Gauge theory

==meaning of the gague invariance

==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

==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 \(G_{\mu\nu}^{a}=\partial_{\mu}W_{\nu}^{a}-\partial_{\nu}W_{\mu}^{a}-gw^{abc}W_{\mu}^{b}W_{\nu}^{c}\)

==Abelian gauge theory

==Non-Abelian gauge 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,

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

목차