Gauge theory

meaning of the gague invariance

• gauge = measure
• gauge invariance = 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
  
• 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.
  

 

 

Principal G-bundle

• principal bundles
  
• topology and vector bundles
  

 

 

 

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
  
•  
  

 

 

related items

• differential geometry
  

 

 

encyclopedia

• http://en.wikipedia.org/wiki/principal_bundle
• http://en.wikipedia.org/wiki/Connection_(vector_bundle)

 

 

books

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

 

 

expositions

• On the Origins of Gauge Theory , Callum Quigley, April 14, 2003
  

• Connections, Gauges and Field Theories
  

• WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG
  

 

 

articles

• Quantum field theory and the Jones polynomial Edward Witten, Comm. Math. Phys. Volume 121, Number 3 (1989), 351-399[1]