"Symmetry and conserved quantitiy : Noether's theorem"의 두 판 사이의 차이
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==field theoretic formulation== | ==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> 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 | * 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> | ||
| 18번째 줄: | 18번째 줄: | ||
:<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 | + | * 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 | * 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=== | ===gauge theory=== | ||
| − | * to each generator | + | * 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> | :<math>j_{a}^{\mu}(x)= \frac{\partial \mathcal{L}}{\partial ( \partial_\mu \phi )}iT_a \phi</math> | ||
| 29번째 줄: | 29번째 줄: | ||
==Local Versus Global Conservation== | ==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. | 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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| 55번째 줄: | 47번째 줄: | ||
==expositions== | ==expositions== | ||
| − | * [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories] | + | * [http://www.thetangentbundle.net/papers/gauge.pdf Connections, Gauges and Field Theories] |
2020년 11월 16일 (월) 10:00 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.
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.