"Gauge theory"의 두 판 사이의 차이

2009년 10월 10일 (토) 15:46 판

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

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
\(A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)}\)

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,

@@ 4번째 줄: / 4번째 줄: @@
 * gauge invariance 란 measurement에 있어서의 invariance를 말함
 * Lagrangian should be gauge invariant.
+<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>
+Now we have a Lagrangian with interaction terms.
+<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">Gauge invariance of the Lagrangian</h5>
+*  local phase transformation of fields<br><math>\psi(x) \to  e^{i\alpha(x)}\psi(x)</math><br>
+*  gauge transformation of electromagnetic field<br><math>A_{\mu}(x) \to A_{\mu}(x)+\frac{1}{e}\partial_{\mu}\alpha(x)}</math><br>