AN INTRODUCTION TO MATHEMATICAL ANALYSIS
CONTENTS PART 1: ANALYSIS ON FOCK SPACES 1 THEORY OF LINEAR OPERATORS LINEAR OPERATORS ON VECTOR SPACE DE?NITIONS EIGENVALUES SUM AND PRODUCT OF LINEAR OPERATORS A PARTIAL ORDERING IN LINEAR OPERATORS COMMUTATOR LINEAR OPERATORS ON BANACH SPACE CONTINUOUS MAPPINGS BOUNDED LINEAR OPERATORS CLOSED OPERATORS RESOLVENT SET AND SPECTRA OF LINEAR OPERATOR COMPACT OPERATORS LINEAR OPERATORS ON HILBERT SPACE DUAL SPACES ADJOINT OPERATORS TOPOLOGIES OF CONVERGENCE OF BOUNDED LINEAR OPERATORS GRAPH AND CORE UNITARY EQUIVALENCE A DECOMPOSITION THEOREM RELATIVE BOUNDEDNESS HERMITIAN, SYMMETRIC AND SELF-ADJOINT OPERATORS TRACE CLASS OPERATOR ORTHOGONAL PROJECTIONS WEAK COMMUTATOR DIRECT SUM OPERATORS SELF-ADJOINT OPERATORS AND RELATED SUBJECTS CRITERIA ON (ESSENTIAL) SELF-ADJOINTNESS FOR SYMMETRIC OPERATORS THE KATO–RELLICH THEOREM MULTIPLICATION OPERATORS SPECTRAL MEASURES AND FUNCTIONAL CALCULUS SPECTRAL MEASURE INDUCED BY THE PAIR OF A SPECTRAL MEASURE AND A MEASURABLE MAPPING SPECTRAL THEOREM SPECTRAL MEASURE AND SPECTRA ESSENTIAL SPECTRUM FUNCTIONAL CALCULUS THE SPECTRAL MEASURE OF A MULTIPLICATION OPERATOR STRONGLY CONTINUOUS ONE-PARAMETER UNITARY GROUPS UNITARY COVARIANCE OF FUNCTIONAL CALCULUS SOME INEQUALITIES OPERATORS OF EXPONENTIAL TYPE ANALYTIC VECTORS ANALYSIS OF STRONGLY COMMUTING SELF-ADJOINT OPERATORS STRONG COMMUTATIVITY JOINT SPECTRAL MEASURE AND JOINT SPECTRUM AN EXAMPLE IN QUANTUM MECHANICS: A FREE RELATIVISTIC QUANTUM PARTICLE FUNCTIONAL CALCULUS OF STRONGLY COMMUTING SELF-ADJOINT OPERATORS STRONGLY CONTINUOUS MULTI-PARAMETER UNITARY GROUPS REDUCTION OF LINEAR OPERATORS CONVERGENCE OF SELF-ADJOINT OPERATORS PROBLEMS 2 TENSOR PRODUCT HILBERT SPACES FORMS BILINEAR FORMS SESQUILINEAR FORMS AND A REPRESENTATION THEOREM CONJUGATE BILINEAR FORMS TENSOR PRODUCT OF TWO HILBERT SPACES CONTRACTION OF TENSOR PRODUCTS TENSOR PRODUCT OF N HILBERT SPACES WITH N ?3 HILBERT SPACE ISOMORPHISMS TENSOR PRODUCT OF L2-SPACES TENSOR PRODUCT OF AN L2-SPACE AND A HILBERT SPACE CONSTANT FIBER DIRECT INTEGRALS PERMUTATION OPERATORS, SYMMETRIC TENSOR PRODUCTS AND ANTI-SYMMETRIC TENSOR PRODUCTS PERMUTATION OPERATORS BASIC ELEMENTS IN GROUP THEORY UNITARY REPRESENTATION OF SN SYMMETRIC AND ANTI-SYMMETRIC TENSOR PRODUCT HILBERT SPACES CONSS OF ?N S H CONSS OF ?N(H ) OCCUPATION NUMBER REPRESENTATIONS THE CASE OF BOSONIC SYSTEMS THE CASE OF FERMIONIC SYSTEMS SYMMETRIC AND ANTI-SYMMETRIC STATE FUNCTIONS IN QUANTUM MECHANICS THE HILBERT SPACE OF SYMMETRIC STATE FUNCTIONS THE HILBERT SPACE OF ANTI-SYMMETRIC STATE FUNCTIONS TENSOR PRODUCT OF DIRECT SUM HILBERT SPACES NOTES 3 TENSOR PRODUCT OF LINEAR OPERATORS ALGEBRAIC TENSOR PRODUCT OF LINEAR OPERATORS TENSOR PRODUCT OF LINEAR OPERATORS TENSOR PRODUCT OF BOUNDED LINEAR OPERATORS REDUCTION OF TENSOR PRODUCT OPERATORS BASIC PROPERTIES OF SIMPLE TENSOR PRODUCT OPERATORS SOME ESTIMATES SPECTRAL PROPERTIES OF A ?I EIGENVALUES OF TENSOR PRODUCT OPERATORS TENSOR PRODUCT OF N LINEAR OPERATORS WITH N ?3 TENSOR PRODUCT OF SELF-ADJOINT OPERATORS POINT SPECTRA OF TENSOR PRODUCT OF SELF-ADJOINT OPERATORS EXPONENTIAL TYPE OPERATORS FORMED OUT OF TENSOR PRODUCTS OF SELF-ADJOINT OPERATORS TENSOR PRODUCT OPERATORS AND DIRECT INTEGRAL OPERATORS 4 FULL FOCK SPACES AND SECOND QUANTIZATION OPERATORS IN?NITE DIRECT SUM HILBERT SPACES IN?NITE DIRECT SUM OPERATORS SUM AND PRODUCT OF TWO IN?NITE DIRECT SUM OPERATORS DIAGONAL AND NON-DIAGONAL OPERATORS ON H SHIFT-TYPE OPERATORS A GENERAL CLASS OF NON-DIAGONAL OPERATORS IN?NITE DIRECT SUM OPERATOR OF SELF-ADJOINT OPERATORS FULL FOCK SPACES SECOND QUANTIZATION OPERATORS GROUND STATE ?-OPERATORS ANTI-UNITARY ?-OPERATORS REPRESENTATIONS OF UNITARY GROUPS AND SEMI-GROUPS RELATIONS BETWEEN ?( ) AND D?( ) COMMUTATION PROPERTIES OF SECOND QUANTIZATION OPERATORS CREATION AND ANNIHILATION OPERATORS 5 BOSON FOCK SPACES INTRODUCTION: SOME PHYSICAL BACKGROUNDS THE BOSON FOCK SPACE OVER A HILBERT SPACE BOSON SECOND QUANTIZATION OPERATORS REDUCTION OF A CLASS OF TENSOR PRODUCT OPERATORS BOSON SECOND QUANTIZATION OPERATORS AND THEIR BASIC PROPERTIES SPECTRAL PROPERTIES OF BOSON SECOND QUANTIZATION OPERATORS EITD?B(T )-INVARIANT VECTORS BOSON NUMBER OPERATOR COMMUTATION PROPERTIES BOSON ?-OPERATORS SPECTRAL PROPERTIES OF ?B(T ) TRACE OF ?B-OPERATOR BOSON ANNIHILATION AND CREATION OPERATORS DE?NITIONS AND BASIC PROPERTIES COMMUTATION RELATIONS IRREDUCIBILITY REDUCTION OF CREATION AND ANNIHILATION OPERATORS OCCUPATION NUMBER REPRESENTATIONS FOR A SYSTEM OF FINITELY MANY BOSONS RELATIONS AMONG THE CREATION AND ANNIHILATION OPERATORS AND THE SECOND QUANTIZATION OPERATORS RELATIVE BOUNDEDNESS OF THE CREATION AND ANNIHILATION OPERATORS WITH RESPECT TO SECOND QUANTIZATION OPERATORS COMMUTATION RELATIONS AMONG D?B(T ) AND A(F)# RELATIVE BOUNDEDNESS OF A(F)#A(G)# WITH RESPECT TO D?B(T ) CONTINUITY OF BOSON SECOND QUANTIZATION OPERATORS IN ONE-PARTICLE OPERATORS REPRESENTATION OF BOSON SECOND QUANTIZATIONS IN TERMS OF CREATION AND ANNIHILATION OPERATORS IN?NITE SERIES REPRESENTATION FOR THE SQUARE ROOT OF A SECOND QUANTIZATION OPERATOR THE SEGAL FIELD OPERATOR BASIC PROPERTIES SELF-ADJOINTNESS OF THE SEGAL ?ELD OPERATOR IRREDUCIBILITY OF SEGAL ?ELD OPERATORS EXPONENTIAL OPERATOR FOR A GENERAL LINEAR OPERATOR BAKER–CAMPBELL–HAUSDOR?FORMULA EXPONENTIAL OPERATORS FOR SEGAL FIELD OPERATORS DECOMPOSITION OF THE SEGAL FIELD OPERATOR TRANSFORMATIONS OF ANNIHILATION AND CREATION OPERATORS BY EXPONENTIAL OPERATORS SPECTRUM OF SEGAL FIELD OPERATORS TRANSFORMATION LAWS OF CREATION AND ANNIHILATION OPERATORS WITH RESPECT TO ?-OPERATORS TIME DEVELOPMENT OF SEGAL FIELD OPERATORS AND FIELD EQUATIONS ABSTRACT FREE BOSE FIELDS VACUUM EXPECTATION VALUES COHERENT VECTORS DE?NITION AND BASIC PROPERTIES RELATION WITH THE SEGAL ?ELD OPERATORS PHYSICAL MEANING OF COHERENT STATES SPECTRA OF A(F) AND A(F)? THE BOSON FOCK SPACE OVER A DIRECT SUM HILBERT SPACE REPRESENTATIONS OF CCR REPRESENTATION OF CCR OVER A VECTOR SPACE SECOND QUANTIZATION ASSOCIATED WITH A REPRESENTATION OF CCR BOSONIC DIAGONALIZATION REPRESENTATION OF HEISENBERG CCR WEYL REPRESENTATIONS OF CCR DE?NITION AND REMARKS A CLASS OF IRREDUCIBLE WEYL REPRESENTATIONS ON THE ABSTRACT BOSON FOCK SPACE GENERAL PROPERTIES A CLASS OF REPRESENTATIONS OF CCR SYMPLECTIC GROUP ON A HILBERT SPACE BOGOLIUBOV TRANSFORMATIONS ANALYSIS OF BOGOLIUBOV TRANSLATIONS A GENERAL CASE OF BOGOLIUBOV TRANSFORMATIONS FUNCTIONAL SCHR¨ODINGER REPRESENTATION OF HEISENBERG CCR WITH IN?NITE DEGREES OF FREEDOM HEURISTIC ARGUMENTS EXISTENCE OF A PROBABILITY SPACE WITH AN IN?NITE-DIMENSIONAL MEASURE REPRESENTATION OF THE HEISENBERG CCR WITH COUNTABLE IN?NITE DEGREES OF FREEDOM FUNCTIONALIZATIONS LP-INTEGRABILITY OF ?(F) WICK PRODUCTS AN ORTHOGONAL DECOMPOSITION A NATURAL ISOMORPHISM WITH A BOSON FOCK SPACE CONCLUDING REMARKS 6 FERMION FOCK SPACES DE?NITIONS AND BASIC PROPERTIES FERMION SECOND QUANTIZATION OPERATORS FERMION ?-OPERATORS COMMUTATION PROPERTIES OF FERMION SECOND QUANTIZATION OPERATORS SPECTRAL PROPERTIES OF FERMION SECOND QUANTIZATION OPERATORS POINT SPECTRUM OF T (Q) F POINT SPECTRUM OF ?QT SPECTRUM OF T (P) F AND ?PT SPECTRA OF D?F(T ) AND ?F(T ) FERMION ANNIHILATION AND CREATION OPERATORS CANONICAL ANTI-COMMUTATION RELATIONS SPECTRA OF B(U)# CHARACTERIZATION OF THE FERMION FOCK VACUUM OCCUPATION NUMBER REPRESENTATION FOR A SYSTEM OF FINITELY MANY FERMIONS RELATIONS BETWEEN THE FERMION SECOND QUANTIZATION OPERATORS AND THE FERMION ANNIHILATION–CREATION OPERATORS UNIFORM DI?ERENTIABILITY OF OPERATOR-VALUED FUNCTIONS THE FERMION FOCK SPACE OVER A DIRECT SUM HILBERT SPACE REPRESENTATIONS OF CAR SECOND QUANTIZATION ASSOCIATED WITH A REPRESENTATION OF CAR FERMIONIC BOGOLIUBOV TRANSFORMATIONS 7 BOSON–FERMION FOCK SPACES AND IN?NITE-DIMENSIONAL DIRAC-TYPE OPERATORS FUNDAMENTAL STRUCTURES OF A BOSON–FERMION FOCK SPACE SECOND QUANTIZATION OPERATORS ON THE BOSON–FERMION FOCK SPACE F(H , K ) OPERATORS OF CO-BOUNDARY TYPE LAPLACE–BELTRAMI OPERATORS QUOTIENT HILBERT SPACE COHOMOLOGY GROUPS AND DE RHAM–HODGE–KODAIRA DECOMPOSITION IDENTI?CATION OF THE COHOMOLOGY GROUPS PROPERTIES OF KERNEL OF SOME OPERATORS KERNEL OF SECOND QUANTIZATION OPERATORS THE KERNEL OF ?S,P AND THE DIMENSION OF HP S IN?NITE-DIMENSIONAL DIRAC-TYPE OPERATORS ON F(H , K ) ANTI-COMMUTATION RELATIONS OF QS AND QT ABSTRACT DIRAC OPERATOR ABSTRACT SUPERSYMMETRIC QUANTUM MECHANICS FREDHOLM OPERATORS OPERATOR MATRIX REPRESENTATION OF A ?-DIRAC OPERATOR SELF-ADJOINT ?-DIRAC OPERATORS FREDHOLMNESS AND THE ?-INDEX OF THE DIRAC OPERATOR QS PART 2: MATHEMATICAL THEORY OF QUANTUM FIELDS 8 GENERAL THEORY OF QUANTUM FIELDS INTRODUCTION OPERATOR-VALUED DISTRIBUTIONS OPERATOR-VALUED FUNCTIONS OPERATOR-VALUED DISTRIBUTIONS TRANSFORMATIONS OF OPERATOR-VALUED DISTRIBUTIONS PARTIAL DERIVATIVES OF OPERATOR-VALUED DISTRIBUTIONS GENERAL CONCEPT OF QUANTUM FIELD TIME-TRANSLATION COVARIANT QUANTUM ?ELD THEORY UNIQUENESS OF HAMILTONIANS SHARP-TIME QUANTUM ?ELDS EQUATIONS FOR QUANTUM FIELDS VACUUM EXPECTATION VALUES AND WIGHTMAN DISTRIBUTIONS A GENERAL FORM OF CANONICAL FORMALISM IN QFT EXAMPLE: A FREE NEUTRAL SCALAR QUANTUM FIELD UNITARY REPRESENTATIONS OF TOPOLOGICAL GROUPS TRANSLATION COVARIANT QFT REVIEW OF SOME ASPECTS RELATED TO THE THEORY OF SPECIAL RELATIVITY MINKOWSKI SPACE-TIME LORENTZ GROUP DUAL OPERATORS ON (M1+D)? THE LIE ALGEBRA OF L ? + POINCAR´E GROUP AXIOMS FOR RELATIVISTIC QFT THE G?ARDING–WIGHTMAN AXIOMS ANGULAR MOMENTUM THE WIGHTMAN AXIOMS THE OSTERWALDER–SCHRADER AXIOMS EUCLIDEAN QUANTUM FIELDS PCT THEOREM HAAG’S THEOREM SCATTERING THEORY AND SPECTRAL ANALYSIS ASYMPTOTIC ANNIHILATION AND CREATION OPERATORS A VANISHING THEOREM COMMUTATION RELATIONS AND REPRESENTATIONS OF CCR SCATTERING OPERATOR SPECTRUM OF H EXISTENCE OF ASYMPTOTIC CREATION AND ANNIHILATION OPERATORS 9 NON-RELATIVISTIC QFT CLASSICAL FIELD THEORY HEURISTIC ARGUMENTS TOWARD CONSTRUCTION OF QFT A BOSONIC QUANTUM DE BROGLIE FIELD A FREE BOSONIC QUANTUM DE BROGLIE ?ELD ENERGY-MOMENTUM OPERATOR CT SYMMETRY AND PARITY SYMMETRY ENERGY-MOMENTUM SPECTRUM GALILEAN SYMMETRY INTERACTION WITH AN EXTERNAL ?ELD THE OPERATOR-VALUED DISTRIBUTION KERNEL OF THE BOSON ANNIHILATION OPERATOR SESQUILINEAR FORMS DE?NED BY ?(X) AND ?(X)? HAMILTONIANS AS SESQUILINEAR FORMS A HAMILTONIAN WITH AN EXTERNAL ?ELD A HAMILTONIAN OF A SELF-INTERACTING QUANTUM DE BROGLIE ?ELD A FERMIONIC QUANTUM DE BROGLIE FIELD HILBERT SPACE FOR STATE VECTORS ANNIHILATION AND CREATION OPERATORS A FERMIONIC QUANTUM DE BROGLIE ?ELD THE OPERATOR-VALUED DISTRIBUTION KERNEL OF THE FERMION ANNIHILATION OPERATOR QFT ON A BOUNDED SPACE REGION QFT ON A CUBIC BOX A NATURAL EMBEDDING OF THE ?NITE VOLUME THEORY INTO AN IN?NITE ONE IN?NITE VOLUME LIMIT TENSOR PRODUCT REPRESENTATION 10 RELATIVISTIC FREE QUANTUM SCALAR FIELDS FREE CLASSICAL KLEIN–GORDON FIELD A SHARP-TIME FREE NEUTRAL QUANTUM KG FIELD REPRESENTATION THEORETIC ASPECTS ENERGY–MOMENTUM SPECTRUM UNITARY REPRESENTATION OF THE (1 + D)-DIMENSIONAL POINCAR´E GROUP ON THE BOSON FOCK SPACE FB(L2(RD K)) A FREE NEUTRAL QUANTUM KG FIELD SMEARED OVER THE SPACE-TIME CYCLICITY OF THE FOCK VACUUM 10 10 COMMUTATION RELATIONS AND MICROSCOPIC CAUSALITY 10 10 1 COMMUTATION RELATIONS 10 10 2 THE PAULI–JORDAN DISTRIBUTION IN THE FOUR-DIMENSIONAL SPACE-TIME 10 10 3 MICROSCOPIC CAUSALITY 10 11 SHARP-TIME DISTRIBUTIONS AND FEYNMAN PROPAGATOR 10 12 MICROSCOPIC CAUSALITY IN THE (1 + D)-DIMENSIONAL SPACE-TIME AND A SUMMARY 10 13 VACUUM EXPECTATION VALUES 10 14 QUANTUM FIELD WITH MOMENTUM CUTO? 10 15 SCHWINGER FUNCTIONS 10 16 CHARGED QUANTUM SCALAR FIELD 10 16 1 HEURISTIC ARGUMENTS 10 16 2 OUTLINE OF A MATHEMATICALLY RIGOROUS CONSTRUCTION 10 16 3 UNITARY EQUIVALENCE TO THE SYSTEM OF TWO NEUTRAL QUANTUM SCALAR ?ELDS 10 16 4 PCT THEOREM 10 17 FINITE VOLUME APPROXIMATIONS AND IN?NITE VOLUME LIMIT 10 18 PROBLEMS 11 QUANTUM THEORY OF ELECTROMAGNETIC FIELDS CLASSICAL THEORY OF ELECTROMAGNETIC FIELDS THE MAXWELL EQUATIONS AND GAUGE INVARIANCE A DI?CULTY IN CONSTRUCTING A QUANTUM ELECTROMAGNETIC ?ELD FREE ELECTROMAGNETIC POTENTIALS IN THE COULOMB GAUGE QUANTUM RADIATION FIELD COMMUTATION RELATIONS QUANTUM ELECTROMAGNETIC FIELDS SHARP-TIME FIELDS CANONICAL CONJUGATE FIELD HAMILTONIAN AND MOMENTUM OPERATOR 11 10 QUANTUM RADIATION FIELD WITH MOMENTUM CUTO? 11 11 EQUIVALENT REPRESENTATION: A NATURAL ISOMORPHISM 11 12 PROBLEMS