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Lecture7.Guage.Invariance.Conserved.Quantities

Lecture7.Guage.Invariance.Conserved.Quantities - Gauge...

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Gauge Invariance and Conserved Quantities “Noether's theorem” was proven by German mathematician, Emmy Noether in 1915 and published in 1918 Noether's theorem has become published in 1918. Noether s theorem has become a fundamental tool of quantum field theory – and has been called "one of the most important mathematical theorems ever proved in guiding the development of modern physics". Amalie Emmy Noether 1882-1935
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Consider the charged scalar field, and the following transformation on , where is a constant.
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=1 Now suppose that can be varied continuously – so that we have i fi it b f ll ti f hi h an infinite number of small, continuous for which  exp(i ) = and L L. The set of all these transformations, U = exp(i ), form a group of operators. It is called U(1) , a unitary group (because U * U = 1 ) Here the Hermitian conjugate is the complex conjugate because U is not a matrix.
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Now, we have a more astounding result: we can vary the (complex) phase of the field operator,
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