Frank Arntzenius
- Published in print:
- 2012
- Published Online:
- May 2012
- ISBN:
- 9780199696604
- eISBN:
- 9780191738333
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199696604.003.0008
- Subject:
- Philosophy, Metaphysics/Epistemology, Philosophy of Science
This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of ...
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This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of mathematical objects such as numbers, sets and functions. In particular it is argued that modern differential geometry can be ‘nominalised’, i.e. be formulated in such a way that it does not entail the existence of mathematical entities, by making judicious use of the geometric structure of the fibre bundles spaces which were encountered in chapter 6.Less
This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of mathematical objects such as numbers, sets and functions. In particular it is argued that modern differential geometry can be ‘nominalised’, i.e. be formulated in such a way that it does not entail the existence of mathematical entities, by making judicious use of the geometric structure of the fibre bundles spaces which were encountered in chapter 6.
Thomas Ryckman
- Published in print:
- 2005
- Published Online:
- April 2005
- ISBN:
- 9780195177176
- eISBN:
- 9780199835324
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195177177.003.0006
- Subject:
- Philosophy, Philosophy of Science
It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. ...
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It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. In response to the Einstein-Pauli objection, Weyl put forth a second version of his theory, arguing on various non-empirical grounds for the conceptual superiority of his gauge-theoretic approach to general relativity. Speculation is made regarding Weyl’s philosophical reasons for opposing Élie Cartan’s “moving frame” generalization of Weyl’s notion of an affine connection, and it is suggested that the basic approach of Weyl’s transcendental phenomenological foundation of differential geometry can be accommodated within the modern fiber bundle formulation of gauge field theories.Less
It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. In response to the Einstein-Pauli objection, Weyl put forth a second version of his theory, arguing on various non-empirical grounds for the conceptual superiority of his gauge-theoretic approach to general relativity. Speculation is made regarding Weyl’s philosophical reasons for opposing Élie Cartan’s “moving frame” generalization of Weyl’s notion of an affine connection, and it is suggested that the basic approach of Weyl’s transcendental phenomenological foundation of differential geometry can be accommodated within the modern fiber bundle formulation of gauge field theories.
Reinhold A. Bertlmann
- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507628
- eISBN:
- 9780191706400
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507628.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
The anomaly, which forms the central part of this book, is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper ...
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The anomaly, which forms the central part of this book, is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper understanding of quantum field theory and has played an increasingly important role in the theory over the past twenty years. This book presents all the different aspects of the study of anomalies in an accessible and self-contained way. Much emphasis is now being placed on the formulation of the theory using the mathematical ideas of differential geometry and topology. This approach is followed here, and the derivations and calculations are given explicitly. Topics discussed include the relevant ideas from differential geometry and topology and the application of these paths (path integrals, differential forms, homotopy operators, etc.) to the study of anomalies. Chapters are devoted to abelian and nonabelian anomalies, consistent and covariant anomalies, and gravitational anomalies.Less
The anomaly, which forms the central part of this book, is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper understanding of quantum field theory and has played an increasingly important role in the theory over the past twenty years. This book presents all the different aspects of the study of anomalies in an accessible and self-contained way. Much emphasis is now being placed on the formulation of the theory using the mathematical ideas of differential geometry and topology. This approach is followed here, and the derivations and calculations are given explicitly. Topics discussed include the relevant ideas from differential geometry and topology and the application of these paths (path integrals, differential forms, homotopy operators, etc.) to the study of anomalies. Chapters are devoted to abelian and nonabelian anomalies, consistent and covariant anomalies, and gravitational anomalies.
Paula Tretkoff
- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ...
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This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.Less
This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number e(X). The Euler number, also known as the Euler-Poincaré characteristic, is an important invariant of a topological space X. It generalizes the notion of the cardinality of a finite set. The chapter presents the simple formulas for computing the Euler-Poincaré characteristic (Euler number) of many of the spaces to be encountered throughout the book. It also discusses fundamental groups and covering spaces and some basics of the theory of complex manifolds and Hermitian metrics, including the concept of real manifold. Finally, it provides some general facts about divisors, line bundles, and the first Chern class on a complex manifold X.
Yvonne Choquet-Bruhat
- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0001
- Subject:
- Mathematics, Applied Mathematics
This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ...
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This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. Topics covered include manifolds, differential mappings, vectors and tensors, pseudo-Riemannian metrics, Riemannian connection, geodesics, curvature, geodesic deviation, maximum length and conjugate points, linearized Ricci and Einstein tensors, and second derivative of the Ricci tensor.Less
This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. Topics covered include manifolds, differential mappings, vectors and tensors, pseudo-Riemannian metrics, Riemannian connection, geodesics, curvature, geodesic deviation, maximum length and conjugate points, linearized Ricci and Einstein tensors, and second derivative of the Ricci tensor.
Clifford Henry Taubes
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.001.0001
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics
Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the ...
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Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. The book uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.Less
Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. The book uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.
Paula Tretkoff
- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0005
- Subject:
- Mathematics, Geometry / Topology
This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally ...
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This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.Less
This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally equivalent to a smooth surface containing no exceptional curves. The latter is known as a minimal surface. Two related birational invariants, the plurigenus and the Kodaira dimension, play an important role in distinguishing between complex surfaces. The chapter first provides an overview of the rough classification of (smooth complex connected compact algebraic) surfaces before presenting two approaches that, in dimension 2, give the Miyaoka-Yau inequality. The first, due to Miyaoka, uses algebraic geometry, whereas the second, due to Aubin and Yau, uses analysis and differential geometry. The chapter also explains why equality in the Miyaoka-Yau inequality characterizes surfaces of general type that are free quotients of the complex 2-ball.
Xin Wei Sha
- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780262019514
- eISBN:
- 9780262318914
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262019514.003.0006
- Subject:
- Philosophy, Aesthetics
This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological ...
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This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological and differentiable dynamical systems (qualitative, topological, and geometrical approaches to systems of ordinary differential equations), and the much more sophisticated perspectives of differential geometry and fiber bundles. Basic poetic concepts introduced include: open (closed) set, neighborhood, map, space, continuity, connectedness, limit, convergence, compactness, and so forth. Introducing the work of Brouwer, Thom, and Petitot prepares the reader for a critical encounter with Petitot's program on ontogenesis. Articulating matter with such anexact concepts seeds the ground for an alternative, nonreductionist approach to ontogenesis. Certain terms used in earlier chapters for their intuitive senses, such as continuous, limit, dense, etc., will now be presented more rigorously, so that they can be used with more precise connotations and conceptual purchase after this chapter.Less
This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological and differentiable dynamical systems (qualitative, topological, and geometrical approaches to systems of ordinary differential equations), and the much more sophisticated perspectives of differential geometry and fiber bundles. Basic poetic concepts introduced include: open (closed) set, neighborhood, map, space, continuity, connectedness, limit, convergence, compactness, and so forth. Introducing the work of Brouwer, Thom, and Petitot prepares the reader for a critical encounter with Petitot's program on ontogenesis. Articulating matter with such anexact concepts seeds the ground for an alternative, nonreductionist approach to ontogenesis. Certain terms used in earlier chapters for their intuitive senses, such as continuous, limit, dense, etc., will now be presented more rigorously, so that they can be used with more precise connotations and conceptual purchase after this chapter.
Araceli Bonifant, Misha Lyubich, and Scott Sutherland
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.001.0001
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, ...
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John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.Less
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put the papers in mathematical and historical perspective, color figures are included, and an index facilitates browsing.
Nathalie Deruelle and Jean-Philippe Uzan
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0040
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + ...
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This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α + β) = dα + dβ, where α and β are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.Less
This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α + β) = dα + dβ, where α and β are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.
Nathalie Deruelle and Jean-Philippe Uzan
- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant ...
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This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.Less
This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.
Steven Carlip
- Published in print:
- 2019
- Published Online:
- March 2019
- ISBN:
- 9780198822158
- eISBN:
- 9780191861215
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822158.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, ...
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The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.Less
The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question ...
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This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.Less
This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? This book asserts that the Riemannian scalar must be a linear combination of three obvious candidates, each of which clearly satisfies the required property: a local conformal invariant, a divergence of a Riemannian vector field, and the Chern–Gauss–Bonnet integrand. The book provides a proof of this conjecture. The result itself sheds light on the algebraic structure of conformal anomalies, which appear in many settings in theoretical physics. It also clarifies the geometric significance of the renormalized volume of asymptotically hyperbolic Einstein manifolds. The methods introduced here make an interesting connection between algebraic properties of local invariants—such as the classical Riemannian invariants and the more recently studied conformal invariants—and the study of global invariants, in this case conformally invariant integrals.
Luciano Rezzolla and Olindo Zanotti
- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780198528906
- eISBN:
- 9780191746505
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528906.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter provides a brief summary of the basic aspects of Einstein’s theory of general relativity. Although not aimed to be comprehensive, a step-by-step presentation of the fundamental ...
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This chapter provides a brief summary of the basic aspects of Einstein’s theory of general relativity. Although not aimed to be comprehensive, a step-by-step presentation of the fundamental principles of relativity is provided. Following this spirit, a pragmatic presentation of all the relativistic concepts and quantities that are used in the rest of the book is resented. Our introduction to general relativity is based on a balance between the differential-geometry approach, which is nowadays regarded as the most elegant and physically appropriate one, and the coordinate-components approach, which is nevertheless fundamental for converting formal and compact equations into relations involving quantities that may be measured or computed.Less
This chapter provides a brief summary of the basic aspects of Einstein’s theory of general relativity. Although not aimed to be comprehensive, a step-by-step presentation of the fundamental principles of relativity is provided. Following this spirit, a pragmatic presentation of all the relativistic concepts and quantities that are used in the rest of the book is resented. Our introduction to general relativity is based on a balance between the differential-geometry approach, which is nowadays regarded as the most elegant and physically appropriate one, and the coordinate-components approach, which is nevertheless fundamental for converting formal and compact equations into relations involving quantities that may be measured or computed.
Spyros Alexakis
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153476
- eISBN:
- 9781400842728
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153476.003.0001
- Subject:
- Mathematics, Geometry / Topology
This introductory chapter first sets out the book's purpose, which is to provide a rigorous proof of the Deser–Schwimmer conjecture. This work is a continuation of the previous two papers of the ...
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This introductory chapter first sets out the book's purpose, which is to provide a rigorous proof of the Deser–Schwimmer conjecture. This work is a continuation of the previous two papers of the author, which established the conjecture in a special case and introduced tools that laid the groundwork for the resolution of the full conjecture. The chapter then provides a formulation of the conjecture, presents some applications, and discusses its close relation with certain questions in index theory and in Cauchy–Riemann and Kähler geometry. Then, it broadly outlines the strategy of the proof and very briefly present the tasks that are undertaken in each of the subsequent chapters.Less
This introductory chapter first sets out the book's purpose, which is to provide a rigorous proof of the Deser–Schwimmer conjecture. This work is a continuation of the previous two papers of the author, which established the conjecture in a special case and introduced tools that laid the groundwork for the resolution of the full conjecture. The chapter then provides a formulation of the conjecture, presents some applications, and discusses its close relation with certain questions in index theory and in Cauchy–Riemann and Kähler geometry. Then, it broadly outlines the strategy of the proof and very briefly present the tasks that are undertaken in each of the subsequent chapters.
Oliver Davis Johns
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780191001628
- eISBN:
- 9780191775161
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780191001628.003.0022
- Subject:
- Physics, Atomic, Laser, and Optical Physics
This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that ...
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This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that sought to introduce quantum ideas into a classical model. Angle-action methods are also used in what is called canonical perturbation theory, the study of systems that deviate only slightly from cyclic ones. That vast subject is best studied using the language of differential geometry. Study of the present chapter will give the reader a firm grasp of the angle-action foundation upon which canonical perturbation theory is constructed. Certain specialised systems of importance in astronomy and quantum theory can be treated by a canonical transformation to angle-action variables.Less
This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that sought to introduce quantum ideas into a classical model. Angle-action methods are also used in what is called canonical perturbation theory, the study of systems that deviate only slightly from cyclic ones. That vast subject is best studied using the language of differential geometry. Study of the present chapter will give the reader a firm grasp of the angle-action foundation upon which canonical perturbation theory is constructed. Certain specialised systems of importance in astronomy and quantum theory can be treated by a canonical transformation to angle-action variables.
