formulas:gauss_law
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formulas:gauss_law [2018/03/30 15:39] jakobadmin [Concrete] |
formulas:gauss_law [2018/05/13 09:19] jakobadmin ↷ Page moved from equations:yang_mills_equations:gauss_law to formulas:yang_mills_equations:gauss_law |
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| ====== Gauss Law ====== | ====== Gauss Law ====== | ||
| + | //also called "Gauss constraint"// | ||
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| * https://www.physicsforums.com/insights/partial-derivation-gausss-law/ | * https://www.physicsforums.com/insights/partial-derivation-gausss-law/ | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| **Gauss' Law for Yang-Mills Theories** | **Gauss' Law for Yang-Mills Theories** | ||
| - | Gauss' law for Yang-Mills theories reads | ||
| - | $$ I(x) \equiv D_i G^{0i} =\partial_i E^i + [A_i,E^i]=0 $$ | + | Gauss law is the $\nu=0$ component of the [[equations:yang_mills_equations|Yang-Mills equation]] |
| - | and is one of the field equations that are derivable from the [[equations:yang_mills_equations|Yang-Mills Lagrangian]]. | + | $$ (\partial_\mu F_{\mu \nu})^a = g j_\nu^a $$ |
| + | $$ \rightarrow (\partial_i F_{i 0})^a = g j_0^a $$ | ||
| + | which is exactly analogous to the inhomogeneous Maxwell equation in the presence of matter fields. | ||
| - | It does not contain second order time derivatatives is therefore not an equation that govers the time development (=an equation of motion), but rather a constraint on the initial conditions. | + | It does not contain second order time derivatives is therefore not an equation that governs the time development (=an equation of motion), but rather a constraint on the initial conditions. |
| - | As in the abelian case discussed above, this equation contradicts the commutator relations and thus is not an operator equation. Instead, we use it as a condition for physical states, which have to satisfy: | + | //(Source: "Classical Theory of Gauge Fields" by Rubakov. See also the discussion there for more details.) |
| + | // | ||
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| + | We can rewrite it in terms of the gauge potentials as | ||
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| + | $$ I(x) \equiv D_i G^{0i} =\partial_i E^i + [A_i,E^i]=0 .$$ | ||
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| + | This equation contradicts the commutator relations and thus is not an operator equation. Instead, we use it as a condition for physical states, which have to satisfy: | ||
| $$ D_i E^i |\Psi\rangle _{phys}=0. $$ | $$ D_i E^i |\Psi\rangle _{phys}=0. $$ | ||
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| ---- | ---- | ||
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| + | **Recommended Reading** | ||
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| + | http://ethesis.helsinki.fi/julkaisut/mat/fysik/vk/salmela/gausssla.pdf | ||
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| + | <tabbox Abstract> | ||
| **Gauss law generates gauge transformations** | **Gauss law generates gauge transformations** | ||
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| ---- | ---- | ||
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| - | **Recommended Reading** | ||
| - | |||
| - | http://ethesis.helsinki.fi/julkaisut/mat/fysik/vk/salmela/gausssla.pdf | ||
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| - | <tabbox Abstract> | ||
| <blockquote>Additional complications arise when defining quantised gauge field theories. This is because the restriction of a theory to be invariant under a gauge group symmetry $\mathcal{G}$, which corresponds to local invariance under some global symmetry group $G$, leads to a strengthened form of the Noether current conservation condition called the //local Gauss law// [11]: | <blockquote>Additional complications arise when defining quantised gauge field theories. This is because the restriction of a theory to be invariant under a gauge group symmetry $\mathcal{G}$, which corresponds to local invariance under some global symmetry group $G$, leads to a strengthened form of the Noether current conservation condition called the //local Gauss law// [11]: | ||
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| <cite>[[https://arxiv.org/abs/1408.3233|Boundary terms in quantum field theory and the spin structure of QCD]] by Peter Lowdon</cite></blockquote> | <cite>[[https://arxiv.org/abs/1408.3233|Boundary terms in quantum field theory and the spin structure of QCD]] by Peter Lowdon</cite></blockquote> | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| + | Gauss law in an important constraint on the initial data in gauge theories. It is particularly important if we want to understand how we can [[advanced_tools:gauge_symmetry:gauge_fixing|fix the gauge]] consistently. | ||
| <blockquote> | <blockquote> | ||
formulas/gauss_law.txt · Last modified: 2018/05/13 09:19 by jakobadmin
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