"In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began." Albert Einstein
The year was 1915, and the young mathematician Emmy Noether had just settled into Göttingen University when Albert Einstein visited to lecture on his nearly finished general theory of relativity. Two leading mathematicians of the day, David Hilbert and Felix Klein, dug into the new theory with gusto, but had difficulty reconciling it with what was known about the conservation of energy. Knowing of her expertise in invariance theory, they requested Noether's help. To solve the problem, she developed a novel theorem, applicable across all of physics, which relates conservation laws to continuous symmetriesone of the most important pieces of mathematical reasoning ever developed.
Noether's "first" and "second" theorem was published in 1918. The first theorem relates symmetries under global spacetime transformations to the conservation of energy and momentum, and symmetry under global gauge transformations to charge conservation. In continuum mechanics and field theories, these conservation laws are expressed as equations of continuity. The second theorem, an extension of the first, allows transformations with local gauge invariance, and the equations of continuity acquire the covariant derivative characteristic of coupled matter-field systems. General relativity, it turns out, exhibits local gauge invariance. Noether's theorem also laid the foundation for later generations to apply local gauge invariance to theories of elementary particle interactions.
In Dwight E. Neuenschwander's new edition of Emmy Noether's Wonderful Theorem, readers will encounter an updated explanation of Noether's "first" theorem. The discussion of local gauge invariance has been expanded into a detailed presentation of the motivation, proof, and applications of the "second" theorem, including Noether's resolution of concerns about general relativity. Other refinements in the new edition include an enlarged biography of Emmy Noether's life and work, parallels drawn between the present approach and Noether's original 1918 paper, and a summary of the logic behind Noether's theorem.
Preface Acknowledgments Questions I. WHEN FUNCTIONALS ARE EXTREMAL 1. Symmetry 1.1. Symmetry, Invariances, and Conservation Laws 1.2. Meet Emmy Noether 2. Functionals 2.1. Single-Integral Functionals 2.2. Formal Definition of a Functional 3. Extremals 3.1. The Euler-Lagrange Equation 3.2. Conservation Laws as Corollariesto the Euler-Lagrange Equation 3.3. On the Equivalence of Hamilton's Principleand Newton's Second Law 3.4. Where Do Functional Extremal PrinciplesCome From? 3.5. Why Kinetic Minus Potential Energy? 3.6. Extremals with External Constraints II. WHEN FUNCTIONALS ARE INVARIANT 4. Invariance 4.1. Formal Definition of Invariance 4.2. The Invariance Identity 4.3. A More Liberal Definition of Invariance 5. Emmy Noether's Elegant (First) Theorem 5.1. Invariance + Extremal = Noether's Theorem 5.2. Executive Summary of Noether's Theorem 5.3. ""Extremal"" or ""Stationary""? 5.4. An Inverse Problem 5.5. Adiabatic Invariance in Noether's Theorem III. THE INVARIANCE OF FIELDS 6. Noether's Theorem and Fields 6.1. Multiple-Integral Functionals 6.2. Euler-Lagrange Equations for Fields 6.3. Canonical Momentum and the HamiltonianTensor for Fields 6.4. Equations of Continuity 6.5. The Invariance Identity for Fields 6.6. Noether's Theorem for Fields 6.7. Complex Fields 6.8. Global Gauge Transformations 7. Local Gauge Transformations of Fields 7.1. Local Gauge Invariance and Minimal Coupling 7.2. Electrodynamics as a Gauge Theory,Part 1 7.3. Pure Electrodynamics, Spacetime Invariances,and Conservation Laws 7.4. Electrodynamics as a Gauge Theory,Part 2 7.5. Local Gauge Invariance and Noether Currents 7.6. Internal Degrees of Freedom 7.7. Noether's Theorem and GaugedInternal Symmetries 8. Emmy Noether's Elegant (Second) Theorem 8.1. Two Noether Theorems 8.2. Noether's Second Theorem 8.3. Parametric Invariance 8.4. Free Fall in a Gravitational Field 8.5. The Gravitational Field Equations 8.6. The Functionals of General Relativity 8.7. Gauge Transformations on Spacetime 8.8. Noether's Resolution of an Enigma inGeneral Relativity IV. TRANS-NOETHER INVARIANCE 9. Invariance in Phase Space 9.1. Phase Space 9.2. Hamilton's Principle in Phase Space 9.3. Noether's Theorem and Hamilton's Equations 9.4. Hamilton-Jacobi Theory 10. The Action as a Generator 10.1. Conservation of Probabilityand Continuous Transformations 10.2. The Poetry of Nature Appendixes A. Scalars, Vectors, and Tensors B. Special Relativity C. Equations of Motion in Quantum Mechanics D. Conjugate Variables and Legendre Transformations E. The Jacobian F. The Covariant Derivative Bibliography Index
""Technical and yet ultimately poetic book on Emmy Neother's wonderful theorems... Neuenschwander's work is recommended for anyone who wants to gain a deeper understanding and appreciation of the physics and mathematics behind Emmy Noether's work, as well as the particular challenges she faced in her life.""
