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?