Symmetry and conserved quantitiy : Noether's theorem

수학노트
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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.


related items



encyclopedia


expositions


articles

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

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Spacy 패턴 목록

  • [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LEMMA': 'theorem'}]
  • [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LOWER': 'first'}, {'LEMMA': 'theorem'}]