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**Aug 24, 2010 12:14 pm**Post7: Spin Networks and Geometric Operators

We have a graph with some nodes and L links joining some of them (for the moment consider this to be embedded in three space. It will eventually become an abstract graph). If we have a function f mapping L copies of SU(2) to the complex numbers then the pair ( Γ, f) defines a functional on the space of connections. To use it to get a complex number from the connection A, just create L SU(2) elements by taking the holonomies of A along each of the L edges, then applying the function f. Given another function f' like f, we showed how to define a scalar product between f and f'.

We defined a basis for functions on SU(2) by expanding in terms of the irreps of SU(2). We used this to define a basis for the functions on SU(2)L.

The next job is to impose the Gauss constraint, i.e. chuck away the states in 𝓗kin that aren't SU(2) gauge invariant. Somewhere a few posts back I said that under gauge transformations, a holonomy he along an link e gets mapped to

gstartheg-1end

where gstart is the group element at the node at the start of the link e, and g-1end is the (inverse of) the group element at node at the termination of link e. Now since typically several links connect to any given node, there are several simultaneous gauge group elements to take into account at that node. We want the overall effect to be such that our functional ψ(Γ, f) is invariant under this gauge transformation.

What we need is some way to connect up, at each node, the collection of SU(2) holonomies consistently into a complex number that's invariant under the SU(2) transformations. The object that enables you to do this is called an "intertwiner", or Clebsch-Gordan coefficient. We thus require an intertwiner at each node.

There is a nice illustration of the operation of the intertwiner given in appendix A1 of Carlo Rovelli's book (a preprint of the book used to be available online, but I can't find this anymore):

Suppose we have 3 links meeting at a node, and the links are assigned spin 5/2, 5/2 and 2 representations. Then we can think of a link with spin 5/2 as bringing 5 fundamental representations (each spin 1/2) to the node - this can be represented by 5 lines coming in. Similarly for the other two links. The intertwiner at the node connects up these fundamentals with each other as shown. The "connecting up" is basically contraction of spinor indices - the intertwiner is an object with indices which are contracted with various indices of the "incoming" spinors at the node.

By taking states corresponding to a graph Γ, an assignment of an SU(2) representation jl to each link l, and an intertwiner in to each node n, we have succeeded in imposing the Gauss constraint. The graph with its set of representations assigned to the links and intertwiners assigned to the nodes is called a spin network.

Spin networks were thought up, essentially out of the blue, by Roger Penrose a long time ago (see this 1971 link ). Penrose suspected at the time that spacetime at a fundamental level was combinatorial in character. LQG seems to have given new life to that idea.

The Hilbert space of functionals 𝓗0kin is the direct sum over all graphs Γ of the Hilbert spaces H0Γ of functionals defined by a given graph Γ. Moreover H0Γ itself decomposes as the direct sum over the intertwiner spaces. (This is the analog of the Fock space decomposition of the free field Hilbert space in standard QFT).

Observables

For a given spin network state , an area operator can be defined (in terms of the flux operator which I mentioned – but did not write down – in post 6). For a given two-surface S it has eigenvalues which satisfy

A(S) ψΓ = Σ hSQRT( γ2 jp(jp+1) ) ψΓ

where the sum is over all points p where the links of the spin network intersect the surface S, and jp is the . (There are some technicalities, but this is the general idea, illustrated in the diagram below). Note that spin network states are eigenstates of this operator.

Analogously, there is a volume operator. Calculation of its eigenvalues is a bit more complicated, but basically if you have a 3 dimensional sub-region R of three space, the volume of R receives a contribution from each node of the spin network that lies inside R. See diagram:

We have a graph with some nodes and L links joining some of them (for the moment consider this to be embedded in three space. It will eventually become an abstract graph). If we have a function f mapping L copies of SU(2) to the complex numbers then the pair ( Γ, f) defines a functional on the space of connections. To use it to get a complex number from the connection A, just create L SU(2) elements by taking the holonomies of A along each of the L edges, then applying the function f. Given another function f' like f, we showed how to define a scalar product between f and f'.

We defined a basis for functions on SU(2) by expanding in terms of the irreps of SU(2). We used this to define a basis for the functions on SU(2)L.

The next job is to impose the Gauss constraint, i.e. chuck away the states in 𝓗kin that aren't SU(2) gauge invariant. Somewhere a few posts back I said that under gauge transformations, a holonomy he along an link e gets mapped to

gstartheg-1end

where gstart is the group element at the node at the start of the link e, and g-1end is the (inverse of) the group element at node at the termination of link e. Now since typically several links connect to any given node, there are several simultaneous gauge group elements to take into account at that node. We want the overall effect to be such that our functional ψ(Γ, f) is invariant under this gauge transformation.

What we need is some way to connect up, at each node, the collection of SU(2) holonomies consistently into a complex number that's invariant under the SU(2) transformations. The object that enables you to do this is called an "intertwiner", or Clebsch-Gordan coefficient. We thus require an intertwiner at each node.

There is a nice illustration of the operation of the intertwiner given in appendix A1 of Carlo Rovelli's book (a preprint of the book used to be available online, but I can't find this anymore):

Suppose we have 3 links meeting at a node, and the links are assigned spin 5/2, 5/2 and 2 representations. Then we can think of a link with spin 5/2 as bringing 5 fundamental representations (each spin 1/2) to the node - this can be represented by 5 lines coming in. Similarly for the other two links. The intertwiner at the node connects up these fundamentals with each other as shown. The "connecting up" is basically contraction of spinor indices - the intertwiner is an object with indices which are contracted with various indices of the "incoming" spinors at the node.

By taking states corresponding to a graph Γ, an assignment of an SU(2) representation jl to each link l, and an intertwiner in to each node n, we have succeeded in imposing the Gauss constraint. The graph with its set of representations assigned to the links and intertwiners assigned to the nodes is called a spin network.

Spin networks were thought up, essentially out of the blue, by Roger Penrose a long time ago (see this 1971 link ). Penrose suspected at the time that spacetime at a fundamental level was combinatorial in character. LQG seems to have given new life to that idea.

The Hilbert space of functionals 𝓗0kin is the direct sum over all graphs Γ of the Hilbert spaces H0Γ of functionals defined by a given graph Γ. Moreover H0Γ itself decomposes as the direct sum over the intertwiner spaces. (This is the analog of the Fock space decomposition of the free field Hilbert space in standard QFT).

Observables

For a given spin network state , an area operator can be defined (in terms of the flux operator which I mentioned – but did not write down – in post 6). For a given two-surface S it has eigenvalues which satisfy

A(S) ψΓ = Σ hSQRT( γ2 jp(jp+1) ) ψΓ

where the sum is over all points p where the links of the spin network intersect the surface S, and jp is the . (There are some technicalities, but this is the general idea, illustrated in the diagram below). Note that spin network states are eigenstates of this operator.

Analogously, there is a volume operator. Calculation of its eigenvalues is a bit more complicated, but basically if you have a 3 dimensional sub-region R of three space, the volume of R receives a contribution from each node of the spin network that lies inside R. See diagram: