"Symmetry and conserved quantitiy : Noether's theorem"의 두 판 사이의 차이

2012년 6월 10일 (일) 16:39 판

\(\alpha_{s}\) continuous symmetry with parameter s
current
\(j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))\)
\(j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) \)

obeys the continuity equation
\(\partial_{\mu} J^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0\)
\(j^{4}(x)\) density of some abstract fluid
\(\mathbf{J}=(j_x,j_y,j_z)\) velocity of this abstract fluid at each space time point
conserved charge
\(Q(t)=\int_V J_0(x) \,d^3 x\)
\(\frac{dQ}{dt}=0\)

@@ 2번째 줄: / 2번째 줄: @@
 * fields
-* the condition extreme of functional lead
+* the condition for the extreme of a functional leads to Euler-Lagrange equation
+* invariance of functional imposes another constraint
+* Noether's theorem : extreme+invariance -> conservation law
+<h5>field theoretic formulation</h5>
 * <math>\alpha_{s}</math> continuous symmetry with parameter s
 *  current <br><math>j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))</math><br><math>j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) </math><br>
 *  obeys the continuity equation<br><math>\partial_{\mu} J^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0</math><br>
@@ 15번째 줄: / 19번째 줄: @@
 * <math>\mathbf{J}=(j_x,j_y,j_z)</math> velocity of this abstract fluid at each space time point
 *  conserved charge<br><math>Q(t)=\int_V J_0(x) \,d^3 x</math><br><math>\frac{dQ}{dt}=0</math><br>  <br>
@@ 31번째 줄: / 33번째 줄: @@
 * [[correlation functions and Ward identity]]
+* [[Emmy Noether’s Wonderful Theorem]]