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Albeverio S.A., Hoegh-Krohn R.J. — Mathematical theory of Feynman path integrals
Albeverio S.A., Hoegh-Krohn R.J. — Mathematical theory of Feynman path integrals



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Название: Mathematical theory of Feynman path integrals

Авторы: Albeverio S.A., Hoegh-Krohn R.J.

Аннотация:

In this work we develop a general theory of oscillatory integrals on real Hilbert spaces and apply it to the mathematical foundation of the so called Feynman path integrals of non relativistic quantum mechanics, quantum statistical mechanics and quantum field theory. The translation invariant integrals we define provide a natural extension of the theory of finite dimensional oscillatory integrals, which has newly undergone an impressive development, and appear to be a suitable tool in infinite imensional analysis. For one example, on the basis of the present work we have extended the methods of stationary phase, Lagrange immersions and orresponding asymptotic expansions to the infinite dimensional case, covering in particular the expansions around the classical limit of quantum mechanics. A particular case of the oscillatory integrals studied in the present work are the Feynman path integrals used extensively in the physical literature, starting with the basic work on quantum dynamics by Dirac and Feynman, in the forties.


Язык: en

Рубрика: Физика/Квантовая теория поля/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1976

Количество страниц: 139

Добавлена в каталог: 26.07.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Action, Hamilton's principle of least action      4 10
Analytic continuation, definition of Feynman integrals by      7 Ref.[10] 79
Analytic continuation, definition of Feynman integrals by of Wiener integrals      7 Ref.[10]
Anharmonic oscillator      11 46 65 78 118
Anharmonic oscillator, dynamics by Fresnel integrals      78 79
Anharmonic oscillator, expectations with respect to Gibbs state of harmonic oscillator      86
Anharmonic oscillator, expectations with respect to ground state of harmonic oscillator      80
Anharmonic oscillator, Feynman — Ito formula, small times      47
Anharmonic oscillator, Fresnel integrals, small times      47
Anharmonic oscillator, solutions of Schroedinger equation by Fresnel integrals      72 74
Annihilation-creation operators      90 91
Automorphisms of Weyl algebra      89 93 95 97 102 103 107 108 109 113
Banach function algebra, over $R^{n}$      16
Banach function algebra, over $\chi$      18
Banach function algebra, over D*      50
Bose fields      10 12 105—114
Brownian motion      7 9
Causal popagator      119
Classical action      10 26 31 80 90 99 105 109
Classical limit      7 8 Ref.[41]
Commutation relations      88 91 92
Complex measure      8
Complex potential      118
Composition, with entire functions      17 18
Cyclic vector      94
Euclidean — Markov fields      11
Euler — Lagrange equations      4
Expectations, for anharmonic oscillator      78
Expectations, for relativistic quantum Bose fields      105—114
Expectations, with respect to Gibbs state of the harmonic oscillator      86 89
Expectations, with respect to ground state of the harmonic oscillator      80 89 101
Expectations, with respect to invariant quasi free states      103—104
Exponential interactions      11 112—114
Feynman history integrals, as Fresnel integrals relative to a quadratic form      105—114
Feynman history integrals, heuristic      10—11
Feynman path integrals, as Fresnel integrals, anharmonic oscillators      65—89
Feynman path integrals, as Fresnel integrals, anharmonic oscillators, small times      46 47
Feynman path integrals, as Fresnel integrals, infinitely many harmonic oscillators      90—104
Feynman path integrals, as Fresnel integrals, non relativistic quantum mechanics      26—45
Feynman path integrals, as Fresnel integrals, relativistic quantum fields      105—114
Feynman path integrals, as Fresnel integrals, scattering operator      42—45
Feynman path integrals, as Fresnel integrals, solutions of Schroedinger equation      26—32
Feynman path integrals, as Fresnel integrals, wave operators      32—41
Feynman path integrals, definition by "analytic continuation"      7 Ref.[10] 79
Feynman path integrals, definition by "sequential limit"      7 Ref.[11]
Feynman path integrals, other definitions      7 9 23—25
Feynman — Ito formula      26 30 31
Feynman — Kac formula      7 26
Fields      see "Bose fields"
Finitely based function      18
Fock representation      91 92 104 110
Free Boson field      105
Fresnel integrable, on $R^{n}$      14—17
Fresnel integrable, on a real Hilbert space      18
Fresnel integrals, computations of      26—32 35—41 68 76 83 85—88 102—104 110—111 114
Fresnel integrals, for anharmonic oscillators, applications      65—114
Fresnel integrals, for anharmonic oscillators, projectics      50—64
Fresnel integrals, for anharmonic oscillators, relative to a non degenerate quadratic form, definition      48—50
Fresnel integrals, for anharmonic oscillators, small times      46—47
Fresnel integrals, in non relativistic quantum mechanics (potential scattering)      26—45
Fresnel integrals, on a real separable Banach space      60—64
Fresnel integrals, on a real separable Hilbert space, definition      17—18
Fresnel integrals, on a real separable Hilbert space, properties      18—25
Fubini theorem, for Fresnel integrals on real Hilbert space      21—23
Fubini theorem, for Fresnel integrals on real separable Banach space      62
Fubini theorem, for Fresnel integrals relative to a quadratic form      53—64
Gaussian measure      24
Gentle perturbations      37
Gibbs states      12 104
Green's function      28 67 75
Hamilton's principle      4 10
Harmonic oscillator, finitely many degrees of freedom      90—90
Harmonic oscillator, infinitely many degrees of freedom      90—104 105—114
Heat equation      6
Homogeneous boundary conditions, transformation to      65
Improper normalized integral      36 41
Indefinite metric      24
Initial value problem      26
Integrals      see "Feynman integrals" "Fresnel
Integration in function spaces      11
Invariance, Fresnel integral, under nearly isometric transformations      24
Invariance, orthogonal, for normalized integral      19—21
Invariance, translation, for normalized integral      15 19—21
Invariance, translation, for normalized integral with respect to quadratic form      53 73
Invariant quasi free state      93
Ito's functional      23
Models of quantum fields      11—12 111—114
Non relativistic quantum mechanics      1—9 26—45 65—104
Normalization      14
Normalized integral      see "Fresnel integral"
Oscillatory integrals      see "Fresnel integrals" "Feynman "Feynman
Parseval relation      18 24
Path integrals      see "Feynman integrals" "Wiener
Polynomial interactions      11
Potential scattering      26—45
Quantum mechanics, non relativistic      26—45 65—104
Quantum mechanics, relativistic      105—114
Quantum mechanics, statistical      86—89
Quasi free states      90 93 97 103 108 113
Scattering amplitude, scattering operator      42
Schroedinger equation, solutions given by Fresnel integrals      26—32 65—74
Semigroup      79
Series expansions      27—45 68—79 85—89 114
Sine-Gordon interactions      112
State, on Weyl algebra      88—112
Stochastic field theory      12
Stochastic mechanics      9
Symplectic space      95 97
Symplectic transformation      95 97 108
Time ordered vacuum expectation values      10
Wave operators      32—42 44
Weyl algebra      88 92 102 107 109
Wiener measure      7 9
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