Coupling lattice eld theory to spin foamseichhorn/Steinhaus_Heidelber… · Coupling lattice eld...
Transcript of Coupling lattice eld theory to spin foamseichhorn/Steinhaus_Heidelber… · Coupling lattice eld...
Coupling lattice field theory to spin foamsw.i.p. with Masooma Ali
Sebastian [email protected]
Perimeter Institute for Theoretical Physics
Workshop “Quantum Gravity and Matter”, HeidelbergSeptember 11th 2019
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 1 / 21
Gravity and quantum matter?
Matter indispensable to describe ouruniverse
Cosmology, early universe, black holes...
Quantum matter on dynamicalspace-time?
Reconcile general relativity withquantum field theory
“Quantum space-time”?
Does quantum gravity affect the matter sector?
Asymptotic safety: restriction on matter sector [Dona, Eichhorn, Percacci ’14]
Incorporating matter is vital to approaches of pure quantum gravity.
Bridge the gap to observable scales with renormalization techniques.
This talk: spin foam quantum gravity perspective
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 2 / 21
Spin foam gravity [Rovelli, Reisenberger, Barrett, Freidel, Livine, Krasnov, Perez, Speziale, ...]
Path integral of geometries
Regulator: Lattice
Finitely many degrees of freedom
Quantum geometric building blocks
(Constrained) topological quantum fieldtheory
Transition amplitudes
Assign an amplitude A ∼ eiSEH to eachgeometry
Single building block ∼ discrete gravity
Derived from general relativity
No reference to background geometry!
Aim to implement diffeomorphism symmetry
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 3 / 21
Spin foam gravity - Basics
Regulator: 2-complex
Vertices v, edges e, faces f
Coloured with group theoretic data
Boundary graph ∼ 3D geometry
Polyhedra ∼ intertwiner ι
Area of face ∼ representation j
Evolution: bulk geometry
History interpolating between boundaries
Sum over all histories
Sum over all ι and j
Assign amplitude to each history
Z =∑
{jf},{ιe}
∏f
Af (jf )∏e
Ae({jf}, ιe)∏v
Av({jf}, {ιe})
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 4 / 21
Open issues
Ambiguities in definition of 4D models
Implementation of simplicity constraints
Lattice is a choice.
Should the lattice be “fine” or “coarse”?
Can we relate the physics on different lattices?
Diffeomorphism symmetry? [Dittrich, Bahr ’09, Bahr, S.St. ’15]
vs. vs. ...
Physics must be independent of choice of regulator!
How do we bridge the enormous gap to observable physics?
Do we recover general relativity in a suitable limit?
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 5 / 21
Matter in spin foams
How to incorporate matter in spin foam quantum gravity?Matter on top of spin foam [Oriti, Pfeiffer ’03, Speziale ’07, Bianchi, Han, Rovelli, Wieland,
Magliaro, Perini ’13]
Unification scenarios [Crane ’00, Smolin ’09]
Massless scalar field [Lewandowski, Sahlmann ’15, Kisielowski, Lewandowski ’18]
Dynamics of combined system?
What is matter at the Planck scale?
More ambiguities!
Towards renormalization of matter and gravity system
Massive scalar field coupled to a restricted 4D spin foam
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 6 / 21
Outline
1 Introduction
2 Cuboid spin foams
3 Lattice field theory – scalar field
4 Interpretation and preliminary results
5 Summary and Outlook
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 7 / 21
Towards a simplified model [Bahr, S.St. ’15]
Strategy: study a subset of the full spin foam path integral
Quantum cuboids: 4D Riemannian EPRL model [Engle, Pereira, Rovelli, Livine
’08] on hypercubic 2-complex [Lewandowski, Kaminski, Kisielowski ’09’]
Restrict shape of intertwiner
Coherent SU(2)-intertwiner [Livine-Speziale ’07]
|ιj1,j2,j3〉 =
∫SU(2)
dg g .
3⊗i=1
|ji, ei〉 ⊗ |ji,−ei〉
Peaked on the shape of a cuboid
ei normal unit vectors in R3.
Drastic restrictions on spins and intertwiners.
Asymptotic expansion of full amplitude.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 8 / 21
Semi-classical spin foam amplitudes [Bahr, S.St. ’15]
Vertex amplitude Av
Av ∼∫
SU(2)8
∏a
dga e∑ab 2jab ln(〈−~nab|g−1
a gb|~nba〉)
Face amplitude Af
Af ∼ (2j + 1)2α
Stationary phase approximation:
Av(j1, j2, j3, j4, j5, j6) ∼ j2αi
(1√D
+1√D∗
)2
Transition from spins to edge lengths
Flat space-time: discrete gravity / Regge action vanishes.
Model: superpostion of hypercuboidal, flat lattices.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 9 / 21
Test case for renormalization [Bahr, S.St. PRL ’16, PRD ’17]
0.55 0.60 0.65 0.70 0.75
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Abelian sub-class of diffeomorphisms
Moving entire hyperplanes
Different subdivisions of flat space-time
Symmetry broken – α dependent
Compute observables on different foams
〈O〉fineα ≈ 〈O〉coarse
α′
Use this to define renormalizationgroup flow α→ α′
Flow from “fine” (UV) to “coarse” (IR)
Indication for a UV-attractive fixed point
and restored diffeomorphism symmetry
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 10 / 21
Outline
1 Introduction
2 Cuboid spin foams
3 Lattice field theory – scalar field
4 Interpretation and preliminary results
5 Summary and Outlook
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 11 / 21
Interlude: scalar field on a regular lattice
Continuum massive, free scalar field:
Z =
∫Dφ e−
∫d4xL(φ,∂µφ), L(φ, ∂µφ) =
1
2∂µφ∂µφ+
M20
2φ2
Physical mass M0
Discretisation: lattice constant a
Z =
∫ ∏i
dφi e− 1
2φiKijφj , Kij = −a2
∑µ
(δi+eµ,j+δi−eµ,j−2δij)+a4M2
0 δij
Theory depends on relation of M0 and a
Lattice mass M(a) = aM0
Correlations:
〈φiφj〉 = K−1ij ∼ exp{−rijM0}
Straightforward continuum limit for a→ 0.
Correlation length diverges for M(a)→ 0.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 12 / 21
Discrete exterior calculus
Generalize lattice field theory to cuboid spin foams
Discrete exterior calculus [Desbrun, Hirani, Leok, Marsden ’05, Arnold, Falk, Winther ’09,
McDonal, Miller ’10, Sorkin ’75, Calcagni, Oriti, Thurigen ’12]
Project p-forms onto p-dim objects |σp〉φ p-form smeared on p-surface S =
∑i |σip〉:
φ(S) = 〈φ|S〉 =∑i
〈φ|σip〉 =∑i
Vσipφσi
p=∑i
∫σip
φ =
∫Sφ
Dual (d− p)-cells defined on dual complex 〈?σp|.〈?σp|σp′〉 = δp,p′ and
∑σ |σ〉〈?σ| = id.
Discrete exterior derivative:
dφ(σp+1) =
∫σp+1
dφ =
∫∂σp+1
φ = φ(∂σp+1) =∑
σp⊂σp+1
sgn(σp, σp+1)φ(σp)
Use this to derive scalar field action for general lattices.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 13 / 21
Scalar field on an irregular lattice
Scalar field action in terms of forms:
S =1
2
∫dφ ∧ ∗dφ+
M20
2
∫φ ∧ ∗φ → S =
1
2
∑σ1
〈dφ|dφ〉+M2
0
2
∑σ0
〈φ|φ〉 .
〈dφ|dφ〉 = 〈?d ? dφ|φ〉 (periodic boundary conditions)
Scalar field action: S = 12φ(σi0)Kij φ(σj0) [Hamber, Williams ’93]:
Kij =
∑
σ1⊃σi0V?σ1Vσ1
+M20 V?σ0 i = j
−∑
σ1⊃σi0V?σ1Vσ1
i 6= j
Notation:
V?σ0 : 4-volume dual to vertex σ0
V?σ1: 3-volume dual to edge σ1.
Vσ1 : length of edge σ1.
Define massive scalar field coupled tocuboid spin foam.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 14 / 21
Scalar field coupled to cuboid spin foams
Partition function of combined system reads:
Z =
∫ ∏σ1
dlσ1
∏σ0
dφ(σ0)∏σ4
Aσ4(α, {lσ1})e−12φ(σi0)Kij(M0)φ(σj0)
Observables of combined system:
Geometric obervables: edge length, volume
Correlations and correlation length of scalar field
〈O〉α,M0 =
∫ ∏σ1
dlσ1
∏σ0
dφ(σ0)O∏σ4
Aσ4(α, {lσ1})e−12φ(σi
0)Kij(M0)φ(σj0)
Massive, free scalar field on coupled to
a superposition of hypercubodial lattices
weighted by spin foam amplitudes.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 15 / 21
Outline
1 Introduction
2 Cuboid spin foams
3 Lattice field theory – scalar field
4 Interpretation and preliminary results
5 Summary and Outlook
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 16 / 21
What can we learn from this model?
How do both systems influence each other?
〈V 〉 or 〈l〉 with or without matter?
〈φiφj〉 on a spin foam? → scale from correlation length?
Physical meaning of “observables”?
Correlations between φ at vertex i and j
Use discrete structure to identify vertices across lattices.
Label dependence an issue?
Diffeomorphism invariance?
Broken – can we restore it?
Lattice dependence! Can we define a continuum / refinement limit?
Conceptual question: What are good observables in quantum gravity?
Hope to gain first insights from this model, e.g. by studying it on differentlattices.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 17 / 21
Numerical setup
3× 3× 3× 3 lattice with periodic boundary conditions
2 Parameters:α: Exponent in face amplitude
M0: scalar field mass
Evaluate observables using Monte Carlo methods / importancesampling:
〈l〉 =1
Z
∫ (∏e
dle
)l
∏?v A?v({li})√
detK
〈φiφj〉 =1
Z
∫ (∏e
dle
)K−1ij
∏?v
A?v({li})
Sample edge lengths, matter part completely integrated out
Cut-off on edge length: 103
Sample size: 104 (not enough!)
Preliminary results, more data and checks are necessary.
Lattice size, sample size, cut-off and thermalization.
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 18 / 21
Correlation length
0 200 400 600 800 1000
0
100
200
300
400
500
1 2 3 4
10- 19
10- 14
10- 9
10- 4
10
Data for α = 0.575 and various M0.
Histogram: almost random sampling
Correlations 〈φiφj〉:Distance d: combinatorial distance
Fit: e−Md
M0 0.01 0.1 1. 10.
M 1.95 4.50 7.73 12.13
Infer lattice constant aeff(α,M0)?
Different correlations for same distanced.
Coincide for regular lattice.
Careful: small lattice!
Effective regular lattice?
Infer a mass dependent length scale?
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 19 / 21
Summary
Massive scalar field coupled to a superposition of hypercuboidlattices
4D cuboid spin foam model
Flat, discrete space-time
Not realistic, but remnant of diffeomorphism symmetry
Matter action derived via discrete exterior calculus
Applicable to more general lattices than hypercuboid
Not unique, depend on choice of dual lattice
First preliminary results: Correlations and correlation length
Typical exponential drop off with (combinatorial) distance
Compare to regular lattice and infer effective length?
More data necessary to study geometric observables and checkconvergence
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 20 / 21
Outlook
Renormalization / coarse graining of matter-spin foam system
First step: study model on different lattices
Adapt methods to coarse grain entire amplitude
Phase diagram, phase transitions, fixed points... [S.St. ’15]
Restoration of diffeomorphism symmetry?
Beyond cuboids and scalar fields
Spin foam: more general configurations and quantum amplitude
Matter: pure Yang-Mills
Conceptual questions:
What are good observables in quantum gravity?
Lorentzian: mismatch between Wick-rotated matter action and spin foam.
Thank you for your attention!
Sebastian Steinhaus (PI) Lattice field theory and spin foams September 11th 2019 21 / 21