Posted: Aug 21, 2010 8:07 pm
Post 6 - Spin Networks - Well almost !
The stuff I've described so far (Ashtekar Connection/Densitized Triad on 3 space, Hamiltonian Constraint, Diffeomorphism Constraint and Gauss Constraint) is all machinery for recasting classical (non quantum) GR as a gauge theory. Now we want to quantize it.
The thing that makes the LQG approach to quantization a bit different is that we're no longer allowed to assume the existence of a background spacetime metric anywhere along the way. The metric is part of the dynamical system we're trying to model. In discussions on LQG you will see lots of mention of "background independence". This is what is meant.
We have our space of connections and to quantize, we somehow need to get a Hilbert space of functionals out of it. (A functional is "just" a function defined on a space of functions, which of course will usually be an infinite dimensional space. In other words a functional maps a function to a number). The space of all connections is just too "big", so what we do is the following:
Define a graph Γ to be a join-the-dots picture on the three-space Σ, with a bunch of links which meet at nodes.
If there are L links, then if we have a function f: SU(2)L->ℂ
(SU(2)L just means L copies of SU(2))
we can use the pair (Γ, f) to define a functional ψ(Γ, f) on the space of connections. It maps the connection A to the complex number
ψ(Γ, f)[A] = f(he1[A], he2[A],.....,heL[A])
where hei[A] is the element of SU(2) given by the holonomy of the connection A along the link ei i=1..L
Now to build a Hilbert space we need to get ourselves a scalar product. If I have another function g: SU(2)L->ℂ then I can define the scalar product of the two functionals to be
<ψ(Γ, f) | ψ(Γ, g)> = ∫f(he1[[A], he2[A],.....,heL[A])*g(he1[[A], he2[A],.....,heL[A]) dhe1dhe2....dheL
* denotes complex conjugate, and dhe is the Haar measure on SU(2).
The cool thing is that the scalar product defined this way is finite, since SU(2) is a compact group. With this scalar product, we have a Hilbert space 𝓗Γ associated with the graph Γ. Define the Hilbert space
𝓗kin = direct sum over all graphs Γ in Σ of 𝓗Γ
OK so we've got a Hilbert space. What next ? Observables are represented as operators on the Hilbert space, and Poisson brackets become commutators. In elementary QM we have position and momentum, q and p. The analogs here are A, E (connection and densitized triad - not writing the indices out). How do A and E act on the Hilbert space ?
To define this, we start by introducing a basis of the Hilbert space. To do this, we use the Peter-Weyl theorem, which states that a basis of functions on a compact group (like SU(2)) is given by the matrix elements of the unitary irreducible representation of the group. See appendix below for a short discussion of reps Djmn of SU(2).
So we can expand a generic function f on SU(2) as
f(g) = Σ fjmnDjmn(g)
This is extendable in an obvious way to our functionals ψ(Γ, f) (too many indices to write out here though).
Using this basis for functionals on 𝓗kin we can define a representation of the Holonomy/Flux algebra that I introduced at the end of post 5. What we want is something like
Aia ψ[A] = Aia ψ[A]
Eai ψ[A] = (𝛿/𝛿Aia) ψ[A]
where the bold entries are the operators which operate on the wave functionals ψ[A] and 𝛿/𝛿 denotes functional differentiation.
Now that we know that we can represent our wave functionals as expansions of matrix elements of group elements (the D's), this becomes straightforward. For example, if h𝛾[A] is the operator representing the holonomy along a curve 𝛾, then its action on a functional is got just by modifying the functional by multiplying each basis element by its value ( an SU(2) element). So for example if Djmn(he[A]) appears in the expansion of the functional, we replace it by Djmn(h𝛾[A]he[A])
The action of the flux can be defined too, but is a bit messier as it involves the functional derivative, so I won't attempt to type it in.
We have a Hilbert space and a representation of the operators. To get to the Hilbert space of physical states we need to impose the constraints. This is done in 3 stages, one for each constraint
𝓗kin ->GaussConstraint->𝓗0kin ->DiffeomorphismConstraint->𝓗Diff -> HamiltonianConstraint->𝓗phys
Next post I'll talk a bit about the implementation of the Gauss constraint. States in the resulting Hilbert space are called spin network states.
Appendix - Irreducible reps of SU(2)
Denote by σ1, σ2, σ3 the Pauli spin matrices (familiar to everybody from angular momentum bollox in elementary QM). Remember that you can find wavefunctions which are simultaneous eigenfunctions of the total angular momementum operator L2 and Lz ? In group theoretic terms this is because σ2 = (σ1)2+(σ2)2+(σ3)2 is a Casimir operator.
So, we can find vectors which are simultaneously eigenvectors of σ2 and (say) σ3. Traditionally denote them by
|jm> where j=0,1/2, 1, 3/2.... and m=-j, -j+1,....-1,0,1, .....j-1, j
j labels the total angular momentum and m the z component of the angular momentum.
A generic rotation on 3 space can be written as
R(α,β,γ) = e-iασ1e-iασ2e-iασ3
where α,β,γ are the Euler angles that parametrize the rotation.
(What's happening here is that the Pauli spin matrices are elements of the Lie algebra of SU(2), and to generate elements of the Lie group, you need to exponentiate).
The irreducible representations of SU(2) are given by the matrices
D[j]mn(g) = <jm|R(α,β,γ)|jn>
where the group element g is parametrised by the Euler angles α,β,γ.
The stuff I've described so far (Ashtekar Connection/Densitized Triad on 3 space, Hamiltonian Constraint, Diffeomorphism Constraint and Gauss Constraint) is all machinery for recasting classical (non quantum) GR as a gauge theory. Now we want to quantize it.
The thing that makes the LQG approach to quantization a bit different is that we're no longer allowed to assume the existence of a background spacetime metric anywhere along the way. The metric is part of the dynamical system we're trying to model. In discussions on LQG you will see lots of mention of "background independence". This is what is meant.
We have our space of connections and to quantize, we somehow need to get a Hilbert space of functionals out of it. (A functional is "just" a function defined on a space of functions, which of course will usually be an infinite dimensional space. In other words a functional maps a function to a number). The space of all connections is just too "big", so what we do is the following:
Define a graph Γ to be a join-the-dots picture on the three-space Σ, with a bunch of links which meet at nodes.
If there are L links, then if we have a function f: SU(2)L->ℂ
(SU(2)L just means L copies of SU(2))
we can use the pair (Γ, f) to define a functional ψ(Γ, f) on the space of connections. It maps the connection A to the complex number
ψ(Γ, f)[A] = f(he1[A], he2[A],.....,heL[A])
where hei[A] is the element of SU(2) given by the holonomy of the connection A along the link ei i=1..L
Now to build a Hilbert space we need to get ourselves a scalar product. If I have another function g: SU(2)L->ℂ then I can define the scalar product of the two functionals to be
<ψ(Γ, f) | ψ(Γ, g)> = ∫f(he1[[A], he2[A],.....,heL[A])*g(he1[[A], he2[A],.....,heL[A]) dhe1dhe2....dheL
* denotes complex conjugate, and dhe is the Haar measure on SU(2).
The cool thing is that the scalar product defined this way is finite, since SU(2) is a compact group. With this scalar product, we have a Hilbert space 𝓗Γ associated with the graph Γ. Define the Hilbert space
𝓗kin = direct sum over all graphs Γ in Σ of 𝓗Γ
OK so we've got a Hilbert space. What next ? Observables are represented as operators on the Hilbert space, and Poisson brackets become commutators. In elementary QM we have position and momentum, q and p. The analogs here are A, E (connection and densitized triad - not writing the indices out). How do A and E act on the Hilbert space ?
To define this, we start by introducing a basis of the Hilbert space. To do this, we use the Peter-Weyl theorem, which states that a basis of functions on a compact group (like SU(2)) is given by the matrix elements of the unitary irreducible representation of the group. See appendix below for a short discussion of reps Djmn of SU(2).
So we can expand a generic function f on SU(2) as
f(g) = Σ fjmnDjmn(g)
This is extendable in an obvious way to our functionals ψ(Γ, f) (too many indices to write out here though).
Using this basis for functionals on 𝓗kin we can define a representation of the Holonomy/Flux algebra that I introduced at the end of post 5. What we want is something like
Aia ψ[A] = Aia ψ[A]
Eai ψ[A] = (𝛿/𝛿Aia) ψ[A]
where the bold entries are the operators which operate on the wave functionals ψ[A] and 𝛿/𝛿 denotes functional differentiation.
Now that we know that we can represent our wave functionals as expansions of matrix elements of group elements (the D's), this becomes straightforward. For example, if h𝛾[A] is the operator representing the holonomy along a curve 𝛾, then its action on a functional is got just by modifying the functional by multiplying each basis element by its value ( an SU(2) element). So for example if Djmn(he[A]) appears in the expansion of the functional, we replace it by Djmn(h𝛾[A]he[A])
The action of the flux can be defined too, but is a bit messier as it involves the functional derivative, so I won't attempt to type it in.
We have a Hilbert space and a representation of the operators. To get to the Hilbert space of physical states we need to impose the constraints. This is done in 3 stages, one for each constraint
𝓗kin ->GaussConstraint->𝓗0kin ->DiffeomorphismConstraint->𝓗Diff -> HamiltonianConstraint->𝓗phys
Next post I'll talk a bit about the implementation of the Gauss constraint. States in the resulting Hilbert space are called spin network states.
Appendix - Irreducible reps of SU(2)
Denote by σ1, σ2, σ3 the Pauli spin matrices (familiar to everybody from angular momentum bollox in elementary QM). Remember that you can find wavefunctions which are simultaneous eigenfunctions of the total angular momementum operator L2 and Lz ? In group theoretic terms this is because σ2 = (σ1)2+(σ2)2+(σ3)2 is a Casimir operator.
So, we can find vectors which are simultaneously eigenvectors of σ2 and (say) σ3. Traditionally denote them by
|jm> where j=0,1/2, 1, 3/2.... and m=-j, -j+1,....-1,0,1, .....j-1, j
j labels the total angular momentum and m the z component of the angular momentum.
A generic rotation on 3 space can be written as
R(α,β,γ) = e-iασ1e-iασ2e-iασ3
where α,β,γ are the Euler angles that parametrize the rotation.
(What's happening here is that the Pauli spin matrices are elements of the Lie algebra of SU(2), and to generate elements of the Lie group, you need to exponentiate).
The irreducible representations of SU(2) are given by the matrices
D[j]mn(g) = <jm|R(α,β,γ)|jn>
where the group element g is parametrised by the Euler angles α,β,γ.