theorems:noethers_theorems:fields
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theorems:noethers_theorems:fields [2017/12/13 12:48] jakobadmin [Layman] |
theorems:noethers_theorems:fields [2018/05/05 12:23] (current) jakobadmin ↷ Links adapted because of a move operation |
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| ====== Noether's Theorem for Fields ====== | ====== Noether's Theorem for Fields ====== | ||
| + | //see also: [[theorems:noethers_theorems]] // | ||
| - | <tabbox Why is it interesting?> | ||
| - | Using Noether's theorem for field theories, we can understand where [[basic_notions:spin|spin]] comes from. | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| Noether's famous theorem states that there is a conserved quantity for every symmetry of the Lagrangian. | Noether's famous theorem states that there is a conserved quantity for every symmetry of the Lagrangian. | ||
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| <note tip>**To summarize:** The Noether theorem in field theories yields for each symmetry a conserved quantity that consists of two parts. One part that corresponds to invariance under the transformation of component functions, and a second part that corresponds to the invariance under the mixing of the components. | <note tip>**To summarize:** The Noether theorem in field theories yields for each symmetry a conserved quantity that consists of two parts. One part that corresponds to invariance under the transformation of component functions, and a second part that corresponds to the invariance under the mixing of the components. | ||
| - | In general only the sum of these two parts is conserved.</note> | + | In general, only the sum of these two parts is conserved.</note> |
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| - | <tabbox Student> | + | <tabbox Concrete> |
| The representations of the Poincare group that act on objects with multiple components, where the components are functions, are known as field representations. They consist of the infinite-dimensional part that acts on functions (the space of functions is infinite-dimensional), and a finite-dimensional part that mixes the components. In the infinite-dimensional representation, the elements of the Poincare group are given by differential operators. For example, the generator of translations is $\partial_x$, because | The representations of the Poincare group that act on objects with multiple components, where the components are functions, are known as field representations. They consist of the infinite-dimensional part that acts on functions (the space of functions is infinite-dimensional), and a finite-dimensional part that mixes the components. In the infinite-dimensional representation, the elements of the Poincare group are given by differential operators. For example, the generator of translations is $\partial_x$, because | ||
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| \begin{equation} \label{eq:boostrotgenDef} M_{\mu \nu}^{\mathrm{inf }}=i( x^\mu \partial^{\nu} - x^\nu \partial^{\mu}) \end{equation} | \begin{equation} \label{eq:boostrotgenDef} M_{\mu \nu}^{\mathrm{inf }}=i( x^\mu \partial^{\nu} - x^\nu \partial^{\mu}) \end{equation} | ||
| - | In contrast, in the finite-dimensional representation the elements of the Poincare group are given by matrices, which have the effect that they mix the elements of objects with multiple elements. | + | In contrast, in the finite-dimensional representation, the elements of the Poincare group are given by matrices, which have the effect that they mix the elements of objects with multiple elements. |
| The complete transformation is then a combination of a transformation generated by the finite-dimensional representation $M_{\mu \nu}^{\mathrm{fin }}$ and a transformation generated by the infinite-dimensional representation $M_{\mu \nu}^{\mathrm{inf }}$ of the generators: | The complete transformation is then a combination of a transformation generated by the finite-dimensional representation $M_{\mu \nu}^{\mathrm{fin }}$ and a transformation generated by the infinite-dimensional representation $M_{\mu \nu}^{\mathrm{inf }}$ of the generators: | ||
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| $S_{\mu \nu}$ is related to the generators of rotations by $S_i = \frac{1}{2} \epsilon_{ijk} S_{jk}$ and to the generators of boosts by $K_i = S_{0i}$. This definition of the quantity $S_{\mu \nu}$ enables us to work with the generators of rotations and boosts at the same time. | $S_{\mu \nu}$ is related to the generators of rotations by $S_i = \frac{1}{2} \epsilon_{ijk} S_{jk}$ and to the generators of boosts by $K_i = S_{0i}$. This definition of the quantity $S_{\mu \nu}$ enables us to work with the generators of rotations and boosts at the same time. | ||
| - | The first part is only important for rotations and boosts, because translations do not lead to a mixing of the field components. For boosts the conserved quantity will not be very enlightening, just as in the particle case, so in fact this term will become only relevant for rotational symmetry. In addition, the first part plays no role for scalars, because these only have one component. | + | The first part is only important for rotations and boosts because translations do not lead to a mixing of the field components. For boosts, the conserved quantity will not be very enlightening, just as in the particle case, so in fact, this term will become only relevant for rotational symmetry. In addition, the first part plays no role for scalars, because these only have one component. |
| In the following, we only discuss the conserved quantity that we get from invariance under rotations. | In the following, we only discuss the conserved quantity that we get from invariance under rotations. | ||
| - | Using Noether's theorem, we can derive that from the invariance under the action of the infinite-dimensional part of the transformation we get the conserved quantitiy.(Again, we skip here the details and only quote the final result. For the details see, for example, "Physics from Symmetry" by J. Schwichtenberg). | + | Using Noether's theorem, we can derive that from the invariance under the action of the infinite-dimensional part of the transformation we get the conserved quantity.(Again, we skip here the details and only quote the final result. For the details see, for example, "Physics from Symmetry" by J. Schwichtenberg). |
| \begin{equation} \label{eq:consrotORBIT} L^i_{\mathrm{orbit}} = \frac{1}{2} \epsilon^{ijk} Q^{jk} = \frac{1}{2} \epsilon^{ijk} \int d^3x ( T^{k0} x^j - T^{j0} x^k), \end{equation} | \begin{equation} \label{eq:consrotORBIT} L^i_{\mathrm{orbit}} = \frac{1}{2} \epsilon^{ijk} Q^{jk} = \frac{1}{2} \epsilon^{ijk} \int d^3x ( T^{k0} x^j - T^{j0} x^k), \end{equation} | ||
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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_theory: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 Researcher> | + | <tabbox Abstract> |
| <note tip> | <note tip> | ||
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| - | <tabbox Examples> | + | <tabbox Why is it interesting?> |
| - | --> Example1# | + | Using Noether's theorem for field theories, we can understand where [[basic_notions:spin|spin]] comes from and why electric charge is conserved. |
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| - | <-- | + | |
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| - | --> Example2:# | + | |
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| - | <-- | + | |
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| - | <tabbox FAQ> | + | |
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| - | <tabbox History> | + | |
| </tabbox> | </tabbox> | ||
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