Cambridge University Press, 2014. - 417 pp. Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of ﬁelds, from quantum mechanics, solid state physics, statistical mechanics, quantum ﬁeld theory, and superstring theory to ﬁnancial modeling, polymers, biology, chemistry, and quantum ﬁnance. Eschewing use of the Schrodinger equation, the powerful and ﬂexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system’s path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics, mathematical ﬁnance, and engineering.Contents: Synopsis. Fundamental principles The mathematical structure of quantum mechanics. Operators. The Feynman path integral. Hamiltonian mechanics. Path integral quantization. Stochastic processes Stochastic systems. Discrete degrees of freedom Ising model. Ising model: magnetic ﬁeld. Fermions. Quadratic path integrals Simple harmonic oscillator. Gaussian path integrals. Action with acceleration Acceleration Lagrangian. Pseudo-Hermitian Euclidean Hamiltonian. Non-Hermitian Hamiltonian: Jordan blocks. Nonlinear path integrals The quartic potential: instantons. Compact degrees of freedom. Conclusions.
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