"Symmetry and conserved quantitiy : Noether's theorem"의 두 판 사이의 차이
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| 12번째 줄: | 12번째 줄: | ||
==field theoretic formulation== | ==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 $\alpha_{s}$ |
| − | * | + | * 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> | :<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>\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 | * <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 | + | * 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 | * conserved charge | ||
:<math>Q(t)=\int_V \rho \,d^3 x</math> | :<math>Q(t)=\int_V \rho \,d^3 x</math> | ||
:<math>\frac{dQ}{dt}=0</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> | ||
| 37번째 줄: | 40번째 줄: | ||
* [[correlation functions and Ward identity]] | * [[correlation functions and Ward identity]] | ||
* [[Emmy Noether’s Wonderful Theorem]] | * [[Emmy Noether’s Wonderful Theorem]] | ||
| − | + | * [[Gauge theory]] | |
| 45번째 줄: | 48번째 줄: | ||
* [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem] | * [http://en.wikipedia.org/wiki/Noether%27s_theorem http://en.wikipedia.org/wiki/Noether's_theorem] | ||
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| + | ==expositions== | ||
| + | * [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories]<br> | ||
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[[분류:개인노트]] | [[분류:개인노트]] | ||
[[분류:physics]] | [[분류:physics]] | ||
[[분류:math and physics]] | [[분류:math and physics]] | ||
2013년 4월 1일 (월) 15:22 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
\[j_{a}^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}iT_a \phi\]
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