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
- <math>\alpha_{s}</math> continuous symmetry with parameter s, i.e. the action does not change by the action of <math>\alpha_{s}</math>
- define the current density <math>j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))</math> by
- <math>j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) </math>
- then it obeys the continuity equation
- <math>\partial_{\mu} j^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0</math>
- <math>j^{0}(x)</math> density of some abstract fluid
- Put <math>\rho:=j_0</math> and <math>\mathbf{J}=(j_x,j_y,j_z)</math> velocity of this abstract fluid at each space time point
- conserved charge
- <math>Q(t)=\int_V \rho \,d^3 x</math>
- <math>\frac{dQ}{dt}=0</math>
gauge theory
- to each generator <math>T_a</math>, associate the current density
- <math>j_{a}^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}iT_a \phi</math>
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.
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.
메타데이터
위키데이터
- ID : Q578555
Spacy 패턴 목록
- [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LEMMA': 'theorem'}]
- [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LOWER': 'first'}, {'LEMMA': 'theorem'}]