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
• 틀:수학노트

 

 

field theoretic formulation

• \(\alpha_{s}\) continuous symmetry with parameter s, i.e. the action does not change by the action of $\alpha_{s}$
• define the current density \(j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))\) by

\[j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) \]

• then it obeys the continuity equation

\[\partial_{\mu} j^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0\]

• \(j^{0}(x)\) density of some abstract fluid
• Put $\rho:=j_0$ and \(\mathbf{J}=(j_x,j_y,j_z)\) velocity of this abstract fluid at each space time point
• conserved charge

\[Q(t)=\int_V \rho \,d^3 x\] \[\frac{dQ}{dt}=0\]

gauge theory

• to each generator $T_a$, associate the current density

\[j_{a}^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}iT_a \phi\]

Local Versus Global Conservation

Equation (10.165) embodies the idea of local conservation, which is stronger than global conservation. Globally, something like energy could well be con-served in that it might disappear in one place only to reappear in another a long way away. But this seems never to be observed in Nature; if energy does disappear in one place and reappear in another, we always observe a current of energy in between those places. That is, energy is conserved locally, which is a much stronger idea than mere global conservation. Even so, it might well be that something can appear from nowhere in an apparent example of nonconser-vation. “Flatlanders” —beings who are confined to a 2-surface—might observe the arrival of a 2-sphere (i.e. a common garden-variety sphere that needs to be embedded in three dimensions) that passes through their world. What will they see? First, a dot appears, which rapidly grows into a circle before growing smaller again to eventually vanish. The Flatlanders have witnessed a higher-dimensional object passing through their world; they might well be perplexed, since the circle seemed to come out of the void before vanishing back into it.

 

history

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

 

 

related items

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

 

 

encyclopedia

• http://en.wikipedia.org/wiki/Noether's_theorem

expositions

• Connections, Gauges and Field Theories
  

articles

• Herman, Jonathan. “Noether’s Theorem Under the Legendre Transform.” arXiv:1409.5837 [math-Ph], September 19, 2014. http://arxiv.org/abs/1409.5837.