Symmetry and conserved quantitiy : Noether's theorem

introduction

• fields
• 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

 

 

field theoretic formulation

• \(\alpha_{s}\) 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
  \(Q(t)=\int_V J_0(x) \,d^3 x\)
  \(\frac{dQ}{dt}=0\)
   
  

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• correlation functions and Ward identity
• Emmy Noether’s Wonderful Theorem