"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==
 
==meaning of the gague invariance==
  
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* Lagrangian should be gauge invariant.
 
* Lagrangian should be gauge invariant.
  
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==gauge field==
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===gauge symmetry and measurement===
  
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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<br>
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*  symmetry implies the existence of something unmeasurable.
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*  phase is one example
  
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* one example is the electromagnetic field<br>
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==gauge field==
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==Gauge invariance of the QED Lagrangian==
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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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<math>\mathcal{L} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}\gamma^\mu \psi A_\mu</math>
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* one example is the electromagnetic field
  
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Now we have a Lagrangian with interaction terms.
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* local phase transformation of fields<br><math>\psi(x) \to  e^{i\alpha(x)}\psi(x)</math><br>
 
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*  gauge transformation of electromagnetic field<br><math>A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)</math><br>
 
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*  Look at the [[QED]] page<br>
 
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* {{수학노트|url=게이지_이론}}
 
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==gauge field tensor==
 
==gauge field tensor==
  
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*  electromagnetic field tensor  <math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!</math><br>
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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><br>
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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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==examples of renormalizable gauge theory==
  
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* [[QED]]<br>
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* [[QED]]
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*  QCD<br>
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*  QCD
 
* [[renormalization]]
 
* [[renormalization]]
  
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==Abelian gauge theory==
 
==Abelian gauge theory==
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* abelian gauge theory has a duality
 
* abelian gauge theory has a duality
  
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==Non-Abelian gauge theory==
 
==Non-Abelian gauge theory==
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* [[Yang-Mills Theory(Non-Abelian gauge theory)|Yang-Mills Theory]]
 
* [[Yang-Mills Theory(Non-Abelian gauge theory)|Yang-Mills Theory]]
  
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==differential geometry formulation==
 
==differential geometry formulation==
  
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*  manifold <math>\mathbb R^{1,3}</math> and having a vector bundle gives a connection<br>
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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 <br>
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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<br>
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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.<br>
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*  Then the Chern class measures the magnetic charge.
  
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==Principal G-bundle==
 
==Principal G-bundle==
  
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* [[principal bundles]]<br>
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* [[principal bundles]]
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* [[topology and vector bundles]]<br>
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* [[topology and vector bundles]]
  
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==3d Chern-Simons theory==
 
==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>.<br>
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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<br>
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*  analogy with class field theory
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*  replace <math>\Sigma</math> by <math>spec O_K</math><br>
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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)<br>
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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, <br>
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*  Now from An's article,  
  
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==메모==
 
==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]<br>
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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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*  <br>
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==related items==
 
==related items==
  
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* [[differential geometry and topology|differential geometry]]<br>
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* [[differential geometry and topology|differential geometry]]
  
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==encyclopedia==
 
==encyclopedia==
123번째 줄: 117번째 줄:
 
* [http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29 http://en.wikipedia.org/wiki/Connection_(vector_bundle)]
 
* [http://en.wikipedia.org/wiki/Connection_%28vector_bundle%29 http://en.wikipedia.org/wiki/Connection_(vector_bundle)]
  
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==books==
 
==books==
  
 
* The Geometry of Physics: An Introduction
 
* The Geometry of Physics: An Introduction
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* An elementary primer for gauge theory
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* An elementary primer for gauge theory
 
* [[2009년 books and articles|찾아볼 수학책]]
 
* [[2009년 books and articles|찾아볼 수학책]]
  
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==expositions==
 
==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.math.toronto.edu/%7Ecolliand/426_03/Papers03/C_Quigley.pdf On the Origins of Gauge Theory] , Callum Quigley, April 14, 2003<br>
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* [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]
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* [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]<br>
 
  
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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<br>
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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==
 
==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]]
  
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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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==메타데이터==
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[[분류:math and physics]]
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q214850 Q214850]
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===Spacy 패턴 목록===
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* [{'LOWER': 'gauge'}, {'LEMMA': 'theory'}]
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* [{'LOWER': 'gauge'}, {'LEMMA': 'symmetry'}]

2021년 2월 17일 (수) 01:24 기준 최신판

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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'gauge'}, {'LEMMA': 'theory'}]
  • [{'LOWER': 'gauge'}, {'LEMMA': 'symmetry'}]
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