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Emmy Noether's Wonderful Theorem - Paperback | Diverse Reads

Emmy Noether's Wonderful Theorem

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One of the most important—and beautiful—mathematical solutions ever devised, Noether’s theorem touches on every aspect of physics.

"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 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.

ISBN-13: 9781421422671

Media Type: Paperback

Publisher: Johns Hopkins University Press

Publication Date: 04-01-2017

Pages: 344

Product Dimensions: 8.90(w) x 5.60(h) x 0.90(d)

Age Range: 18 Years

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 Emmy Noether's Wonderful Theorem, also published by Johns Hopkins, and How to Involve Undergraduates in Research: A Field Guide for Faculty.

What People are Saying About This

Dr. Cliff Chancey

Neuenschwander displays the instincts of a good teacher and writes clearly. Using Noether's Theorem as an overarching principle across areas of theoretical physics, he helps students gain a more integrated picture of what sometimes seem to be independent courses—an ever-important thing for undergraduate physics education.

— Dr. Cliff Chancey, University of Northern Iowa

Table of Contents

Preface xi

Acknowledgments xiii

Questions xv

I When Functionals are Extremal

1 Symmetry 3

1.1 Symmetry, Invariances, and Conservation Laws 3

1.2 Meet Emmy Noether 8

2 Functionals 21

2.1 Single-Integral Functional 21

2.2 Formal Definition of a Functional 26

3 Extremals 31

3.1 The Euler-Lagrange Equation 31

3.2 Conservation Laws as Corollaries to the Euler-Lagrange Equation 43

3.3 On the Equivalence of Hamilton's Principle and Newton's Second Law 46

3.4 Where Do Functional Extremal Principles Come From? 49

3.5 Why Kinetic Minus Potential Energy? 53

3.6 Extremals with External Constraints 55

II When Functionals are Invariant

4 Invariance 71

4.1 Formal Definition of Invariance 71

4.2 The Invariance Identity 77

4.3 A More Liberal Definition of Invariance 79

5 Emmy Noether's Elegant (First) Theorem 84

5.1 Invariance + Extremal = Noether's Theorem 84

5.2 Executive Summary of Noether's Theorem 88

5.3 "Extremal" or "Stationary"? 90

5.4 An Inverse Problem: Finding Invariances 94

5.5 Adiabatic Invariance in Noether's Theorem 98

III The Invariance of Fields

6 Noether's Theorem and Fields 111

6.1 Multiple-Integral Functionals 111

6.2 Euler-Lagrange Equations for Fields 115

6.3 Canonical Momentum and the Hamiltonian Tensor for Fields 119

6.4 Equations of Continuity 122

6.5 The Invariance Identity for Fields 124

6.6 Noether's Theorem for Fields 128

6.7 Complex Fields 129

6.8 Global Gauge Transformations 133

7 Local Gauge Transformations of Fields 147

7.1 Local Gauge Invariance and Minimal Coupling 147

7.2 Electrodynamics as a Gauge Theory, Part 1: Field Tensors 153

7.3 Pure Electrodynamics, Spacetime Invariances, and Conservation Laws 159

7.4 Electrodynamics as a Gauge Theory, Part 2: Matter-Field Interactions 163

7.5 Local Gauge Invariance and Noether Currents 168

7.6 Internal Degrees of Freedom 171

7.7 Noether's Theorem and Gauged Internal Symmetries 180

8 Emmy Noether's Elegant (Second) Theorem 194

8.1 Two Noether Theorems 194

8.2 Noether's Second Theorem 199

8.3 Parametric Invariance 205

8.4 Free Fall in a Gravitational Field 211

8.5 The Gravitational Field Equations 218

8.6 The Functionals of General Relativity 226

8.7 Gauge Transformations on Spacetime 229

8.8 Noether's Resolution of an Enigma in General Relativity 231

IV Trans-Noether Invariance

9 Invariance in Phase Space 241

9.1 Phase Space 241

9.2 Hamilton's Principle in Phase Space 243

9.3 Noether's Theorem and Hamilton's Equations 245

9.4 Hamilton-Jacobi Theory 246

10 The Action as a Generator 260

10.1 Conservation of Probability and Continuous Transformations 261

10.2 The Poetry of Nature 265

Appendixes 271

A Scalars, Vectors, and Tensors 273

B Special Relativity 279

C Equations of Motion in Quantum Mechanics 286

D Conjugate Variables and Legendre Transformations 291

E The Jacobian 295

F The Covariant Derivative 299

Bibliography 305

Index 311