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

수학노트
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==introduction==
  
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* fields
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* the condition for the extreme of a functional leads to Euler-Lagrange equation
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* invariance of functional imposes another constraint
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* Noether's theorem : extreme+invariance -> conservation law
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* {{수학노트|url=연속_방정식}}
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==field theoretic formulation==
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* <math>\alpha_{s}</math> continuous symmetry with parameter s, i.e. the action does not change by the action of $\alpha_{s}$
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* define the current density <math>j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))</math> by
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:<math>j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) </math>
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* then it obeys the continuity equation
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:<math>\partial_{\mu} j^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0</math>
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* <math>j^{0}(x)</math> density of some abstract fluid
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* Put $\rho:=j_0$ and <math>\mathbf{J}=(j_x,j_y,j_z)</math> velocity of this abstract fluid at each space time point
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* conserved charge
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:<math>Q(t)=\int_V \rho \,d^3 x</math>
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:<math>\frac{dQ}{dt}=0</math>
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===gauge theory===
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* to each generator $T_a$, associate the current density
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:<math>j_{a}^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}iT_a \phi</math>
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==Local Versus Global Conservation==
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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.
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==history==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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==related items==
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* [[correlation functions and Ward identity]]
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* [[Emmy Noether’s Wonderful Theorem]]
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* [[Gauge theory]]
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==encyclopedia==
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* [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem]
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==expositions==
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* [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]<br>
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==articles==
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* 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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[[분류:개인노트]]
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[[분류:physics]]
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[[분류:math and physics]]
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[[분류:classical mechanics]]

2020년 11월 16일 (월) 10:46 판

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

 

 

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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