The Feynman Integral and Feynman's Operational Calculus

by Johnson, Gerald W.; Lapidus, Michel L.

• ISBN13: 9780198515722
• ISBN10: 0198515723
• Edition: Reprint
• Format: Paperback
• Copyright: 2002-03-28
• Publisher: Clarendon Press

Summary

The aim of this book is to make accessible to mathematicians, physicists and other scientists interested in quantum theory, the mathematically beautiful but difficult subjects of the Feynman integral and Feynman's operational calculus. Some advantages of the four approaches to the Feynman integral which are given detailed treatment in this book are the following: the existence of the Feynman integral is established for very general potentials in all four cases; under more restrictive but still broad conditions, three of these Feynman integrals agree with one another and with the unitary group from the usual approach to quantum dynamics; these same three Feynman integrals possess pleasant stability properties. Much of the material covered here was previously only in the research literature, and the book also contains some new results. The background material in mathematics and physics that motivates the study of the Feynman integral and Feynman's operational calculus is discussed and detailed proofs are provided for the central results.

Table of Contents

Introduction	1	(23)
General introductory comments	1	(7)
Feynman's path integral	1	(3)
Feynman's operational calculus	4	(1)
Feynman's operational calculus via the Feynman and Wiener integrals	5	(2)
Feynman's operational calculus and evolution equations	7	(1)
Further work on or related to the Feynman integral: Chapter 20	8	(1)
Recurring themes and their connections with the Feynman integral and Feynman's operational calculus	8	(9)
Product formulas and applications to the Feynman integral	8	(2)
Feynman--Kac formula: Analytic continuation in time and mass	10	(2)
The role of operator theory	12	(1)
Connections between the Feynman--Kac and Trotter product formulas	13	(1)
Evolution equations	13	(2)
Functions of noncommuting operators	15	(1)
Time-ordered perturbation series	15	(1)
The use of measures	16	(1)
Relationship with the motivating physical theories: background and quantum-mechanical models	17	(7)
Physical background	17	(1)
Highly singular potentials	18	(1)
Time-dependent potentials	19	(1)
Phenomenological models: complex and nonlocal potentials	19	(2)
Prerequisites, new material, and organization of the book	21	(3)
The physical phenomenon of Brownian motion	24	(7)
A brief historical sketch	24	(4)
Einstein's probabilistic formula	28	(3)
Wiener measure	31	(31)
There is no reasonable translation invariant measure on Wiener space	32	(2)
Construction of Wiener measure	34	(8)
Wiener's integration formula and applications	42	(9)
Finitely based functions	43	(2)
Applications	45	(6)
Axiomatic description of the Wiener process	51	(1)
Nondifferentiability of Wiener paths	51	(6)
d-dimensional Wiener measure and Wiener process	57	(1)
Appendix: Converse measurability results	57	(3)
Appendix: B(X x Y) = B(X) B(Y)	60	(2)
Scaling in Wiener space and the analytic Feynman integral	62	(27)
Quadratic variation of Wiener paths	63	(4)
Scale change in Wiener space	67	(7)
Translation pathologies	74	(3)
Scale-invariant measurable functions	77	(2)
The scalar-valued analytic Feynman integral	79	(3)
The nonexistence of Feynman's ``measure''	82	(3)
Appendix: Some useful Gaussian-type integrals	85	(2)
Appendix: Proof of formula (4.2.3a)	87	(2)
Stochastic processes and the Wiener process	89	(5)
Stochastic processes and probability measures on function spaces	89	(1)
The Kolmogorov consistency theorem	90	(2)
Two realizations of the Wiener process	92	(2)
Quantum dynamics and the Schrodinger equation	94	(5)
Hamiltonian approach to quantum dynamics	94	(1)
Transition amplitudes and measurement	95	(1)
The Heisenberg uncertainty principle	96	(1)
Hamiltonian for a system of particles	97	(2)
The Feynman integral: heuristic ideas and mathematical difficulties	99	(22)
Introduction	99	(2)
Feynman's formula	101	(5)
Connections with classical mechanics: The method of stationary phase	105	(1)
Heuristic derivation of the Schrodinger equation	106	(3)
Feynman's approximation formula	109	(2)
Nelson's approach via the Trotter product formula	111	(4)
The Trotter product formula	114	(1)
The approach via analytic continuation	115	(6)
Semigroups of operators: an informal introduction	121	(6)
Linear semigroups of operators	127	(25)
Infinitesimal generator	127	(7)
Integral equation	129	(1)
Evolution equation	130	(1)
Closed unbounded operators	130	(4)
Examples of semigroups and their generators	134	(4)
The translation semigroup	134	(1)
The heat semigroup	135	(2)
The Poisson semigroup	137	(1)
The resolvent	138	(2)
Generation theorems	140	(3)
The Hille--Yosida theorem	140	(1)
Dissipative operators and the Lumer--Phillips theorem	141	(2)
Uniformly continuous and weakly continuous semigroups	143	(1)
Self-adjoint operators, unitary groups and Stone's theorem	144	(3)
Perturbation theorems	147	(5)
Unbounded self-adjoint operators and quadratic forms	152	(45)
Spectral theorem for unbounded self-adjoint operators	153	(8)
Multiplication operators	153	(2)
Three useful forms of the spectral theorem	155	(6)
Applications of the spectral theorem	161	(15)
The free Hamiltonian H0	161	(2)
The heat semigroup and unitary group	163	(8)
Standard cores for the free Hamiltonian	171	(3)
Imaginary resolvents	174	(2)
Representation theorems for unbounded quadratic forms	176	(17)
Basic definitions and properties	176	(7)
Representation theorems for quadratic forms	183	(7)
The form sum of operators	190	(3)
Conditions on the potential V for H0-form boundedness	193	(4)
Product formulas with applications to the Feynman integral	197	(75)
Trotter and Chernoff product formulas	198	(6)
Product formula for unitary groups	203	(1)
Feynman integral via the Trotter product formula	204	(16)
Criteria for essential self-adjointness of positive operators	204	(1)
A brief outline of distribution theory	205	(1)
Kato's distributional inequality	206	(4)
Essential self-adjointness of the Hamiltonian H = H0 + V	210	(5)
Conditions on the potential V for H0-operator boundedness	215	(2)
Feynman integral via the Trotter product formula for unitary groups	217	(3)
Product formula for imaginary resolvents	220	(12)
Hypotheses and statement of the main result	220	(2)
Proof of the product formula	222	(7)
Consequences, extensions and open problems	229	(3)
Application to the modified Feynman integral	232	(13)
Modified Feynman integral and Schrodinger equation with singular potential	233	(8)
Extensions: Riemannian manifolds and magnetic vector potentials	241	(4)
Dominated convergence theorem for the modified Feynman integral	245	(14)
Preliminaries	247	(1)
Perturbation of form sums of self-adjoint operators	248	(4)
Application to a general dominated convergence theorem for Feynman integrals	252	(7)
The modified Feynman integral for complex potentials	259	(7)
Product formula for imaginary resolvents of normal operators	260	(3)
Application to dissipative quantum systems	263	(3)
Appendix: Extended Vitali's theorem with application to unitary groups	266	(6)
Extension of Vitali's theorem for sequences for analytic functions	266	(3)
Analytic continuation and product formula for unitary groups	269	(3)
The Feynman-Kac formula	272	(21)
The Feynman-Kac formula, the heat equation and the Wiener integral	273	(3)
Proof of the Feynman--Kac formula	276	(14)
Bounded potentials	276	(6)
Monotone convergence theorems for forms and integrals	282	(2)
Unbounded potentials	284	(6)
Consequences	290	(3)
Analytic-in-time or-mass operator-valued Feynman integrals	293	(81)
Introduction	293	(5)
The analytic-in-time operator-valued Feynman integral	298	(2)
Proof of existence	300	(3)
The Feynman integrals compared with one another and with the unitary group. Application to stability theorems	303	(5)
The analytic-in-mass operator-valued Feynman integral	308	(23)
Definition of the analytic-in-mass operator-valued Feynman integral	309	(3)
Nelson's results	312	(9)
Haugsby's result for time-dependent, complex-valued potentials	321	(2)
Further extensions via a product formula for semigroups	323	(8)
The analytic-in-mass modified Feynman integral	331	(27)
Existence of the analytic-in-mass modified Feynman integral	334	(3)
Product formula for resolvents: The case of imaginary mass	337	(8)
Comparison with other analytic-in-mass Feynman integrals	345	(1)
Highly singular central potentials---the attractive inverse-square potential	346	(12)
The analytic-in-time operator-valued Feynman integral via additive functionals of Brownian motion	358	(16)
Introductory remarks	359	(1)
The parallel with Section 13.3	359	(2)
Generalized signed measures	361	(1)
The generalized Kato class	361	(1)
Capacity on Rd	362	(1)
Smooth measures	363	(1)
Positive continuous additive functionals of Brownian motion	364	(1)
The relationship between smooth measures and PCAFs	365	(2)
The analytic-in-time operator-valued Feynman integral exists for μ = μ+ - μ_ S -- GKd	367	(1)
Examples	368	(6)
Feynman's operational calculus for noncommuting operators: an introduction	374	(30)
Functions of operators	375	(1)
The rules for Feynman's operational calculus	376	(7)
Feynman's time-ordering convention	377	(1)
Feynman's heuristic rules	377	(1)
Two elementary examples	378	(5)
Time-ordered Perturbation series	383	(11)
Perturbation series via Feynman's operational calculus	383	(6)
Perturbation series via a path integral	389	(4)
The origins of Feynman's operational calculus rigorous	393	(1)
Making Feynman's operational calculus rigorous	394	(3)
Rigor via path integrals	394	(1)
Well-defined and useful formulas arrived at via Feynman's heuristic rules	395	(1)
A general theory of Feynman's operational calculus with computations which are rigorous at each stage	396	(1)
Feynman's operational calculus via Wiener and Feynman integrals: Comments on Chapter 15--18	397	(7)
Generalized Dyson series, the Feynman integral and Feynman's operational calculus	404	(58)
Introduction	404	(3)
The analytic operator-valued Feynman integral	407	(9)
Notation and definitions	407	(3)
The analytic (in mass) operator-valued Feynman integral Ktλ(·)	410	(3)
Preliminary results	413	(3)
A simple generalized Dyson series (η = μ + ωδτ)	416	(10)
The classical Dyson series	424	(2)
Generalized Dyson series: The general case	426	(8)
Disentangling via perturbation expansions: Examples	434	(12)
A single measure and potential	435	(7)
Several measures and potentials	442	(4)
Generalized Feynman diagrams	446	(5)
Commutative Banach algebras of functionals: The disentangling algebras	451	(11)
The disentangling algebras At	452	(3)
The time-reversal map on At and the natural physical ordering	455	(4)
Connections with Feynman's operational calculus	459	(3)
Stability results	462	(15)
Stability in the potentials	462	(2)
Stability in the measures	464	(13)
The Feynman--Kac formula with a Lebesgue--Stieltjes measure and Feynman's operational calculus	477	(53)
Introduction	477	(3)
Notation and hypotheses	478	(2)
The Feynman--Kac formula with a Lebesgue--Stieltjes measure: Finitely supported discrete part v	480	(6)
Integral equation (Integrated form of the evolution equation)	480	(1)
Differential equation (differential form of the evolution equation)	481	(1)
Discontinuities (in time) of the solution	482	(1)
Propagator and explicit solution	483	(3)
Derivation of the integral equation in a simple case (η = μ + ωδτ)	486	(10)
Sketch of the proof when v is finitely supported	495	(1)
Discontinuities of the solution to the evolution equation	496	(3)
The time discontinuities	496	(1)
Differential equation and change of initial condition	497	(2)
Explicit solution and physical interpretations	499	(8)
Continuous measure: Uniqueness of the solution	500	(1)
Measure with finitely supported discrete part: Propagator and explicit solution	501	(2)
Physical interpretations in the quantum-mechanical case	503	(1)
Physical interpretations in the diffusion case	504	(1)
Further connections with Feynman's Operational calculus	505	(2)
The Feynman--Kac formula with a Lebesgue--Stieltjes measure: The general case (arbitrary measure η)	507	(23)
Integral equation (integrated form of the evolution equation)	507	(2)
Basic properties of the solution to the integral equation	509	(2)
Quantum-mechanical case: Reformulation in the interaction (or Dirac) picture	511	(3)
Product integral representation of the solution	514	(3)
Distributional differential equation (true differential form of the evolution equation)	517	(2)
Unitary propagators	519	(1)
Scattering matrix and improper product integral	520	(1)
Sketch of the proof of the integral equation	521	(9)
Noncommutative operations on Wiener functionals, disentangling algebras and Feynman's operational calculus	530	(32)
Introduction	530	(2)
Preliminaries: maps, measures and measurability	532	(3)
The noncommutative operations * and +	535	(5)
The functional integrals Ktλ(·) and the operations * and +	540	(4)
The disentangling algebras At, the operations * and +, and the disentangling process	544	(9)
Examples: Trigonometric, binomial and exponential formulas	552	(1)
Appendix: Quantization, axiomatic Feynman's operational calculus, and generalized functional integral	553	(9)
Algebraic and analytic axioms	554	(2)
Consequences of the axioms	556	(3)
Examples: the disentangling algebras and analytic Feynman integrals	559	(3)
Feynman's operational calculus and evolution equations	562	(47)
Introduction and hypotheses	562	(6)
Feynman's operational calculus as a generalized path integral	562	(1)
Exponentials of sums of noncommuting operators	563	(1)
Disentangling exponentials of sums via perturbation series	563	(2)
Local and nonlocal potentials	565	(1)
Hypotheses	566	(2)
Disentangling exp{--tα + ∫t0 β(s)μ(ds)}	568	(5)
Disentangling exp{-tα + ∫t0 β1(s)μ1(ds) + ... + ∫t0 βn(s)μn(ds)}	573	(8)
Convergence of the disentangled series	581	(6)
The evolution equation	587	(9)
Uniqueness of the solution to the evolution equation	596	(3)
Further examples of the disentangling process	599	(10)
Nonlocal potentials relevant to phenomenological nuclear theory	604	(5)
Further work on or related to the Feynman integral	609	(88)
Transform approaches to the Feynman integral. References to further approaches	609	(28)
The Fresnel integral and other transform approaches to the Feynman integral	610	(1)
The Fresnel integral	610	(1)
Properties of the Fresnel integral	611	(2)
An approach to the Feynman integral via the Fresnel integral	613	(1)
Advantages and disadvantages of Fresnel integral approaches to the Feynman integral	613	(2)
The Feynman map	615	(1)
The Poison process and transforms	616	(1)
A ``Fresnel integral'' on classical Wiener space	616	(4)
The Banach algebras S and F(H1) are the same	620	(1)
Consequences of the close relationship between S and F(H1)	621	(1)
More functions in F(H1)	622	(1)
A unified theory of Fresnel integrals: Introductory remarks	623	(1)
Background material	624	(2)
A unified theory of Fresnel integrals (continued)	626	(3)
The Fresnel classes along with quadratic forms	629	(1)
The classes Gq(H) and Gq(B)	630	(1)
Quadratic forms extended	631	(1)
Functions in the Fresnel class of an abstract Wiener space: Examples of abstract Wiener spaces	632	(4)
Fourier--Feynman transforms, convolution, and the first variation for functions in S	636	(1)
References to further approaches to the Feynman integral	636	(1)
The influence of heuristic Feynman integrals on contemporary mathematics and physics: Some examples	637	(60)
The heuristic Feynman path integral	638	(1)
Knot invariants and low-dimentional topology	639	(1)
The Jones polynomial invariant for knots and links	639	(2)
Witten's topological invariants via Feynman path integrals	641	(13)
Further developments: Vassiliev invariants and the Kontsevich integral	654	(5)
Further comments and references on subjects related to the Feynman integral	659	(1)
Supersymmetric Feynman path integrals and the Atiyah--Singer index theorem	659	(15)
Deformation quantization: Star products and perturbation series	674	(8)
Gauge field theory and Feynman path integrals	682	(6)
String theory, Feynman--Polyakov integrals, and dualities	688	(7)
What lies ahead? Towards a geometrization of Feynman path integrals?	695	(2)
References	697	(48)
Index of symbols	745	(5)
Author index	750	(6)
Subject index	756