, the Hamiltonian is inequivalent to any Hermitian operator, and is shown in Chapter 15 as being equal to an infinite-dimensional block diagonal matrix, with each block being a Jordan matrix.