Gauge theory

meaning of the gague invariance

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

 

 

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 invariance of the QED Lagrangian

\(\mathcal{L} = \bar{\psi} (i\gamma^\mu \partial_\mu -m)\psi - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e\bar{\psi}\gamma^\mu \psi A_\mu\)

Now we have a Lagrangian with interaction terms.

• local phase transformation of fields
  \(\psi(x) \to e^{i\alpha(x)}\psi(x)\)
  
• gauge transformation of electromagnetic field
  \(A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)}\)
  
• Look at the QED page
  

 

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

• QED
  
• QCD
  
• Particle physics : Quirks and the Standard Models
  
• renormalization

 

 

Abelian gauge theory

• abelian gauge theory has a duality

 

 

Non-Abelian gauge theory

• Yang-Mills 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.
  

 

 

 

 

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

• http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Quigley.pdf
  
• http://www.google.com/search?hl=en&rls=IBMA,IBMA:2008-50,IBMA:en&q=brief+introduction+to+principal+bundles+connections&aq=f&oq=&aqi=
  

 

 

 

 

 

encyclopedia

• http://en.wikipedia.org/wiki/principal_bundle
• http://en.wikipedia.org/wiki/Connection_(vector_bundle)
• http://viswiki.com/en/
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• 다음백과사전 http://enc.daum.net/dic100/search.do?q=

 

 

books

• The Geometry of Physics: An Introduct
• An elementary primer for gauge theory
• 찾아볼 수학책
• http://gigapedia.info/1/gauge
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

• 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/