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Applications in Rigorous Quantum Field Theory 2nd rev. ed. [Hardback]

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  • Formāts: Hardback, 557 pages, height x width: 240x170 mm, weight: 1062 g
  • Sērija : De Gruyter Studies in Mathematics
  • Izdošanas datums: 09-Mar-2020
  • Izdevniecība: De Gruyter
  • ISBN-10: 3110403501
  • ISBN-13: 9783110403503
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Applications in Rigorous Quantum Field Theory 2nd rev. ed.Palielināt
  • Formāts: Hardback, 557 pages, height x width: 240x170 mm, weight: 1062 g
  • Sērija : De Gruyter Studies in Mathematics
  • Izdošanas datums: 09-Mar-2020
  • Izdevniecība: De Gruyter
  • ISBN-10: 3110403501
  • ISBN-13: 9783110403503
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9783110403503.html
Keywords:
This is the second updated and extended edition of the successful book on Feynman-Kac Theory. It offers a state-of-the-art mathematical account of functional integration methods in the context of self-adjoint operators and semigroups using the concepts and tools of modern stochastic analysis. In the second volume, these ideas are applied principally to a rigorous treatment of some fundamental models of quantum field theory.
Contents of Volume 1 ix
Preface to the second edition xv
1 Free Euclidean quantum field and Ornstein-Uhlenbeck processes
1(130)
1.1 Background
1(2)
1.2 Boson Fock space
3(40)
1.2.1 Second quantization
3(6)
1.2.2 The case W = L2(Rd)
9(3)
1.2.3 The case W = C1
12(2)
1.2.4 Segal--Bargmann space
14(2)
1.2.5 Segal fields
16(4)
1.2.6 Wick product
20(1)
1.2.7 Exponentials of creation and annihilation operators
21(6)
1.2.8 The case W = L2(Rd)
27(16)
1.3 L-spaces
43(14)
1.3.1 Gaussian random processes
43(3)
1.3.2 Wiener--Ito--Segal isomorphism and positivity improving
46(6)
1.3.3 Hypercontractivity
52(4)
1.3.4 Lorentz covariant quantum fields
56(1)
1.4 Existence of L-spaces
57(9)
1.4.1 Countable product spaces
57(2)
1.4.2 Bochner theorem and Minlos theorem
59(7)
1.5 Functional integral representation of the Euclidean quantum field
66(16)
1.5.1 Basic results in Euclidean quantum field theory
66(9)
1.5.2 Markov property of projections
75(3)
1.5.3 Feynman--Kac--Nelson formula
78(2)
1.5.4 Van Hove Hamiltonian
80(2)
1.6 Infinite dimensional Ornstein--Uhlenbeck processes
82(29)
1.6.1 Abstract theory of Gaussian measures on Hilbertspaces
82(8)
1.6.2 Abstract theory of Borel measures on Hilbert spaces
90(7)
1.6.3 Fock space as a function space
97(3)
1.6.4 Infinite dimensional Ornstein--Uhlenbeck process
100(6)
1.6.5 Markov property
106(2)
1.6.6 Regular conditional Gaussian probability measures
108(2)
1.6.7 Feynman--Kac--Nelson formula by path measures
110(1)
1.7 Connection with infinite dimensional stochastic analysis and Malliavin calculus
111(20)
1.7.1 Finite dimensional case
111(4)
1.7.2 Stochastic derivative and Cameron--Martin space
115(3)
1.7.3 Malliavin derivative and divergence operator on L2(X)
118(3)
1.7.4 Wiener--Ito chaos expansion
121(3)
1.7.5 Malliavin derivative and divergence operator on Wiener chaos
124(2)
1.7.6 Infinite dimensional Ornstein--Uhlenbeck semigroup
126(1)
1.7.7 Malliavin derivative on white noise space
127(4)
2 The Nelson model by path measures
131(186)
2.1 Preliminaries
131(1)
2.2 The Nelson model in Fock space
132(5)
2.2.1 Definition of the Nelson model
132(3)
2.2.2 Infrared and ultraviolet divergences
135(1)
2.2.3 Embedded eigenvalues
136(1)
2.3 The Nelson model in function space
137(14)
2.3.1 Infinite dimensional Ornstein--Uhlenbeck processes and P(Φ)1-processes
137(6)
2.3.2 Euclidean field and Brownian motion
143(5)
2.3.3 Extension to general external potential
148(3)
2.4 Nelson model with Kato-class potential
151(4)
2.5 Existence and uniqueness of the ground state
155(9)
2.5.1 Uniqueness
155(2)
2.5.2 Existence
157(7)
2.6 Ground state expectations
164(20)
2.6.1 General expressions
164(6)
2.6.2 Ground state expectations for second quantized operators
170(6)
2.6.3 Ground state expectation for fractional powers of the number operator
176(4)
2.6.4 Ground state expectations of field operators
180(2)
2.6.5 Gaussian domination
182(2)
2.7 Infrared divergence
184(5)
2.8 Gibbs measure associated with the ground state
189(18)
2.8.1 Local convergence and Gibbs measures
189(7)
2.8.2 P(Φ)1-process associated with the Nelson Hamiltonian
196(3)
2.8.3 Applications to ground state expectations
199(8)
2.9 Carmona-type estimates
207(4)
2.9.1 Exponential decay of bound states: upper bound
207(1)
2.9.2 Exponential decay of bound states: lower bound
208(3)
2.10 Martingale properties and applications
211(4)
2.10.1 Martingale properties
211(2)
2.10.2 Exponential decay of bound states
213(2)
2.11 Ultraviolet divergence
215(52)
2.11.1 Energy renormalization
215(4)
2.11.2 Regularized interaction
219(13)
2.11.3 Removal of ultraviolet cutoff on Fockvacuum
232(5)
2.11.4 Uniform lower bound and removal of ultraviolet cutoff
237(2)
2.11.5 Functional integral representation of the ultraviolet renormalized Nelson model
239(14)
2.11.6 Gibbs measures and applications
253(9)
2.11.7 Weak coupling limit and removal of ultraviolet cutoff
262(5)
2.12 Translation invariant Nelson model
267(24)
2.12.1 Definition of translation invariant Nelson model
267(3)
2.12.2 Functional integral representation
270(3)
2.12.3 Existence of ground state
273(2)
2.12.4 Gibbs measure associated with the ground state of Nelson model with zero total momentum
275(2)
2.12.5 P(Φ)1-process associated with Nelson Hamiltonian with zero total momentum
277(3)
2.12.6 Removal of ultraviolet cutoff
280(5)
2.12.7 Ground state energy and ultraviolet renormalization term
285(3)
2.12.8 Gibbs measures and applications
288(3)
2.13 Polaron model
291(5)
2.13.1 Definition of the polaron model
291(1)
2.13.2 Functional integral representation
292(2)
2.13.3 Removal of ultraviolet cutoff
294(2)
2.14 Functional central limit theorem
296(21)
2.14.1 Gibbs measures with no external potential
296(10)
2.14.2 Diffusive behavior
306(8)
2.14.3 Diffusion matrix and effective mass
314(3)
3 The Pauli--Fierz model by path measures
317(162)
3.1 Preliminaries
317(7)
3.1.1 Introduction
317(1)
3.1.2 Lagrangian QED
318(4)
3.1.3 Classical variant of nonrelativistic QED
322(2)
3.2 The Pauli--Fierz model in nonrelativistic QED
324(13)
3.2.1 The Pauli--Fierz model in Fock space
324(5)
3.2.2 The Pauli--Fierz model in function space
329(5)
3.2.3 Markov property
334(3)
3.3 Functional integral representation for the Pauli--Fierz Hamiltonian
337(14)
3.3.1 Hilbert space-valued stochastic integrals
337(4)
3.3.2 Functional integral representation
341(8)
3.3.3 Extension to general external potential
349(2)
3.4 The Pauli--Fierz model with Kato-class potential
351(4)
3.5 Applications of functional integral representations
355(16)
3.5.1 Self-adjointness of the Pauli--Fierz Hamiltonian
355(9)
3.5.2 Positivity improving and uniqueness of the ground state
364(6)
3.5.3 Spatial decay of bound states
370(1)
3.6 Path measure associated with the ground state of Pauli--Fierz Hamiltonian
371(11)
3.6.1 Path measure with double stochastic integrals
371(4)
3.6.2 Expression in terms of iterated stochastic integrals
375(3)
3.6.3 Weak convergence and Gibbs measures
378(3)
3.6.4 Gaussian domination of ground states
381(1)
3.7 Translation invariant Pauli--Fierz model
382(7)
3.8 The Pauli--Fierz model with spin
389(50)
3.8.1 Counting measures
389(1)
3.8.2 Review of classical cases
390(2)
3.8.3 Definition of the Pauli--Fierz Hamiltonian with spin
392(3)
3.8.4 Symmetry and polarization
395(6)
3.8.5 Scalar representations
401(4)
3.8.6 Fock representations
405(2)
3.8.7 Preparation of functional integral representations
407(14)
3.8.8 Functional integral representations
421(11)
3.8.9 Translation invariant Pauli--Fierz Hamiltonian with spin
432(7)
3.9 Relativistic Pauli--Fierz model
439(40)
3.9.1 Definition of relativistic Pauli--Fierz Hamiltonian
439(3)
3.9.2 Functional integral representations
442(5)
3.9.3 Self-adjointness
447(10)
3.9.4 Nonrelativistic limit of relativistic Pauli--Fierz Hamiltonian
457(3)
3.9.5 Relativistic Pauli--Fierz model with relativistic Kato-class potential
460(5)
3.9.6 Martingale properties
465(4)
3.9.7 Spatial decay of bound states
469(2)
3.9.8 Gaussian domination of ground states
471(2)
3.9.9 Path measure associated with the ground state of relativistic Pauli--Fierz Hamiltonian
473(1)
3.9.10 Translation invariant relativistic Pauli--Fierz model
474(4)
3.9.11 Nonrelativistic limit of translation invariant relativistic Pauli--Fierz Hamiltonian
478(1)
4 Spin-boson model by path measures
479(26)
4.1 Definitions
479(4)
4.2 Functional integral representation for the spin-boson Hamiltonian
483(5)
4.2.1 Preliminaries
483(1)
4.2.2 Spin process
484(2)
4.2.3 Functional integral representation
486(2)
4.3 Existence and uniqueness of ground state
488(2)
4.4 Gibbs measure associated with the ground state
490(11)
4.4.1 Local convergence and Gibbs measures
490(1)
4.4.2 Ground state properties
491(8)
4.4.3 Van Hove representation
499(2)
4.5 Rabi Hamiltonian
501(4)
5 Notes and references
505(16)
Notes to
Chapter 1
505(2)
Notes to
Chapter 2
507(7)
Notes to
Chapter 3
514(5)
Notes to
Chapter 4
519(2)
Bibliography 521(12)
Index 533
József Lorinczi, Loughborough University, UK; Fumio Hiroshima, University of Kyushu, Fukuoka, Japan.