Gauge theory

수학노트
http://bomber0.myid.net/ (토론)님의 2010년 1월 26일 (화) 14:03 판
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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

 

 

 

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, 

 

 

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