Coarse graining and renormalization: the bottom up approach[35] “On the role of the...

[Picture by R. Raussendorf]

Bianca Dittrich, Perimeter Institute, Canada

Asymptotic Safety Seminar, April 2014

Coarse graining and renormalization:

the bottom up approach

Raussendorf

Copyright: The M. C. Escher Company B. V.1

LiteratureConceptual.

June 2013

Publication List – Bianca Dittrich

Preprints

[45] “On the path integral measure for 4D discrete gravity,”B. Dittrich, W. Kaminski, S. Steinhaus,to appear

[44] “A new vacuum for loop quantum gravity,”B. Dittrich, M. Geiller,arXiv:1401.6441[gr-qc]

[43] “Ising model from intertwiners,”B. Dittrich, J. Hnybida,arXiv:1312.5646 [gr-qc]

[42] “Quantum group spin net models: refinement limit and relation to spin foams,”B. Dittrich, M. Martin-Benito, S. Steinhaus,arXiv:1312.0905[gr-qc]

[41] “Time evolution as refining, coarse graining and entangling,”B. Dittrich, S. Steinhaus,arXiv:1311.7565[gr-qc]

[40] “Topological lattice field theories from intertwiner dynamics,”B. Dittrich, W. Kaminski,arXiv:1311.1798[gr-qc]

Refereed

[39] “Coarse graining of spin net models: dynamics of intertwiners,”B. Dittrich, M. Martin-Benito, E. Schnetter,New J. Phys. 15 (2013) 103004 (38pp)arXiv:1306.2987[gr-qc]

[38] “Bubble divergences and gauge symmetries in spin foams,”V. Bonzom, B. Dittrich,Phys. Rev. D88 (2013) 124021 (23pp)arXiv:1304.6632 [gr-qc]

[37] “Dirac’s discrete hypersurface deformation algebras,”V. Bonzom, B. Dittrich,Class. Quant. Grav. 30 (2013) 205013 (35pp)arXiv:1304.5983 [gr-qc]

[36] “Constraint analysis for variational discrete systems,”B. Dittrich, P.A. Hohn,J. Math. Phys. 54 (2013) 093505 (43pp)arXiv:1303.4294 [math-ph]

[35] “On the role of the Barbero–Immirzi parameter in discrete quantum gravity,”B. Dittrich, J.P. Ryan,Class. Quant. Grav. 30 (2013) 095015 (30pp)arXiv:1209.4892 [gr-qc]

1

[34] “Holonomy spin foam models: boundary Hilbert spaces and time evolution operators,”B. Dittrich, F. Hellmann, W. KaminskiClass. Quant. Grav. 30 (2013) 085005 (41pp)arXiv:1209.4539 [gr-qc]

[33] “Holonomy spin foam models: definition and coarse graining,”B. Bahr, B. Dittrich, F. Hellmann, W. KaminskiPhys. Rev. D87 (2013) 044048 (18pp)arXiv:1208.3388 [gr-qc]

[32] “On the space of generalized fluxes for loop quantum gravity,”B. Dittrich, C. Guedes, D. OritiClass. Quant. Grav. 30 (2013) 055008 (24pp)arXiv:1205.6166 [gr-qc]

[31] “From the discrete to the continuous – towards a cylindrically consistent dynamics,”B. Dittrich,New J. Phys. 14 (2012) 123004 (28pp)arXiv:1205.6127 [gr-qc]

[30] “How to construct di↵eomorphism symmetry on the lattice,”B. Dittrich,PoS (QGQGS 2011) 012 (23pp)arXiv:1201.3840 [gr-qc]

[29] “Towards computational insights into the large–scale structure of spin foams,”B. Dittrich, F.C. Eckert,J. Phys. Conf. Ser. 360 (2012) 012004 (10pp)arXiv:1111.0967 [gr-qc]

[28] “Path integral measure and triangulation independence in discrete gravity,”B. Dittrich, S. SteinhausPhys. Rev. D85 (2012) 044032 (21pp)arXiv:1110.6866 [gr-qc]

[27] “Coarse graining methods for spin net and spin foam models,”B. Dittrich, F.C. Eckert, M. Martin-BenitoNew J. Phys. 14 (2012) 025008 (43pp),arXiv:1109.4927 [gr-qc]

[26] “Canonical simplicial gravity,”B. Dittrich, P.A. Hohn,Class. Quant. Grav. 29 (2012) 115009 (52pp),arXiv:1108.1974 [gr-qc]

[25] “Spin foam models with finite groups,”B. Bahr, B. Dittrich, J.P. Ryan,Journal of Gravity 2013 (2013) 549824 (28pp),arXiv:1103.6264 [gr-qc]

[24] “Perfect discretization of reparametrization invariant path integrals,”B. Bahr, B. Dittrich, S. Steinhaus,Phys. Rev. D83 (2011) 105026 (19pp),arXiv:1101.4775 [gr-qc]

[23] “Coarse graining theories with gauge symmetries: the linearized case,”B. Bahr, B. Dittrich, S. He,New J. Phys. 13 (2011) 045009 (34pp),arXiv:1011.3667 [gr-qc]

[22] “Simplicity in simplicial phase space,”B. Dittrich, J.P. Ryan,Phys. Rev. D82 (2010) 064026 (19pp),arXiv:1006.4295 [gr-qc]

2

(Latest) Coarse graining results for spin foams / spin nets.

June 2013


Preprints







Refereed






1

New representation for LQG.

June 2013


Preprints







Refereed






1

Tensor network method in condensed matter. (plus many many more)

[6] F. Markopoulou, “An algebraic approach to coarse graining,” arXiv:hep-th/0006199. R. Oeckl, “Renor-malization of discrete models without background,” Nucl. Phys. B 657 (2003) 107 [arXiv:gr-qc/0212047].J. A. Zapata, “Loop quantisation from a lattice gauge theory perspective,” Class. Quant. Grav. 21 (2004)L115-L122, [arXiv:gr-qc/0401109].

[7] B. Bahr, B. Dittrich, F. Hellmann and W. Kaminski, “Holonomy Spin Foam Models: Definition and CoarseGraining,” Phys. Rev. D 87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. Kamin-ski, “Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators,” Class. Quant.Grav. 30 (2013) 085005 [arXiv:1209.4539 [gr-qc]].

[8] B. Dittrich, M. Martın-Benito and E. Schnetter, “Coarse graining of spin net models: dynamics of inter-twiners,” New J. Phys. 15 (2013) 103004 [arXiv:1306.2987 [gr-qc]].

[9] B. Dittrich, M. Martın-Benito, S. Steinhaus, “The refinement limit of quantum group spin net models,” toappear

[10] M. Levin, C. P. Nave, “Tensor renormalization group approach to 2D classical lattice models,” Phys. Rev.Lett. 99 (2007) 120601, [arXiv:cond-mat/0611687 [cond-mat.stat-mech]].

[11] Z. -C. Gu and X. -G. Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Pro-tected Topological Order,” Phys. Rev. B 80 (2009) 155131 [arXiv:0903.1069 [cond-mat.str-el]].

[12] B. Dittrich, “From the discrete to the continuous: Towards a cylindrically consistent dynamics,” New J.Phys. 14 (2012) 123004 [arXiv:1205.6127 [gr-qc]].

[13] B. Dittrich and P. A. Hohn, “Canonical simplicial gravity,” Class. Quant. Grav. 29 (2012) 115009[arXiv:1108.1974 [gr-qc]].

[14] B. Dittrich and P. A. Hohn, “Constraint analysis for variational discrete systems,” J. Math. Phys. 54 (2013)093505 [arXiv:1303.4294 [math-ph]].

[15] G. Ponzano; T. Regge, “Semiclassical limit of Racah coe�cients,” p. 1-58, in: Spectroscopic and grouptheoretical methods in physics, ed. F. Bloch, (North-Holland Publ. Co., Amsterdam, 1968). J. W. Barrettand I. Naish-Guzman, “The Ponzano-Regge model,” Class. Quant. Grav. 26 (2009) 155014 [arXiv:0803.3319[gr-qc]].

[16] A. Perez, “The Spin Foam Approach to Quantum Gravity,” Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019[gr-qc]].

[17] C. Rovelli, “Quantum Gravity” (Cambridge University Press, Cambridge 2004).

[18] J. W. Barrett and L. Crane, “Relativistic spin networks and quantum gravity,” J. Math. Phys. 39, 3296 (1998)[gr-qc/9709028]. J. Engle, E. Livine, R. Pereira and C. Rovelli, “LQG vertex with finite Immirzi parameter,”Nucl. Phys. B 799 (2008) 136 [arXiv:0711.0146 [gr-qc]]. E. R. Livine and S. Speziale, “A new spinfoam vertexfor quantum gravity,” Phys. Rev. D 76, 084028 (2007), [arXiv:0705.0674 [gr-qc]]. L. Freidel and K. Krasnov,“A New Spin Foam Model for 4d Gravity,” Class. Quant. Grav. 25 (2008) 125018 [arXiv:0708.1595 [gr-qc]].A. Baratin and D. Oriti, “Quantum simplicial geometry in the group field theory formalism: reconsideringthe Barrett-Crane model,” New J. Phys. 13 (2011) 125011 [arXiv:1108.1178 [gr-qc]].

[19] M. Rocek and R. M. Williams, “Quantum Regge Calculus,” Phys. Lett. B 104 (1981) 31. “The QuantizationOf Regge Calculus,” Z. Phys. C 21 (1984) 371.

[20] L. Freidel and D. Louapre, “Di↵eomorphisms and spin foam models,” Nucl. Phys. B 662 (2003) 279 [arXiv:gr-qc/0212001].

[21] B. Dittrich, “Di↵eomorphism symmetry in quantum gravity models,” Adv. Sci. Lett. 2 (2009) 151[arXiv:0810.3594 [gr-qc]].

[22] B. Bahr and B. Dittrich, “(Broken) Gauge Symmetries and Constraints in Regge Calculus,” Class. Quant.Grav. 26 (2009) 225011 [arXiv:0905.1670 [gr-qc]]. B. Bahr and B. Dittrich, “Breaking and restoring ofdi↵eomorphism symmetry in discrete gravity,” AIP Conference Proceedings Volume 1196, ed. by J. Kowalski-Glikman et. al. ,pp. 10-17 arXiv:0909.5688 [gr-qc].

[23] B. Bahr, B. Dittrich and S. Steinhaus, “Perfect discretization of reparametrization invariant path integrals,”Phys. Rev. D 83 (2011) 105026 [arXiv:1101.4775 [gr-qc]].

[24] J. J. Halliwell and J. B. Hartle, “Wave functions constructed from an invariant sum over histories satisfyconstraints,” Phys. Rev. D 43 (1991) 1170. C. Rovelli, “The projector on physical states in loop quantumgravity,” Phys. Rev. D 59 (1999) 104015 [arXiv:gr-qc/9806121].

29

[6] F. Markopoulou, “An algebraic approach to coarse graining,” arXiv:hep-th/0006199. R. Oeckl, “Renor-malization of discrete models without background,” Nucl. Phys. B 657 (2003) 107 [arXiv:gr-qc/0212047].J. A. Zapata, “Loop quantisation from a lattice gauge theory perspective,” Class. Quant. Grav. 21 (2004)L115-L122, [arXiv:gr-qc/0401109].

[7] B. Bahr, B. Dittrich, F. Hellmann and W. Kaminski, “Holonomy Spin Foam Models: Definition and CoarseGraining,” Phys. Rev. D 87 (2013) 044048 [arXiv:1208.3388 [gr-qc]]. B. Dittrich, F. Hellmann and W. Kamin-ski, “Holonomy Spin Foam Models: Boundary Hilbert spaces and Time Evolution Operators,” Class. Quant.Grav. 30 (2013) 085005 [arXiv:1209.4539 [gr-qc]].

[8] B. Dittrich, M. Martın-Benito and E. Schnetter, “Coarse graining of spin net models: dynamics of inter-twiners,” New J. Phys. 15 (2013) 103004 [arXiv:1306.2987 [gr-qc]].

[9] B. Dittrich, M. Martın-Benito, S. Steinhaus, “The refinement limit of quantum group spin net models,” toappear

[10] M. Levin, C. P. Nave, “Tensor renormalization group approach to 2D classical lattice models,” Phys. Rev.Lett. 99 (2007) 120601, [arXiv:cond-mat/0611687 [cond-mat.stat-mech]].

[11] Z. -C. Gu and X. -G. Wen, “Tensor-Entanglement-Filtering Renormalization Approach and Symmetry Pro-tected Topological Order,” Phys. Rev. B 80 (2009) 155131 [arXiv:0903.1069 [cond-mat.str-el]].

[12] B. Dittrich, “From the discrete to the continuous: Towards a cylindrically consistent dynamics,” New J.Phys. 14 (2012) 123004 [arXiv:1205.6127 [gr-qc]].

[13] B. Dittrich and P. A. Hohn, “Canonical simplicial gravity,” Class. Quant. Grav. 29 (2012) 115009[arXiv:1108.1974 [gr-qc]].

[14] B. Dittrich and P. A. Hohn, “Constraint analysis for variational discrete systems,” J. Math. Phys. 54 (2013)093505 [arXiv:1303.4294 [math-ph]].

[15] G. Ponzano; T. Regge, “Semiclassical limit of Racah coe�cients,” p. 1-58, in: Spectroscopic and grouptheoretical methods in physics, ed. F. Bloch, (North-Holland Publ. Co., Amsterdam, 1968). J. W. Barrettand I. Naish-Guzman, “The Ponzano-Regge model,” Class. Quant. Grav. 26 (2009) 155014 [arXiv:0803.3319[gr-qc]].

[16] A. Perez, “The Spin Foam Approach to Quantum Gravity,” Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019[gr-qc]].

[17] C. Rovelli, “Quantum Gravity” (Cambridge University Press, Cambridge 2004).

[18] J. W. Barrett and L. Crane, “Relativistic spin networks and quantum gravity,” J. Math. Phys. 39, 3296 (1998)[gr-qc/9709028]. J. Engle, E. Livine, R. Pereira and C. Rovelli, “LQG vertex with finite Immirzi parameter,”Nucl. Phys. B 799 (2008) 136 [arXiv:0711.0146 [gr-qc]]. E. R. Livine and S. Speziale, “A new spinfoam vertexfor quantum gravity,” Phys. Rev. D 76, 084028 (2007), [arXiv:0705.0674 [gr-qc]]. L. Freidel and K. Krasnov,“A New Spin Foam Model for 4d Gravity,” Class. Quant. Grav. 25 (2008) 125018 [arXiv:0708.1595 [gr-qc]].A. Baratin and D. Oriti, “Quantum simplicial geometry in the group field theory formalism: reconsideringthe Barrett-Crane model,” New J. Phys. 13 (2011) 125011 [arXiv:1108.1178 [gr-qc]].

[19] M. Rocek and R. M. Williams, “Quantum Regge Calculus,” Phys. Lett. B 104 (1981) 31. “The QuantizationOf Regge Calculus,” Z. Phys. C 21 (1984) 371.

[20] L. Freidel and D. Louapre, “Di↵eomorphisms and spin foam models,” Nucl. Phys. B 662 (2003) 279 [arXiv:gr-qc/0212001].

[21] B. Dittrich, “Di↵eomorphism symmetry in quantum gravity models,” Adv. Sci. Lett. 2 (2009) 151[arXiv:0810.3594 [gr-qc]].

[22] B. Bahr and B. Dittrich, “(Broken) Gauge Symmetries and Constraints in Regge Calculus,” Class. Quant.Grav. 26 (2009) 225011 [arXiv:0905.1670 [gr-qc]]. B. Bahr and B. Dittrich, “Breaking and restoring ofdi↵eomorphism symmetry in discrete gravity,” AIP Conference Proceedings Volume 1196, ed. by J. Kowalski-Glikman et. al. ,pp. 10-17 arXiv:0909.5688 [gr-qc].

[23] B. Bahr, B. Dittrich and S. Steinhaus, “Perfect discretization of reparametrization invariant path integrals,”Phys. Rev. D 83 (2011) 105026 [arXiv:1101.4775 [gr-qc]].

[24] J. J. Halliwell and J. B. Hartle, “Wave functions constructed from an invariant sum over histories satisfyconstraints,” Phys. Rev. D 43 (1991) 1170. C. Rovelli, “The projector on physical states in loop quantumgravity,” Phys. Rev. D 59 (1999) 104015 [arXiv:gr-qc/9806121].

29

2

Bottom-up design

Your input of “fundamental excitation/

fundamental theory”

(auxiliary discrete?)

building blocks

continuum/ refinement limit:

extract low energy/ large scale physics

?

causal setsgroup field theory

tensor models(causal) dynamical

triangulationsloop quantum

gravity / spin foamsemergent gravity

string theoryRenormalization

toolsKEY

PROBLEM

3

Overview

Motivation.

Spin foams and space time atoms.

Coarse graining without a scale.

The importance of refining states. Tensor network coarse graining algorithms.

Application to spin foams / spin nets.

Classifying (symmetry protected) phases.

New representations and vacua in loop quantum gravity

Expanding the theory around different vacua corresponding to different ways of refinement.

4

Why loop quantum gravity / spin foams?(My personal short list of reasons)

•“minimal assumptions”: theories result from quantizing gravity (as a geometric theory), taking background independence seriously

[gives you a higher probability to obtain a quantum theory of gravity]

•choice of (connection) variables: originally motivated by “convenience”,

•allowed first rigorous quantization of gravity (kinematical)

•so far leads to the most advanced inclusion of diffeomorphism symmetry

[kinematics: for instance LOST uniqueness theorem, but also BD, Geiller 14]

[conceptually very important in particular for coarse graining / renormalization, key role in regaining gravity]

[dynamics: we know what we are looking for! [for instance in the formulation BD, Steinhaus 13]]

•(in particular spin foams): deep connection to topological field theories,

(because there is) no Wick rotation involved

[topological field theories are fixed points of renormalization flow]

[Wick rotation leads to action unbounded from below: prevents useful continuum limit]

5

What are spin foams?

Embeddings determined by the dynamics of the system. Represent the physical vacuum forfiner degrees of freedom.

S3, permutation of 3 elements, has 6 elements: unit element, three 2-cycles, two 3-cylces.

E–functions invariant under Z2 generated by first 2-cycle element:

E(g) = δ(unit, g) + a (δ(1. 2-cycle, g)) + b (δ(2. 2-cycle, g) + δ(3. 2-cylce, g)) +

c (δ(1. 3-cylce, g) + δ(2. 3-cycle, g))

⇒ Phase space parametrized by a, b, c.

If a = b, models can be rewritteninto standard ‘edge models’.

Obvious fixed points:

• zero temp (BF, weak coupling):a = b = c = 0

• BF on quotient group Z2 = S3/Z3:a = b = 0, c = 1

• high temp (strong coupling):a = b = c = 1

a "= b

• Barrett Crane analogue model:a = 1, b = c = 0(not a fixed point)

a = b c

b = 0 0.1 × a

1/2 1 3/2

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28












a "= b


a = b c

b = 0 0.1 × a

1/2 1 3/2 0

28

ψvac(EGauss) ≡ 1 , F (A) ≡ 0

∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l

e} = fklmXme (0.172)

π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},

χ{α!} := R{Adt!(α!)}ηBF =

∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF

∫exp(iS(geom)) Dgeom (0.175)

∑

geom labels

A( geom labels) (0.176)

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels


29

discretization / quantization/reformulation into generalized

lattice gauge theory

We keep the i !


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels


A( geom labels) =∏

fund building blocks

Afbb( geom labels) (0.177)

29

with local (i.e. factorizing) amplitude:

years of research:Reisenberger,

Rovelli, Barrett- Crane, Barbieri,Freidel-Krasnov, Engle-Pereira-

Rovelli, Livine, ...

6

Space time atoms: relation to gravity action


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29

3D


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29


∫dT e−S(T ) (0.168)

hγ (0.169)

Xπ , Xσ (0.170)

{Xke , ge} = geT

k (0.171)

{Xke ,X l


π E(π)

ηAL(A) ≡ 1

ψ{j}(A) = ψ{j}(A) · ηAL(A)

µ(ψ{j}) = δ{j},


∏

%

δ(g%α%). (0.173)

µBF (χ{α!}) =∏

%

δ(α%) (0.174)

∫R{α′−1

! α!}

∏

%

δ(g%)∏

%′

δ(g%′) d|L|g% =∏

%

δ(α′−1% α%

),

RCe ∼ EF


∑

geom labels





j1 j2 j3 j4 j5 j6

29




j1 j2 j3 j4 j5 j6

j10 i1 i5

30




j1 j2 j3 j4 j5 j6

j10 i1 i5

30




j1 j2 j3 j4 j5 j6

j10 i1 i5

30

4D

sum over orientations of space time atoms

Large j (semi-classical) limit for single building blocks gives discrete GR action (for flat building blocks)!

[ Ponzano-Regge, Barrett et al, Conrady-Freidel, ... ]

∑

geom labels





j1 j2 j3 j4 j5 j6

j10 i1 i5

A(j) ∼j>>1

exp(iSdiscr grav) + exp(−iSdiscr grav) (0.179)

ψ(j) ∼j>>1

cos(Sdiscr grav) (0.180)

ψ(j) ∼j>>1

cos(Sdiscr grav) (0.181)

30

7

Many space time atoms?

Is there a phase describing smooth space time?

Do we get General Relativity at “large scales”?

What are the phases of spin foam theories (and in loop quantum gravity)?

8

Spin foams as lattice theories•there is an underlying (auxiliary) lattice: refinement limit with respect to this lattice

•use tools developed in condensed matter, in particular tensor network renormalization.

[Cirac,Levin, Nave, Gu, Wen, Verstraete, Vidal, ...] [‘Emergent gravity’]

•Can deal with complex amplitudes: (we do not Wick rotate!)

•However many conceptional differences:

•result should be independent on choice of lattice (discretization independence)

•diffeomorphism symmetry should emerge (at fixed points of renormalization flow)

•(well supported) conjecture: discrete notion of diffeomorphism symmetry equivalent to discretization independence - should emerge in refinement limit

[BD 08, Bahr, BD 09, Bahr, BD, Steinhaus 11, BD 12] .

•There is no lattice scale, instead notion of scale included in dynamical variables.

•Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once.

9

Coarse graining without a scale

10

Fixed point model encodes all scales!•There is no lattice scale, instead notion of scale included in dynamical variables.

•Hope to flow to perfect discretization, mirroring exactly continuum theory at all scales at once.

•Scale encoded in boundary state - not in renormalization step.

•The initial model is an approximation (via discretization), which we hope to be valid in a small curvature regime [determined by choice of boundary states and number of building blocks].

•Models obtained by coarse graining improve this approximation and increase domain of validity.

•Refinement limit corresponds to a ``perfect discretization’’.

View point:

11

Fixed point as a continuum theoryIf we do not have a lattice scale, how do we know that we reached the continuum limit?

Answer:

•Fixed points of renormalization flow correspond to continuum limit.

(a) “local” amplitudes: topological (discretization independent) theories correspond to phases

(b) “non-local” amplitudes (infinite bond dimension) correspond to phase transitions

6.2.2 Phase diagram for k = 8

For the quantum group with k = 8, we discuss the linear combintation of four fixed point intertwiners, each labelledwith a maximal (even) spin 1 J 4, where we neglect J = 0 as argued above. Together with the requirementthat

PJ ↵J = 1, we have three free parameters. In figure 9 we show the full parameter space, with a raster of

coloured points indicating the fixed point they flow to. In figure 10 we show the interesting slice, where ↵3 = 0.

0.0

0.5

1.0

a1

0.00.5

1.0

a 2

0.0

0.5

1.0

a 3

Figure 9: Phase diagram for k = 8 with ↵0 = 0. The coloured dots indicate to which fixed point the respectiveinitial models flow to: The green dots show the factorizing models, lighter green for J = 1 (area that starts at thevertex (↵1,↵2,↵3) = (1, 0, 0)), darker for J = 2 (area that starts at the vertex (0,1,0)). Analogue BF theory isblue (area that starts at the vertex (0,0,1)). The so-called ‘mixed’ fixed point is orange (area that starts at thevertex (0,0,0)).

Ê ÊÊÊ

ÊÊÊ

Ê Ê ÊÊÊÊÊÊÊ

Ê

Ê

Ê

ÊÊ

Ê Ê

0.0 0.2 0.4 0.6 0.8 1.0a10.0

0.2

0.4

0.6

0.8

1.0a 2

Figure 10: Slice of the phase diagram for k = 8 with ↵0 = ↵3 = 0. The colouring is the same as in the previousdiagram, namely the right corner corresponds to models flowing to the factorizing fixed point with J = 1, the upperleft corner corresponds to models that flow to the factorizing fixed point with J = 2, the models at the bottom leftcorner flow to the ‘mixed’ fixed point, and the phase in between these three phases corresponds to analogue BF.

As in the previous diagram, we find extended phases for all fixed points, here the two factorizing fixed points forJ = 1 and J = 2, a phase for analogue BF theory and one for the ‘mixed’ fixed point. Again, the two dominatingphases are analogue BF theory and the factorizing fixed point with J = 1. Of particular interest is the specialslice that we picked in figure 10 because of the following two observations: First this slice shows clearly that theanalogue BF fixed point is very attractive, since in this slice its associated fixed point intertwiner is not excited,↵3 = 0. Even if we stay on the line given by ↵1 + ↵2 = 1, i.e. the diagonal boundary in figure 10, the systemflows to BF for an intermediate region between the two phases and spoils a direct phase transition between the

26

Phase space: (some) parameters in initial model determine end points of coarse graining flow encoded in different colours

12

Fixed point model still looks discrete. How to reconstruct continuum theory?

Fixed point as a continuum theory

Answer (here a short summary, expanded upon in the next slides):

•Build continuum theory as inductive limit: standard technique of loop quantum gravity [Ashtekar,

Isham, Lewandowski 92+]

•Based on embedding (or refining) coarser (boundary) states into Hilbert space describing finer boundary states

•In this way any coarse (discrete looking) state can be understood as a state of the continuum Hilbert space.

•(Cylindrically) consistency requirements on observables, amplitudes, inner product.

•Surprisingly (condensed matter) tensor network renormalization methods are well adapted to this technique and can provide embedding maps - determined by dynamics of the system. [BD 12]

•Allows to formulate continuum theory of spin foams based on discrete boundary states

[BD, Steinhaus 13]

13

Inductive limit techniques and

Tensor network coarse graining algorithms.

14

Boundary Hilbert spaces and transition amplitudes

A

A

AA

AAA

AA

A A

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

22

(local) amplitudes depending on(boundary) variables, represented by blue edges

boundary Hilbert space (tensor product over local sites)

summing over (bulk) variables in path integral

This represents a transition amplitude built from local amplitudes. The boundary Hilbert space has two components.

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) =∑

xbulka(x1, x2, x3, x4, xbulk)

where x are boundary data

ψ(x1, x2, x3, x4) is a boundary wave function

A is an (anti-)linear functional on space of ψ’s,defines transition amplitudes

State sum models associate amplitudes to space time regions with boundary (data)

18

This represents a one-component boundary Hilbert space.

[generalized boundary formalism: Oeckl 03]

15

25

coarser boundary (with four sites) withattached boundary Hilbert space(tensor product over four sites)

Embedding Hilbert spaces

finer boundary (16 sites) with attached boundary Hilbert space

(local) embedding map

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk

a(x1, x2, x3, x4, xbulk)


ψ(x1, x2, x3, x4)is a boundary wave function

A is an (anti-)linear functionalon bdry Hilbert space H1,

A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)

defines (transition) amplitudes


Amplitude for a ‘larger’ regionglued from amplitudes of smaller regions,acts on ‘refined’ bdry Hilbert space H2

18

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk





A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)




18

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk





A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)




18

embedding map identifies coarse states with statesin finer boundary Hilbert space

16

Good embedding maps?What are the embedding maps good for?

Answer:

•efficiency!

•do computation on the most coarse grained level possible

•can represent continuum theory by discrete (boundary) data

➡ To be really efficient, embedding maps have to be adjusted to dynamics of the system:

➡Coarse Hilbert space should represent ``most typical states”

(state describing smooth geometry / low energy)

Whereas initial boundary states may rather describe fundamental excitations, and one expects a large number of these to describe smooth geometry.

17

Amplitudes encode the dynamics of the system

associate Hilbert spaces Hb

. As for the classical case we have to choose injection maps ◆bb

0 thatembed the Hilbert space H

b

into Hbb

0 . Such injections can be naturally constructed if we haveprojections ⇡

b

0b

: Cb

0 ! Cb

on the configuration spaces at our disposal: assuming a polarizationin which the quantum states are functions on the configuration space we can use the pullbackof the projections to define injections:

◆bb

0 : Hb

! Hb

0

b

7! b

0 where b

0(cb

0) = b

(⇡b

0b

(cb

0)) . (7.2) {q2}

The reader will note that in the classical case we also used injections instead of projections.In the case we discussed here we can however obtain projections by ‘forgetting’ the fluctuationvariables. I.e. in the case that coarse graining was based on piecewise linear fields and fora refinement which doubled the number of fields, {�

i

} to {�i

,�i,i+1

}, we perform a variabletransformation �

ij

= �ij

� 1

2

(�i

+ �j

) and define the projection as

⇡b

0b

(�1

, �12

,�2

, . . .) = (�1

,�2

, . . .) . (7.3) {q3}

Such a projection satisfies ⇡b

0b

� ◆class

bb

0 = Idb

where here the ◆class

b

is the classical embedding mapfor the configuration spaces.

A measure (which here will be just understood as linear functional) on a family of inductiveHilbert spaces can be defined by providing a representation on each of the Hilbert spaces H

b

.Such a family of measures {µ}

b

is cylindrically consistent if

µb

( b

) = µb

0(◆bb

0( b

)) . (7.4) {q4}

The Ashtekar–Lewandowki measure [?, ?] for the kinematical Hilbert spaces in loop quantumgravity satisfies such consistency relations. One way to define the dynamics (and the physicalinner product) would be via the path integrals acting as functionals (rigging maps []) on thesekinematical Hilbert spaces (technically on some dense subspace of these Hilbert spaces) as in(7.1). Cylindrical consistent dynamics would then mean to have a cylindrical consistent familyof (anti–linear) maps

Pb

: Hb

! C . (7.5) {q5}

As an example we can again consider the massless free scalar field. The kinematical Hilbertspace associated to the boundary with N vertices will be L2(RN ), here we will have the caseN = 4 and N = 8. We define the path integral map for the boundary b of the basic square as4

Pb

( b

) =1

Nb

Z Y

i=1,...,4

d�i

eiSb(�i) b

(�i

) (7.6) {q6}

where Nb

is a normalization factor. The path integral map for a refinement b0 that subdividesa squares into four would be given by

Pb

0( b

0) =1

Nb

0

Zd�

0

Y

i=1,...,4

d�i

d�i,i+1

eiSb0 (�i,�i,i+1,�0) b

0(�i

, �i,i+1

) (7.7) {q7}

where �0

is the scalar field on the inner vertex and �i,i+1

the variables introduced in (7.3) andS

b

0 is here the action for four squares obtained by summing the basic action Sb

over the four4The complex conjugation of the kinematical wave function can be interpreted as the kinematical wave function

being associated with the other orientation of the boundary [?].

12

•embedding maps:

ι : (φ1,φ2, . . .) !→ (φ1,34φ1 + 1

4φ2,12 (φ1 + φ2),

14φ1 + 3

4φ2,φ2, . . .) (0.130)

S′4 = (φ1 + φ2)

2 + (φ2 + φ3)2 + (φ3 + φ4)

2 + (φ4 + φ1)2 − α(φ1 − φ2 + φ3 − φ4)

2 (0.131)

S4 = (φ1 + φ2)2 + (φ2 + φ3)

2 + (φ3 + φ4)2 + (φ4 + φ1)

2 (0.132)

α =2

3(0.133)

S∗8 = S∗

4(φi) +

φ1γ12 + . . . − φ1γ23 + . . . +

2.28γ212 + . . . − 1.20γ12γ23 + . . . − 0.34γ12γ34 + . . . (0.134)

S∗16 = S∗

4(φi) +

φ1γ12 + . . . − φ1γ23 + . . . +

2.20γ212 + . . . − 1.18γ12γ23 + . . . 0.36γ12γ34 + . . . +

φ · κ + γ · κ + κ · κ –terms (0.135)

S = . . . λa2(φ41 + φ4

2 + φ43 + φ4

4)

S∗4 = . . . − 0.0039λ2a4φ6

1 + . . .

S∗8 = . . . − 0.0050λ2a4φ6

1 + . . .

cb ∼ cb′

A family of functions {Sb}b∈B is cylindrically consistent on the inductive family defined by(Cb, ιb), if

Sb = ι∗bb′Sb′ , i.e. Sb(c) = Sb′(ιbb′(c)) ∀c ∈ Cb (0.136)

where ι∗bb′ is the pullback of ιbb′ . This just implements that the value of the function F shouldnot depend on the representative on which one chooses to evaluate.

Ab : Hb !→ C . (0.137)

ιbb′ : ψb !→ ψb′ where ψb′(φ1, γ12,φ2, . . .) = ψb(φ1,φ2, . . .) . (0.138)

15

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk





A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)




18

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk





A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)




18

•(transition) amplitudes encode dynamics:

[transition amplitudes if there are two boundary components]

•in fact we have a family of (transition) amplitudes labelled by boundaries b

•order boundaries into coarser and finer

ψ1

ψ1

ψ1

ψ2

ψ2

A

A

A

A

AA

AA

A

A

ψψ

A′

A′A′

A(x1, x2, x3, x4) = (0.147)∑

xbulk




A is an (anti-)linear functionalon space of ψ’s,

A(ψ) =∑

xi

A(xi)ψ(xi) (0.148)



18

•(transition) amplitudes for instance defined by path integral (blue edges represent sum over variables)

•Cylindrical consistency conditions relate amplitudes for different boundaries b and b’.

18

Cylindrically consistent amplitudes

Computing the amplitude for a coarse state should give the same result, as•embedding coarse state to a finer state•using the amplitude map for finer states.

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

generalizes correspondence between4D lattice gauge theories and

2D Ising like models

T = K · W · K (0.160)

blocking of finer fieldvariables

B(ψ) = Ψ (0.161)

Z =∑

ψ

a(ψ) =∑

Ψ

∑

ψ: B(ψ)=Ψ

a(ψ) =∑

Ψ

a′(Ψ) (0.162)

ι"bb′Sbb′ = S∗

b (0.163)

Sb = ι"bb′Sb′ i.e. Sb(c) = Sb′(ιbb′(c)) ∀c ∈ Cb (0.164)

Ab(ψb) = Ab′(ιbb′(ψb)) (0.165)

coarse field variables

amplitude function

effective amplitudeincludes sum overfiner field variables

Localize truncations,diagonalize only subpartsof transfer operator

iteration procedure

determine embedding maps

embedding map after 3 iterationsPlateau (scale free dynamics) of almost constant embedding maps around phase transition

25

Thus, computation for coarse states give already continuum results.Cylindrically consistent amplitudes define the continuum limit of the theory.

[BD 12, BD, Steinhaus 13]

19

What are good choices for the embedding maps (for the basis of boundary states)?

20

Motivation: transfer operator technique

A

A

AA

AAA

AA

A A

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

AA

AAA

AA

A A

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

AA

AAA

AA

A A

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

22

free boundary fields

eom’s have been solved for these fields

approximated fields in first iteration

approximated fields in second iteration


S′4(φ1, . . . ,φ4) =

S8(φ1, . . . ,12(φ1 + φ2), . . .) (0.150)

iterate

diagonal couplings

flow in parameter α:

fixed point: α∗ = 23

After N iteration find an approximation to Hamilton’s function for square with 2N basic squaresand ‘edge wise’ linear boundary fields.

Fixed point: approximation to continuum Hamilton’s function evaluated on ‘edge wise’ linearboundary data.

For free massless scalarfield actually exact!

The same procedure for squares with refined boundary data will in general give a correction tothis approximation.

set γ233, . . . = 0

Transition amplitude betweentwo states 〈ψ|A|ψ2〉

insert id =∑

ONB |ψ〉〈ψ|

21






S′4(φ1, . . . ,φ4) =

S8(φ1, . . . ,12(φ1 + φ2), . . .) (0.150)

iterate

diagonal couplings







set γ233, . . . = 0

Transition amplitude betweentwo states 〈ψ|A|ψ2〉

insert id =∑

ONB |ψ〉〈ψ|

A = TN

21






S′4(φ1, . . . ,φ4) =

S8(φ1, . . . ,12(φ1 + φ2), . . .) (0.150)

iterate

diagonal couplings







set γ233, . . . = 0

Transition amplitude betweentwo states 〈ψ1|A|ψ2〉

insert id =∑

ONB |ψ〉〈ψ|

A = TN

21

A = TN

Truncate by restricting∑

ONBto the eigenvectors of T with theχ largest (in mod) eigenvalues.

23

A

A

AA

AAA

AA

A A

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

22

Motivation: transfer operator technique

Here coarse states correspond to low energy states.Results (typically) in non-local embedding maps (example: Fourier transform for free theories).Explicit diagonalization difficult for large systems.

21

Localized embedding maps

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

A

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

A

∑

ONB

T

T

T

T

A′A′A′

= id

22

a

bc

d

ef

AA

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

C2C2

C2

C2C2

C2

HH

H

22

A = TN



Expect good approximation if ψ1,ψ2

are in span of these eigenvectors.

But: explicit diagonalization of T difficult.

Determined by (generalized)EV-decomposition.

blocking

embedding

23

A = TN







blocking

embedding

23

A = TN







blocking

embedding

23

A = TN



23

amplitude function

effective amplitude


27

amplitude function

effective amplitude


iteration procedure

27

amplitude function

effective amplitude


iteration procedure

determine embedding maps

embedding map after 3 iterations

27

22

Relation to vacuumHere coarse states correspond to low energy states / states with few excitations from vacuum.

Embedding maps depend on the dynamics of the system.

However in General Relativity: What is vacuum? Hamiltonian is a constraint and hence zero (and we do not Wick rotate).Transfer matrix should be a projector if diffeomorphism symmetry is realized (with eigenvalues one and zero).NB: In 4D diffeomorphism symmetry is broken, eigenvalues take other values too.

So we can hope that even for GR, unphysical states (eigenvalue zero) are truncated away.

There is however a further mechanism that differentiates states even in GR: radial evolution.

[BD, Steinhaus 13: Time evolution as refining, coarse graining and entangling]

23

• •• •

•

•

•

•

•

•

Figure 6: Illustration of radial time evolution in tensor networks: By adding eight additional tensor (ingray) we perform one time evolution step. The boundary data grows exponentially from �4 to �48.

The radial time evolution can be split into steps with time evolution operators

T (R1, R2) = exp(�Z R2

R1

Hrdr) . (6.1)

This is a refining time evolution in the sense that the Hilbert space H2 at R2 will have more kinematicaldegrees of freedom than the Hilbert space H1 at R1. The main idea is to truncate the degrees of freedomin H2. Ideally one would perform a singular value decomposition of T (R1, R2). This would give a maximalnumber of dim(H1) singular values. Hence T (R1, R2) will have a non–trivial co–image in H2, which canbe projected out.

Such a scheme might be indeed worthwhile to investigate further (for statistical systems), in order toobtain an intuition about the truncations. One would expect that the reorganization of the degrees offreedom via the singular value decomposition will be highly non–local, as we have seen for the examplein section 2.1. The time evolution considered there exactly corresponds to T (R1, R2).

Even such a classical investigation might be worthwhile: the radial evolution as a refining timeevolution will result in post constraints. Below we will introduce coarse graining time evolution steps,that will implement a truncation, and should classically lead to pre–constraints. This suggest that for agood choice of truncation, the pre–constraints should match to the post–constraints.

As a remark, we now see why tensor network coarse graining should also work for theories in whichevolution between equal size Hilbert space is actually unitary. (For statistical systems one usually workswith exp(�H), so the path integral is Wick rotated. Here we refer to a non–Wick rotated path integral.)In such a case one might argue that all singular values of the transfer matrix should be equal to one.However adopting the picture of radial evolution, we do not have a unitary evolution – rather one expectsan isometric embedding of coarser states into a Hilbert space of finer states. Thus the transfer matrixshould lead to a set of singular values equal to one and others vanishing. A good truncation would thenidentify these vanishing singular values. In this way the singular value decomposition and the associatedembeddings V can rather be understood as a field redefinition, which automatically defines appropriatecoarse grained observables of the theory.

6.2 Embedding maps via singular value decomposition

In praxis, other schemes are preferred due to computational e�ciency. These involve rather local trun-cation maps. The basic idea is as follows. Imagine two space time regions or e↵ective vertices connectedwith each other by two edges, representing the summation over a certain set of variables, see figure 7. Wewould like to replace these edges carrying an index pair {↵,�} of size �2 with an e↵ective edge carryingonly a number � of indices. We choose an optimal truncation for the summation over the index pair{↵,�}, which is given by the singular value decomposition of MA↵� :

MA↵� =�2X

i=1

UAi�iVi ↵� (6.2)

17

Radial time evolution and coarse states

Consider a “radial time evolution” from a ``smaller” to a “larger” Hilbert space.

(We can easily construct such evolutions for systems based on simplicial discretizations, such as spin foams).

This time evolution itself defines an embedding map: The image of the time evolution defines coarser states in the larger Hilbert space.

Conjecture: This definition leads to states describing coarse grained excitations. (Radial) time evolution can be used to define useful embedding maps. [BD, Steinhaus 13]

[also BD, Hoehn 11,13, Hoehn 14, BD, Hoehn, Jacobson wip]

NB: Again this definition would lead in general to non-local embedding maps.

24

Tensor network renormalization methods... rather using local embedding maps (to keep algorithms efficient) but different versions are possible.

bare/initial amplitude depending on four variables

2 4 6 8 10

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

AA

AA

A′

32

Contract initial amplitudes (sum over bulk variables).Obtain “effective amplitude” with more boundary variables.

2 4 6 8 10

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

AA

AA

A′

32

2 4 6 8 10

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

AA

AA

A′

32

Find an approximation (embedding map) that would minimize the error as compared to full summation (dotted lines). For instance using singular value decomposition, keeping only the largest ones.Leads to field redefinition, and ordering of fields into more and less relevant.

AA

AA

A′

25

AA

AA

A′

25

Use embedding maps to define coarse grained amplitude with the same (as initial) number of boundary variables.

25

25

25

Tensor network renormalization methods

Iteration. Summation over bulk variables are truncated using the embedding maps.

Associated embedding maps for boundary Hilbert space.

26

Fixed points give cylindrical consistent amplitudes

AA

AA

A′

25

a

bc

d

ef

A

AA

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑

ONB

T

T

T

T

A′A′A′

= id

C2C2

C2

C2C2

C2

HH

H

g1 g2

ω(g1g−12 )

k

ω(k)Gauss

22

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

Condition for cylindrically consistent amplitudes

satisfied for

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

2 4 6 8 10

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

0.2

0.4

0.6

0.8

AA

AA

A′

32

ψb

ψb

ψb

ιbb′(ψb)

ιbb′(ψb)

ιbb′(ψb)

A

A

A

A

A

AA

AA

A

ψψ

Ab

AbAb

Ab′



A(ψ) =∑

xi

A(xi)ψ(xi) (0.152)




19

h2h1

A

AA

A

A

A A

AAA A

AAA

AA

AAAAAAA

AAAA

AAAA

AAAA

AAAA

AAAA

∑ONB

T

T

T

T

A′A′A′

= id

C2C2

C2

C2C2

C2

HHH

g1

g1

g2

g2

hehe′

he′′he′′′

ω(g1g−12 )

k

ω(k)Gauss

10

1

01

1

31

27

Fixed points corresponding to topological field theories or interacting field theories

•method works particularly well for convergence to a “gapped phase”

(lead to topological field theories)

•finitely many ground states separated from excited states: determine bond dimension

(number of non-vanishing singular values)

•phase transition: usually involve ‘infinite bond dimension’

•expect to find interacting theories / propagating degrees of freedom

•truncation can be nevertheless quite good, and provides a method to determine phase transition (and sometimes even to calculate critical exponents)

•However argument about radial time evolution suggest that better truncations (near phase transitions) might be possible if non-local embedding maps are incorporated.

•Might even not need infinite bond dimension.

28

Applications: (analogue) spin foam models

•4D spin foam models are very very complicated

•devised analogue models capturing the essential dynamical ingredients of spin foams

[similar to 4D lattice gauge and 2D spin system duality]

•these spin nets can actually be interpreted as spin foams based on very special

discretizations

•unlike spin foams the spin net models are non-trivial in 2D

•investigated the phase diagrams for such models (with quantum group structure)

•these phase diagrams have a very rich structure

(The details of this and the adaption of the tensor network methods

take easily up another talk.)

29

Lattice gauge theory/ Ising like systems

deconfining phase(topological phase)

coupling

confining phase

Spin nets

‘no space’ phase

topological phase(gives 3D gravity!)





0.0

0.5

1.0

a1

0.00.5

1.0

a 2

0.0

0.5

1.0

a 3


Ê ÊÊÊ

ÊÊÊ


Ê

Ê

Ê

ÊÊ

Ê Ê

0.0 0.2 0.4 0.6 0.8 1.0a10.0

0.2

0.4

0.6

0.8

1.0a 2



26

Much richer phase structure![ BD, Martin-Benito, Schnetter 2013, BD, Martin-Benito,

Steinhaus 2013]

Applications: (analogue) spin foam models

Interpretation: different phases describe uncoupled space time atoms (green) and coupled space time atoms (orange,blue).

30

Towards spin foam coarse graining

•So far encouraging results for ‘spin foam analogue’ models.

Conclusions:

•Relevant parameters related to (SU(2)) intertwiners (also appearing in spin/anyon chains) leading to rich phase space structure. This is expected from the gravity dynamics.

•Need to implement a weak version of discretization independence to uncover these rich phase spaces (escape the two lattice gauge theory phases).

•Positive indication for finding a geometric phase in spin foams.

•Are now looking at spin foam analogue and actual spin foam models in higher dimensions.

•Need to extend tensor network tools (applicable to lattice gauge theories) to this end. [wip]

31

Defines embedding maps and associated vacua.

Each phase corresponds to a toplogical field theory.

New representations (Hilbert spaces) for

Loop quantum gravity.

32

Loop quantum gravity•succeeded in constructing a quantum representation of geometric operators

[Ashtekar, Lewandowski, Isham, Rovelli, Smolin, Corichi, ... 90’s]

•spatial geometric operators (area, volume operators) have discrete eigenvalues

•spin networks based on (finite) graphs give a basis in the (non-separable) LQG Hilbert space

•despite this (discrete) basis the theory is defined in the continuum: via embedding maps

•dynamics based on Hamiltonian constraints [Thiemann ’96]

So far the theory is based on the

Ashtekar-Lewandowski representation

based on a (AL) vacuum describing

• zero volume spatial geometry

• maximal uncertainty in conjugated variable.

33

What is vacuum?

•Ashtekar-Lewandowski representation /vacuum corresponds to confining phase for lattice gauge theories (i.e. embedding maps coincide).

Question: Can we base the representation on a different vacuum corresponding to one of the other phases?

34

Loop quantum gravity vacua

Ashtekar - Lewandowski representation (90’s)

geometric variables:

j = 5j = 0

T = exp(−H)

TN →N→∞

j

j′

j′′

∑

j,j′,j′′,...

(0.166)

︸︷︷︸︷︸︸︷︷︸︸︷

Hb = (⊕j Vj)N (0.167)

a(j1, j2, j3)Z(bdry) =

∑bulk j

∏∆ a(j, j′, j′)

⊃Hcoarser Hfiner

|j1,m1; j2,m2; j3,m3, . . .〉

Z ′finer = (dim - factors) × Zfiner

Zcylb1,b2

=√

dim(j)

Zcyl∞,∞[ιb1∞(ψb1), ιb2,∞(ψb2)] = Zcyl

b1,b2[ψb1 ,ψb2 ]

C = I2−

{A , E} = δ

ψvac(A) ≡ 1 , E ≡ 0

28

connection flux: spatial geometry

j = 5j = 0

T = exp(−H)

TN →N→∞

j

j′

j′′

∑

j,j′,j′′,...

(0.166)

︸︷︷︸︷︸︸︷︷︸︸︷

Hb = (⊕j Vj)N (0.167)

a(j1, j2, j3)Z(bdry) =

∑bulk j

∏∆ a(j, j′, j′)

⊃Hcoarser Hfiner

|j1,m1; j2,m2; j3,m3, . . .〉

Z ′finer = (dim - factors) × Zfiner

Zcylb1,b2

=√

dim(j)

Zcyl∞,∞[ιb1∞(ψb1), ιb2,∞(ψb2)] = Zcyl

b1,b2[ψb1 ,ψb2 ]

C = I2−

{A , E} = δ

ψvac(A) ≡ 1 , E ≡ 0

28peaked on degenerate (spatial) geometry

maximal uncertainty in connection

excitations: spin network states supported on graphs

BF (topological) theory representation

[BD, Geiller 2014,BD, Geiller to appear]


29

peaked on flat connectionsmaximal uncertainty in spatial geometry

excitations: flux states supported on (d-1) D-surfaces

(representation) labels for edges

(group) labels for faces

35

We applied this to BF (topological field theory), corresponding to unconfined phase. [BD, Geiller 14]

But construction can be generalized to other topological field theories:

36

(discretized)

topological field

theory

excitations

vacuum

state

(dynamics)

‘inductive limit’ construction

continuum

Hilbert space

for excitations

determine observables stable under refining time evolution

(non-trivial step)

Excitations and vacuum

37

Advantages•This new construction allows to expand loop quantum gravity around different vacua corresponding to the different phases (fixed points) for spin foams.

•Ashtekar-Lewandowski representation: peaked on completely degenerate spatial geometry, maximal fluctuation in conjugated variable: difficult to built up states corresponding to smooth geometry

•New representation [BD, Geiller 2014] : peaked on flat connection, maximal fluctuation in spatial geometry. Corresponds to the physical state in (2+1)D gravity and in general to condensate states with respect to Ashtekar-Lewandowski representation.

•Facilitates construction of states corresponding to smooth geometries and should be useful for discussions of i.e. black hole entropy in loop quantum gravity [Sahlmann 2011].

•New representation much easier to interpret geometrically: new handle on the dynamics of the theory. [need substitute of Thiemanns (1996) quantization of Hamiltonian constraints]

[BD, Geiller 2014,BD, Geiller to appear]

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Phase transitions?





0.0

0.5

1.0

a1

0.00.5

1.0

a 2

0.0

0.5

1.0

a 3


Ê ÊÊÊ

ÊÊÊ


Ê

Ê

Ê

ÊÊ

Ê Ê

0.0 0.2 0.4 0.6 0.8 1.0a10.0

0.2

0.4

0.6

0.8

1.0a 2



26

•This new construction allows to expand loop quantum gravity around different vacua corresponding to the different phases (fixed points) for spin foams.

•What about phase transitions (‘non-trivial’ fixed points)?

•Instead of topological theory expect conformal theory / propagating degrees of freedom.

•Expect such fixed point amplitudes to be non-local non-local embedding maps

(also given by time evolution?)

•Do we regain triangulation independence / diffeomorphism symmetry there?

39

Summary

Coarse graining is about refining boundary states.

Tensor network methods construct embeddings of coarse states, representing “low energy/coarse degrees of freedom”,

into continuum Hilbert space.

40

Summary and Outlook:

Quantum Space Time Engineering

•We are on a good way to understand the ‘many body physics’ of spin foams and loop quantum gravity.

•Corresponds to the ‘continuum limit’ of these models.

•In the path integral approach (spin foams): mapping out the phase diagrams

•connections to condensed matter physics / (new?) topological field theories and phases

•challenge: going to higher dimensions

•challenge: understanding symmetries of phase transition fixed points

•In the canonical approach (loop quantum gravity): expanding around different vacua

•facilitates investigation/construction of physical states describing extended/smooth geometries

•each vacuum comes with its own set of excitations: investigate dynamics of these.

41

Thank you!

42

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