The Feynman Integral and Feynman's Operational Calculus

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

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ISBN: 9780198515722 | 0198515723
Cover: Paperback
Copyright: 3/28/2002

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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.

Introduction

(23)

General introductory comments

(7)

Feynman's path integral

(3)

Feynman's operational calculus

(1)

Feynman's operational calculus via the Feynman and Wiener integrals

(2)

Feynman's operational calculus and evolution equations

(1)

Further work on or related to the Feynman integral: Chapter 20

(1)

Recurring themes and their connections with the Feynman integral and Feynman's operational calculus

(9)

Product formulas and applications to the Feynman integral

(2)

Feynman--Kac formula: Analytic continuation in time and mass

(2)

The role of operator theory

(1)

Connections between the Feynman--Kac and Trotter product formulas

(1)

Evolution equations

(2)

Functions of noncommuting operators

(1)

Time-ordered perturbation series

(1)

The use of measures

(1)

Relationship with the motivating physical theories: background and quantum-mechanical models

(7)

Physical background

(1)

Highly singular potentials

(1)

Time-dependent potentials

(1)

Phenomenological models: complex and nonlocal potentials

(2)

Prerequisites, new material, and organization of the book

(3)

The physical phenomenon of Brownian motion

(7)

A brief historical sketch

(4)

Einstein's probabilistic formula

(3)

Wiener measure

(31)

There is no reasonable translation invariant measure on Wiener space

(2)

Construction of Wiener measure

(8)

Wiener's integration formula and applications

(9)

Finitely based functions

(2)

Applications

(6)

Axiomatic description of the Wiener process

(1)

Nondifferentiability of Wiener paths

(6)

d-dimensional Wiener measure and Wiener process

(1)

Appendix: Converse measurability results

(3)

Appendix: B(X x Y) = B(X) B(Y)

(2)

Scaling in Wiener space and the analytic Feynman integral

(27)

Quadratic variation of Wiener paths

(4)

Scale change in Wiener space

(7)

Translation pathologies

(3)

Scale-invariant measurable functions

(2)

The scalar-valued analytic Feynman integral

(3)

The nonexistence of Feynman's ``measure''

(3)

Appendix: Some useful Gaussian-type integrals

(2)

Appendix: Proof of formula (4.2.3a)

(2)

Stochastic processes and the Wiener process

(5)

Stochastic processes and probability measures on function spaces

(1)

The Kolmogorov consistency theorem

(2)

Two realizations of the Wiener process

(2)

Quantum dynamics and the Schrodinger equation

(5)

Hamiltonian approach to quantum dynamics

(1)

Transition amplitudes and measurement

(1)

The Heisenberg uncertainty principle

(1)

Hamiltonian for a system of particles

(2)

The Feynman integral: heuristic ideas and mathematical difficulties

(22)

Introduction

(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

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The Feynman Integral and Feynman's Operational Calculus

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