Spin foam model of 3d gravity

There are essentially two methods for constructing the spin foam sum of 3d gravity. One of them departs from a canonical quantization, and the sum over spin foams results from the imposition of the scalar constraint [38]. The second method is covariant in the sense that it does not involve a 2+1 decomposition: it starts from the BF-formulation (with d = 3, G=SU(2)), and reaches the spin foam sum by means of a dual transformation¹².

Here, we will sketch the second of the two methods. Note that the same construction exists also for Lorentzian 3d gravity, i.e. for gauge group SO(2,1).

We begin by replacing the continuous 3-manifold by a triangulation κ. We also choose a complexκ^∗ that is dual toκ. In the following, only the 0-, 1- and 2-cells ofκ^∗will be relevant.

Thus, we may remove the higher-dimensional cells and think ofκ^∗ as a 2-complex. The one-form B is translated into a Lie algebra element B_l for each edge¹³ l of the triangulation κ.

As in canonical LQG, the connection A is replaced by holonomies—group elements g_l^∗ that are associated to each dual edge l^∗ of the dual complex κ^∗. Since edges are in one-to-one correspondence to dual faces f^∗, we can write the B-variable also as B_f^∗. The BF action (1.176) is represented as

S = X

f^∗∈κ^∗

tr (B_f^∗U_f^∗) , (1.186)

where U_f^∗ denotes the product g_l^∗_n· · ·g_l^∗

1 of group elements around the dual face f^∗. The rational behind this is the naive continuum limit: for

U_f^∗ ≡1+F_f^∗ ≈1 (1.187)

we can find smooth fields B and A such that X

The partition function of the quantum theory is defined by the path integral Z(κ) = Lebesgue measure on R³. It can be shown that the integration over each B_f^∗ yields a delta function of the holonomy U_f^∗, i.e.

At this point, the theory has the form of a pure lattice gauge theory, and we can apply the dual transformation to the spin foam model: by that we mean that we expand the amplitude for each dual face in characters, and subsequently integrate over group variables.

12For any pure gauge theory, there exists a dual transformation to a spin foam model, which we explain in more detail in chapter 4.

13In this section, we use the letterl for edges to avoid confusion with the triade.

The character expansion is¹⁴:

δ(Uf^∗) = X

j∈N⁰/2

(2j+ 1)χj(Uf^∗), (1.192)

and by inserting this in (1.191) we get Z(κ) =

Z Q

l^∗∈κ^∗

dg_l^∗

X

{j_f∗}

Y

f^∗∈κ^∗

(2j_f^∗+ 1)χ_j(U_f^∗). (1.193) The holonomy around a dual face is a product g_l^∗_n· · ·g_l^∗

1, so we can rewrite the character as a contraction of representation matrices of the gl^∗_i:

χ_j(U_f^∗) =R_j(g_ln^∗)· · ·R_j(g_l^∗

1) (1.194)

The next step is to interchange the order of integral and sum, and to integrate out the group variables. Here, we only state the result, as it is explained in more detail in chapter 4. The integration produces a sum over assignments of invariant tensors I_l^∗ to dual edges, and an amplitude factor for each vertex of the dual complex (or tetrahedron of the triangulation):

Z(κ) = X

{j_f∗}

X

{I_l∗}

Y

f^∗∈κ^∗

(2jf^∗+ 1) Y

v^∗∈κ^∗

Av^∗. (1.195)

The admissible tensorsI_l^∗come from an orthonormal basis of invariant tensors inV_j1⊗V_j2⊗. . . , wherej₁, j₂, . . . are the labels of the dual faces incident onl^∗. The vertex amplitude results from contracting all invariant tensors from edges that run into the vertex: we symbolize this by

A_v^∗ =

j12

j14

j23

j24

j34

I

^l∗1

I

^l∗2

I

^l∗3

I

^l∗4 . (1.196)

Here, the j_ij’s denote the labels of the dual faces that are bounded by pairs of edges l^∗_i, l_j^∗. In the present case (d = 3, G =SU(2)), each dual edge has only three incident dual faces, and the invariant tensor is uniquely fixed by the labels. The sum over Il^∗ in (1.195) drops out, and (1.196) is simply the 6j-symbol.

Each term in (1.195) corresponds to a labelling of the 2-complex κ^∗ with irreducible representations and invariant tensors. We call such a labelled 2-complex a spin foam, and regard (1.195) as a sum over spin foams. In general, spin foams may have different underlying 2-complexes, but in the sum (1.195) the complex is fixed to be κ^∗.

More generally, the transition from path integral to spin foam sum can be also done for transition amplitudes, i.e. for path integrals that are weighted with boundary functionals.

As in canonical LQG, the states translate into superpositions of spin networks, and the admissible spin foams in the sum are those that interpolate between the boundary spin networks.

We should mention that there appear divergences in the BF spin foam sum, which can be interpreted as a consequence of a symmetry in the original BF action. There is a prescription

14Strictly speaking, the integration overBgives˜δ(U) =P

j∈N0(2j+ 1)χ_j(U_f∗), where the sum runs only overevenj (see Appendix B in [39]). It is customary, however, to include also half-integer representations.

for dividing out the infinite gauge volume associated to this symmetry, which makes tran-sition amplitudes mathematically well-defined [40]. The resulting “renormalized” partition function Z_r(κ) is still divergent, but independent of the original choice of triangulation κ, i.e.

Z_r(κ) = Z_r(κ⁰) (1.197)

In that sense, Zr is a topological invariant.

The above technique provides a rigorous quantization of 3d gravity, and it can be shown to be equivalent with the quantization via Chern-Simons theory [41]. Furthermore, it is possible to include fermionic matter in the model and describe scattering between spinning massive particles. This has been done in a first-quantized approach at first [42], and later been extended to a combined perturbative quantization of scalar field theory and 3d gravity¹⁵ [43]. The associated amplitudes equal that of an effective field theory without gravity that lives on a non-commutative spacetime.

1.5.3 4d gravity as a constrained BF theory

When we come to 4-dimensional gravity, the situation is expected to be profoundly different:

now the theory has local degrees of freedom. The action can no longer be written in the BF form, and the dual transformation cannot be applied directly, since it would require the integration over the tetrad fields.

What we can do, however, is to reexpress the gravity action by a BF action with additional constraints. The strategy is to start from this constrained BF action, and construct a modified BF spin foam model from it.

We begin by discussing the classical theory. In the next subsection, we sketch the transi-tion to the quantum theory and spin foams, which yields the so-called Barret-Crane model.

The latter is the most studied 4d model in the literature, but there exist also other propos-als. Here, we restrict the discussion to the Barret-Crane model, and follow the derivation described in [44]. The signature is positive, i.e. G=SO(4).

The idea of constraining BF theory originates from the Plebanski formulation of gravity:

it has the action

S = Z

B^IJ ∧F_IJ(A)−1

2φ_{IJ KL}B^IJ ∧B^KL (1.198)

where the Lagrange multiplier φ satisfies φ_{IJ KL}=−φ_{J IKL} =−φ_{IJ LK} =φ_KLIJ and

^{IJ KL}φIJ KL = 0. (1.199)

The equations of motion read

DB = 0, (var. of ω) (1.200)

F_IJ(A) = φ^{IJ KL}B_KL, (var. of B) (1.201)

B^IJ ∧B^KL =e ^{IJ KL}, (var. of φ) (1.202)

where we abbreviated

e:= 1

4!IJ KLB^IJ ∧B^KL. (1.203)

For configurations that have e6= 0, equation (1.202) is equivalent to

C:=_{IJ KL}B_αβ^IJB_γδ^KL−˜e _αβγδ = 0 (1.204)

15One expands w.r.t. the coupling between scalar field and gravity. The treatment of the pure gravity term remains non-perturbative.

with

˜ e:= 1

4!_{IJ KL}B_αβ^IJB_γδ^KLαβγδ. (1.205)

One can show that the solutions to this equation areB’s that are formed from co-triad fields e^I. More precisely, there are two sectors of solutions, namely,

(I) B^IJ =±^{IJ KL}e_K∧e_L, (1.206)

and

(II) B^IJ =±e^I ∧e^J. (1.207)

By plugging (I) back into (1.198), together with the “solution” µ = 0, we arrive at the Hilbert-Palatini action

S = Z

_{IJ KL}e^I∧e^J ∧F^KL(A). (1.208)

This tells us that the Plebanski action admits three sectors of solutions,

e= 0, (1.209)

e6= 0, (I), (1.210)

e6= 0, (II), (1.211)

and the second one corresponds to gravity. Clearly, the last two sectors could be also obtained by using instead the BF action

S = Z

B^IJ ∧F_IJ(A) (1.212)

with the additional constraints

e6= 0, and C = 0. (1.213)

Im Dokument Semiclassical analysis of loop quantum gravity (Seite 36-39)