Baaquie B.E. Path Integrals and Hamiltonians: Principles and Methods

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.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.