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1.

図書

図書
M. Göckeler, T. Schücker
出版情報: Cambridge ; New York : Cambridge University Press, 1989  xii, 230 p. ; 23 cm
シリーズ名: Cambridge monographs on mathematical physics
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Preface
Exterior algebra / 1:
Differential forms on open subsets of Rn / 2:
Metric structures / 3:
Gauge theories / 4:
Einstein-Cartan theory / 5:
The Lie derivative / 6:
Manifolds / 7:
Lie groups / 8:
Fibre bundles / 9:
Monopoles, instantons, and related fibre bundles / 10:
Spin / 11:
An algebraic approach to anomalies / 12:
Anomalies from graphs / 13:
References
Bibliography
Notation
Index
Preface
Exterior algebra / 1:
Differential forms on open subsets of Rn / 2:
2.

図書

図書
Ronald B. Guenther and John W. Lee
出版情報: Englewood Cliffs, N.J. : Prentice-Hall, c1988  xii, 544 p. ; 25 cm
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3.

図書

図書
Siegfried Flügge
出版情報: Berlin ; New York : Springer-Verlag, 1971  2 v. ; 24 cm
シリーズ名: Die Grundlehren der mathematischen Wissenschaften ; Bd. 177-178
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4.

図書

図書
Yvonne Choquet-Bruhat, Cécile DeWitt-Morette, and Margaret Dillard-Bleick
出版情報: Amsterdam ; New York : North Holland Pub., 1977  xvii, 544 p. ; 25 cm
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Preface
Preface to the second edition
Contents
Conventions
Review of fundamental notions of analysis / I:
Differential calculus on banach spaces / II:
Differentiable manifolds / III:
Integration on manifolds / IV:
Riemannian manifolds, Kahlerian manifolds / V:
bis. Connections on a principle fibre bundle
Distributions / VI:
Supplements and additional problems
Subject Index
Errata to Part I
Preface
Preface to the second edition
Contents
5.

図書

図書
Robert D. Richtmyer
出版情報: New York ; Tokyo : Springer Verlag, c1978-1981  2 v. ; 25 cm
シリーズ名: Texts and monographs in physics
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6.

図書

図書
Victor Guillemin, Shlomo Sternberg
出版情報: Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1984  xi, 468 p. ; 24 cm
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Preface
Introduction / I:
Gaussian optics / 1:
Hamilton's method in Gaussian optics / 2:
Fermat's principle / 3:
From Gaussian optics to linear optics / 4:
Geometrical optics, Hamilton's method, and the theory of geometrical aberrations / 5:
Fermat's principle and Hamilton's principle / 6:
Interference and diffraction / 7:
Gaussian integrals / 8:
Examples in Fresnel optics / 9:
The phase factor / 10:
Fresnel's formula / 11:
Fresnel optics and quantum mechanics / 12:
Holography / 13:
Poisson brackets / 14:
The Heisenberg group and representation / 15:
The Groenwald-van Hove theorem / 16:
Other quantizations / 17:
Polarization of light / 18:
The coadjoint orbit of a semidirect product / 19:
Electromagnetism and the determination of symplectic structures / 20:
Epilogue: Why symplectic geometry?
The geometry of the moment map / II:
Normal forms / 21:
The Darboux-Weinstein theorem / 22:
Kaehler manifolds / 23:
Left-invariant forms and Lie algebra cohomology / 24:
Symplectic group actions / 25:
The moment map and some of its properties / 26:
Group actions and foliations / 27:
Collective motion / 28:
Cotangent bundles and the moment map for semidirect products / 29:
More Euler-Poisson equations / 30:
The choice of a collective Hamiltonian / 31:
Convexity properties of toral group actions / 32:
The lemma of stationary phase / 33:
Geometric quantization / 34:
Motion in a Yang-Mills field and the principle of general covariance / III:
The equations of motion of a classical particle in a Yang-Mills field / 35:
Curvature / 36:
The energy-momentum tensor and the current / 37:
The principle of general covariance / 38:
Isotropic and coisotropic embeddings / 39:
Symplectic induction / 40:
Symplectic slices and moment reconstruction / 41:
An alternative approach to the equations of motion / 42:
The moment map and kinetic theory / 43:
Complete integrability / IV:
Fibrations by tori / 44:
Collective complete integrability / 45:
Collective action variables / 46:
The Kostant-Symes lemma and some of its variants / 47:
Systems of Calogero type / 48:
Solitons and coadjoint structures / 49:
The algebra of formal pseudodifferential operators / 50:
The higher-order calculus of variations in one variable / 51:
Contractions of symplectic homogeneous spaces / V:
The Whitehead lemmas / 52:
The Hochschild-Serre spectral sequence / 53:
Galilean and Poincare elementary particles / 54:
Coppersmith's theory / 55:
References
Index
Preface
Introduction / I:
Gaussian optics / 1:
7.

図書

図書
by Eugene R. Speer
出版情報: Princeton, N.J. : Princeton University Press , [Tokyo] : University of Tokyo Press, 1969  120 p. ; 24 cm
シリーズ名: Annals of mathematics studies ; no. 62
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8.

図書

図書
Jamal Nazrul Islam
出版情報: Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1985  vi, 122 p. ; 24 cm
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Preface
Introduction / 1:
The Einstein equations for a rotating metric and some classes of solutions / 2:
The Kerr and Timimatsu-Sato solutions / 3:
Rotating neutral dust / 4:
rotating Einstein-Maxwell fields / 5:
Rotating charged dust / 6:
Appendix
References
Index
Preface
Introduction / 1:
The Einstein equations for a rotating metric and some classes of solutions / 2:
9.

図書

図書
Bryce DeWitt
出版情報: Cambridge [Cambridgeshire] ; New York : Cambridge University Press, 1984  xiv, 316 p. ; 24 cm
シリーズ名: Cambridge monographs on mathematical physics
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Preface to the first edition
Preface to the second edition
Analysis over supernumbers / 1:
Supernumbers and superanalytic functions / 1.1:
Grassmann algebras
Supernumbers
c-numbers and a-numbers
Superanalytic functions of supernumbers
Integration of superanalytic functions of supernumbers
Real supernumbers. Differentiable functions of real c-numbers and their integrals / 1.2:
Complex conjugation
Functions, distributions and integrals over R[subscript c]
Fourier transforms over R[subscript c]
Functions and integrals over R[subscript a] / 1.3:
Basic definitions
Fourier transforms over R[subscript a]
Integrals over R[superscript n][subscript a]
Supervector spaces / 1.4:
Definition
Bases
Pure bases
Pure real bases
Standard bases
Linear transformations, supertranspositions and dual supervector spaces / 1.5:
Change of basis
Shifting indices. The supertranspose
Extensions of the supertransposition rules
Dual supervector spaces
Dual bases
Further index-shifting conventions
The supertrace and the superdeterminant / 1.6:
The supertrace
The superdeterminant
The superdeterminant in special cases
The superdeterminant in the general case
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a] / 1.7:
Notation
Integration
Homogeneous linear transformations of the a-number coordinates
Homogeneous linear transformations of all the coordinates
Nonlinear transformations
Gaussian integrals over R[superscript m][subscript c] [times] R[superscript n][subscript a]
Exercises
Comments on chapter 1
Supermanifolds / 2:
Definition and structure of supermanifolds / 2.1:
Topology of R[superscript m][subscript c] [times] R[superscript n][subscript a]. Differentiable mappings
Supermanifolds, charts and atlases
Scalar fields and supercurves
Diffeomorphisms and embeddings
Ordinary manifolds. Skeleton and body of a supermanifold
Projectively Hausdorff, compact, paracompact and orientable supermanifolds. Realizations of the body
Supervector structures on supermanifolds / 2.2:
Scalar fields as supervectors
Contravariant vector fields
Alternative presentation of contravariant vector fields
Components
Tangent spaces
Tangents to supercurves
Super Lie brackets, local frames and covariant vector fields / 2.3:
Supercommutators and antisupercommutators
A matter of notation
The super Lie bracket
Local frames
Super Lie brackets of local frame fields
Covariant vector fields
Differentials
Tensor fields / 2.4:
Tensors at a point
The supervector space T[superscript r] [subscript s](p)
Tensor products
Tensor and multitensor fields
Index-shifting conventions. Contractions
The unit tensor field
The Lie derivative / 2.5:
Explicit forms
Lie derivations as supervectors
The derivative mapping
Integral supercurves. Congruences
Dragging of tensor fields
Forms / 2.6:
The exterior product
Bases for forms
Derivations of forms
The exterior derivative
The inner product
Connections / 2.7:
The connection components
Multiple covariant derivatives. The torsion
The Riemann tensor field
The super Bianchi identity
Parallel transport. Supergeodesics
Distant parallelism
Riemannian supermanifolds / 2.8:
The metric tensor field
Canonical form of the metric tensor at a point
Canonical or orthosymplectic bases
Riemannian connections
The curvature tensor field
The Ricci tensor field
Flat Riemannian supermanifolds
Conformally related Riemannian supermanifolds. The Weyl tensor field
Conformally flat Riemannian supermanifolds
Killing vector fields
Conformal Killing vector fields
The global conformal group
Integration over supermanifolds / 2.9:
Integration over R[superscript m][subscript c] [times] R[superscript n][subscript a]. Measure functions
Locally finite atlases and partitions of unity
Integration over paracompact orientable supermanifolds
Integration over Riemannian supermanifolds
Integrals of total divergences
The compact case
An example
Comments on chapter 2
Super Lie groups. General theory / 3:
Definition and structure of super Lie groups / 3.1:
Canonical diffeomorphisms
Left- and right-invariant vector fields
Left- and right-invariant local frame fields
Left- and right-invariant congruences
One-parameter Abelian subgroups
The exponential mapping. Canonical coordinates
The super Lie algebra
The structure constants
The right and left auxiliary functions
Identities satisfied by the auxiliary functions
Construction of a super Lie group from its super Lie algebra
Realizations of super Lie groups / 3.2:
Orbits
Transitive realizations
Isotropy subgroups
Coset spaces
Killing flows
Properties of the coordinate components of the Q[subscript a]
A special canonical coordinate system
Coordinates for the coset spaces
Classification of transitive realizations
Matrix representations of super Lie groups
Contragredient representations
Inner automorphisms. The adjoint representation
Matrix representations of the super Lie algebra
Geometry of coset spaces / 3.3:
Invariant tensor fields
Differential equations for geometrical structures
Integrability of the differential equations
A special coordinate system
Condition for the existence of a group-invariant measure function
Condition for the existence of a group-invariant metric tensor field
Condition for the existence of a group-invariant connection
Solutions of the differential equations
Geometry of the group supermanifold
Identity of the left- and right-invariant connections
Parallelism at a distance in the group supermanifold
Integration over the group
A special class of super Lie groups
Comments on chapter 3
Super Lie groups. Examples / 4:
Construction of super Lie algebras and super Lie groups / 4.1:
Properties of the structure constants
Conventional super Lie groups, Z[subscript 2]-graded algebras
Unconventional super Lie groups
Structure of conventional super Lie Groups. The extending representation
Construction of a class of super Lie algebras
The classical super Lie groups / 4.2:
The group GL (m, n)
The group SL (m, n)
The group SL (m, m)/GL (1, 0)
The orthosymplectic group OSp (m, n)
The Kac notation
The group P(m)
The group Q(m)
The exceptional simple super Lie groups / 4.3:
The groups D(2, 1, [alpha])
The group F(4)
The structure of F(4)
Pseudorepresentation of F(4)
The group G(3)
The structure of G[subscript 2]
The structure of G(3)
Pseudorepresentation of G(3)
Super Lie groups of basic importance in physics / 4.4:
The super de Sitter group
The super Poincare group
The coset space: super Poincare group/SO(1, 3)
Killing flows and invariant connections
Riemannian geometry of the coset space
The super Lorentz group
The Cartan super Lie groups / 4.5:
The diffeomorphism group Diff(M)
The group SDiff(M, [mu])
The canonical transformation group Can(M, [omega])
The group of contact transformations
The case m = 0
The group W(n)
The groups S(n) and S(n)
The groups H(n) and H(n)
Comments on chapter 4
Selected applications of supermanifold theory / 5:
Superclassical dynamical systems / 5.1:
Configuration spaces
Supermanifolds as configuration spaces
Space of histories
The action functional and the dynamical equations
Infinitesimal disturbances and Green's functions
Reciprocity relations
The Peierls bracket
Peierls bracket identities
Super Hilbert spaces / 5.2:
Linear operators
Physical observables
Quantum systems / 5.3:
Transition to the quantum theory
The Schwinger variational principle
External sources
Chronologically ordered form of the operator dynamical equations
The Feynman functional integral
A simple Fermi system / 5.4:
Action functional and Green's functions
Eigenvectors of x
The energy
A pure basis
An alternative representation
The functional integral representation of [x", t"|x', t']
Evaluation of the functional integral
The average superclassical trajectory
Propagator for x[subscript av](t)
The Fermi oscillator / 5.5:
Mode functions and Hamiltonian
Basic supervectors
Eigenvectors of x[subscript 1] and x[subscript 2]. Choice of pure basis
Coherent states
The functional integral representation of [a"*, t"
Direct evaluation of the functional integral
The importance of endpoint contributions
The stationary trajectory as a matrix element
The Feynman propagator
The Bose oscillator / 5.6:
Energy eigenvectors
Hamilton-Jacobi theory
The amplitude [x', t'|x', t'] and its functional integral representation
The functional-integral representation of [a"*, t"
The stationary path between coherent states
Energy eigenfunctions
Bose-Fermi supersymmetry / 5.7:
The simplest model
New conserved quantities
The Bose-Fermi supersymmetry group
Eigenvectors of Q[subscript 1] and Q[subscript 2]
The supersymmetry group as a transformation group
Auxiliary variable
Nonlinear Bose-Fermi supersymmetry
The supersymmetry group
The energy spectrum
Spontaneously broken supersymmetry
Comments on chapter 5
Applications involving topology / 6:
Nontrivial configuration spaces / 6.1:
Standard canonical systems
Green's functions
Equivalence of Peierls and Poisson brackets
Quantization / 6.2:
Problems with the naive quantization rule
Operator-valued forms. The projection m-form
The position operator
Vector operators
The momentum operator
Restriction to a local chart
Lack of uniqueness of the momentum operator
Overlapping charts. Transformation of coordinates
The position representation
The momentum operator in the position representation
The Schrodinger equation
The position representation of the projection m-form
Curved configuration spaces / 6.3:
A special class of systems
Covariant variation
Covariant differentiation with respect to t
The dynamical equations
Covariant functional differentiation
The Feynman functional integral and its meaning / 6.4:
Formal computation of det G[superscript +][x]
The functional integral
Normalization
Ambiguity in the functional integral
Homotopy
Homotopy mesh
The total amplitude
Change of homotopy mesh
The role of homology
The universal covering space
The total amplitude revisited
The Hamiltonian operator: a nonlattice derivation / 6.5:
Integration over phase space
Evaluation of the chronologically ordered Hamiltonian
Approximate evaluation of the path integral / 6.6:
Brief review of Hamilton-Jacobi theory
The Van Vleck-Morette determinant
Jacobi fields and the Green's function for the trajectory x[subscript c]
Determinantal relations
The loop expansion
The WKB approximation
The heat kernel expansion
Role of the two-loop term in the independent verification of (6.5.25)
New variables
Computation of the two-loop term
Supersymmetry and the Euler-Poincare characteristic / 6.7:
Inclusion of a-type dynamical variables
Green's functions and Peierls brackets
Energy and supersymmetry group
Basis supervectors
Differential representation of operators
The Euler-Poincare characteristic
Functional integral for the coherent-state transition amplitude
The Chern-Gauss-Bonnet formula
Comments on chapter 6
References
Index
Preface to the first edition
Preface to the second edition
Analysis over supernumbers / 1:
10.

図書

図書
George Papanicolaou, editor
出版情報: New York ; Tokyo : Springer-Verlag, c1987  xi, 321 p. ; 25 cm
シリーズ名: The IMA volumes in mathematics and its applications ; v. 7
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