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Rigorous Time Slicing Approach to Feynman Path Integrals (Hardcover, 1st ed. 2017)
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Rigorous Time Slicing Approach to Feynman Path Integrals (Hardcover, 1st ed. 2017)
Series: Mathematical Physics Studies
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This book proves that Feynman's original definition of the path
integral actually converges to the fundamental solution of the
Schroedinger equation at least in the short term if the potential
is differentiable sufficiently many times and its derivatives of
order equal to or higher than two are bounded. The semi-classical
asymptotic formula up to the second term of the fundamental
solution is also proved by a method different from that of
Birkhoff. A bound of the remainder term is also proved.The Feynman
path integral is a method of quantization using the Lagrangian
function, whereas Schroedinger's quantization uses the Hamiltonian
function. These two methods are believed to be equivalent. But
equivalence is not fully proved mathematically, because, compared
with Schroedinger's method, there is still much to be done
concerning rigorous mathematical treatment of Feynman's method.
Feynman himself defined a path integral as the limit of a sequence
of integrals over finite-dimensional spaces which is obtained by
dividing the time interval into small pieces. This method is called
the time slicing approximation method or the time slicing
method.This book consists of two parts. Part I is the main part.
The time slicing method is performed step by step in detail in Part
I. The time interval is divided into small pieces. Corresponding to
each division a finite-dimensional integral is constructed
following Feynman's famous paper. This finite-dimensional integral
is not absolutely convergent. Owing to the assumption of the
potential, it is an oscillatory integral. The oscillatory integral
techniques developed in the theory of partial differential
equations are applied to it. It turns out that the
finite-dimensional integral gives a finite definite value. The
stationary phase method is applied to it. Basic properties of
oscillatory integrals and the stationary phase method are explained
in the book in detail.Those finite-dimensional integrals form a
sequence of approximation of the Feynman path integral when the
division goes finer and finer. A careful discussion is required to
prove the convergence of the approximate sequence as the length of
each of the small subintervals tends to 0. For that purpose the
book uses the stationary phase method of oscillatory integrals over
a space of large dimension, of which the detailed proof is given in
Part II of the book. By virtue of this method, the approximate
sequence converges to the limit. This proves that the Feynman path
integral converges. It turns out that the convergence occurs in a
very strong topology. The fact that the limit is the fundamental
solution of the Schroedinger equation is proved also by the
stationary phase method. The semi-classical asymptotic formula
naturally follows from the above discussion.A prerequisite for
readers of this book is standard knowledge of functional analysis.
Mathematical techniques required here are explained and proved from
scratch in Part II, which occupies a large part of the book,
because they are considerably different from techniques usually
used in treating the Schroedinger equation.
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