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

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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
-* {{수학노트|url=연속_방정식}}
-==field theoretic formulation==
-* <math>\alpha_{s}</math> continuous symmetry with parameter s, i.e. the action does not change by the action of $\alpha_{s}$
-* 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 $\rho:=j_0$ 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 $T_a$, 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.
-==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%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem]
-==expositions==
-* [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]<br>
-==articles==
-* Herman, Jonathan. “Noether’s Theorem Under the Legendre Transform.” arXiv:1409.5837 [math-Ph], September 19, 2014. http://arxiv.org/abs/1409.5837.
-[[분류:개인노트]]
-[[분류:physics]]
-[[분류:math and physics]]
-[[분류:classical mechanics]]

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

2020년 11월 12일 (목) 22:03 판

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