Fujiwara D. Rigorous Time Slicing Approach to Feynman Path Integrals

Springer Japan KK, 2017. — 333 p. — (Mathematical Physics Studies) — ISBN: 978-4-431-56551-2.The Feynman path integral is a method of quantization using the Lagrangian function, while Schrödinger’s quantization uses the Hamiltonian function. Since it provides a different view point from Schrödinger’s, it is a very useful basic tool in quantum physics. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger’s method, there is still much to be done concerning rigorous mathematical treatment of Feynman’s method. The difficulty lies in the fact that the Feynman path integral is not an integral by means of a countably additive measure.
Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces. To construct this approximating sequence he divided the time interval into small pieces. This method is called the time slicing approximation method or sequential method.Convergence of Time Slicing Approximation of Feynman
Path Integrals
Feynman’s Idea
Assumption on Potentials
Path Integrals and Oscillatory Integrals
Statement of Main Results
Convergence of Feynman Path Integrals
Feynman Path Integral and Schrödinger Equation
Supplement–Some Results of Real Analysis
Kumano-go–Taniguchi Theorem
Stationary Phase Method for Oscillatory Integrals over a Space of Large Dimension
L2-boundedness of Oscillatory Integral Operators