The canonical operator in many-partiсle problems and quantum field theory:
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| Language: | English |
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De Gruyter
[2022]
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| Series: | De Gruyter expositions in mathematics
volume 71 |
| Online Access: | DE-1043 DE-1046 DE-858 DE-898 DE-859 DE-860 DE-91 DE-706 DE-739 Volltext Inhaltsverzeichnis |
| Physical Description: | 1 Online-Ressource (XXVI, 448 Seiten) |
| ISBN: | 9783110762709 9783110762747 |
| DOI: | 10.1515/9783110762709 |
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| 100 | 1 | |a Maslov, Viktor P. |d 1930- |e Verfasser |0 (DE-588)123340853 |4 aut | |
| 245 | 1 | 0 | |a The canonical operator in many-partiсle problems and quantum field theory |c Victor P. Maslov and Oleg Yu. Shvedov |
| 264 | 1 | |a Berlin ; Boston |b De Gruyter |c [2022] | |
| 264 | 4 | |c © 2022 | |
| 300 | |a 1 Online-Ressource (XXVI, 448 Seiten) | ||
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| 490 | 1 | |a De Gruyter expositions in mathematics |v volume 71 | |
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| 700 | 1 | |a Švedov, Oleg Jur'evič |d 1973-2015 |e Verfasser |0 (DE-588)1262692083 |4 aut | |
| 710 | 2 | |a Walter de Gruyter GmbH & Co. KG |0 (DE-588)10095502-2 |4 pbl | |
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CONTENTS
PREFACE
TO
THE
ENGLISH
EDITION
-
VII
PREFACE
-
XI
LIST
OF
NOTATION
----
XXVII
1
1.1
1.1.1
ABSTRACT
CANONICAL
OPERATOR
AND
SYMPLECTIC
GEOMETRY
-
1
INTRODUCTION
----
1
CONSTRUCTION
OF
APPROXIMATE
SOLUTIONS
OF
VARIOUS
EQUATIONS:
COMPARISON
OF
METHODS
----
1
1.1.2
1.1.3
ABSTRACT
CANONICAL
OPERATOR
AND
GEOMETRY
OF
THE
PHASE
SPACE
-
5
PROBLEM
OF
EXISTENCE
OF
EQUATIONS
OF
MOTION
AND
THE
CONCEPT
OF
ASYMPTOTIC
QUANTIZATION
----
8
1.2
ABSTRACT
CANONICAL
OPERATOR
AND
INDUCED
GEOMETRIC
STRUCTURES
ON
THE
PHASE
SPACE
----
9
1.2.1
1.2.2
1.2.3
AUXILIARY
NOTIONS
-
9
DEFINITION
OF
AN
ABSTRACT
CANONICAL
OPERATOR
-
11
WELL-DEFINEDNESS
OF
THE
CANONICAL
OPERATOR
-
THE
ACTION
1-FORM
ON
THE
PHASE
SPACE
-
12
1.2.4
1.2.5
1.2.6
OPERATOR-VALUED
1-FORM
INDUCED
BY
THE
CANONICAL
OPERATOR
-
16
CANONICAL
TRANSFORMATION
OF
THE
ABSTRACT
CANONICAL
OPERATOR
----
18
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
-
21
1.2.7
1.3
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
----
21
ABSTRACT
COMPLEX
GERM
AND
CONSTRUCTION
OF
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
----
22
1.3.1
1.3.2
1.3.3
1.3.4
1.4
1.4.1
DEFINITION
OF
COMPLEX
GERM
----
22
PROPERTIES
OF
THE
ABSTRACT
COMPLEX
GERM
-
24
CANONICAL
TRANSFORMATION
OF
THE
ABSTRACT
COMPLEX
GERM
-----26
CANONICAL
OPERATOR
CORRESPONDING
TO
A
COMPLEX
GERM
----
28
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
30
CANONICAL
TRANSFORMATIONS
OF
THE
CANONICAL
OPERATOR
AND
CONSTRUCTION
OF
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
30
1.4.2
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
MODULO
O^/
2
)
----
33
1.5
THEORY
OF
COMPLEX
GERM
AT
A
POINT
IN
FINITE-DIMENSIONAL
QUANTUM
MECHANICS
----
35
1.5.1
1.5.2
DEFINITION
OF
THE
CANONICAL
OPERATOR
AND
CHECKING
THE
AXIOMS
-
35
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
----
36
XX
-
CONTENTS
1.5.3
1.5.4
1.5.5
COMPLEX
GERM
-
37
CANONICAL
TRANSFORMATIONS
DEPENDING
ON
TIME
-
42
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
THE
HAMILTONIAN
AND
PROOF
OF
THE
ASYMPTOTIC
FORMULA
-
45
1.A
CLASSICAL
AND
QUANTUM
MECHANICS:
THE
MAIN
DEFINITIONS
[1,
20,53]
----
52
1.B
1.B.1
1.B.2
1.B.3
SOME
RECOLLECTIONS
FROM
DIFFERENTIAL
GEOMETRY
-
54
SMOOTH
MAPPINGS
OF
HILBERT
SPACES
-
54
SMOOTH
MANIFOLDS
AND
DIFFERENTIAL
FORMS
-
55
UNIVERSAL
COVERING,
HOMOLOGY,
AND
COHOMOLOGY
-
58
2
2.1
2.1.1
MULTIPARTICLE
CANONICAL
OPERATOR
AND
ITS
PROPERTIES
-
61
INTRODUCTION
-
61
PHYSICAL
PROBLEMS
GIVING
RISE
TO
THE
STUDY
OF
FUNCTIONS
OF
LARGE
NUMBER
OF
ARGUMENTS
-
61
2.1.2
PHYSICAL
ARGUMENTS
LEADING
TO
THE
CHOICE
OF
A
NORM
IN
THE
SPACE
OF
FUNCTIONS
OF
LARGE
NUMBER
OF
ARGUMENTS
-
62
2.1.3
2.1.4
METHOD
OF
BBGKY
HIERARCHIES
----
64
NONCONSERVATION
OF
CHAOS
FOR
M-PARTICLE
WAVE
FUNCTION:
STATEMENT
OF
THE
THEOREM
-
66
2.1.5
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM:
STATEMENT
OF
THE
THEOREM
----
68
2.2
2.2.1
2.2.2
DEFINITION
OF
MULTIPARTICLE
CANONICAL
OPERATOR
-
73
EXAMPLES
OF
MULTIPARTICLE
FUNCTIONS
SATISFYING
THE
CHAOS
PROPERTY
-
73
DEFINITION
AND
SIMPLEST
PROPERTIES
OF
THE
MULTIPARTICLE
CANONICAL
OPERATOR
-
74
2.3
GEOMETRIC
STRUCTURES
ON
THE
ONE-PARTICLE
SPACE
INDUCED
BY
THE
MULTIPARTICLE
CANONICAL
OPERATOR
----
76
2.4
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
MANIFOLD
M
----
83
2.4.1
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
AND
THEIR
PROPERTIES
-
83
2.4.2
2.5
2.5.1
2.5.2
2.5.3
2.6
2.6.1
PROPER
CANONICAL
TRANSFORMATIONS
-
87
COMPLEX
GERM
----
89
COMPLEX
GERM
CORRESPONDING
TO
A
GAUSSIAN
VECTOR
-
89
CANONICAL
TRANSFORMATION
OF
A
COMPLEX
GERM
-
94
CONSTRUCTION
OF
THE
OPERATOR
W
----
95
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
-
97
SOME
EXAMPLES
OF
FORMAL
ASYMPTOTIC
SOLUTIONS
AND
THE
PROBLEM
OF
CONSERVATION
OF
CHAOS
----
97
2.6.2
INVARIANT
FORMAL
ASYMPTOTIC
SOLUTIONS
-
100
CONTENTS
-
XXI
2.6.3
EQUATIONS
FOR
A
CANONICAL
TRANSFORMATION
AND
A
COMPLEX
GERM
DEPENDING
ON
TIME
----
102
2.6.4
2.6.5
2.7
EQUATIONS
FOR
THE
OPERATOR
IV
----
104
FORMAL
ASYMPTOTIC
SOLUTION
SATISFYING
A
GIVEN
INITIAL
CONDITION
----
108
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
THE
HAMILTONIAN
AND
THE
MAIN
THEOREM
-
112
2.7.1
2.7.2
2.A
2.A.1
2.A.2
2.A.3
2
.A.
4
2.A.5
2.B
2.C
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
OTHER
OPERATORS
----
112
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
119
METHOD
OF
SECOND
QUANTIZATION
[6]
----
120
FOCK
SPACE
AND
CREATION
AND
ANNIHILATION
OPERATORS
----
120
COHERENT
STATES
AND
THEIR
PROPERTIES
----
123
GENERATING
FUNCTIONALS
----
123
NORM
OF
A
GAUSSIAN
VECTOR
-
125
LINEAR
CANONICAL
TRANSFORMATIONS
----
127
SOME
PROPERTIES
OF
THE
UNIT
SPHERE
IN
THE
SPACE
L
2
----
129
PROOF
OF
EXISTENCE
OF
THE
SOLUTIONS
OF
SOME
EQUATIONS
----
133
3
3.1
3.1.1
3.1.2
3.1.3
ASYMPTOTIC
SOLUTIONS
OF
THE
MANY-BODY
PROBLEM
-
139
INTRODUCTION
----
139
VARIOUS
PHYSICAL
APPLICATIONS
OF
THE
METHOD
-
139
PROBABILITY
DENSITY
DISTRIBUTION:
THE
CHOICE
OF
A
NORM
----
143
HALF-DENSITY
FUNCTION
AND
APPLICABILITY
OF
THE
METHOD
IN
CHAPTER
2
-
146
3.1.4
CHAOS
CONSERVATION
PROBLEM
IN
SYSTEMS
OF
MANY
CLASSICAL
PARTICLES
----
148
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
ASYMPTOTIC
FORMULAS
FOR
THE
MULTIPARTICLE
DENSITY
MATRIX
----
149
CHOICE
OF
A
NORM
ON
THE
SPACE
OF
DENSITY
MATRICES
----
149
HALF-DENSITY
MATRIX
AND
ITS
PROPERTIES
----
150
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATION
FOR
THE
HALF-DENSITY
MATRIX
----
152
REPRESENTATION
VIA
THE
GERM
CREATION
AND
ANNIHILATION
OPERATORS
----
154
HERMITIAN
PROPERTY
OF
THE
HALF-DENSITY
MATRIX
----
156
HALF-DENSITY
MATRIX
FOR
PURE
STATES
----
156
EXISTENCE
OF
SOLUTIONS
OF
CERTAIN
AUXILIARY
EQUATIONS
AND
ASYMPTOTICS
OF
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
THE
MULTIPARTICLE
WIGNER
EQUATION
----
158
3.3
ASYMPTOTIC
SOLUTIONS
OF
THE
W-PARTICLE
SCHRODINGER
EQUATION
AS
N
-
OO
AND
SUPERFLUIDITY
----
161
3.3.1
3.3.2
3.3.3
3.4
ASYMPTOTIC
SOLUTIONS
OF
THE
W-PARTICLE
SCHRODINGER
EQUATION
----
161
N.
N.
BOGOLIUBOV
'
S
SUPERFLUIDITY
THEORY
----
166
GENERALIZATION
OF
THE
NOTION
OF
SUPERFLUIDITY
----
168
ASYMPTOTICS
OF
SOLUTION
OF
THE
W-PARTICLE
LIOUVILLE
EQUATION
AND
VIOLATION
OF
THE
CHAOS
CONJECTURE
FOR
DENSITY
FUNCTION
----
169
XXII
-
CONTENTS
3.4.1
3.4.2
EXISTENCE
OF
SOLUTIONS
OF
THE
VLASOV
AND
RICCATI
EQUATIONS
-
169
ASYMPTOTICS
OF
SOLUTION
OFTHE
CAUCHY
PROBLEM
FOR
THE
W-PARTICLE
LIOUVILLE
EQUATION
----
174
3.4.3
3.5
STATIONARY
ASYMPTOTIC
SOLUTIONS
OFTHE
MANY-BODY
PROBLEM
-
175
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATION
CORRESPONDING
TO
THE
UNIFORMIZATION
OF
A
FUNCTIONAL
ON
AN
ABSTRACT
HAMILTONIAN
ALGEBRA
-
177
3.5.1
3.5.2
3.5.3
DEFINITION
OF
ABSTRACT
HAMILTONIAN
ALGEBRA
-
178
ABSTRACT
HALF-DENSITY
-
181
TENSOR
PRODUCTS
OF
HAMILTONIAN
ALGEBRASAND
OF
THEIR
REPRESENTATIONS.
THE
NOTION
OF
UNIFORMIZATION
-
183
3.5.4
3
.A
3.A.1
CONSTRUCTION
OF
ASYMPTOTICS
OF
SOLUTION
OFTHE
CAUCHY
PROBLEM
-
186
EXISTENCE
OF
SOLUTIONS
OF
THE
HARTREE
AND
RICCATI
EQUATIONS
-
188
EXISTENCE
AND
UNIQUENESS
OF
SOLUTIONSOFA
HARTREE-TYPE
EQUATION
-
188
3
.A.
2
EXISTENCE
OF
SOLUTIONS
OF
THE
VARIATIONAL
EQUATION
AND
THE
RICCATI
EQUATION
-
189
4
4.1
4.1.1
4.1.2
COMPLEX
GERM
METHOD
IN
THE
FOCK
SPACE
-
193
INTRODUCTION
-
193
SYSTEMS
WITH
VARIABLE
NUMBER
OF
PARTICLES
-
193
APPLICATION
OFTHE
COMPLEX
GERM
METHOD
TO
SYSTEMS
WITH
FINITELY
MANY
DEGREES
OF
FREEDOM
-
194
4.1.3
PROJECTION
ONTO
THE
W-PARTICLE
SUBSPACE
AND
LAGRANGIAN
MANIFOLDS
WITH
COMPLEX
GERM
-
197
4.2
4.2.1
4.2.2
4.2.3
4.2.4
COMPLEX
GERM
AT
A
POINT
IN
THE
FOCK
SPACE
-
198
AUXILIARY
NOTIONS
-
198
CHECKING
THE
AXIOMS
OF
ABSTRACT
CANONICAL
OPERATOR
-
199
CANONICAL
TRANSFORMATION
OFTHE
PHASE
SPACE
-
201
PROPER
CANONICAL
TRANSFORMATION
OF
INFINITE-DIMENSIONAL
PHASE
SPACE
----
202
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.3
COMPLEX
GERM
----
203
FORMAL
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATIONS
OF
MOTION
-
208
EQUATION
FOR
THE
OPERATOR
W
-
209
COMMUTATION
OFTHE
CANONICAL
OPERATOR
WITH
OTHER
OPERATORS
-
210
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATIONS
OF
MOTION
-
213
CORRECTIONS
TO
THE
ASYMPTOTIC
FORMULA
-
214
SUPERPOSITION
OF
WAVE
PACKETS
IN
FINITE-DIMENSIONAL
QUANTUM
MECHANICS
----
216
4.3.1
4.3.2
4.3.3
STATEMENT
OF
THE
PROBLEM
-
216
SUPERPOSITION
OF
WAVE
PACKETS
IN
THE
ONE-DIMENSIONAL
CASE
-
217
SUPERPOSITION
OF
WAVE
PACKETS
IN
THE
MULTIDIMENSIONAL
CASE
-
221
CONTENTS
-
XXIII
4.3.4
NORM
OF
THE
WAVE
FUNCTION
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
----
225
4.4
CANONICAL
OPERATOR
CORRESPONDING
TO
A
LAGRANGIAN
MANIFOLD
WITH
A
COMPLEX
GERM
----
228
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
4.4.6
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
228
CASE
OF
A
LAGRANGIAN
MANIFOLD
OF
FULL
DIMENSION
-
231
HOLOMORPHIC
REPRESENTATION
OF
DISTRIBUTIONS
IN
S'
-
235
GENERAL
CASE
-
236
COMPLEX
GERM
IN
THE
HOLOMORPHIC
REPRESENTATION
-
244
DEFINITION
OF
CANONICAL
OPERATOR
ON
A
LAGRANGIAN
MANIFOLD
WITH
COMPLEX
GERM
----
252
4.4.7
SOLUTION
OF
THE
CAUCHY
PROBLEM:
ASYMPTOTICS
CORRESPONDING
TO
ISOTROPIC
MANIFOLDS
-
256
4.5
SUPERPOSITION
OF
WAVE
FUNCTIONS
CORRESPONDING
TO
AN
ABSTRACT
CANONICAL
OPERATOR
----
259
4.5.1
4.5.2
4.5.3
CALCULATION
OF
THE
NORM
AS
E
-
0
-
260
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
262
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
CORRESPONDING
TO
ISOTROPIC
MANIFOLDS
-
264
4.6
SPECIFIC
FEATURES
OF
STATEMENT
OF
THE
CAUCHY
PROBLEM
FORA
TOPOLOGICALLY
INVARIANT
ISOTROPIC
MANIFOLD
-
265
4.6.1
4.6.2
4.6.3
4.7
ISOTROPIC
MANIFOLD
DIFFEOMORPHIC
TO
A
CIRCLE
-
265
GENERAL
CASE:
MAIN
DIFFICULTIES
-
268
DEFINITION
OF
THE
OPERATOR
/C
R
*
----
270
ASYMPTOTIC
FORMULAS
IN
THE
FOCK
SPACE
CORRESPONDING
TO
FINITE-DIMENSIONAL
ISOTROPIC
MANIFOLDS
-
272
4.7.1
4.7.2
4.7.3
VERIFICATION
OF
PROPERTIES
OF
THE
CANONICAL
OPERATOR
-
272
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
274
ISOTROPIC
MANIFOLD
OF
SPECIAL
FORM
AND
FIXING
THE
NUMBER
OF
PARTICLES
----
277
4.7.4
4.7.5
4.7.6
CASE
OF
TWO
TYPES
OF
PARTICLES
-
280
COMPLEX
GERM
-
282
CANONICAL
TRANSFORMATION
OF
COMPLEX
GERM
AND
FORMAL
ASYMPTOTIC
SOLUTIONS
----
286
4.7.7
4.
A
ASYMPTOTIC
SOLUTIONS
OF
THE
CAUCHY
PROBLEM
-
288
SEPARATION
OF
THE
CYCLIC
VARIABLE
AND
CONSTRUCTION
OF
THE
TUNNEL
ASYMPTOTICS
----
289
4.A.1
4.A.2
4.A.3
4.A.4
ADDITIVE
ASYMPTOTICS:
ADVANTAGES
AND
DRAWBACKS
-
289
EQUATION
IN
THE
REPRESENTATION
OF
GENERATING
FUNCTIONALS
-
289
DEFINITION
OF
TUNNEL
ASYMPTOTICS
-
291
EQUATION
FOR
THE
EXPONENTIAL
FACTOR,
PREEXPONENTIAL
FACTOR,
AND
CORRECTIONS
-
291
XXIV
-
CONTENTS
4.A.5
4.A.6
4.A.7
EQUATIONS
OF
CHARACTERISTICS
----
292
RELATIONSHIP
TO
THE
GERM
ASYMPTOTICS
----
295
APPLICATION
TO
THE
MULTIPARTICLE
SCHRODINGER
EQUATION
----
296
5
5.1
5.1.1
ASYMPTOTIC
METHODS
IN
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
-
299
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
IN
QUANTUM
MECHANICS
-
299
PHYSICAL
PROBLEMS
RESULTING
IN
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
----
299
5.1.2
ANALOG
OF
THE
EHRENFEST
THEOREM
FOR
THE
OPERATOR-VALUED
CASE.
"
MULTIVALENCE
"
OF
THE
HAMILTONIAN
FUNCTION
-
301
5.1.3
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
-
303
5.2
ABSTRACT
CANONICAL
OPERATOR
IN
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
----
313
5.3
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
IN
THE
MANY-PARTICLE
PROBLEM
----
316
5.3.1
5.3.2
5.3.3
SEVERAL
EXAMPLES
OF
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
-
316
ANALOG
OF
THE
HARTREE
EQUATION
FOR
THE
OPERATOR-VALUED
CASE
-
318
NONPRESERVATION
OF
CHAOS
FOR
CORRELATION
FUNCTIONS
IN
THE
OPERATOR-VALUED
CASE
-
321
5.3.4
CONSTRUCTION
OF
ASYMPTOTICS
FOR
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
323
5.3.5
5.3.6
ANALYSIS
OF
THE
SYSTEM
OF
RECURSIVE
RELATIONS
-
325
CASE
OF
THE
VARIABLE
NUMBER
OF
PARTICLES
-
327
6
6.1
6.2
SEMICLASSICAL
FIELD
THEORY
IN
THE
HAMILTONIAN
FORMALISM
-
331
INTRODUCTION
----
331
LAGRANGIAN
MANIFOLDS
WITH
THE
COMPLEX
GERM
IN
QUANTUM
FIELD
THEORY
----
332
6.2.1
6.2.2
SCALAR
FIELD
THEORY:
GERM
AT
A
POINT
----
332
SCALAR
FIELD
THEORY:
GERM
ON
A
MANIFOLD
AND
QUANTIZATION
NEAR
SOLITONS
----
334
6.2.3
6.3
6.3.1
6.3.2
6.3.3
6.4
6.4.1
SCALAR
QUANTUM
ELECTRODYNAMICS
----
335
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
IN
QUANTUM
FIELD
THEORY
----
337
SCALAR
FIELD
----
337
ELECTROMAGNETIC
FIELD
-
339
YANG-MILLS
FIELD
----
342
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
----
343
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
IN
THE
FUNCTIONAL
REPRESENTATION
----
343
6.4.2
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
IN
THE
FOCK
REPRESENTATION
-
347
CONTENTS
-
XXV
6.5
GENERAL
SCHEME
OF
RENORMALIZATION
OF
THE
HAMILTONIAN
FIELD
THEORY
----
352
6.5.1
6.5.2
6.6
CONDITIONS
ON
THE
HAMILTONIAN
AND
THE
INITIAL
DATA
----
352
AXIOMS
OF
REGULARIZED
FIELD
THEORY
----
355
FADDEEV
TRANSFORMATION
AND
REMOVAL
OF
THE
STIICKELBERG
DIVERGENCES
----
358
6.6.1
6.6.2
6.6.3
6.7
FADDEEV
TRANSFORMATION
----
358
S-MATRIXAND
NONUNIQUENESS
OF
THE
FADDEEV
TRANSFORMATION
----
363
FADDEEV
TRANSFORMATION
AND
PARTICLE
DECAY
RATE
----
365
BOGOLIUBOV
S-MATRIXAND
RENORMALIZATION
OF
THE
EQUATIONS
OF
MOTION
----
367
6.7.1
6.7.2
6.8
6.8.1
6.8.2
6.8.3
6.8.4
6.9
6.9.1
6.9.2
6.9.3
6.9.4
BOGOLIUBOV
S-MATRIX
AND
THE
EQUATION
OF
MOTION
-
367
BOGOLIUBOV
S-MATRIX
IN
THE
FIRST
ORDER
OF
PERTURBATION
THEORY
----
370
RENORMALIZATION
IN
SEMICLASSICAL
FIELD
THEORY
-
371
SEMICLASSICAL
APPROXIMATION
IN
REGULARIZED
FIELD
THEORY
----
371
PROBLEM
OF
REMOVAL
OF
REGULARIZATION
IN
HAMILTONIAN
FORMALISM
-
376
CONDITION
ON
THE
GAUSSIAN
STATE
VECTOR
-
380
SOME
CONCLUSIONS
----
383
INVARIANCE
OF
THE
CONDITIONS
ON
THE
COMPLEX
GERM
----
384
TRANSFORMATION
OF
THE
RICCATI
EQUATION
----
384
ON
OPERATORS
WITH
SMOOTH
WEYL
SYMBOL
----
388
STUDY
OF
THE
TRANSFORMED
EQUATION
----
392
CONSTRUCTION
OF
AN
APPROXIMATE
SOLUTION
OF
THE
RICCATI
EQUATION
-
397
7
7.1
7.2
ASYMPTOTIC
METHODS
FOR
SYSTEMS
OF
A
LARGE
NUMBER
OF
FIELDS
-
403
INTRODUCTION
----
403
0(W)-SYMMETRIC
ANHARMONIC
OSCILLATOR
AS
AN
ANALOG
OF
A
MANY-FIELD
SYSTEM
----
404
7.2.1
7.2.2
7.2.3
7.2.4
7.2.5
7.3
COLLECTIVE
FIELD
METHOD
-
405
CLASSICAL
EQUATIONS
IN
THE
SECOND-QUANTIZED
APPROACH
-
406
RELATION
BETWEEN
CLASSICAL
EQUATIONS
UNDER
VARIOUS
APPROACHES
-
406
SEMICLASSICAL
WAVE
FUNCTIONS
UNDER
VARIOUS
APPROACHES
-
407
ASYMPTOTIC
SPECTRUM
AS
N
-
OO
----
410
FORMALISM
OF
THE
THIRD
QUANTIZATION
AND
THE
SEMICLASSICAL
APPROXIMATION
----
412
7.3.1
ASYMPTOTIC
METHODS
IN
THE
THEORY
OF
LARGE
NUMBER
OF
FIELDS
ON
A
LATTICE
-
412
7.3.2
CLASSICAL
EQUATIONS
IN
THE
THIRD-QUANTIZED
APPROACH
TO
THE
THEORY
OF
LARGE
NUMBER
OF
FIELDS
-
415
7.3.3
7.3.4
7.3.5
ON
THE
CONSTRUCTION
OF
ASYMMETRIC
SOLUTIONS
-
417
QXPX
MODEL
----
418
CASE
OF
SPONTANEOUS
SYMMETRY
BREAKING
-
419
XXVI
-
-
CONTENTS
7.3.6
7.4
7.4.1
7.4.2
7.4.3
7.4.4
7.4.5
7.5
O(/V)-ASYMMETRIC
THEORIES
-
420
ON
RENORMALIZATIONS
OF
THE
CLASSICAL
EQUATIONS
-
421
ON
REGULARIZATION
AND
RENORMALIZATION
-
421
ON
GAUSSIAN
AND
NON-GAUSSIAN
SOLUTIONS
OF
CLASSICAL
EQUATIONS
-
422
ON
SINGULARITIES
OF
THE
GAUSSIAN
QUADRATIC
FORM
-
423
RENORMALIZATION
OF
THE
MASS
AND
THE
COUPLING
CONSTANT
-
424
RENORMALIZATION
OF
THE
COSMOLOGICAL
CONSTANT
-
426
ASYMPTOTIC
SPECTRUM
OF
THE
HAMILTONIAN
OF
A
LARGE
NUMBER
OF
FIELDS
----
428
7.5.1
7.5.2
7.5.3
7.5.4
STATIONARY
SOLUTIONS
OF
THE
HARTREE-TYPE
EQUATION
-
428
CLASSICAL
ENERGIES
OF
VARIOUS
SERIES
-
430
NEAR-VACUUM
SOLUTIONS
OF
THE
/V-FIELD
EQUATION
-
432
OTHER
SERIES
OF
ASYMPTOTICS
-
437
CONCLUDING
REMARKS
-
439
BIBLIOGRAPHY
-
443
INDEX
-
447 |
| adam_txt |
CONTENTS
PREFACE
TO
THE
ENGLISH
EDITION
-
VII
PREFACE
-
XI
LIST
OF
NOTATION
----
XXVII
1
1.1
1.1.1
ABSTRACT
CANONICAL
OPERATOR
AND
SYMPLECTIC
GEOMETRY
-
1
INTRODUCTION
----
1
CONSTRUCTION
OF
APPROXIMATE
SOLUTIONS
OF
VARIOUS
EQUATIONS:
COMPARISON
OF
METHODS
----
1
1.1.2
1.1.3
ABSTRACT
CANONICAL
OPERATOR
AND
GEOMETRY
OF
THE
PHASE
SPACE
-
5
PROBLEM
OF
EXISTENCE
OF
EQUATIONS
OF
MOTION
AND
THE
CONCEPT
OF
ASYMPTOTIC
QUANTIZATION
----
8
1.2
ABSTRACT
CANONICAL
OPERATOR
AND
INDUCED
GEOMETRIC
STRUCTURES
ON
THE
PHASE
SPACE
----
9
1.2.1
1.2.2
1.2.3
AUXILIARY
NOTIONS
-
9
DEFINITION
OF
AN
ABSTRACT
CANONICAL
OPERATOR
-
11
WELL-DEFINEDNESS
OF
THE
CANONICAL
OPERATOR
-
THE
ACTION
1-FORM
ON
THE
PHASE
SPACE
-
12
1.2.4
1.2.5
1.2.6
OPERATOR-VALUED
1-FORM
INDUCED
BY
THE
CANONICAL
OPERATOR
-
16
CANONICAL
TRANSFORMATION
OF
THE
ABSTRACT
CANONICAL
OPERATOR
----
18
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
-
21
1.2.7
1.3
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
----
21
ABSTRACT
COMPLEX
GERM
AND
CONSTRUCTION
OF
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
----
22
1.3.1
1.3.2
1.3.3
1.3.4
1.4
1.4.1
DEFINITION
OF
COMPLEX
GERM
----
22
PROPERTIES
OF
THE
ABSTRACT
COMPLEX
GERM
-
24
CANONICAL
TRANSFORMATION
OF
THE
ABSTRACT
COMPLEX
GERM
-----26
CANONICAL
OPERATOR
CORRESPONDING
TO
A
COMPLEX
GERM
----
28
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
30
CANONICAL
TRANSFORMATIONS
OF
THE
CANONICAL
OPERATOR
AND
CONSTRUCTION
OF
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
30
1.4.2
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
MODULO
O^/
2
)
----
33
1.5
THEORY
OF
COMPLEX
GERM
AT
A
POINT
IN
FINITE-DIMENSIONAL
QUANTUM
MECHANICS
----
35
1.5.1
1.5.2
DEFINITION
OF
THE
CANONICAL
OPERATOR
AND
CHECKING
THE
AXIOMS
-
35
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
----
36
XX
-
CONTENTS
1.5.3
1.5.4
1.5.5
COMPLEX
GERM
-
37
CANONICAL
TRANSFORMATIONS
DEPENDING
ON
TIME
-
42
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
THE
HAMILTONIAN
AND
PROOF
OF
THE
ASYMPTOTIC
FORMULA
-
45
1.A
CLASSICAL
AND
QUANTUM
MECHANICS:
THE
MAIN
DEFINITIONS
[1,
20,53]
----
52
1.B
1.B.1
1.B.2
1.B.3
SOME
RECOLLECTIONS
FROM
DIFFERENTIAL
GEOMETRY
-
54
SMOOTH
MAPPINGS
OF
HILBERT
SPACES
-
54
SMOOTH
MANIFOLDS
AND
DIFFERENTIAL
FORMS
-
55
UNIVERSAL
COVERING,
HOMOLOGY,
AND
COHOMOLOGY
-
58
2
2.1
2.1.1
MULTIPARTICLE
CANONICAL
OPERATOR
AND
ITS
PROPERTIES
-
61
INTRODUCTION
-
61
PHYSICAL
PROBLEMS
GIVING
RISE
TO
THE
STUDY
OF
FUNCTIONS
OF
LARGE
NUMBER
OF
ARGUMENTS
-
61
2.1.2
PHYSICAL
ARGUMENTS
LEADING
TO
THE
CHOICE
OF
A
NORM
IN
THE
SPACE
OF
FUNCTIONS
OF
LARGE
NUMBER
OF
ARGUMENTS
-
62
2.1.3
2.1.4
METHOD
OF
BBGKY
HIERARCHIES
----
64
NONCONSERVATION
OF
CHAOS
FOR
M-PARTICLE
WAVE
FUNCTION:
STATEMENT
OF
THE
THEOREM
-
66
2.1.5
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM:
STATEMENT
OF
THE
THEOREM
----
68
2.2
2.2.1
2.2.2
DEFINITION
OF
MULTIPARTICLE
CANONICAL
OPERATOR
-
73
EXAMPLES
OF
MULTIPARTICLE
FUNCTIONS
SATISFYING
THE
CHAOS
PROPERTY
-
73
DEFINITION
AND
SIMPLEST
PROPERTIES
OF
THE
MULTIPARTICLE
CANONICAL
OPERATOR
-
74
2.3
GEOMETRIC
STRUCTURES
ON
THE
ONE-PARTICLE
SPACE
INDUCED
BY
THE
MULTIPARTICLE
CANONICAL
OPERATOR
----
76
2.4
CANONICAL
AND
PROPER
CANONICAL
TRANSFORMATIONS
OF
THE
MANIFOLD
M
----
83
2.4.1
CANONICAL
TRANSFORMATIONS
OF
THE
PHASE
SPACE
AND
THEIR
PROPERTIES
-
83
2.4.2
2.5
2.5.1
2.5.2
2.5.3
2.6
2.6.1
PROPER
CANONICAL
TRANSFORMATIONS
-
87
COMPLEX
GERM
----
89
COMPLEX
GERM
CORRESPONDING
TO
A
GAUSSIAN
VECTOR
-
89
CANONICAL
TRANSFORMATION
OF
A
COMPLEX
GERM
-
94
CONSTRUCTION
OF
THE
OPERATOR
W
----
95
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
-
97
SOME
EXAMPLES
OF
FORMAL
ASYMPTOTIC
SOLUTIONS
AND
THE
PROBLEM
OF
CONSERVATION
OF
CHAOS
----
97
2.6.2
INVARIANT
FORMAL
ASYMPTOTIC
SOLUTIONS
-
100
CONTENTS
-
XXI
2.6.3
EQUATIONS
FOR
A
CANONICAL
TRANSFORMATION
AND
A
COMPLEX
GERM
DEPENDING
ON
TIME
----
102
2.6.4
2.6.5
2.7
EQUATIONS
FOR
THE
OPERATOR
IV
----
104
FORMAL
ASYMPTOTIC
SOLUTION
SATISFYING
A
GIVEN
INITIAL
CONDITION
----
108
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
THE
HAMILTONIAN
AND
THE
MAIN
THEOREM
-
112
2.7.1
2.7.2
2.A
2.A.1
2.A.2
2.A.3
2
.A.
4
2.A.5
2.B
2.C
COMMUTATION
OF
THE
CANONICAL
OPERATOR
WITH
OTHER
OPERATORS
----
112
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
119
METHOD
OF
SECOND
QUANTIZATION
[6]
----
120
FOCK
SPACE
AND
CREATION
AND
ANNIHILATION
OPERATORS
----
120
COHERENT
STATES
AND
THEIR
PROPERTIES
----
123
GENERATING
FUNCTIONALS
----
123
NORM
OF
A
GAUSSIAN
VECTOR
-
125
LINEAR
CANONICAL
TRANSFORMATIONS
----
127
SOME
PROPERTIES
OF
THE
UNIT
SPHERE
IN
THE
SPACE
L
2
----
129
PROOF
OF
EXISTENCE
OF
THE
SOLUTIONS
OF
SOME
EQUATIONS
----
133
3
3.1
3.1.1
3.1.2
3.1.3
ASYMPTOTIC
SOLUTIONS
OF
THE
MANY-BODY
PROBLEM
-
139
INTRODUCTION
----
139
VARIOUS
PHYSICAL
APPLICATIONS
OF
THE
METHOD
-
139
PROBABILITY
DENSITY
DISTRIBUTION:
THE
CHOICE
OF
A
NORM
----
143
HALF-DENSITY
FUNCTION
AND
APPLICABILITY
OF
THE
METHOD
IN
CHAPTER
2
-
146
3.1.4
CHAOS
CONSERVATION
PROBLEM
IN
SYSTEMS
OF
MANY
CLASSICAL
PARTICLES
----
148
3.2
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
3.2.6
3.2.7
ASYMPTOTIC
FORMULAS
FOR
THE
MULTIPARTICLE
DENSITY
MATRIX
----
149
CHOICE
OF
A
NORM
ON
THE
SPACE
OF
DENSITY
MATRICES
----
149
HALF-DENSITY
MATRIX
AND
ITS
PROPERTIES
----
150
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATION
FOR
THE
HALF-DENSITY
MATRIX
----
152
REPRESENTATION
VIA
THE
GERM
CREATION
AND
ANNIHILATION
OPERATORS
----
154
HERMITIAN
PROPERTY
OF
THE
HALF-DENSITY
MATRIX
----
156
HALF-DENSITY
MATRIX
FOR
PURE
STATES
----
156
EXISTENCE
OF
SOLUTIONS
OF
CERTAIN
AUXILIARY
EQUATIONS
AND
ASYMPTOTICS
OF
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
THE
MULTIPARTICLE
WIGNER
EQUATION
----
158
3.3
ASYMPTOTIC
SOLUTIONS
OF
THE
W-PARTICLE
SCHRODINGER
EQUATION
AS
N
-
OO
AND
SUPERFLUIDITY
----
161
3.3.1
3.3.2
3.3.3
3.4
ASYMPTOTIC
SOLUTIONS
OF
THE
W-PARTICLE
SCHRODINGER
EQUATION
----
161
N.
N.
BOGOLIUBOV
'
S
SUPERFLUIDITY
THEORY
----
166
GENERALIZATION
OF
THE
NOTION
OF
SUPERFLUIDITY
----
168
ASYMPTOTICS
OF
SOLUTION
OF
THE
W-PARTICLE
LIOUVILLE
EQUATION
AND
VIOLATION
OF
THE
CHAOS
CONJECTURE
FOR
DENSITY
FUNCTION
----
169
XXII
-
CONTENTS
3.4.1
3.4.2
EXISTENCE
OF
SOLUTIONS
OF
THE
VLASOV
AND
RICCATI
EQUATIONS
-
169
ASYMPTOTICS
OF
SOLUTION
OFTHE
CAUCHY
PROBLEM
FOR
THE
W-PARTICLE
LIOUVILLE
EQUATION
----
174
3.4.3
3.5
STATIONARY
ASYMPTOTIC
SOLUTIONS
OFTHE
MANY-BODY
PROBLEM
-
175
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATION
CORRESPONDING
TO
THE
UNIFORMIZATION
OF
A
FUNCTIONAL
ON
AN
ABSTRACT
HAMILTONIAN
ALGEBRA
-
177
3.5.1
3.5.2
3.5.3
DEFINITION
OF
ABSTRACT
HAMILTONIAN
ALGEBRA
-
178
ABSTRACT
HALF-DENSITY
-
181
TENSOR
PRODUCTS
OF
HAMILTONIAN
ALGEBRASAND
OF
THEIR
REPRESENTATIONS.
THE
NOTION
OF
UNIFORMIZATION
-
183
3.5.4
3
.A
3.A.1
CONSTRUCTION
OF
ASYMPTOTICS
OF
SOLUTION
OFTHE
CAUCHY
PROBLEM
-
186
EXISTENCE
OF
SOLUTIONS
OF
THE
HARTREE
AND
RICCATI
EQUATIONS
-
188
EXISTENCE
AND
UNIQUENESS
OF
SOLUTIONSOFA
HARTREE-TYPE
EQUATION
-
188
3
.A.
2
EXISTENCE
OF
SOLUTIONS
OF
THE
VARIATIONAL
EQUATION
AND
THE
RICCATI
EQUATION
-
189
4
4.1
4.1.1
4.1.2
COMPLEX
GERM
METHOD
IN
THE
FOCK
SPACE
-
193
INTRODUCTION
-
193
SYSTEMS
WITH
VARIABLE
NUMBER
OF
PARTICLES
-
193
APPLICATION
OFTHE
COMPLEX
GERM
METHOD
TO
SYSTEMS
WITH
FINITELY
MANY
DEGREES
OF
FREEDOM
-
194
4.1.3
PROJECTION
ONTO
THE
W-PARTICLE
SUBSPACE
AND
LAGRANGIAN
MANIFOLDS
WITH
COMPLEX
GERM
-
197
4.2
4.2.1
4.2.2
4.2.3
4.2.4
COMPLEX
GERM
AT
A
POINT
IN
THE
FOCK
SPACE
-
198
AUXILIARY
NOTIONS
-
198
CHECKING
THE
AXIOMS
OF
ABSTRACT
CANONICAL
OPERATOR
-
199
CANONICAL
TRANSFORMATION
OFTHE
PHASE
SPACE
-
201
PROPER
CANONICAL
TRANSFORMATION
OF
INFINITE-DIMENSIONAL
PHASE
SPACE
----
202
4.2.5
4.2.6
4.2.7
4.2.8
4.2.9
4.2.10
4.3
COMPLEX
GERM
----
203
FORMAL
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATIONS
OF
MOTION
-
208
EQUATION
FOR
THE
OPERATOR
W
-
209
COMMUTATION
OFTHE
CANONICAL
OPERATOR
WITH
OTHER
OPERATORS
-
210
ASYMPTOTIC
SOLUTIONS
OFTHE
EQUATIONS
OF
MOTION
-
213
CORRECTIONS
TO
THE
ASYMPTOTIC
FORMULA
-
214
SUPERPOSITION
OF
WAVE
PACKETS
IN
FINITE-DIMENSIONAL
QUANTUM
MECHANICS
----
216
4.3.1
4.3.2
4.3.3
STATEMENT
OF
THE
PROBLEM
-
216
SUPERPOSITION
OF
WAVE
PACKETS
IN
THE
ONE-DIMENSIONAL
CASE
-
217
SUPERPOSITION
OF
WAVE
PACKETS
IN
THE
MULTIDIMENSIONAL
CASE
-
221
CONTENTS
-
XXIII
4.3.4
NORM
OF
THE
WAVE
FUNCTION
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
----
225
4.4
CANONICAL
OPERATOR
CORRESPONDING
TO
A
LAGRANGIAN
MANIFOLD
WITH
A
COMPLEX
GERM
----
228
4.4.1
4.4.2
4.4.3
4.4.4
4.4.5
4.4.6
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
228
CASE
OF
A
LAGRANGIAN
MANIFOLD
OF
FULL
DIMENSION
-
231
HOLOMORPHIC
REPRESENTATION
OF
DISTRIBUTIONS
IN
S'
-
235
GENERAL
CASE
-
236
COMPLEX
GERM
IN
THE
HOLOMORPHIC
REPRESENTATION
-
244
DEFINITION
OF
CANONICAL
OPERATOR
ON
A
LAGRANGIAN
MANIFOLD
WITH
COMPLEX
GERM
----
252
4.4.7
SOLUTION
OF
THE
CAUCHY
PROBLEM:
ASYMPTOTICS
CORRESPONDING
TO
ISOTROPIC
MANIFOLDS
-
256
4.5
SUPERPOSITION
OF
WAVE
FUNCTIONS
CORRESPONDING
TO
AN
ABSTRACT
CANONICAL
OPERATOR
----
259
4.5.1
4.5.2
4.5.3
CALCULATION
OF
THE
NORM
AS
E
-
0
-
260
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
262
FORMAL
ASYMPTOTIC
SOLUTIONS
OF
THE
EQUATIONS
OF
MOTION
CORRESPONDING
TO
ISOTROPIC
MANIFOLDS
-
264
4.6
SPECIFIC
FEATURES
OF
STATEMENT
OF
THE
CAUCHY
PROBLEM
FORA
TOPOLOGICALLY
INVARIANT
ISOTROPIC
MANIFOLD
-
265
4.6.1
4.6.2
4.6.3
4.7
ISOTROPIC
MANIFOLD
DIFFEOMORPHIC
TO
A
CIRCLE
-
265
GENERAL
CASE:
MAIN
DIFFICULTIES
-
268
DEFINITION
OF
THE
OPERATOR
/C
R
*
----
270
ASYMPTOTIC
FORMULAS
IN
THE
FOCK
SPACE
CORRESPONDING
TO
FINITE-DIMENSIONAL
ISOTROPIC
MANIFOLDS
-
272
4.7.1
4.7.2
4.7.3
VERIFICATION
OF
PROPERTIES
OF
THE
CANONICAL
OPERATOR
-
272
CANONICAL
OPERATOR
CORRESPONDING
TO
AN
ISOTROPIC
MANIFOLD
-
274
ISOTROPIC
MANIFOLD
OF
SPECIAL
FORM
AND
FIXING
THE
NUMBER
OF
PARTICLES
----
277
4.7.4
4.7.5
4.7.6
CASE
OF
TWO
TYPES
OF
PARTICLES
-
280
COMPLEX
GERM
-
282
CANONICAL
TRANSFORMATION
OF
COMPLEX
GERM
AND
FORMAL
ASYMPTOTIC
SOLUTIONS
----
286
4.7.7
4.
A
ASYMPTOTIC
SOLUTIONS
OF
THE
CAUCHY
PROBLEM
-
288
SEPARATION
OF
THE
CYCLIC
VARIABLE
AND
CONSTRUCTION
OF
THE
TUNNEL
ASYMPTOTICS
----
289
4.A.1
4.A.2
4.A.3
4.A.4
ADDITIVE
ASYMPTOTICS:
ADVANTAGES
AND
DRAWBACKS
-
289
EQUATION
IN
THE
REPRESENTATION
OF
GENERATING
FUNCTIONALS
-
289
DEFINITION
OF
TUNNEL
ASYMPTOTICS
-
291
EQUATION
FOR
THE
EXPONENTIAL
FACTOR,
PREEXPONENTIAL
FACTOR,
AND
CORRECTIONS
-
291
XXIV
-
CONTENTS
4.A.5
4.A.6
4.A.7
EQUATIONS
OF
CHARACTERISTICS
----
292
RELATIONSHIP
TO
THE
GERM
ASYMPTOTICS
----
295
APPLICATION
TO
THE
MULTIPARTICLE
SCHRODINGER
EQUATION
----
296
5
5.1
5.1.1
ASYMPTOTIC
METHODS
IN
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
-
299
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
IN
QUANTUM
MECHANICS
-
299
PHYSICAL
PROBLEMS
RESULTING
IN
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
----
299
5.1.2
ANALOG
OF
THE
EHRENFEST
THEOREM
FOR
THE
OPERATOR-VALUED
CASE.
"
MULTIVALENCE
"
OF
THE
HAMILTONIAN
FUNCTION
-
301
5.1.3
ASYMPTOTICS
OF
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
FOR
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
-
303
5.2
ABSTRACT
CANONICAL
OPERATOR
IN
PROBLEMS
WITH
OPERATOR-VALUED
SYMBOL
----
313
5.3
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
IN
THE
MANY-PARTICLE
PROBLEM
----
316
5.3.1
5.3.2
5.3.3
SEVERAL
EXAMPLES
OF
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
-
316
ANALOG
OF
THE
HARTREE
EQUATION
FOR
THE
OPERATOR-VALUED
CASE
-
318
NONPRESERVATION
OF
CHAOS
FOR
CORRELATION
FUNCTIONS
IN
THE
OPERATOR-VALUED
CASE
-
321
5.3.4
CONSTRUCTION
OF
ASYMPTOTICS
FOR
THE
SOLUTION
OF
THE
CAUCHY
PROBLEM
----
323
5.3.5
5.3.6
ANALYSIS
OF
THE
SYSTEM
OF
RECURSIVE
RELATIONS
-
325
CASE
OF
THE
VARIABLE
NUMBER
OF
PARTICLES
-
327
6
6.1
6.2
SEMICLASSICAL
FIELD
THEORY
IN
THE
HAMILTONIAN
FORMALISM
-
331
INTRODUCTION
----
331
LAGRANGIAN
MANIFOLDS
WITH
THE
COMPLEX
GERM
IN
QUANTUM
FIELD
THEORY
----
332
6.2.1
6.2.2
SCALAR
FIELD
THEORY:
GERM
AT
A
POINT
----
332
SCALAR
FIELD
THEORY:
GERM
ON
A
MANIFOLD
AND
QUANTIZATION
NEAR
SOLITONS
----
334
6.2.3
6.3
6.3.1
6.3.2
6.3.3
6.4
6.4.1
SCALAR
QUANTUM
ELECTRODYNAMICS
----
335
EQUATIONS
WITH
OPERATOR-VALUED
SYMBOL
IN
QUANTUM
FIELD
THEORY
----
337
SCALAR
FIELD
----
337
ELECTROMAGNETIC
FIELD
-
339
YANG-MILLS
FIELD
----
342
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
----
343
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
IN
THE
FUNCTIONAL
REPRESENTATION
----
343
6.4.2
DIFFICULTIES
WITH
THE
HAMILTONIAN
FIELD
THEORY
IN
THE
FOCK
REPRESENTATION
-
347
CONTENTS
-
XXV
6.5
GENERAL
SCHEME
OF
RENORMALIZATION
OF
THE
HAMILTONIAN
FIELD
THEORY
----
352
6.5.1
6.5.2
6.6
CONDITIONS
ON
THE
HAMILTONIAN
AND
THE
INITIAL
DATA
----
352
AXIOMS
OF
REGULARIZED
FIELD
THEORY
----
355
FADDEEV
TRANSFORMATION
AND
REMOVAL
OF
THE
STIICKELBERG
DIVERGENCES
----
358
6.6.1
6.6.2
6.6.3
6.7
FADDEEV
TRANSFORMATION
----
358
S-MATRIXAND
NONUNIQUENESS
OF
THE
FADDEEV
TRANSFORMATION
----
363
FADDEEV
TRANSFORMATION
AND
PARTICLE
DECAY
RATE
----
365
BOGOLIUBOV
S-MATRIXAND
RENORMALIZATION
OF
THE
EQUATIONS
OF
MOTION
----
367
6.7.1
6.7.2
6.8
6.8.1
6.8.2
6.8.3
6.8.4
6.9
6.9.1
6.9.2
6.9.3
6.9.4
BOGOLIUBOV
S-MATRIX
AND
THE
EQUATION
OF
MOTION
-
367
BOGOLIUBOV
S-MATRIX
IN
THE
FIRST
ORDER
OF
PERTURBATION
THEORY
----
370
RENORMALIZATION
IN
SEMICLASSICAL
FIELD
THEORY
-
371
SEMICLASSICAL
APPROXIMATION
IN
REGULARIZED
FIELD
THEORY
----
371
PROBLEM
OF
REMOVAL
OF
REGULARIZATION
IN
HAMILTONIAN
FORMALISM
-
376
CONDITION
ON
THE
GAUSSIAN
STATE
VECTOR
-
380
SOME
CONCLUSIONS
----
383
INVARIANCE
OF
THE
CONDITIONS
ON
THE
COMPLEX
GERM
----
384
TRANSFORMATION
OF
THE
RICCATI
EQUATION
----
384
ON
OPERATORS
WITH
SMOOTH
WEYL
SYMBOL
----
388
STUDY
OF
THE
TRANSFORMED
EQUATION
----
392
CONSTRUCTION
OF
AN
APPROXIMATE
SOLUTION
OF
THE
RICCATI
EQUATION
-
397
7
7.1
7.2
ASYMPTOTIC
METHODS
FOR
SYSTEMS
OF
A
LARGE
NUMBER
OF
FIELDS
-
403
INTRODUCTION
----
403
0(W)-SYMMETRIC
ANHARMONIC
OSCILLATOR
AS
AN
ANALOG
OF
A
MANY-FIELD
SYSTEM
----
404
7.2.1
7.2.2
7.2.3
7.2.4
7.2.5
7.3
COLLECTIVE
FIELD
METHOD
-
405
CLASSICAL
EQUATIONS
IN
THE
SECOND-QUANTIZED
APPROACH
-
406
RELATION
BETWEEN
CLASSICAL
EQUATIONS
UNDER
VARIOUS
APPROACHES
-
406
SEMICLASSICAL
WAVE
FUNCTIONS
UNDER
VARIOUS
APPROACHES
-
407
ASYMPTOTIC
SPECTRUM
AS
N
-
OO
----
410
FORMALISM
OF
THE
THIRD
QUANTIZATION
AND
THE
SEMICLASSICAL
APPROXIMATION
----
412
7.3.1
ASYMPTOTIC
METHODS
IN
THE
THEORY
OF
LARGE
NUMBER
OF
FIELDS
ON
A
LATTICE
-
412
7.3.2
CLASSICAL
EQUATIONS
IN
THE
THIRD-QUANTIZED
APPROACH
TO
THE
THEORY
OF
LARGE
NUMBER
OF
FIELDS
-
415
7.3.3
7.3.4
7.3.5
ON
THE
CONSTRUCTION
OF
ASYMMETRIC
SOLUTIONS
-
417
QXPX
MODEL
----
418
CASE
OF
SPONTANEOUS
SYMMETRY
BREAKING
-
419
XXVI
-
-
CONTENTS
7.3.6
7.4
7.4.1
7.4.2
7.4.3
7.4.4
7.4.5
7.5
O(/V)-ASYMMETRIC
THEORIES
-
420
ON
RENORMALIZATIONS
OF
THE
CLASSICAL
EQUATIONS
-
421
ON
REGULARIZATION
AND
RENORMALIZATION
-
421
ON
GAUSSIAN
AND
NON-GAUSSIAN
SOLUTIONS
OF
CLASSICAL
EQUATIONS
-
422
ON
SINGULARITIES
OF
THE
GAUSSIAN
QUADRATIC
FORM
-
423
RENORMALIZATION
OF
THE
MASS
AND
THE
COUPLING
CONSTANT
-
424
RENORMALIZATION
OF
THE
COSMOLOGICAL
CONSTANT
-
426
ASYMPTOTIC
SPECTRUM
OF
THE
HAMILTONIAN
OF
A
LARGE
NUMBER
OF
FIELDS
----
428
7.5.1
7.5.2
7.5.3
7.5.4
STATIONARY
SOLUTIONS
OF
THE
HARTREE-TYPE
EQUATION
-
428
CLASSICAL
ENERGIES
OF
VARIOUS
SERIES
-
430
NEAR-VACUUM
SOLUTIONS
OF
THE
/V-FIELD
EQUATION
-
432
OTHER
SERIES
OF
ASYMPTOTICS
-
437
CONCLUDING
REMARKS
-
439
BIBLIOGRAPHY
-
443
INDEX
-
447 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Maslov, Viktor P. 1930- Švedov, Oleg Jur'evič 1973-2015 |
| author_GND | (DE-588)123340853 (DE-588)1262692083 |
| author_facet | Maslov, Viktor P. 1930- Švedov, Oleg Jur'evič 1973-2015 |
| author_role | aut aut |
| author_sort | Maslov, Viktor P. 1930- |
| author_variant | v p m vp vpm o j š oj ojš |
| building | Verbundindex |
| bvnumber | BV048243034 |
| classification_rvk | SK 540 |
| classification_tum | MAT 000 |
| collection | ZDB-23-DMA ZDB-23-DGG |
| ctrlnum | (ZDB-23-DMA)9783110762709 (OCoLC)1334029739 (DE-599)DNB1251573509 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| doi_str_mv | 10.1515/9783110762709 |
| format | Electronic eBook |
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| id | DE-604.BV048243034 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T19:54:54Z |
| indexdate | 2025-03-31T10:01:49Z |
| institution | BVB |
| institution_GND | (DE-588)10095502-2 |
| isbn | 9783110762709 9783110762747 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033623479 |
| oclc_num | 1334029739 |
| open_access_boolean | |
| owner | DE-706 DE-91 DE-BY-TUM DE-1043 DE-1046 DE-858 DE-859 DE-860 DE-739 DE-898 DE-BY-UBR DE-11 |
| owner_facet | DE-706 DE-91 DE-BY-TUM DE-1043 DE-1046 DE-858 DE-859 DE-860 DE-739 DE-898 DE-BY-UBR DE-11 |
| physical | 1 Online-Ressource (XXVI, 448 Seiten) |
| psigel | ZDB-23-DMA ZDB-23-DGG ZDB-23-DGG FAB_PDA_DGG ZDB-23-DGG FAW_PDA_DGG ZDB-23-DGG FCO_PDA_DGG ZDB-23-DMA ZDB-23-DMA22 ZDB-23-DGG FKE_PDA_DGG ZDB-23-DGG FLA_PDA_DGG ZDB-23-DMA TUM_Paketkauf_2022 ZDB-23-DGG UPA_PDA_DGG |
| publishDate | 2022 |
| publishDateSearch | 2022 |
| publishDateSort | 2022 |
| publisher | De Gruyter |
| record_format | marc |
| series | De Gruyter expositions in mathematics |
| series2 | De Gruyter expositions in mathematics |
| spelling | Maslov, Viktor P. 1930- Verfasser (DE-588)123340853 aut The canonical operator in many-partiсle problems and quantum field theory Victor P. Maslov and Oleg Yu. Shvedov Berlin ; Boston De Gruyter [2022] © 2022 1 Online-Ressource (XXVI, 448 Seiten) txt rdacontent c rdamedia cr rdacarrier De Gruyter expositions in mathematics volume 71 Aus dem Russischen übersetzt Švedov, Oleg Jur'evič 1973-2015 Verfasser (DE-588)1262692083 aut Walter de Gruyter GmbH & Co. KG (DE-588)10095502-2 pbl Erscheint auch als Druck-Ausgabe 978-3-11-076238-9 (DE-604)BV048615303 De Gruyter expositions in mathematics volume 71 (DE-604)BV044998893 71 https://doi.org/10.1515/9783110762709 Verlag URL des Erstveröffentlichers Volltext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033623479&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p vlb 20220211 DE-101 https://d-nb.info/provenance/plan#vlb |
| spellingShingle | Maslov, Viktor P. 1930- Švedov, Oleg Jur'evič 1973-2015 The canonical operator in many-partiсle problems and quantum field theory De Gruyter expositions in mathematics |
| title | The canonical operator in many-partiсle problems and quantum field theory |
| title_auth | The canonical operator in many-partiсle problems and quantum field theory |
| title_exact_search | The canonical operator in many-partiсle problems and quantum field theory |
| title_exact_search_txtP | The canonical operator in many-partiсle problems and quantum field theory |
| title_full | The canonical operator in many-partiсle problems and quantum field theory Victor P. Maslov and Oleg Yu. Shvedov |
| title_fullStr | The canonical operator in many-partiсle problems and quantum field theory Victor P. Maslov and Oleg Yu. Shvedov |
| title_full_unstemmed | The canonical operator in many-partiсle problems and quantum field theory Victor P. Maslov and Oleg Yu. Shvedov |
| title_short | The canonical operator in many-partiсle problems and quantum field theory |
| title_sort | the canonical operator in many partiсle problems and quantum field theory |
| url | https://doi.org/10.1515/9783110762709 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033623479&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV044998893 |
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