# Ising Spin Network States for Loop Quantum Gravity by Feller and Livine

This week I have been studying a great paper by Feller and Livine on Ising Spin Network States for Loop Quantum Gravity. In the context of loop quantum gravity, quantum states of geometry are defined as
spin networks. These are graphs decorated with spin and intertwiners, which represent quantized excitations of areas and volumes of the space geometry. In this paper the authors develop the condensed matter point of view on extracting the physical and geometrical information out of spin network states: they
introduce  Ising spin network states, both in 2d on a square lattice and in 3d on a hexagonal lattice, whose correlations map onto the usual Ising model in statistical physics. They construct these states from the basic holonomy operators of loop gravity and derive a set of local Hamiltonian constraints which entirely characterize the states. By studying their phase diagram distance can be reconstructed from the correlations in the various phases.

The line of research pursued in this paper is at the interface between condensed matter and quantum information and quantum gravity on the other: the aim is to understand how the distance can be recovered from correlation and entanglement between sub-systems of the quantum gravity state. The core of the investigation is the correlations and entanglement entropy on spin network states. Correlations and especially entropy are of special importance
for the understanding of black holes dynamics. Understanding the microscopic origin of black holes entropy is one the major test of any attempt to quantify gravity and entanglement between the horizon and its environment degrees of freedom appears crucial

A spin network state is defined on a graph, dressed
with spins on the edge and intertwiners at the vertices.
A spin on an edge e is a half-integer je 2 N=2 giving an
irreducible representation of SU(2) while an intertwiner
at a vertex v is an invariant tensor, or singlet state, between
the representations living on the edges attached
to that vertex. Spins and intertwiners respectively carry
the basic quanta of area and volume. The authors build the spin
network states based on three clear simplifications:

1.  Use a fixed graph, discarding graph superposition and graph changing dynamics and  work with a fixed regular
lattice.
2. Freeze all the spins on all the graph edges. Fix to smallest possible value, ½, which correspond to the most basic excitation of geometry in loop quantum gravity, thus representing a quantum geometry directly at the Planck scale.
3. Restricted  to 4-valent vertices, which represent the basic quanta of volume in loop quantum gravity, dual to quantum tetrahedra.

These simplifications provide us with the perfect setting to map spin network states, describing the Planck scale quantum geometry, to qubit-based condensed matter models. Such models have been extensively studied in statistical physics and much is known on their phase diagrams and correlation functions, and we hope to be able to import these results to the context of loop quantum gravity. One of the most useful model is the Ising model whose relevance goes from modeling binary mixture to the magnetism of matter. We thus naturally propose to construct and investigate Ising spin
network states.

The paper reviews the definition of spin network and analyzes the structure of 4-valent intertwiners between spins ½ leading to the
effective two-state systems used to define the Ising spin network states. Different equivalent definitions are given in terms of the high and low temperature expansions of the Ising model. The loop representation of the spin network is then obtained and studied as well as the associated density which gives information about
parallel transport in the classical limit. Section III introduces

It then  introduces a set of local Hamiltonian constraints for which
the Ising state is a unique solution and elaborates on their usefulness for understanding the coarse-graining of
spin network  and the dynamic of loop quantum gravity.

After this, it discusses the phase diagram of the Ising
states and their continuum limit as well as the distance
from correlation point of view.

Ising Spin Network State

Spin network basis states  define the basic excitations of the quantum geometry and they are provided with a natural interpretation in terms of discrete geometry with the spins giving the quanta of area and the intertwiners giving the quanta of volume.

These 4-valent vertices will be organized along a regular lattice. The 3d diamond lattice and the 2d square lattice are considered . Looking initially at the 2d square the square lattice: in this setting, the space of 4-valent interwiners between four spins ½ is two dimensional – it can be decomposed into spin 0 and spin 1 states by combining the spins by pairs, as

Different such decompositions exist and are shown as  a graphical representation below. There are three such decompositions, depending on which spins are paired together, the  s, t and u channels.

The spin 0 and 1 states in the s channel  can be explicitly written in terms of the up and down states of the four spins:

Those two states form a basis of the intertwiner space.  There transformation matrices between this basis and the two other channels:

Let’s look at the  intertwiner basis  defined in terms of the square volume operator U of loop quantum gravity. Since the spins, and the area quanta, are fixed, the only freedom left in the spin network states are the volume quanta defined by the intertwiners. This will provide the geometrical interpretation of our spin network states as
excitations of volumes located at each lattice node. For a 4-valent vertex, this operator is defined as:

where J are the spin operators acting on the i link.

The volume itself can then obtained by taking the square root of the absolute value of U . Geometrically, 4-valent intertwiners are interpreted as representing quantum tetrahedron, which becomes the building block of the quantum geometry in loop quantum gravity  and spinfoam models. U takes the following form in the s channel basis:

The smallest  possible value of a chunk of space is the square volume ±√3/4 in Plank units.

The two oriented volume states of û,  | u ↑,↓〉 , can be considered as
the two levels of an effective qubit. Let’s now define a pure spin network state which maps its quantum fluctuations on the thermal fluctuations of a given classical statistical model such as the Ising model by

This state represents a particular configuration of the spin network and the full state is a quantum superposition of them all. Defined as such, the state is unnormalized but its norm is easily computed using the Ising partition function ZIsing:

The intertwiner states living at each vertex are now entangled and carry non-trivial correlations. More precisely this state exhibits Ising correlations between two vertex i, j:

Those correlations are between two volume operators at different vertex which are in fact components of the 2- point function of the gravitational field. So understanding how those correlations can behave in a non-trivial way is a first step toward understanding the behavior of the full 2-point gravity correlations and for instance recover the inverse square law of the propagator.

The generalization to 3d is straightforward. Keeping the requirement that the lattice be 4-valent the natural
regular lattice is the diamond lattice

Using the usual geometrical interpretation of loop quantum gravity, this lattice can be seen as dual to a triangulation of the 3d space in terms of tetrahedra dual to each vertex. This can be seen as an extension of the more used cubic lattice better suited to loop quantum gravity. The Ising spin network state and the whole
set of results which followed are then identical :

• The wave function

• The Hamiltonian constraints and their algebra  are the same.

In 3d, the Ising model also exhibits a phase transition

Information about the 2-points correlation functionssuch as long distance behavior at the phase transition or near it can be obtained using methods of quantum field theory. In d dimensions we have

where K(r) are modified Bessel functions and ξ is the
correlation length. For the three-dimensional case, we
have the simple and exact expression

Conclusions

In this paper, the authors have introduced a class of spin network
states for loop quantum gravity on 4-valent graph. Such 4-valent graph allows for a natural geometrical interpretation in terms of quantum tetrahedra glued together into a 3d triangulation of space, but it also allows them to be map the degrees of freedom of those states to effective qubits. Then we can define spin network states corresponding to known statistical spin models, such as the Ising model, so that the correlations living on the spin network are exactly the same as those models.
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# Polyhedra in spacetime from null vectors by Neiman

This week I have been studying a nice paper about Polyhedra in spacetime.

The paper considers convex spacelike polyhedra oriented in Minkowski space. These are classical analogues of spinfoam intertwiners. There is  a parametrization of these shapes using null face normals. This construction is dimension-independent and in 3+1d, it provides the spacetime picture behind the property of the loop quantum gravity intertwiner space in spinor form that the closure constraint is always satisfied after some  SL(2,C) rotation.These  variables can be incorporated in a 4-simplex action that reproduces the large-spin behaviour of the Barrett–Crane vertex amplitude.

In loop quantum gravity and in spinfoam models, convex polyhedra are fundamental objects. Specifically, the intertwiners between rotation-group representations that feature in these theories can be viewed as the quantum versions of convex polyhedra. This makes the parametrization of such shapes a subject of interest for  LQG.
In kinematical LQG, one deals with the SU(2) intertwiners, which correspond to 3d polyhedra in a local 3d Euclidean frame. These polyhedra are naturally parametrized in terms of area-normal vectors: each face i is associated with a vector xi, such that its norm
equals the face area Ai, and its direction is orthogonal to the face. The area normals must satisfy a ‘closure constraint’:

Minkowski’s reconstruction theorem guarantees a one-to-one correspondence between space-spanning sets of vectors xi that satisfy (1) and convex polyhedra with a spatial orientation. In
LQG, the vectors xi correspond to the SU(2) fluxes. The closure condition  then encodes the Gauss constraint, which also generates spatial rotations of the polyhedron.

In the EPRL/FK spinfoam, the SU(2) intertwiners get lifted into SL(2,C) and are acted on by SL(2,C) ,Lorentz, rotations. Geometrically, this endows the polyhedra with an orientation in the local 3+1d Minkowski frame of a spinfoam vertex. The polyhedron’s
orientation is now correlated with those of the other polyhedra surrounding the vertex, so that together they define a generalized 4-polytope. In analogy with the spatial case, a polyhedron with spacetime orientation can be parametrized by a set of area-normal
simple bivectors Bi. In addition to closure, these bivectors must also satisfy a cross-simplicity
constraint:

In this paper, the author presents a different parametrization of convex spacelike polyhedra with spacetime orientation. Instead of bivectors Bi, they associate null vectors i to the polyhedron’s
faces. This parametrization does not require any constraints between the variables on different faces. It is unusual in that both the area and the full orientation of each face are functions of the data on all the faces. This construction, like the area-vector and
area-bivector constructions above, is dimension-independent. So we can parametrize d-dimensional convex spacelike polytopes with (d − 1)-dimensional faces, oriented in a (d + 1)-dimensional Minkowski spacetime.  These variables can be to construct an action principle for a Lorentzian 4-simplex. The action principle reproduces the large spin behaviour of the Barrett–Crane spinfoam vertex. In particular, it recovers the Regge action for the classical simplicial gravity, up to a possible sign and the existence of additional,degenerate solutions.

In d = 2, 3 spatial dimensions, the parametrization is  contained in the spinor-based description of the LQG intertwiners. There, the face normals are constructed as squares of spinors. It was observed that the closure constraint in these variables can always be satisfied by acting on the spinors with an SL(2,C) boost.  The simple spacetime picture presented in this paper is new. Hopefully, it will contribute to the geometric interpretation of the modern spinor and twistor variables in LQG.

The parametrization
Consider a set of N null vectors liμ in the (d + 1)-dimensional Minkowski space Rd,1, where i = 1, 2, . . . ,N and d ≥2.  Assume the following conditions on the null vectors liμ.

• The liμ span the Minkowski space and  N ≥  d + 1.
• The  liμ  are either all future-pointing or all past-pointing.

The central observation in this paper is that such sets of null vectors are in one-to-one correspondence with convex d-dimensional spacelike polytopes oriented in Rd,1.

Constructing the polytrope

Consider a set {liμ} ,take the sum of the liμ normalized
to unit length:

The unit vector nμ is timelike, with the same time orientation as the liμ. Now take nμ to be the unit normal to the spacelike polytope. To construct the polytope in the spacelike hyperplane ∑ orthogonal to nμ define the projections of the null vectors liμinto this hyperplane:

The spacelike vectors siμ  automatically sum up to zero. Also, since the liμ span the spacetime, the siμ must span the hyperplane ∑ . By the Minkowski reconstruction theorem, it follows that the siμ are the (d − 1)-area normals of a unique convex d-dimensional polytope in . In this way, the null vectors li define a d-polytope oriented in spacetime.

Basic features of the parametrization.

The vectors  are liμ  associated to the polytope’s (d −1)-dimensional faces and are null normals to these faces. The orientation of a spacelike (d − 1)-plane in Rd,1 is in one-to-one correspondence with the directions of its two null normals. So each liμ carries partial information about the orientation of the ith face. The second null normal to the face is a function of all the liμ. It can be expressed as:

where  nμ is given by

Similarly, the area Ai of each face is a function of the
null normals liμ to all the faces:

The total area of the faces has the simple expression:

A (d+1)-simplex action

To construct a (d + 1)-simplex action that reproduces in the d = 3 case the large-spin behaviour of the Barrett–Crane spinfoam vertex.

At the level of degree-of-freedom counting, the shape of a (d +1)-simplex is determined by the (d + 1)(d + 2)/2 areas Aab of its (d − 1)-faces. These areas are directly analogous to the spins that appear in the Barrett–Crane spinfoam. Let us fix a set of values for Aab and consider the action:

Then restrict to the variations where:

The stationary points of the action  have the following properties. For each a, the vectors  labμ define a d-simplex with unit normal naμ
and (d − 1)-face areas Aab.

A (d − 1)-face in a (d + 1)-simplex, shared by two d-simplices a and b. The diagram depicts the 1+1d plane orthogonal to the face. The dashed lines are the two null rays in this normal plane.

The d-simplices automatically agree on the areas of their shared (d −1)-faces. The two d-simplices agree not only on the area of their shared (d − 1)-face, but also on the orientation of its (d − 1)-plane in spacetime. In other words, they agree on the face’s area-normal bivector:

The area bivectors defined  automatically satisfy closure and cross-simplicity:

We conclude that the stationary points are in one-to-one correspondence with the bivector geometries of the Barrett-Crane model with an action of the form:

# Group field theory as the 2nd quantization of Loop Quantum Gravity by Daniele Oriti

This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference  paper – Group field theory: A quantum field theory for the atoms of space  and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:

Introduction

We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,
there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus
define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be  the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical
dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators  and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is  obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization  and Effective quantum cosmology from GFT condensates.

Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise

The group elements Gij are assigned to each link of the graph, with Gij=Gij-1. Assume  gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

Spin networks in 2nd quantization

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum –  the natural background independent, diffeo-invariant vacuum state.

The  2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible
with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field  quantum is the spin network vertex or tetrahedron – the so called building block of space.

A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties  from results in quantum simplicial geometry.

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# 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.

Introduction
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:

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

The Quantum Tetrahedron

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

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:

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

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:

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”

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

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

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,

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

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

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:

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:

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.

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

and for two choices of φ, corresponding to eigenvalues ±1:

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

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 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:

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.

The state can be written as:

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

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.

# Background independence in a nutshell: the dynamics of a tetrahedron by Rovelli et al

This week I’ve been looking at the dynamics of the quantum tetrahedron. So I’ve been reading a couple of papers and doing calcualtions in sagemath which I’ll post later.

The first paper I looked at was this paper, in which  the authors study how physical information can be extracted from a background independent quantum system. They use an extremely simple system that models a finite region of 3d euclidean quantum spacetime with a single equilateral tetrahedron. They show that the physical information can be expressed as a boundary amplitude and how the notions of ‘evolution’ in a boundary proper-time and ‘vacuum’ can be extracted from the background independent dynamics. The paper discusses the  classical theory, classical time evolution, the quantum theory and  the quantum time evolution.

Introduction
In a background independent field theory the distance and time separation must be extracted from the dynamical variables.An idea for solving this problem is to study the quantum propagator of a finite spacetime region, as a function of the boundary data.The key observation is that in gravity the boundary data include the gravitational field, the geometry of the boundary and so all relevant relative distances and time separations. The boundary formulation realizes very elegantly in the quantum context the complete
identification between spacetime geometry and dynamical fields.
Formally, the idea consists in extracting the physical information from a background independent quantum field theory in terms of the quantity;

The particle scattering amplitudes can be effectively computed from W[] in quantum gravity. This equation becomes a generalized Wheeler-DeWitt equation in the background independent context. Theboundary picture is appealing, but its implementation in the full 4d quantum gravity theory is difficult because of the technical complexity of the theory. It is useful to test and illustrate it in a simple context.That is is what is done in this paper. The authors consider riemannian general relativity in three dimensions.

To further simplify the context, the authors triangulate spacetime, reducing the field variables to a finite number. Taking a minimalist triangulation: a single tetrahedron with four equal edges. The number of variables dealt with is reduced to a bare minimum. The result is an extremely simple system, which is sufficient to realize the conceptual complexity of a background independent theory of spacetime geometry.

The authors show that this simple system has in fact a background independent classical and quantum dynamics. The classical dynamics is governed by the relativistic Hamilton function the quantum dynamics is governed by the relativistic propagator, both these
functions explicitly computed . The classical dynamics, which is equivalent to the Einstein equations, fixes relations between quantities that can be measured on the boundary of the tetrahedron. The quantum dynamics gives probability amplitudes for ensembles of boundary measurements.
The model and its interpretation are well-defined with no need of picking a particular
variable as a time variable. However, it is posssible to identify an elapsed proper time T among the boundary variables, and reinterpret the background independent theory as a theory describing evolution in the observable time T.

Two  interpretations of the model are described , in the classical as well in the quantum theory. The distinction between the nonperturbative vacuum state and the Minkowski vacuum that minimizes the energy associated with the evolution in T are illustrated , and it is shown that the usual  technique suggested for computing the Minkowski vacuum state from the nonperturbative vacuum state works in this context. This system captures the essence of background independent physics in a nutshell.

Elementary geometry of an equilateral tetrahedron

Consider a tetrahedron immersed in euclidean three-dimensional space. Let a be the length of the top edge and b the length of the bottom edge, assume that the other four side edges have equal length c. Such a tetrahedron is called an equilateral tetrahedron. There are “bottom”, “top” and “side” dihedral angles at the edges with length a, b, c. Elementary geometry gives;

Classical theory

Regge action
Consider the action of general relativity, in the case of a simply connected finite spacetime region R. In the presence of a boundary

have to add a boundary term to the Einstein–Hilbert action, in order to have well defined equations of motion. The full action reads;

Here g is the metric field, R is the Ricci scalar, n is the number of spacetime dimensions, while q is the metric, and k the trace of the extrinsic curvature.

In general, the Hamilton function of a finite dimensional dynamical system is the value of the action of a solution of the equations of motion, viewed as a function of the initial and final coordinates; the general solution of the equations of motion can be obtained from the Hamilton function. Since the bulk action vanishes on a vacuum solution of the equations of motion, the Hamilton function of general relativity reads

where the extrinsic curvature k[q] is a nonlocal function, determined by the Ricci-flat metric g bounded by q.

In the paper only the three-dimensional riemannian case is considered, where n = 3 and the signature of g is [+ + +]. The discretization of the theory is provided by a Regge triangulation. Let i be the index labelling the links of the triangulation and call li the length of the link i. In three dimensions, the bulk Regge action is;

where theta(i,t) the dihedral angle of the tetrahedron t at the link i, and the angle in the parenthesis is therefore the deficit angle at i. The boundary term is;

Notice that the angle in the parenthesis is the angle formed by the boundary, which can be seen as a discretization of the extrinsic curvature.

Choosing the minimalist triangulation formed by a single tetrahedron, and considering only the case in which the tetrahedron is equilateral. Then there are no internal links, the Regge action is the same as the Regge Hamilton function. The expression for the dihedral angles as functions of the edges length, for a flat interior geometry, gives the Hamilton function;

The dynamical model and its physical meaning
The Hamilton function (defines a simple relativistic dynamical model. The model has three variables, a, b and c, these are partial observables. That is, they include both the independent -‘time’ and the dependent-dynamical variables, all treated on equal footing.

The equations of motion are obtained following the general algorithm of the relativistic Hamilton– Jacobi theory: define the momenta;

These equations give the dynamics, namely the solution of the equations of motion. Explicitly, the calculation of the momenta is simplified by the observation that the action is a homogeneous function of degree one, so this gives;

The evolution equations are;

Time evolution
In the description given so far, no reference to evolution in a preferred time variable was considered. To introduce regard the direction of the axis of the equilateral tetrahedron as a temporal direction. In particular interpret b as an initial variable and a as a final variable. The length c of the side links can then be regarded as a proper length measured in the temporal direction, namely as the physical time elapsed from the measurement of a to the measurement of b.

Renaming c as T . The Hamilton function reads then S(a, b, T ) and can now be interpreted as the Hamilton function that determines the evolution in T of a variable a. The variable b is interpreted as measured at time T = 0 and the variable a at time T ; therefore b can be viewed as an integration constant for
the evolution of a in T .

In this system, the hamiltonian that evolves the system in the time T , which is called ‘proper-time hamiltonian’, can obtained from the energy

Notice that the angle theta can vary between 0 and pi/2, and therefore so does the arccos. Therefore the energy can vary between 2pi and 4pi. The fact that the domain of the energy is bounded has important consequences. For instance –  should expect time to become discrete in the quantum theory.

The relativistic background independent system can be reinterpreted as an evolution system, where the ‘proper time’ on the boundary of the region of interest is taken as the independent time variable. The Hamilton equation generated by the hamiltonian for a(T ) and pa(T ) are:

In the large T limit we have the behaviour;

Quantum equilateral tetrahedron

Specialize the formalism to the case of an equilateral tetrahedron. The simplest way to do so is to restrict attention to the states where four of the six edge lengths are equal. More precisely, put:

ja ≡ j13,
jb ≡ j24,
jc ≡ j12 = j23 = j34 = j41
and consider only the states

|ja, jb, jc> = |jc, ja, jc, jc, jb, jc>.

The states are restricted to the subset of (SU(2))^6. The boundary Hilbert state K is spanned by the states |ja, jb, jc>. The boundary observables a, b, c, pa, pb, pc that measure the length of the edges of the tetrahedron and the external angles are represented by Casimir and trace operators, and the dynamics is given by the propagator

which expresses the probability amplitude of measuring the lengths determined by ja, jb, jc. The predictions of the theory are given by the
quantization of the lengths and by the relative probability amplitude, W() above.

# Disappearance and emergence of space and time in quantum gravity by D. Oriti

This week I’ve been studying GFT condensates, Bose-Einstein condensates and the Gross-Pitaevskii Equation and the emergence of spacetime. I have posted on spacetime as a GFT condensate and numerical work with python on Bose-Einstein condensates. This post is based on D. Oriti’s paper, ‘Disappearance and emergence of space and time in quantum gravity’

In this paper he looks at the disappearance of continuum space and time at a microscopic scale. These include arguments for the discrete spacetimes and non-locality in a quantum theory of gravity. He also discusses how these ideas are realized in specific quantum gravity approaches. He then considers the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space such as quantum tetrahedra, and for understanding spacetime as a kind of condensate and presents the case for this emergence process being the result of a phase transition, called ‘geometrogenesis’. Oriti then discusses some conceptual issues of this scenario and of the idea of emergent spacetime in general. A concrete example is given in the form of the GFT framework for quantum gravity, and he illustrates a procedure for the
emergence of spacetime in this framework.
GEOMETROGENESIS
An example of emergent spacetime in the context of the GFT framework is given in this paper. It aims at extracting cosmological dynamics directly from microscopic GFT models, using the idea of continuum spacetime as a condensate, possibly emerging from a big bang phase transition.

GFTs are defined usually in perturbative expansion around the Fock vacuum. In this approximation, they describe the interaction of quantized simplices and spin networks, in terms of spin foam models and simplicial gravity. The true ground state of the system, however, for non-zero couplings and for generic choices of the macroscopic parameters, will not be the Fock vacuum. The interacting system will organize itself around a new, non-trivial state, as in the case of standard Bose condensates. The relevant ground states for different values of the parameters will correspond to the different macroscopic, continuum phases of the theory, with the dynamical transitions
from one to the other being phase transitions of the physical system called spacetime.
The fact that the relevant ground state for a proper continuum geometric phase would probably not be the GFT Fock vacuum can be argued also on the basis of the pregeomet ”meaning of it: it is a quantum state in which no pregeometric excitations at all are present, no simplices, no spin networks. It is a no space state, the absolute void. It can be the full non-perturbative, diffeo-invariant quantum state around which one defines the theory – in fact, it is analogous to the diffeoinvariant vacuum state of loop quantum gravity, but it is not where to look for effective continuum physics. Hence the need to change vacuum and study the effective geometry and dynamics of a different
one.

As described in my last post it is possible to define an approximation procedure that associates an approximate continuum geometry to the set of data encoded in a generic GFT state. This applies to GFT models whose group and Lie algebra variables admit an interpretation in terms of discrete geometries, i.e. in which the group chosen is SO(3, 1) in the Lorentzian setting or SO(4) in the Riemannian setting and additional simplicity conditions are imposed, in the model, to reduce generic group and Lie algebra elements to discrete counterparts of a discrete tetrad and a discrete gravity connection.

A generic GFT state with a fixed number N of GFT quanta will be associated to a set of 4N Lie algebra elements: BI(m) , with m = 1, …,N running over the set of tetrahedra/vertices, I = 1, …, 4 indicating the four triangles of each tetrahedron. In turn, the geometricity conditions imply that only three elements are independent for each tetrehadron.

If the tetrahedra are embedded in a spatial 3-manifold M with a transitive
group action. The embedding is defined by specifying a location of the tetrahedra, i.e. associating for example one of their vertices with a point on the manifold, and three tangent vectors defining a local frame and specifying the directions of the three edges incident at that vertex. The vectors eA canbe interpreted  as continuum tetrad vectors integrated over paths in M corresponding to the edges of the tetrahedron. Then, the variables gij(m) can be used to define the coefficients of continuum metric at a finite number N of points, as: gij(m) = g(xm)(vi(m), vj(m)), invariant under the action of the group SO(4).

The reconstruction of metric coefficients gij(m), at a finite number of points, from the variables associated to a state of N GFT quanta – such as quantum tetrahedra, depends only on the topology of the assumed symmetric manifold M and on the choice of group action H. The approximate metric will be homogenous if it has the same coefficients gij(m) at any m. This captures the notion of the metric being the same at every point. It also implies that the same metric would be also isotropic if H = R3 or H = SU(2).

The quantum GFT states obtained using the above procedure can be interpreted as continuum homogeneous quantum geometries. In such a second quantized setting, the definition of states involving varying and even infinite numbers of discrete degrees of freedom is straightforward, and field theory formalism is well adapted to dealing with their dynamics.

The crucial point, from the point of view of emergent spacetime and of the idea of spacetime as a condensate of quantum pregeometric and not spatio-temporal building blocks is that quantum states corresponding to homogeneous continuum geometries are exactly GFT condensate states. The hypothesis of spacetime as a condensate, as a quantum fluid, is realized in a literal way. The simplest state of this type -one-particle GFT condensate, for which we assume a bosonic quantum statistics, is

This describes a coherent superposition of quantum states of arbitrary number of GFT quanta, all of them described by the same distribution ϕo of pregeometric variables. The function ϕo is a collective variable characterizing such continuum geometry, and it depends only on invariant homogeneous geometric data.  It is a second quantized state characterized by the fact that the mean value of the fundamental quantum operator φ is non-zero:

<ϕo|φ(gi)|ϕo> = ϕo(gi),

contrary to what happens in the Fock vacuum.

The effective dynamical equations for the condensate can be extracted directly from the fundamental GFT quantum dynamics.The generic form of the dynamics for the condensate ϕo is, schematically:

where Keff and Veff are modified versions of the kinetic and interaction kernels entering the fundamental GFT dynamics, reflecting the approximations needed to interpret ϕo as a cosmological condensate.This is a non-linear and non-local, Gross-Pitaevskii-like equation for the spacetime condensate function ϕo. In the simple case in which:

and we assume that the function ϕo depends on four such SU(2) variables. The final equation one gets for non-degenerate geometries) is:

that is the Friedmann equation for a homogeneous universe with constant curvature k.