Benjamin BahrII. Institute for Theoretical Physics

University of HamburgLuruper Chaussee 149

22761 Hamburg

674 WE Heraeus Seminar Bad Honnef, 18th June 2018“Quantum spacetime and the Renormalization Group“

in collab with Sebastian Steinhaus (PI), Giovanni Rabuffo (I02), Vadim Belov (I02)

Exploring the RG flow of truncated Spin Foam Models

I Motivation

Loop Quantum Gravity:

Attempt at a quantization of GR, which is

* minimal:

a priori no SUSY, extra dimensions, or even matter → unification of forces not a primary goal

* background-independent & non-perturbative:

no a priori choice of metric, rather full geometry fluctuates

Rests on quantization of GR in Ashtekar variables:

connection & (densitised) triad → quantization of gauge theories

[Ashtekar, Isham ‘92, Ashtekar, Lewandowksi ‘94, Rovelli, Reisenberger '94, Ashtekar, Lewandowski Marolf, Mourão, Thiemann ‘95]

I Motivation

States of Loop Quantum Gravity:

spin network functions (“quantized Cauchy data”)

dynamics (two connected possibilities): * canonical(constraints, ~Wheeler-deWitt equation)* covariant(path integral , transition amplitudes via “spin foam models”)

[Penrose ‘71, Rovelli, Smolin ‘95] see also: [Delcamp,Dittrich, Riello ‘16]

I Motivation

Theory rests on discrete structures (graphs, polyhedral decompositions)

→ open problem: continuum limit? ( ↔ RG flow )

→ crucial to:

* UV-complete the theory

* make contact with continuum approaches to QGR, or even QFT(e.g. asymptotic safety, EFT “where is Minkowski space?”)

* check whether continuum limit is actually GR

In general, not much is known about the RG flow of LQG in 4d.

This talk:

Look at a specific spin foam model numerically, within some truncations

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

II Spin Foam Models

Boundary data: spin network functions ( quantized 3d Cauchy data)

labels:

graph: (nodes , links ) spins:

intertwiners:

interpretations as 3d polyhedra:

spins correspond to 2d areas

intertwiners as 3d shapes

[Ashtekar, Lewandowski ‘92, Rovelli, Smolin ‘95, Livine, Speziale ‘07, Bianchi, Dona, Speziale ‘11]

I Motivation

boundary data fixed by

boundary:3d polyhedrabulk: 4d polyhedra

[Reisenberger '94, Barrett, Crane '99, Livine, Speziale '07, Engle, Pereira, Rovelli, Livine '07, Freidel, Krasnov '07, Oriti Baratin '11, … Kaminski, Kisielowski, Lewandowski ‘09]

spin foam models:

boundary data: spins & intertwiners associated to 2d & 3d polyhedra in bdybulk data: spins & intertwiners associated to 2d & 3d polyhedra in bulk

Theory as boundary amplitudes:

II Spin Foam Models

EPRL-FK spin foam model: certain choices for amplitudes

Very popular, for several reasons

e.g.: asymptotic formula for vertex amplitude (4-simplex):

Regge action (discrete GR)

[Barrett, Dowdall, Fairbairn, Gomez, Hellmann, '07, Freidel, Krasnov, ‘08]

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

II Spin Foam Models

Coarse graining idea: the polyhedral decompositions are scales

physical motivation: scales ↔ amount of detail that can be measured

mathematical concept: form a partially ordered set (refinement of polyhedra)

Not just one parameter: RG flow runs along a poset, not an ordered sequence.→ RG scale ~ # of building blocks

Note: boundary polyhedral decomposition also to be refined

[Manrique, Oeckl, Weber, Zapata '95, Oeckl ‘02, Smerlak, Rovelli '11, BB ‘11, Dittrich ‘12, Dttrich, Steinhaus ‘13, BB’ 14]

II Spin Foam Models

Embedding maps:

for , one needs an isometry which relates the boundary Hilbert spaces

embedding maps contain physical information about coarse graining of degreesof freedom: “coarse state” → “fine state”

embedding maps add d.o.f. in the vacuum state → “dynamical embedding maps”

[Dittrich, Steinhaus ‘13 → see talk by Dittrich tomorrow!]

II Spin Foam Models

amplitudes describe effective theory at some discretization scale

cylindrical consistency of amplitudes:

→ RG flow equation of amplitudes

note:

* scale not given by number, but bydiscretization itself

* scales carry no geometry. rather:sum over = sum over geometries

* here: no sum over triangulations, but consistency between differenttriangulations → amplitudes = effective theories, compare to GFT, tensor models, CDT

→ see e.g. talks by Carrozza, Oriti, and Laiho, Loll

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

II Truncations

1.) Truncation of state space: (e.g.) symmetry restriction

Attempt at simplifying state sum: restrict sum over spins / intertwinerse.g. to only those which

satisfying certain symmetry restrictions,allow fluctuation in certain degrees of freedom

→ truncation of the model!

pro: model simplified!interpretation of geometries (Livine-Speziale intertwiners)

con: hard to justify physically: only applicable where truncated states can be neglected

Still: gives insight into some aspects of the model

Ultimately: Relax truncations and check whether found features persist

[BB, Steinhaus ‘15, BB, Rabuffo, Steinhaus ‘17]

II Truncations

2.) Truncation of theory space

In general, RG flow in general amplitude functionsEPRL-FK model not necessarily form-invariant.

Still, we restrict to (deformed) EPRL-FK model, just vary coupling constants

choice for amplitudes:

coupling constants:

Barbero-Immirzi parameter fixed:

II Truncations

2.) Truncation of theory space to parameterized by

→ RG flow projected to that subset of effective amplitudes

consistency conditions ↔ all coarse observables agree with their fine counterparts

relax this condition:

choose finitely many observables , only demand error minimization

used to define flow

Numerical comparison via MC methods

[BB ‘14, BB, Steinhaus ‘17, BB, Rabuffo, Steinhaus ‘18]

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

III Numerical investigation: hypercuboids

4d “hypercubic lattices”

restriction of allowed spins & intertwiners: quantum cuboids:

[BB, Steinhaus ‘15, BB, Steinhaus ‘16, BB, Steinhaus ‘17]

III Numerical investigation: hypercuboids

Coarse graining step: 2x2x2x2 → 1 hypercuboid

→ iterate

embedding map (unfortunately, not dynamical):

EPRL model amplitudes, large j-asymptotical formula (integral over spins)

only coupling constant in this case:flow in

III Numerical investigation: hypercuboids

Isochoric RG flow: 32 → 2 vertices

boundary state const 4-volume const

flow:

flow has a fixed point!

→ unstable (UV-attravtive)

→ splits phase diagraminto two regions

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

III Numerical investigation: hyperfrusta

Vertex geometries: hyperfrusta

Allowed spins/intertwiners: quantum cube & quantum frustum

“cosmological” transition

allows for curvature fluctuations: → couplings and also interesting!

[BB, Klöser, Rabuffo ‘17]

III Numerical investigation: hyperfrusta

Isochoric flow:

total 4-volume constant,

initial & final spin fixed

time-like and bulk spins fluctuate

keep fixed:

→ flow

find fixed point:

similar to hypercubic case,but shifted (0.63 → 0.69)

III Numerical investigation: hyperfrusta

Higher-dimensional flow:

keep fixed, flow in

isotemporal gauge: fix “time” steps

fix boundary spins (but not 4-volume)

observables: 3-volume, 4-volume, fluctuations

coarse graining step: 3x4x4x4 → 2x3x3x3

III Numerical investigation: hyperfrusta

2d flow, fixed

arrow colours: relative error in cylindrical consistency

III Numerical investigation: hyperfrusta

letting all three parameters flow:

arrow colours: relative error in cylindrical consistency

fixed point around

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

III Numerical investigation: hypercuboids and finite size scaling

Fluctuations in hypercuboidal setting:

* non-geometric configurations volume simplicity not implemented

* vertex translations:

similar to simplicial case: remnant of diffeomorphisms on lattice

III Numerical investigation: hypercuboids and finite size scaling

(nearly) vertex displacement symmetry at fixed point!

Fixed point separates tworegions with qualitatively differentamplitude behaviour

regular subdivisions favoured irregular subdivisions favoured

[BB, Dittrich ‘08, BB Dittrich ‘09]

III Numerical investigation: hypercuboids and finite size scaling

Consider larger lattices!

Reduced volume fluctuations increase with larger lattices:

total volume and bdy state kept constant

due to symmetry:

reduced volume:

III Numerical investigation: hypercuboids and finite size scaling

Finite size scaling: fluctuations are similar for different :

reduced coupling constant:

fluctuations for different lattice sizes:

read off critical exponents by collapsing data for different

III Numerical investigation: hypercuboids and finite size scaling

Collapsing to read off critical exponents ( , large error bars so far):

Increasing fluctuations at the fixed point.

Careful for the interpretation: fluctuations diverge not because of divergence incorrelation lengths, but because of restoration of diffeomorphism symmetry!

Natural: how much volumeis in the “left half” is notdiffeo-invariant

→ expectation value has huge fluctuations

I Motivation

II Spin foam models

– the EPRL-FK model– background-independent renormalization– truncations

III Numerical investigations– hypercuboids: RG flow – hyperfrusta: RG flow– hypercuboids: finite size scaling

IV Summary & conclusion

IV Summary & conclusion

RG flow of the truncated EPRL model numerically

→ truncations of state space→ truncation of theory space

truncations hard to control, but several lessons can be learned:

* diffeo symmetry broken in EPRL-FK model, even for flat configurations* truncated RG flow can be computed numerically* NG RG fixed point: mechanism for diffeo symmetry restoration

→ fluctuations of diff-variant observables diverge* fixed point persistent under some relaxing of truncation* finite size scaling at fixed point: critical exponents of truncated theory

IV Summary & conclusion

Many open questions:

* Does fixed point persist in full (un-truncated) theory?* Physical meaning of fixed point? Is it the only one?* If it persists, does it specify a phase transition? Which order? * Connection to asymptotic safety scenario, (Causal) dynamical triangulations?