"Gauge theory"의 두 판 사이의 차이

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==introduction==
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* {{수학노트|url=게이지_이론}}
  
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==meaning of the gague invariance==
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* gauge = measure
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* gauge invariance = measurement에 있어서의 invariance를 말함
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* Lagrangian should be gauge invariant.
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===gauge symmetry and measurement===
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*  symmetry implies the existence of something unmeasurable.
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*  phase is one example
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==gauge field==
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*  a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one
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*  one example is the electromagnetic field
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==gauge field tensor==
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*  electromagnetic field tensor  <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math>
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*  general gauge fields tensor  <math>G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}</math>
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==examples of renormalizable gauge theory==
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* [[QED]]
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*  QCD
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* [[renormalization]]
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==Abelian gauge theory==
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* abelian gauge theory has a duality
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==Non-Abelian gauge theory==
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* [[Yang-Mills Theory(Non-Abelian gauge theory)|Yang-Mills Theory]]
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==differential geometry formulation==
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*  manifold <math>\mathbb R^{1,3}</math> and having a vector bundle gives a connection
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*  connection <math>A</math> = special kind of 1-form
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* <math>dA</math> = 2-form which measures the electromagnetic charge
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*  Then the Chern class measures the magnetic charge.
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==Principal G-bundle==
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* [[principal bundles]]
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* [[topology and vector bundles]]
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==3d Chern-Simons theory==
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*  3d Chern-Simons theory on <math>\Sigma\times \mathbb R^{1}</math> with gauge choice <math>A_0=0</math> is the moduli space of flat connections on <math>\Sigma</math>.
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*  analogy with class field theory
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*  replace <math>\Sigma</math> by <math>spec O_K</math>
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*  then flat connection on <math>spec O_K</math> is given by Homomorphism group the absolute Galois group Gal(\barQ/K)->U(1)
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*  Now from An's article,
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==메모==
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* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf]
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* 
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==related items==
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* [[differential geometry and topology|differential geometry]]
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==encyclopedia==
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* http://en.wikipedia.org/wiki/principal_bundle
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* [http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29 http://en.wikipedia.org/wiki/Connection_(vector_bundle)]
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==books==
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* The Geometry of Physics: An Introduction
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* An elementary primer for gauge theory
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* [[2009년 books and articles|찾아볼 수학책]]
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==expositions==
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* Wilczek, Frank. “Unification of Force and Substance.” arXiv:1512.02094 [hep-Ph, Physics:hep-Th, Physics:physics], December 7, 2015. http://arxiv.org/abs/1512.02094.
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* [http://www.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf On the Origins of Gauge Theory] , Callum Quigley, April 14, 2003
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* [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]
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* [http://www.math.cornell.edu/%7Egoldberg/Notes/AboutConnections.pdf WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR?] TIMOTHY E. GOLDBERG
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==articles==
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* Slavnov, A. A. “60 Years of Gauge Fields.” arXiv:1511.05713 [hep-Th], November 18, 2015. http://arxiv.org/abs/1511.05713.
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* Weatherall, James Owen. ‘Fiber Bundles, Yang-Mills Theory, and General Relativity’. arXiv:1411.3281 [gr-Qc, Physics:hep-Th, Physics:math-Ph, Physics:physics], 12 November 2014. http://arxiv.org/abs/1411.3281.
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* [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104178138 Quantum field theory and the Jones polynomial] Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399[http://www.zentralblatt-math.org/zmath/en/ ]
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[[분류:math and physics]]
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[[분류:gauge theory]]
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[[분류:migrate]]

2020년 11월 13일 (금) 17:21 판

introduction


meaning of the gague invariance

  • gauge = measure
  • gauge invariance = measurement에 있어서의 invariance를 말함
  • Lagrangian should be gauge invariant.


gauge symmetry and measurement

  • symmetry implies the existence of something unmeasurable.
  • phase is one example


gauge field

  • a gauge field is defined as a four-vector field with the freedom of gauge transformation, and it corresponds to massless particlas of spin one
  • one example is the electromagnetic field


gauge field tensor

  • electromagnetic field tensor \(F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!\)
  • general gauge fields tensor \(G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}\)



examples of renormalizable gauge theory



Abelian gauge theory

  • abelian gauge theory has a duality



Non-Abelian gauge theory



differential geometry formulation

  • manifold \(\mathbb R^{1,3}\) and having a vector bundle gives a connection
  • connection \(A\) = special kind of 1-form
  • \(dA\) = 2-form which measures the electromagnetic charge
  • Then the Chern class measures the magnetic charge.



Principal G-bundle




3d Chern-Simons theory

  • 3d Chern-Simons theory on \(\Sigma\times \mathbb R^{1}\) with gauge choice \(A_0=0\) is the moduli space of flat connections on \(\Sigma\).
  • analogy with class field theory
  • replace \(\Sigma\) by \(spec O_K\)
  • then flat connection on \(spec O_K\) is given by Homomorphism group the absolute Galois group Gal(\barQ/K)->U(1)
  • Now from An's article,



메모



related items



encyclopedia



books

  • The Geometry of Physics: An Introduction
  • An elementary primer for gauge theory
  • 찾아볼 수학책



expositions

articles

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