Physical boundary state for the quantum tetrahedron by Livine and Speziale

In this paper Levine and Specziale consider the stability under evolution as a criterion to select a  physical boundary state for the spinfoam formalism. They apply this to the simplest spinfoam defined by a single quantum tetrahedron and solve the associated eigenvalue problem at leading  order in the large spin limit.

They show that this fixes uniquely the free parameters entering the boundary state. The state obtained this way gives a correlation between edges which varies inversely with the distance  between the edges, in agreement with the linearized continuum theory. They argue that this  correlator represents the propagation of a pure gauge which is consistent with the absence of  physical degrees of freedom in 3d general relativity.

The covariant spinfoam formulation of Loop Quantum Gravity gives a regularized expression for the path integral of the full theory. The state of the art in spinfoams is the proposal for computing the graviton propagator in non-perturbative quantum gravity by working with a bounded region of space-time and introducing a semiclassical state peaked on a given boundary geometry. This boundary state gauge-fixes the spinfoam amplitude and allows to compute the
(two-point) correlations for the gravitational field ,so inducing a non-trivial bulk geometry. The boundary state is a taken as a Gaussian state with a phase factor in the Hilbert space of boundary spin networks.

This paper addresses issues in 3d Riemannian quantum gravity, because  the theory is much simpler in three dimensions. Gravity is topological and we know how to spinfoam quantize it exactly as the Ponzano-Regge model. We can solve this toy model explicitly and we know the physical states.

In this paper the authors consider the smallest 3d triangulation, that is a single tetrahedron. The corresponding Ponzano-Regge spinfoam amplitude is simply Wigner’s {6j} symbol for the unitary group SU(2).

Following the setting introduced in the post Towards the graviton from spinfoams: The 3d toy model, the authors consider the
‘time-fixed’ tetrahedron: out of its six edges, study the correlations between the fluctuations of two opposite edges while freezing the lengths of the remaining four edges. The fixed-length edges define the time interval associated to that piece of 3d spacetime, while the two fluctuating edges are distinguished as the initial and final edges.

By defining a physical state as a wave function unaffected by time evolution,  the phased Gaussian ansatz is shown to be a physical state at first order in the asymptotical regime. The physical state criterion uniquely fixes the width of the Gaussian. From a covariant point of view looking at the tetrahedron, there is a unique physical state, namely the flat connection boundary state.



From the propagation kernel to the boundary state

Consider a scalar φ in the Schrodinger picture , and introduce two spacelike hyperplanes in Minkowski spacetime, separed by a time T. Denote by  φ1 and φ2 two classical field configurations associated with these planes, and ψn[φ] a complete basis of energy eigenstates. Given a state:

physical boundary equ 1

on the initial plane, the evolution to the final state can be written in terms of the propagation kernel K[φ1, φ2, T] through:physical boundary equ 2

The Quantum Tetrahedron

Consider the quantum tetrahedron introduced in the post Towards the graviton from spinfoams: The 3d toy model

physical boundary fig 1
Given a tetrahedron of edge lengths ℓe, fix four opposite ones to j, and the remaining two to a and b., orient it in such a way that we can think of it as representing the evolution of the edge a into the edge b, in a time j.

The dynamics of this model was studied at both the classical and quantum level in the post: Background independence in a nutshell: The dynamics of a tetrahedron.

The classical dynamics is encoded in the Regge action:

physical boundary equ 2a

where θe are the dihedral angles of the tetrahedron. The quantum dynamics can be studied as by Ponzano-Regge, by making the assumption that the lengths in the quantum theory can only have half-integer values ℓe = je+ 1/2 , and associating with the tetrahedron the amplitude

physical boundary equ 4

In this expression da ≡ 2a + 1 is the dimension of the spin-a representation of SU(2) and the {6j} is Wigner’s 6j-symbol for the recoupling theory of SU(2).

With this simple form of amplitude K[φ1, φ2, T] the stability condition for physical states  simply is:

physical boundary equ 4a

If we view this tetrahedron as part of a triangulation of flat 3d space between the two planes, we can also introduce the “asymptotic time”

physical boundary equ 5

In the isosceles p case a = b = jo, we have

physical boundary equ 6

Diagonalizing the kernel
Consider the kernel as a dj -by-dj matrix Kab[j] = K[a, b, j]. This matrix
satisfies K^2 ≡ 1 for all j, thus the evolution generated by it is unitary, K is diagonalizable and all its eigenvalues are ±1. Using the notation ψn(a) = <a|n> to indicate the n-th eigenvector in the basis a, we have

physical boundary equ 7
The presence here of two possible eigenvalues ±1, is because the  Ponzano-Regge model sums over both orientation of the tetrahedron.

The next step is to work out the eigenvectors solving the associate eigenvalue problem,

physical boundary equ 8

The general explicit solutions to this equation are not known at the moment. So at this stage we can not study exact physical semiclassical states as linear combinations of eigenstates. The equation can however  be solved approximately.

Semiclassical states
The perturbative expansion considered is the large spin limit , where the {6j} symbol is dominated by exponentials of the Regge action

physical boundary equ 9
This is the property that makes it possible to show that the quantum theory based upon the {6j} has a  semiclassical limit.

To the lowest order of the wavefunction should satisfy

physical boundary equ 8

with the linearized kernel and be peaked around q. In the simple setting considered here, the intrinsic and extrinsic geometry of q are specified giving a value jo for edge length and a value θo for its dihedral angle. The latter is chosen  in such a way that the complete background tetrahedron is given by the isosceles configuration a = b = jo. For the exterior dihedral angles associated to jo and j, elementary geometry gives respectively:

physical boundary equ 10

By analogy with the continuum, we expect a physical semiclassical state to be implemented in this approximation by a Gaussian state around q = (jo, θo), for which we make the following assumption:

physical boundary equ 12

Here N is the normalization, σ the width and φ is a phase that is undetermined for the moment. For σ scaling linearly with the spins, this Gaussian is peaked on q in the large spin limit as seen in the post: A semiclassical tetrahedron

These kind of states have been extensively used as ansatz for leading order semiclassical states in the recent spinfoam literature. A semiclassical state typically has a precise width uniquely fixed by the dynamics (e.g. the factor ω in the exponent of the vacuum Gaussian of the harmonic oscillator).

Thus a crucial question is whether also in spinfoams the width can be fixed requiring the state to solve the dynamics.

physical boundary equ 12

Is an eigenstate of the kernel for a unique choice of the width, given by:

physical boundary equ 12a

and for two choices of φ, corresponding to eigenvalues ±1:
physical boundary equ 12b

Here ~ means at first order in the large spin limit. There is a clear restriction that should be kept in mind:

physical boundary equ 9

holds only if the volume V appearing in it is real. For a generic configuration (a, b, j) this is not always the case. The quantum range of a and b is [0, 2j], and the condition V real is violated when the endpoints are approached by one of the variables. Specifically,  have V ~0 for a and/or b close to zero, and V unreal for a and/or b close to 2j. The calculations here only apply in the regime a ~b ~ j.

The end result of these calculations being that the physical boundary state is given by:

physical boundary equ 13

Physical state in the general boundary formalism

In the so-called general boundary formalism, the four bulk edges are varied freely.  The boundary state then has to carry information on the background value of the (intrinsic and  extrinsic) geometry of all six edges, and correlations between all six of them can be computed.
Such a general boundary formalism  is used for the 4d spinfoam graviton calculations.

In the particular 3d case the fact that the theory is topological strongly simplifies this analysis,  because once the topology and the triangulation are fixed, there is a single physical state.  Assuming trivial topology, this is given in the group representation by:

physical boundary equ 14

where gf, represents the gravitational holonomy on a closed path ∂f. The product is over the independent faces, and the condition F = 0 is ensured everywhere.

In the case considered here, the triangulation is given by a single tetrahedron. Then gf is a product of four deltas ensuring the flatness of each face. Only three faces are independent, so can get rid of one delta. Fixing the orientation of the edges as in the diagram below.

physical boundary fig 2

The state can be written as:

physical boundary equ 15

The physical boundary state coincides with the kernel, so that correlations now read:

physical boundary equ 16

This result  can be understood as follows. The boundary state ψo(j1…j6) is the state induced by the (exterior) bulk geometry onto the tetrahedron. Having assumed a trivial boundary topology (homomorphic to S2) and a trivial bulk topology – obtain a spinfoam amplitude in {6j} which is naturally associated to the triangulation of the closed S3 manifold with two tetrahedra.

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