*Anthony Duncan*

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the ...
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This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the canonical Lagrangian/Hamiltonian formalism discussed in Section 12.3 of Chapter 12 in order to handle the presence of constraints entailed by the presence of local symmetries. The chapter is organized as follows. Section 15.1 introduces the concept of a local symmetry with a simple example from classical mechanics. Section 15.2 describes the Dirac constrained Hamiltonian theory, and the Faddeev–deWitt functional quantization method for such systems. The quantization of gauge theories using this functional (path-integral) method is then explained, first using abelian gauge theory in Section 15.3, where the technical complications are minimal. In Section 15.4 the extension to non-abelian gauge theories is performed, again using path-integral methods applied to the constrained Hamiltonian, leading to the Feynman rules for general (unbroken) non-abelian gauge theories. Section 15.5 explores the existence of quantum anomalies in the chiral currents of internal global symmetries. It shows that the classical current conservation implied by Noether's theorem may be violated by quantum effects, yielding a non-vanishing divergence of the Noether current explicitly proportional to Planck's constant. Section 15.6 focuses on the features of spontaneous symmetry breaking in the presence of local gauge symmetry. The chapter then explains the famous ‘Higgs phenomenon’ in the context of the electroweak sector of the Standard Model and outlines the derivation of the Feynman rules for a general spontaneously broken local gauge theory.Less

This chapter examines the additional rich structure introduced when a local quantum field theory displays a local gauge symmetry. It shows how such symmetries require a generalization of the canonical Lagrangian/Hamiltonian formalism discussed in Section 12.3 of Chapter 12 in order to handle the presence of constraints entailed by the presence of local symmetries. The chapter is organized as follows. Section 15.1 introduces the concept of a local symmetry with a simple example from classical mechanics. Section 15.2 describes the Dirac constrained Hamiltonian theory, and the Faddeev–deWitt functional quantization method for such systems. The quantization of gauge theories using this functional (path-integral) method is then explained, first using abelian gauge theory in Section 15.3, where the technical complications are minimal. In Section 15.4 the extension to non-abelian gauge theories is performed, again using path-integral methods applied to the constrained Hamiltonian, leading to the Feynman rules for general (unbroken) non-abelian gauge theories. Section 15.5 explores the existence of quantum anomalies in the chiral currents of internal global symmetries. It shows that the classical current conservation implied by Noether's theorem may be violated by quantum effects, yielding a non-vanishing divergence of the Noether current explicitly proportional to Planck's constant. Section 15.6 focuses on the features of spontaneous symmetry breaking in the presence of local gauge symmetry. The chapter then explains the famous ‘Higgs phenomenon’ in the context of the electroweak sector of the Standard Model and outlines the derivation of the Feynman rules for a general spontaneously broken local gauge theory.

*Ta-Pei Cheng*

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199669912
- eISBN:
- 9780191744488
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669912.003.0016
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

We review gauge invariance in classical electrodynamics and in quantum mechanics, showing that the gauge transformation must involve a spacetime-dependent phase change of the wavefunction for a ...
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We review gauge invariance in classical electrodynamics and in quantum mechanics, showing that the gauge transformation must involve a spacetime-dependent phase change of the wavefunction for a charged particle. If we reverse this procedure and start first with the phase transformation in quantum mechanics, the gauge principle would guide us to a new gauge field (hence new dynamics) when changing a global symmetry into a spacetime-dependent one. We demonstrate in detail how Maxwell’s electrodynamics can then be “derived” from the requirement of a local U ( 1 ) symmetry in the internal charge space. In this way we understand special relativity and gauge invariance as the essence of Maxwell’s theory. The gauge symmetry idea has advanced much beyond quantum electrodynamics (QED), culminating in the formulation of quantum chromodynamics (QCD) and the electroweak theory of the Standard Model of particle interactions, based on the S U ( 3 ) × S U ( 2 ) × U ( 1 ) symmetry group. We provide a narrative account of this subsequent development.Less

We review gauge invariance in classical electrodynamics and in quantum mechanics, showing that the gauge transformation must involve a spacetime-dependent phase change of the wavefunction for a charged particle. If we reverse this procedure and start first with the phase transformation in quantum mechanics, the gauge principle would guide us to a new gauge field (hence new dynamics) when changing a global symmetry into a spacetime-dependent one. We demonstrate in detail how Maxwell’s electrodynamics can then be “derived” from the requirement of a local U ( 1 ) symmetry in the internal charge space. In this way we understand special relativity and gauge invariance as the essence of Maxwell’s theory. The gauge symmetry idea has advanced much beyond quantum electrodynamics (QED), culminating in the formulation of quantum chromodynamics (QCD) and the electroweak theory of the Standard Model of particle interactions, based on the S U ( 3 ) × S U ( 2 ) × U ( 1 ) symmetry group. We provide a narrative account of this subsequent development.

*Thomas Ryckman*

- Published in print:
- 2005
- Published Online:
- April 2005
- ISBN:
- 9780195177176
- eISBN:
- 9780199835324
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0195177177.003.0008
- Subject:
- Philosophy, Philosophy of Science

It is shown how Eddington’s 1921 generalization of Weyl’s theory of gravitation and electromagnetism stemmed from similar transcendental idealist epistemological concerns. While Eddington’s affine ...
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It is shown how Eddington’s 1921 generalization of Weyl’s theory of gravitation and electromagnetism stemmed from similar transcendental idealist epistemological concerns. While Eddington’s affine field theory served as a template for numerous attempts within Einstein’s own unified field theory program, Eddington’s understanding of a geometrized physics remained epistemological in origin and motivation, seeking to provide a graphical representation of the most general conditions of possible experience of a world constructed from the point of view of “no one in particular.” This represents Eddington’s belief that local symmetries have a particularly important constitutive role in fundamental physical theory.Less

It is shown how Eddington’s 1921 generalization of Weyl’s theory of gravitation and electromagnetism stemmed from similar transcendental idealist epistemological concerns. While Eddington’s affine field theory served as a template for numerous attempts within Einstein’s own unified field theory program, Eddington’s understanding of a geometrized physics remained epistemological in origin and motivation, seeking to provide a graphical representation of the most general conditions of possible experience of a world constructed from the point of view of “no one in particular.” This represents Eddington’s belief that local symmetries have a particularly important constitutive role in fundamental physical theory.

*David Blow*

- Published in print:
- 2002
- Published Online:
- November 2020
- ISBN:
- 9780198510512
- eISBN:
- 9780191919244
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198510512.003.0017
- Subject:
- Chemistry, Crystallography: Chemistry

At this stage, we have derived a model from an electron-density map and have interpreted it as closely as we can in terms of molecular structure. Provided the job has been done well enough, the ...
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At this stage, we have derived a model from an electron-density map and have interpreted it as closely as we can in terms of molecular structure. Provided the job has been done well enough, the next task of improving that interpretation can be left to computational procedures known as structural refinement. If further uninterpreted features of the structure are revealed, it will be necessary to go back to the methods of Chapter 11 to improve the interpretation. The purpose of structural refinement is to adjust a structure to give the best possible fit to the crystallographic observations. The intensities of the Bragg reflections constitute the observations, and the various quantities that define the structure are adjusted to give the best fit. Box 12.1 gives an outline of what is meant by refinement of quantities to fit observations. In structural refinement, a measure of the discrepancies between the calculated X-ray scattering by the model structure and the observed intensities is defined: this is called the refinement parameter. The purpose of the refinement procedure is to alter the model to give the lowest possible refinement parameter. Box 12.1 uses a simple example to bring out some important general points: 1. My model will be specified by a number of variables. In a diffraction experiment, they are usually the coordinates and B factor of every atom. If the number of observed quantities is less than the number of variables, the results can have no validity. 2. If the number of observations equals the number of variables, a perfect fit can be obtained, irrespective of the accuracy of the observations or of the model. (This is true of so-called linear problems, and approximately so in non-linear cases.) 3. If the number of observations exceeds the number of variables by only a small quantity, the estimate of the reliability of the model is questionable. In practice, refinement procedures can only work when there is a sufficient number of observations which are sufficiently accurate. Also, the model must already be good enough to make the refinement procedure meaningful.
Less

At this stage, we have derived a model from an electron-density map and have interpreted it as closely as we can in terms of molecular structure. Provided the job has been done well enough, the next task of improving that interpretation can be left to computational procedures known as structural refinement. If further uninterpreted features of the structure are revealed, it will be necessary to go back to the methods of Chapter 11 to improve the interpretation. The purpose of structural refinement is to adjust a structure to give the best possible fit to the crystallographic observations. The intensities of the Bragg reflections constitute the observations, and the various quantities that define the structure are adjusted to give the best fit. Box 12.1 gives an outline of what is meant by refinement of quantities to fit observations. In structural refinement, a measure of the discrepancies between the calculated X-ray scattering by the model structure and the observed intensities is defined: this is called the refinement parameter. The purpose of the refinement procedure is to alter the model to give the lowest possible refinement parameter. Box 12.1 uses a simple example to bring out some important general points: 1. My model will be specified by a number of variables. In a diffraction experiment, they are usually the coordinates and B factor of every atom. If the number of observed quantities is less than the number of variables, the results can have no validity. 2. If the number of observations equals the number of variables, a perfect fit can be obtained, irrespective of the accuracy of the observations or of the model. (This is true of so-called linear problems, and approximately so in non-linear cases.) 3. If the number of observations exceeds the number of variables by only a small quantity, the estimate of the reliability of the model is questionable. In practice, refinement procedures can only work when there is a sufficient number of observations which are sufficiently accurate. Also, the model must already be good enough to make the refinement procedure meaningful.