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Emmy Noether's Wonderful Theorem (Paperback, revised and updated edition)
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Emmy Noether's Wonderful Theorem (Paperback, revised and updated edition)
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"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 Gottingen
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.
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