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{{Unreferenced stub|auto=yes|date=December 2009}}
{{technical|date=August 2016}}
In [[Quantum gauge theory|quantum gauge theories]], in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] formulation ([[Hamiltonian system]]), the [[wave function]] is a [[functional (mathematics)|functional]] of the gauge [[Connection (vector_bundle)|connection]] <math>\,A</math> and matter fields <math>\,\phi</math>. Being a quantum gauge theory, one has to impose [[first class constraint]]s in the form of [[functional differential equation]]s—basically, the [[Gauss constraint]].

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. Indeed, 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.

The Gauss constraints mean that the value of the wave function functional is constant along the [[orbit]]s of small gauge transformation.

i.e.,

:<math>\Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=\Psi[\mathbf{A},\phi]</math>

for all small gauge transformations U. But this is not true in general for large gauge transformations.

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.

The Hilbert space decomposes into [[superselection sector]]s labeled by a '''theta angle''' θ such that 

:<math>\Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=e^{i\theta}\Psi[\mathbf{A},\phi]</math>

==See also==
* [[Instanton]]
* [[Strong CP problem]]

{{DEFAULTSORT:Vacuum Angle}}
[[Category:Quantum field theory]]
[[Category:Quantum chromodynamics]]


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