theorems:noethers_theorems:fields
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theorems:noethers_theorems:fields [2018/03/28 15:23] jakobadmin |
theorems:noethers_theorems:fields [2018/05/05 12:23] (current) jakobadmin ↷ Links adapted because of a move operation |
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| \begin{equation} L^i = L^{i}_{\mathrm{spin}}+ L^{i}_{\mathrm{orbit}}. \end{equation} | \begin{equation} L^i = L^{i}_{\mathrm{spin}}+ L^{i}_{\mathrm{orbit}}. \end{equation} | ||
| - | The first part is something new, but needs to be similar to the usual orbital angular momentum we previously considered, because the two terms are added and appear when we consider the same invariance. The standard point of view is that the first part of this conserved quantity is some-kind of internal angular momentum. (In [[theories:quantum_field_theory|quantum field theory]] fields create and destroy particles. A spin $\frac{1}{2}$ field creates spin $\frac{1}{2}$ particles, which is an unchangeable property of an elementary particle. Hence the usage of the word "internal". Orbital angular momentum is a quantity that describes how two or more particles revolve around each other.) | + | The first part is something new, but needs to be similar to the usual orbital angular momentum we previously considered, because the two terms are added and appear when we consider the same invariance. The standard point of view is that the first part of this conserved quantity is some-kind of internal angular momentum. (In [[theories:quantum_field_theory:canonical|quantum field theory]] fields create and destroy particles. A spin $\frac{1}{2}$ field creates spin $\frac{1}{2}$ particles, which is an unchangeable property of an elementary particle. Hence the usage of the word "internal". Orbital angular momentum is a quantity that describes how two or more particles revolve around each other.) |
| One effect of a rotation is that the arguments of the components get transformed $\Phi_1(x) \to \Phi_1(x'),\Phi_2(x) \to \Phi_2(x') ,\ldots $. From the invariance under this effect on the spatial coordinates $x\to x'$, we get the part of the conserved quantitiy that we call orbital angular momentum. | One effect of a rotation is that the arguments of the components get transformed $\Phi_1(x) \to \Phi_1(x'),\Phi_2(x) \to \Phi_2(x') ,\ldots $. From the invariance under this effect on the spatial coordinates $x\to x'$, we get the part of the conserved quantitiy that we call orbital angular momentum. | ||
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| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | Using Noether's theorem for field theories, we can understand where [[basic_notions:spin|spin]] comes from. | + | Using Noether's theorem for field theories, we can understand where [[basic_notions:spin|spin]] comes from and why electric charge is conserved. |
| </tabbox> | </tabbox> | ||
theorems/noethers_theorems/fields.1522243423.txt.gz · Last modified: 2018/03/28 13:23 (external edit)
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