A quantum entity is intrinsically indeterminate and is not in a determinate (classical) state when it is not being observed. The quantum entity is found to be in a determinate state only if an experiment is carried out to observe it, as discussed in Baaquie (2013e). The indeterminate behavior of the quantum entity is described by the formalism of quantum mathematics discussed in Chapter 2. The measure of quantum indeterminacy and uncertainty is Planck's constant ħ. Randomness also exists in classical systems; however, a classical system's randomness is in principle different from quantum indeterminacy, since a priori, a classical random system is intrinsically in a determinate state. Classical randomness arises entirely due to our lack of complete knowledge of the determinate state. The theory of classical probability describes a classical system about which one has incomplete knowledge. There are many ways of introducing randomness in a classical system, depending on its nature. A classical random system in thermal equilibrium is described by statistical mechanics. A classical system undergoing random time evolution can be described by taking the classical equations of motion and adding a random force to it. The evolution of the classical particle is then described by a stochastic process. The random force is usually represented by Gaussian white noise. Classical random processes driven by white noise, such as those that arise in finance and other areas, form a wide class of problems that can be described by classical probability theory. It is shown in this chapter that the mathematical framework of quantum mechanics, and in particular, the Hamiltonian operator and path integrals, provide an appropriate mathematical formalism for describing processes that are driven by Gaussian white noise.
A quantum entity is intrinsically indeterminate and is not in a determinate (classical) state when it is not being observed. The quantum entity is found to be in a determinate state only if an experiment is carried out to observe it, as discussed in Baaquie (2013e). The indeterminate behavior of the quantum entity is described by the formalism of quantum mathematics discussed in Chapter 2. The measure of quantum indeterminacy and uncertainty is Planck's constant ħ. Randomness also exists in classical systems; however, a classical system's randomness is in principle different from quantum indeterminacy, since a priori, a classical random system is intrinsically in a determinate state. Classical randomness arises entirely due to our lack of complete knowledge of the determinate state. The theory of classical probability describes a classical system about which one has incomplete knowledge. There are many ways of introducing randomness in a classical system, depending on its nature. A classical random system in thermal equilibrium is described by statistical mechanics. A classical system undergoing random time evolution can be described by taking the classical equations of motion and adding a random force to it. The evolution of the classical particle is then described by a stochastic process. The random force is usually represented by Gaussian white noise. Classical random processes driven by white noise, such as those that arise in finance and other areas, form a wide class of problems that can be described by classical probability theory. It is shown in this chapter that the mathematical framework of quantum mechanics, and in particular, the Hamiltonian operator and path integrals, provide an appropriate mathematical formalism for describing processes that are driven by Gaussian white noise.
In this chapter, we consider the way quantum theory represents deterministic, statistical, and probabilistic aspects of Nature. After a discussion in Section 3.1 of uncertainty in general, we discuss in Sections 3.2-3.4 several formal notions of classical probability, their relation to the probability concept used in applied statistics, and their dependence on the description used. Section 3.5 briefly discusses various examples, where probabilistic features emerge from fully deterministic situations-a theme to be taken up in much more detail in Chapter 11. Section 3.6 then shows how the statistical aspects of the quantum formalism naturally follow from the weak law of large numbers. The notions of c-probability in the classical case and of q-probability in the quantum case are formally defined in Section 3.7. Historically, the concept of classical probability (including its use in stochastic processes) was given an undisputed formal mathematical foundation in 1933 in terms of the measure-theoretic setting of Kolmogorov [164]. Apart from this traditional axiomatic foundation of classical probability theory, there is a less well-known equivalent axiomatic treatment by Whittle [302] in terms of expectations. Here probabilities appear as the expectations of statements, {0, 1}-valued random variables. von Plato [240] discusses the history of the concept of probability. Krüger et al. [169] discuss the history of probability in the various fields of application. Sklar [275] discusses the philosophical problems of the probability concept, with an emphasis on statistical mechanics.
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.
Introduction The apparent generality of the title to this chapter is appropriately reduced by sufficiently precisely defining the terms used. We have in mind the natural extension to quantum mechanical settings of the ideas usually covered in a discussion of stochastic processes in classical mechanical settings. In classical mechanical settings, the fundamental process is Brownian motion, which describes the influence of external fluctuating forces on a small particle. The mathematical apparatus developed for the study of Brownian motion has been generalized and applied to irreversible thermodynamics (Onsager and Machlup, 1953), fluctuating hydrodynamics (Fox and Uhlenbeck, 1970), light scattering from fluids and mixtures (Berne and Pecora, 1976) and chemical reactions (McQuarrie and Keizer, 1981). In each case, the stochastic process involves fluctuations generated by thermal motions. We are not considering the stochastic trajectories (chaos) of classically deterministic motion in non-linear systems (Lichtenberg and Lieberman, 1983), which is an entirely separate context for stochastic processes. Our goal in this chapter is to study the effects of thermal fluctuations in quantum mechanical settings. Therefore, we exclude a discussion of interpreting quantum mechanics as a classical stochastic motion (Nelson, 1966), an idea which should have been put to rest by Grabert, Hänggi and Talkner (1979). We also exclude a discussion of quantum chaos, which is currently a very active area of research (Zaslavsky, 1981). Just as Brownian motion provides the paradigm for stochastic processes in classical mechanical settings, the Anderson–Kubo oscillator (Anderson, 1954; Kubo, 1954; Kubo, Toda and Nashitsume, 1985) is the paradigm for stochastic processes in quantum mechanical settings.
This chapter discusses the mathematical formalism of the theory of quantum mechanics and present the differences between the theory of classical mechanics and quantum mechanics. The mathematics that is used to account for the indeterminacy of quantum mechanics renders the mathematical approach of quantum mechanics distinct from that of classical mechanics. In quantum mechanics, any dynamical variable that is an observable that can be measured experimentally is described as a linear Hermitian operator. In quantum mechanics, it is not only necessary to define an observable, but it is also necessary to define the states associated with it. When a system is in a linear superposition of states of an observable, the number of different values of an observable observed in an experiment is equal to the number of states that a quantum mechanical system can jump into upon measurement of an observable.
Introduction This chapter gives the physical and mathematical basis of decoherence in quantum systems, with particular emphasis on its importance in quantum information theory and quantum computation. We assume that the reader is already familiar with the basics of quantum mechanics, such as wave functions, the Schrödinger equation, and Hilbert spaces, which are covered in standard textbooks. Nevertheless, all basic concepts will be at least briefly defined and described. A working knowledge of graduate-level linear algebra and calculus is also assumed. In quantum information processing, the term decoherence is often used loosely to describe any kind of noise that can affect a quantum system. This can include sources of noise that seem, qualitatively, to be quite different from each other. These noise sources include: (i) Random driving forces from the environment (e.g., Brownian motion). (ii) Interactions that produce entanglement between the system and the environment. (iii) Statistical imprecision in the experimental controls on the system (e.g., timing errors, frequency fluctuations, etc.). Remarkably, all of these sources of noise can be described by an almost identical mathematical framework, in which pure quantum states can evolve into mixed quantum states, and the quantum effects necessary for information processing – interference, entanglement, and reversible evolution – are disrupted. The chapter begins by examining a highly schematic model of decoherence, involving systems of qubits, but uses this model to present the mathematical formalism used in describing more realistic systems as well.
Classical mechanics have been used to study Newtonian mechanics including conventional mechanics, electrodynamics, cosmology, and relativity physics of Einstein. Compared to quantum mechanics, classical mechanics is deterministic, in which the processes are well determined and foreseeable. On the contrary, processes in quantum mechanics are undeterministic, uncertain, nonlocal, and intrinsically random. Therefore, quantum mechanics deals with probability and uncertainty principles to study the atomic and subatomic world. Due to their extremely small sizes of the quantum particles, the capabilities of apparatuses are quite limited to measure the undeterministic and uncertain behaviors of the particles. Therefore, interactions between these very small particles in the quantum world, such as electrons and photons are simulated using mathematical abstractions that will be discussed in the following sections. Mathematics of quantum physics describes nonlocal correlations between the particles with the intrinsic randomness (e.g. Schrödinger’s cat) in the combined system. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.