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| {{Unreferenced stub|auto=yes|date=December 2009}}
| | Hello! <br>My name is Lynette and I'm a 19 years old girl from Vaulx-En-Velin.<br><br>Feel free to visit my website ... [http://www.trachome.com/Utilisateur:MurielGoble Fifa 15 Coin Generator] |
| In [[Quantum gauge theory|quantum gauge theories]], in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] formulation, the [[wave function]] is a [[functional (mathematics)|functional]] of the gauge [[Connection (vector_bundle)|connection]] A and matter fields <math>\,\phi</math>. Being a quantum gauge theory, we have to impose [[first class constraint]]s in the form of functional differential equations—basically, the [[Gauss constraint]].
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| In flat spacetime, space is noncompact '''R'''<sup>3</sup>. Since the Gauss constraints are local, it suffices to consider [[gauge transformation]]s U which approach 1 at spatial infinity. Alternatively, we can assume space is a very large three sphere S<sup>3</sup> or that space is a compact 3-ball B<sup>3</sup> with a S<sup>2</sup> boundary where the values of the fields are fixed so that the gauge transformations occur only in the interior of the ball. At any rate, we can see that there are gauge transformations U [[homotopic]] to the trivial gauge transformation. These gauge transformations are called [[small gauge transformation]]s. All the other gauge transformations are called [[big gauge transformation]]s, which are classified by the [[homotopy group]] π<sub>3</sub>(G) where G is the gauge group.
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| The Gauss constraints mean that the value of the wave function functional is constant along the [[orbit]]s of small gauge transformation.
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| i.e.,
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| :<math>\Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=\Psi[\mathbf{A},\phi]</math> | |
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| for all small gauge transformations U. But this is not true in general for large gauge transformations.
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| It turns out that if G is some [[simple Lie group]], then π<sub>3</sub>(G) is '''Z'''. Let U be any representative of a gauge transformation with [[winding number]] 1.
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| The Hilbert space decomposes into [[superselection sector]]s labeled by a '''theta angle''' θ such that
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| :<math>\Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=e^{i\theta}\Psi[\mathbf{A},\phi]</math>
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| ==See also==
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| * [[Instanton]]
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| * [[Strong CP problem]]
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| {{DEFAULTSORT:Vacuum Angle}}
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| [[Category:Quantum field theory]]
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| [[Category:Quantum chromodynamics]]
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| {{Quantum-stub}}
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Hello!
My name is Lynette and I'm a 19 years old girl from Vaulx-En-Velin.
Feel free to visit my website ... Fifa 15 Coin Generator