Vacuum angle
Encyclopedia
In quantum gauge theories
, in the Hamiltonian
formulation, the wave function is a functional
of the gauge Connection
A and matter fields
. Being a quantum gauge theory, we have to impose first class constraint
s 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 orbit
s of small gauge transformation.
i.e.,
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 sector
s labeled by a theta angle θ such that
Quantum gauge theory
In quantum physics, in order to quantize a gauge theory, like for example Yang-Mills theory, Chern-Simons or BF model, one method is to perform a gauge fixing. This is done in the BRST and Batalin-Vilkovisky formulation...
, in the Hamiltonian
Hamiltonian (quantum mechanics)
In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...
formulation, the wave function is a functional
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
of the gauge Connection
Connection (vector bundle)
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
A and matter fields
. Being a quantum gauge theory, we have to impose first class constraintFirst class constraint
In a constrained Hamiltonian system, a dynamical quantity is called a first class constraint if its Poisson bracket with all the other constraints vanishes on the constraint surface .-Poisson brackets:In Hamiltonian mechanics, consider a symplectic manifold M with a smooth Hamiltonian over...
s 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
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
π3(G) where G is the gauge group.
The Gauss constraints mean that the value of the wave function functional is constant along the orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s of small gauge transformation.
i.e.,
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
Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...
, then π3(G) is Z. Let U be any representative of a gauge transformation with winding number
Winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
1.
The Hilbert space decomposes into superselection sector
Superselection sector
In Quantum mechanics, superselection extends the concept of selection rules.Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables....
s labeled by a theta angle θ such that