"In the judgment of the most competent living mathematicians, Fraulein 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 Goettingen 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 symmetries-one 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.
Dwight E. Neuenschwander is a professor of physics at Southern Nazarene University. He is a columnist for the Observer, the magazine of the Society for Physics Students, and the author of Tensor Calculus for Physics: A Concise Guide.
PrefaceAcknowledgmentsQuestionsI. WHEN FUNCTIONALS ARE EXTREMAL1. Symmetry1.1. Symmetry, Invariances, and Conservation Laws1.2. Meet Emmy Noether2. Functionals2.1. Single-Integral Functionals2.2. Formal Definition of a Functional3. Extremals3.1. The Euler-Lagrange Equation3.2. Conservation Laws as Corollariesto the Euler-Lagrange Equation3.3. On the Equivalence of Hamilton's Principleand Newton's Second Law3.4. Where Do Functional Extremal PrinciplesCome From?3.5. Why Kinetic Minus Potential Energy?3.6. Extremals with External ConstraintsII. WHEN FUNCTIONALS ARE INVARIANT4. Invariance4.1. Formal Definition of Invariance4.2. The Invariance Identity4.3. A More Liberal Definition of Invariance5. Emmy Noether's Elegant (First) Theorem5.1. Invariance + Extremal = Noether's Theorem5.2. Executive Summary of Noether's Theorem5.3. "Extremal" or "Stationary"?5.4. An Inverse Problem5.5. Adiabatic Invariance in Noether's TheoremIII. THE INVARIANCE OF FIELDS6. Noether's Theorem and Fields6.1. Multiple-Integral Functionals6.2. Euler-Lagrange Equations for Fields6.3. Canonical Momentum and the HamiltonianTensor for Fields6.4. Equations of Continuity6.5. The Invariance Identity for Fields6.6. Noether's Theorem for Fields6.7. Complex Fields6.8. Global Gauge Transformations7. Local Gauge Transformations of Fields7.1. Local Gauge Invariance and Minimal Coupling7.2. Electrodynamics as a Gauge Theory,Part 17.3. Pure Electrodynamics, Spacetime Invariances,and Conservation Laws7.4. Electrodynamics as a Gauge Theory,Part 27.5. Local Gauge Invariance and Noether Currents7.6. Internal Degrees of Freedom7.7. Noether's Theorem and GaugedInternal Symmetries8. Emmy Noether's Elegant (Second) Theorem8.1. Two Noether Theorems8.2. Noether's Second Theorem8.3. Parametric Invariance8.4. Free Fall in a Gravitational Field8.5. The Gravitational Field Equations8.6. The Functionals of General Relativity8.7. Gauge Transformations on Spacetime8.8. Noether's Resolution of an Enigma inGeneral RelativityIV. TRANS-NOETHER INVARIANCE9. Invariance in Phase Space9.1. Phase Space9.2. Hamilton's Principle in Phase Space9.3. Noether's Theorem and Hamilton's Equations9.4. Hamilton-Jacobi Theory10. The Action as a Generator10.1. Conservation of Probabilityand Continuous Transformations10.2. The Poetry of NatureAppendixesA. Scalars, Vectors, and TensorsB. Special RelativityC. Equations of Motion in Quantum MechanicsD. Conjugate Variables and Legendre TransformationsE. The JacobianF. The Covariant DerivativeBibliographyIndex