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Monograph series on nonlinear science and complexity

Chapter 23 Path Integral for Non-Hamiltonian Systems

Vasily Evgenevich Tarasov

Published Jan 1, 2008• 0 citations
Publisher
DOI
10.1016/s1574-6917(07)00023-2

Related papers

Hamiltonian path integrals

Ashok Kumar Kapoor

•Jan 1, 1988•0 citations

Algebraic Generalization

Tarō Kashiwa, Yoshio Ohnuki, Masuo Suzuki

Abstract The path integral method discussed up to the preceding section is based on Trotter’s formula (1.10) and its accuracy is of the order of Δt for the time slice Δt. This is sufficient for discussing quantum systems qualitatively. Higher accuracy of the Quantum Monte method is required for studying quantum systems numerically. In the present chapter, we explain the algebraic generalization of the path integral method. By applying the integral representations given in chapters 1−4 to such generalized product formulae, some new kinds of path integral of higher accuracy can be obtained. This is crucial when the essence of a phenomenon can be found only by increasing the accuracy of calculations. It should be emphasized that such higher order decomposition formulae to be explained in the present chapter can also be applied not only to the path integral method but to many other problems such as non-linear dynamics in Hamiltonian systems.

•Jul 3, 1997•0 citations

Synopsis

Belal Ehsan Baaquie

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.

Cambridge University Press eBooks•Mar 27, 2014•0 citations

Chapter 9: Path integrals and quantum mechanics

De Gruyter eBooks•Mar 14, 2013•0 citations

Quantum mechanics and path integrals

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Institute of Physics eBooks•Oct 1, 2022•1 citation

Conclusions

Belal Ehsan Baaquie

All the chapters of the book have examined and studied various aspects of path integrals and Hamiltonians, which in turn exemplified different aspects of quantum mathematics. The principles of quantum mathematics, stated in Chapter 2, can be summarized as follows: • The fundamental degrees of freedom F form the bedrock of the quantum system. • A linear vector state space V based on the degrees of freedom F provides an exhaustive description of the quantum system. • Operators O, which includes the Hamiltonian H , represent the physical properties of the degree of freedom and act on the state space V . Observable quantities are the matrix elements of the physical operators. • A spacetime description of quantum indeterminacy is encoded in the Lagrangian L and the Dirac–Feynman formula relates it to the Hamiltonian. • The path integral provides a representation of all the physical properties of a quantum system. In particular, the path integral yields the correlation functions of the degrees of freedom as well as the probability amplitudes for quantum transitions. • The interconnection of the path integral with the underlying Hamiltonian and state space is a specific feature of quantum mathematics that distinguishes path integration from functional integration in general.

Cambridge University Press eBooks•Mar 27, 2014•0 citations
Abstract The path integral method discussed up to the preceding section is based on Trotter’s formula (1.10) and its accuracy is of the order of Δt for the time slice Δt. This is sufficient for discussing quantum systems qualitatively. Higher accuracy of the Quantum Monte method is required for studying quantum systems numerically. In the present chapter, we explain the algebraic generalization of the path integral method. By applying the integral representations given in chapters 1−4 to such generalized product formulae, some new kinds of path integral of higher accuracy can be obtained. This is crucial when the essence of a phenomenon can be found only by increasing the accuracy of calculations. It should be emphasized that such higher order decomposition formulae to be explained in the present chapter can also be applied not only to the path integral method but to many other problems such as non-linear dynamics in Hamiltonian systems.
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.
All the chapters of the book have examined and studied various aspects of path integrals and Hamiltonians, which in turn exemplified different aspects of quantum mathematics. The principles of quantum mathematics, stated in Chapter 2, can be summarized as follows: • The fundamental degrees of freedom F form the bedrock of the quantum system. • A linear vector state space V based on the degrees of freedom F provides an exhaustive description of the quantum system. • Operators O, which includes the Hamiltonian H , represent the physical properties of the degree of freedom and act on the state space V . Observable quantities are the matrix elements of the physical operators. • A spacetime description of quantum indeterminacy is encoded in the Lagrangian L and the Dirac–Feynman formula relates it to the Hamiltonian. • The path integral provides a representation of all the physical properties of a quantum system. In particular, the path integral yields the correlation functions of the degrees of freedom as well as the probability amplitudes for quantum transitions. • The interconnection of the path integral with the underlying Hamiltonian and state space is a specific feature of quantum mathematics that distinguishes path integration from functional integration in general.