# Vacuum angle

In quantum gauge theories, in the Hamiltonian formulation (Hamiltonian system), the wave function is a functional of the gauge connection ${\displaystyle \,A}$ and matter fields ${\displaystyle \,\phi }$. Being a quantum gauge theory, we have to impose first class constraints in the form of functional differential equations—basically, the Gauss constraint.

In flat spacetime, space is noncompact R3. Since the Gauss constraints are local, it suffices to consider gauge transformations U which approach 1 at spatial infinity. Alternatively, we can assume space is a very large three sphere S3 or that space is a compact 3-ball B3 with a S2 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 transformations. All the other gauge transformations are called big gauge transformations, which are classified by the homotopy group π3(G) where G is the gauge group.

The Gauss constraints mean that the value of the wave function functional is constant along the orbits of small gauge transformation.

i.e.,

${\displaystyle \Psi [U\mathbf {A} U^{-1}-(dU)U^{-1},U\phi ]=\Psi [\mathbf {A} ,\phi ]}$

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 π3(G) is Z. Let U be any representative of a gauge transformation with winding number 1.

The Hilbert space decomposes into superselection sectors labeled by a theta angle θ such that

${\displaystyle \Psi [U\mathbf {A} U^{-1}-(dU)U^{-1},U\phi ]=e^{i\theta }\Psi [\mathbf {A} ,\phi ]}$