ABSTRACT (original article): Although spin foams arose as quantizations of the length metric degrees of freedom, the quantum configuration space is rather based on areas as more fundamental variables. This is also highlighted by the semi-classical limit of four-dimensional spin foam models, which is described by the Area Regge action. Despite its central importance to spin foams the dynamics encoded by the Area Regge action is only poorly understood, in particular in the continuum limit. We perform here a systematic investigation of the dynamics defined by the Area Regge action on a regular centrally subdivided hypercubical lattice. This choice of lattice avoids many problems of the non-subdivided hypercubical lattice, for which the Area Regge action is singular. The regularity of the lattice allows to extract the continuum limit and its corrections, order by order in the lattice constant. We show that, contrary to widespread expectations which arose from the so-called flatness problem of spin foams, the continuum limit of the Area Regge action does describe to leading order the same graviton dynamics as general relativity. The next-to-leading order correction to the effective action for the length metric is of second order in the lattice constant, and is given by a quadratic term in the Weyl curvature tensor. This correction can be understood to originate from an underlying dynamics of area metrics. This suggests that the continuum limit of spin foam dynamics does lead to massless gravitons, and that the leading order quantum corrections can be understood to emerge from a generalization of the configuration space from length to area metrics. CITATION (original article): Bianca Dittrich and Athanasios Kogios (2023), From spin foams to area metric dynamics to gravitons, Class. Quantum Grav. 40 095011. “Setup” partially provided by Jose Padua Arguelles. https://doi.org/10.1088/1361-6382/acc5d9
Please cite the paper if you use parts of the notebook.
GitHub: https://github.com/star-pirate/Centrally_Subdivided_Area_Regge
Please cite the paper if you use parts of the notebook.
GitHub: https://github.com/star-pirate/Centrally_Subdivided_Area_Regge
Introduction
Introduction
This notebook computes the linearized Area Regge action, defined on centrally subdivided hyper-cubical lattice. This allows to determine the (perturbative) continuum limit of Area Regge calculus.
The Length Regge action constitutes a coordinate-free discretization of the Einstein Hilbert action, which encodes the dynamics of general relativity. The Area Regge action is a generalization of the Length Regge action and plays a central role in spin foams, in particular for the so-called effective spin foam models. Spin foams are a non-perturbative path integral approach to quantum gravity, which incorporates the quantum geometry and rigorous quantization techniques of Loop quantum gravity. The Area Regge action is also of independent interest; it is the most natural candidate, if one considers a generalization of gravity to area metric variables, and asks for a discretized action.
A main open question for spin foams is to determine their continuum limit. Determining the (perturbative) continuum limit of the Area Regge action is a key step for understanding the continuum limit of spin foams. The key questions are whether we obtain linearized general relativity to leading order, and what are the sub-leading corrections.
Defining the linearized Area Regge action on a regular hyper-cubical lattice, or a subdivision thereof, allows to take the continuum limit and to determine corrections in a systematic manner. We find in this notebook that (a) the leading order of the continuum limit is indeed given by the linearized Einstein-Hilbert action for general relativity and (b) the leading order correction to this continuum limit, which we show (c) to arise from integrating out area metric fluctuations, which are not induced by length metric fluctuations.
The Length Regge action constitutes a coordinate-free discretization of the Einstein Hilbert action, which encodes the dynamics of general relativity. The Area Regge action is a generalization of the Length Regge action and plays a central role in spin foams, in particular for the so-called effective spin foam models. Spin foams are a non-perturbative path integral approach to quantum gravity, which incorporates the quantum geometry and rigorous quantization techniques of Loop quantum gravity. The Area Regge action is also of independent interest; it is the most natural candidate, if one considers a generalization of gravity to area metric variables, and asks for a discretized action.
A main open question for spin foams is to determine their continuum limit. Determining the (perturbative) continuum limit of the Area Regge action is a key step for understanding the continuum limit of spin foams. The key questions are whether we obtain linearized general relativity to leading order, and what are the sub-leading corrections.
Defining the linearized Area Regge action on a regular hyper-cubical lattice, or a subdivision thereof, allows to take the continuum limit and to determine corrections in a systematic manner. We find in this notebook that (a) the leading order of the continuum limit is indeed given by the linearized Einstein-Hilbert action for general relativity and (b) the leading order correction to this continuum limit, which we show (c) to arise from integrating out area metric fluctuations, which are not induced by length metric fluctuations.
In our setup we use A, a, L, l and Th, th as the background and the perturbation of the area, the length and the dihedral angle respectively. The symbol s together with the aforementioned refer to the squared versions eg AsLs is the Jacobian from the area squared background to the length squared background.
Setup
Setup
Simplex and Complex Definitions
Simplex and Complex Definitions
Volumes
Volumes
Here we define the signed squared volumes of simplices of various dimensions, using the Caley-Menger determinant, as function of the squared edge lengths sij. The signed squared volume is positive for Euclidean (or spacelike) simplices, which are considered in this notebook, and negative for timelike simplices.
Signed Square solume of a 4 - simplex in terms of square lengths
V4s[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=-Det
1
4
2
2
(4!)
0 | 1 | 1 | 1 | 1 | 1 |
1 | 0 | s12 | s13 | s14 | s15 |
1 | s12 | 0 | s23 | s24 | s25 |
1 | s13 | s23 | 0 | s34 | s35 |
1 | s14 | s24 | s34 | 0 | s45 |
1 | s15 | s25 | s35 | s45 | 0 |
Signed Square volume of a tetrahedron (3 - simplex) in terms of square lengths
In[]:=
V3s[s12_,s13_,s14_,s23_,s24_,s34_]:=Det
1
3
2
2
(3!)
0 | 1 | 1 | 1 | 1 |
1 | 0 | s12 | s13 | s14 |
1 | s12 | 0 | s23 | s24 |
1 | s13 | s23 | 0 | s34 |
1 | s14 | s24 | s34 | 0 |
Signed Square area of a triangle (2 - simplex) in terms of square lengths
In[]:=
V2s[s12_,s13_,s23_]:=-Det
1
2
2
2
(2!)
0 | 1 | 1 | 1 |
1 | 0 | s12 | s13 |
1 | s12 | 0 | s23 |
1 | s13 | s23 | 0 |
Area square and Area
Area square and Area
Here we define areas and area squares, as well as their derivative with respect to lengths for a given 4-simplex. These geometric data are expressed as functions of squared edge lengths.
We will need this to compute the Area Regge action and associated Hessian first for a given 4-simplex. The results will then be added in order to obtain the Hessian for a (centrally subdivided) hypercube, and then the Hessian for the entire lattice.
We will need this to compute the Area Regge action and associated Hessian first for a given 4-simplex. The results will then be added in order to obtain the Hessian for a (centrally subdivided) hypercube, and then the Hessian for the entire lattice.
Area - length square derivative for one simplex - in canonical ordering of triangles and lengths
Squared area of every triangle in a simplex . The order is starting from triangle 123.
SmplxArsq[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:={V2s[s12,s13,s23],V2s[s12,s14,s24],V2s[s12,s15,s25],V2s[s13,s14,s34],V2s[s13,s15,s35],V2s[s14,s15,s45],V2s[s23,s24,s34],V2s[s23,s25,s35],V2s[s24,s25,s45],V2s[s34,s35,s45]}
Area of every triangle in a simplex . The order is starting from triangle 123.
SmplxAr[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=Sqrt@{V2s[s12,s13,s23],V2s[s12,s14,s24],V2s[s12,s15,s25],V2s[s13,s14,s34],V2s[s13,s15,s35],V2s[s14,s15,s45],V2s[s23,s24,s34],V2s[s23,s25,s35],V2s[s24,s25,s45],V2s[s34,s35,s45]}
The derivative of the areas with respect to the length squared.
DADlsM[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=Table[D[SmplxAr[ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45][[i]],{{ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45}}],{i,1,10}]/.{ls12s12,ls13s13,ls14s14,ls15s15,ls23s23,ls24s24,ls25s25,ls34s34,ls35s35,ls45s45}
The derivative of the areas squared with respect to the length squared.
DAsDlsM[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=Table[D[SmplxArsq[ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45][[i]],{{ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45}}],{i,1,10}]/.{ls12s12,ls13s13,ls14s14,ls15s15,ls23s23,ls24s24,ls25s25,ls34s34,ls35s35,ls45s45}
Dihedral angle and derivative
Dihedral angle and derivative
We also need to compute all the dihedral angles in a given 4-simplex and its length square derivative.
Compute dihedral angle at triangle 123
ArcCosD[V4s[s12,s13,s14,s15,s23,s24,s25,s34,s35,s45],s45]//Simplify;
2
4
V3s[s12,s13,s14,s23,s24,s34]V3s[s12,s13,s15,s23,s25,s35]
Dihedral angle at triangle 123
θ123[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=ArcCos(-(s24+s25-s45)-(s34+s35-s45)+s13(s23s24+s23s25-2s24s25+s15(s23+s24-s34)+s25s34+s14(s23+s25-s35)+s24s35-2s23s45)+s12(s15s23-s15s24+s15s34+s23s34+s25s34+s23s35+s24s35-2s34s35+s14(s23-s25+s35)+s13(-2s23+s24+s25+s34+s35-2s45)-2s23s45)+s23(-s25s34+s15(-s23+s24+s34)-s24s35+s14(-2s15-s23+s25+s35)+s23s45))2((s24+s34+s23(+s14(s23-s24-s34)+s24s34)-s13(s14(s23+s24-s34)+s24(s23-s24+s34))-s12((s23+s24-s34)s34+s14(s23-s24+s34)+s13(-s23+s24+s34)))(s25+s35+s23(+s15(s23-s25-s35)+s25s35)-s13(s15(s23+s25-s35)+s25(s23-s25+s35))-s12((s23+s25-s35)s35+s15(s23-s25+s35)+s13(-s23+s25+s35))));
2
s13
2
s12
2
s13
2
s12
2
s14
2
s13
2
s12
2
s15
List of dihedral angles in a 4 - simplex . The order is starting from triangle 123.
ThS[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:={θ123[s12,s13,s14,s15,s23,s24,s25,s34,s35,s45],θ123[s12,s14,s13,s15,s24,s23,s25,s34,s45,s35],θ123[s12,s15,s13,s14,s25,s23,s24,s35,s45,s34],θ123[s13,s14,s12,s15,s34,s23,s35,s24,s45,s25],θ123[s13,s15,s12,s14,s35,s23,s34,s25,s45,s24],θ123[s14,s15,s12,s13,s45,s24,s34,s25,s35,s23],θ123[s23,s24,s12,s25,s34,s13,s35,s14,s45,s15],θ123[s23,s25,s12,s24,s35,s13,s34,s15,s45,s14],θ123[s24,s25,s12,s23,s45,s14,s34,s15,s35,s13],θ123[s34,s35,s13,s23,s45,s14,s24,s15,s25,s12]}
The derivative of the angles with respect to the length squared .
DthDlsM[s12_,s13_,s14_,s15_,s23_,s24_,s25_,s34_,s35_,s45_]:=Table[D[ThS[ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45][[i]],{{ls12,ls13,ls14,ls15,ls23,ls24,ls25,ls34,ls35,ls45}}],{i,1,10}]/.{ls12s12,ls13s13,ls14s14,ls15s15,ls23s23,ls24s24,ls25s25,ls34s34,ls35s35,ls45s45}
Triangulation
Triangulation
Here we define the basic cell of the centrally divided hyper-cubic lattice, that is the centrally divided hypercube. We start with a hypercube (HCubeA), which is triangulated into 4-simplices, so that we do not have a bulk vertex. We then apply a subdivision procedure, which inserts a vertex in the centre of the hypercube and subdivides all the 4-simplices into two 4-simplices.
This defines a triangulation of a hypercube with 16 vertices 0, ..., 15. All 4 - simplices share the hyperdiagonal of the hypercube {0, 15}.
HCubeA={{0,1,3,7,15},{0,1,3,11,15},{0,1,5,7,15},{0,1,5,13,15},{0,1,9,11,15},{0,1,9,13,15},{0,2,3,7,15},{0,2,3,11,15},{0,2,6,7,15},{0,2,6,14,15},{0,2,10,11,15},{0,2,10,14,15},{0,4,5,7,15},{0,4,5,13,15},{0,4,6,7,15},{0,4,6,14,15},{0,4,12,13,15},{0,4,12,14,15},{0,8,9,11,15},{0,8,9,13,15},{0,8,10,11,15},{0,8,10,14,15},{0,8,12,13,15},{0,8,12,14,15}};
This subdivides a given 4 - simplex into two 4 - simplices with a new vertex, which subdivides the edge {i, p} . We will apply this with {i, p} = {0, 15} being the hyperdiagonal .
DivSimpA[i_,j_,k_,o_,p_]={{i,j,k,o,{i,p}},{j,k,o,p,{i,p}}};
In[]:=
HCubeB=Flatten[DivSimpA@@@HCubeA,1]
Out[]=
{{0,1,3,7,{0,15}},{1,3,7,15,{0,15}},{0,1,3,11,{0,15}},{1,3,11,15,{0,15}},{0,1,5,7,{0,15}},{1,5,7,15,{0,15}},{0,1,5,13,{0,15}},{1,5,13,15,{0,15}},{0,1,9,11,{0,15}},{1,9,11,15,{0,15}},{0,1,9,13,{0,15}},{1,9,13,15,{0,15}},{0,2,3,7,{0,15}},{2,3,7,15,{0,15}},{0,2,3,11,{0,15}},{2,3,11,15,{0,15}},{0,2,6,7,{0,15}},{2,6,7,15,{0,15}},{0,2,6,14,{0,15}},{2,6,14,15,{0,15}},{0,2,10,11,{0,15}},{2,10,11,15,{0,15}},{0,2,10,14,{0,15}},{2,10,14,15,{0,15}},{0,4,5,7,{0,15}},{4,5,7,15,{0,15}},{0,4,5,13,{0,15}},{4,5,13,15,{0,15}},{0,4,6,7,{0,15}},{4,6,7,15,{0,15}},{0,4,6,14,{0,15}},{4,6,14,15,{0,15}},{0,4,12,13,{0,15}},{4,12,13,15,{0,15}},{0,4,12,14,{0,15}},{4,12,14,15,{0,15}},{0,8,9,11,{0,15}},{8,9,11,15,{0,15}},{0,8,9,13,{0,15}},{8,9,13,15,{0,15}},{0,8,10,11,{0,15}},{8,10,11,15,{0,15}},{0,8,10,14,{0,15}},{8,10,14,15,{0,15}},{0,8,12,13,{0,15}},{8,12,13,15,{0,15}},{0,8,12,14,{0,15}},{8,12,14,15,{0,15}}}
In[]:=
BracketsB[i_,j_,k_,o_,p_]={{i},{j},{k},{o},p};
In[]:=
HCube=BracketsB@@@HCubeB
Out[]=
{{{0},{1},{3},{7},{0,15}},{{1},{3},{7},{15},{0,15}},{{0},{1},{3},{11},{0,15}},{{1},{3},{11},{15},{0,15}},{{0},{1},{5},{7},{0,15}},{{1},{5},{7},{15},{0,15}},{{0},{1},{5},{13},{0,15}},{{1},{5},{13},{15},{0,15}},{{0},{1},{9},{11},{0,15}},{{1},{9},{11},{15},{0,15}},{{0},{1},{9},{13},{0,15}},{{1},{9},{13},{15},{0,15}},{{0},{2},{3},{7},{0,15}},{{2},{3},{7},{15},{0,15}},{{0},{2},{3},{11},{0,15}},{{2},{3},{11},{15},{0,15}},{{0},{2},{6},{7},{0,15}},{{2},{6},{7},{15},{0,15}},{{0},{2},{6},{14},{0,15}},{{2},{6},{14},{15},{0,15}},{{0},{2},{10},{11},{0,15}},{{2},{10},{11},{15},{0,15}},{{0},{2},{10},{14},{0,15}},{{2},{10},{14},{15},{0,15}},{{0},{4},{5},{7},{0,15}},{{4},{5},{7},{15},{0,15}},{{0},{4},{5},{13},{0,15}},{{4},{5},{13},{15},{0,15}},{{0},{4},{6},{7},{0,15}},{{4},{6},{7},{15},{0,15}},{{0},{4},{6},{14},{0,15}},{{4},{6},{14},{15},{0,15}},{{0},{4},{12},{13},{0,15}},{{4},{12},{13},{15},{0,15}},{{0},{4},{12},{14},{0,15}},{{4},{12},{14},{15},{0,15}},{{0},{8},{9},{11},{0,15}},{{8},{9},{11},{15},{0,15}},{{0},{8},{9},{13},{0,15}},{{8},{9},{13},{15},{0,15}},{{0},{8},{10},{11},{0,15}},{{8},{10},{11},{15},{0,15}},{{0},{8},{10},{14},{0,15}},{{8},{10},{14},{15},{0,15}},{{0},{8},{12},{13},{0,15}},{{8},{12},{13},{15},{0,15}},{{0},{8},{12},{14},{0,15}},{{8},{12},{14},{15},{0,15}}}
Computation of the Hessian
Computation of the Hessian
Background geometry of the hyper-cubic cell
Background geometry of the hyper-cubic cell
Here we define the background geometry of the hyper-cube by embedding it into Euclidean flat space, that is by providing coordinates for the vertices. The embedding into flat space determines their background edge lengths. The then compute some basic geometric entities for the triangulation of the hypercube.
The lattice constant λ is positive.
In[]:=
$Assumptions=λ>0;
We define the lattice vectors for a hypercubic lattice
ed[1]={0,0,0,λ};ed[2]={0,0,λ,0};ed[4]={0,λ,0,0};ed[8]={λ,0,0,0};
Define vertex coordinates of hypercube embedded into flat space . The vertex in the the center of the cube is denoted by {0, 15}
Compute background length square of a given edge {X, Y}
Area Square of the triangle spanned by vertices ijk
All simplices have the same geometry - the following list the length square of the edges in each 4 - simplex.
Compute area square - length square derivatives for each 4 - simplex.
All null spaces of the area - length Jacobian are empty . This allows as to invert these Jacobians, which we need to do to compute the Area Regge action Hessian.
Compute inverse of the Jacobians for each 4 - simplex - this is needed to compute the Area Regge Hessian.
Compute derivatives of dihedral angles - for each 4 - simplex - this is needed to compute the Area Regge Hessian .
List of Areas for each 4 - simplex in the hypercube - this is needed to compute the Area Regge Hessian
Area Regge Hessian
Area Regge Hessian
Compute Hessian for the Hypercube
Compute Hessian for the Hypercube
Here we compute the Hessian for the Hypercube by adding up the Hessians for the 4-simplices belonging to the hypercube. We do that by
a) computing the Hessians for each 4-simplex
b) constructing the linearized action for each 4-simplex from the simplex Hessian
c) adding up these simplex contributions to the action, and then
d) constructing the Hessian for the hypercube from the coefficients of the monomials (in area square variables) in the action.
a) computing the Hessians for each 4-simplex
b) constructing the linearized action for each 4-simplex from the simplex Hessian
c) adding up these simplex contributions to the action, and then
d) constructing the Hessian for the hypercube from the coefficients of the monomials (in area square variables) in the action.
Integrate out bulk areas from hypercube action
Integrate out bulk areas from hypercube action
The triangles in the hypercube split into boundary and bulk set. The bulk triangles all contain the central vertex {0,15}, and we associated all these triangles to {0}. Thus the Fourier transformed Hessian will not have any phase factors for entries in the bulk-bulk block. This allows us to integrate out the bulk variables without needing to worry about inverting an ω-dependent matrix. However, the bulk-bulk block will feature four null vectors, which extend to the full matrix. These four null vectors arise from a vertex translation symmetry (aka diffeomorphism symmetry) of the linearized Area Regge action associated to the central vertex. We add a gauge fixing term, which allows us to invert the bulk-bulk block and in this way to integrate out the bulk variables.
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
From spin foams to area metric dynamics to gravitons: Hessian, Regge calculus, hypercubical lattice
by Bianca Dittrich, Athanasios Kogios
“Setup” partially provided by Jose Padua Arguelles
Wolfram Community, STAFF PICKS, April 21, 2025
https://community.wolfram.com/groups/-/m/t/3448109
by Bianca Dittrich, Athanasios Kogios
“Setup” partially provided by Jose Padua Arguelles
Wolfram Community, STAFF PICKS, April 21, 2025
https://community.wolfram.com/groups/-/m/t/3448109