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

2021년 2월 17일 (수) 03:16 기준 최신판

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

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.

메타데이터

위키데이터

ID : Q578555

Spacy 패턴 목록

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

@@ 5번째 줄: / 5번째 줄: @@
 * 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
+* <math>\alpha_{s}</math> continuous symmetry with parameter s, i.e. the action does not change by the action of <math>\alpha_{s}</math>
-*  current <br><math>j(x)=(j^0(x),j^1(x),j^2(x),j^3(x))</math><br><math>j^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}\left(\frac{\partial\alpha_{s}(\phi)}{\partial s} \right) </math><br>
+* 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>
-*  obeys the continuity equation<br><math>\partial_{\mu} J^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0</math><br>
+* then it obeys the continuity equation
-* <math>j^{4}(x)</math> density of some abstract fluid
+:<math>\partial_{\mu} j^{\mu}=\sum_{\mu=0}^{3}\frac{\partial j^{\mu}}{\partial x^{\mu}}=0</math>
-* <math>\mathbf{J}=(j_x,j_y,j_z)</math> velocity of this abstract fluid at each space time point
+* <math>j^{0}(x)</math> density of some abstract fluid
-*  conserved charge<br><math>Q(t)=\int_V J_0(x) \,d^3 x</math><br><math>\frac{dQ}{dt}=0</math><br>  <br>
+* 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>
-==history==
+===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>
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
+==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==
@@ 34번째 줄: / 36번째 줄: @@
 * [[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]
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* [http://eom.springer.de/ http://eom.springer.de]
-* http://www.proofwiki.org/wiki/
-* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
-==books==
-* [[Emmy Noether’s Wonderful Theorem]]
-* [[2011년 books and articles]]
-* http://library.nu/search?q=
-* http://library.nu/search?q=
 ==expositions==
+* [http://www.thetangentbundle.net/papers/gauge.pdf 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.
-* http://www.ams.org/mathscinet
+[[분류:개인노트]]
-* http://www.zentralblatt-math.org/zmath/en/
+[[분류:physics]]
-* http://arxiv.org/
+[[분류:math and physics]]
-* http://www.pdf-search.org/
+[[분류:classical mechanics]]
-* http://pythagoras0.springnote.com/
-* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
-* http://dx.doi.org/
-==question and answers(Math Overflow)==
-* http://mathoverflow.net/search?q=
-* http://math.stackexchange.com/search?q=
-* http://physics.stackexchange.com/search?q=
-==blogs==
-*  구글 블로그 검색<br>
-**  http://blogsearch.google.com/blogsearch?q=<br>
-** http://blogsearch.google.com/blogsearch?q=
-* http://ncatlab.org/nlab/show/HomePage
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q578555 Q578555]
+===Spacy 패턴 목록===
-==experts on the field==
+* [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LEMMA': 'theorem'}]
+* [{'LOWER': 'noether'}, {'LOWER': "'s"}, {'LOWER': 'first'}, {'LEMMA': 'theorem'}]
-* http://arxiv.org/
-==links==
-* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
-* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
-[[분류:개인노트]]