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

[email protected]

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


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


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})


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?


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


Outline

1 Introduction

2 Cuboid spin foams

3 Lattice field theory – scalar field

4 Interpretation and preliminary results

5 Summary and Outlook


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.


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.


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


Outline

1 Introduction

2 Cuboid spin foams





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.


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.


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.


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.


Outline

1 Introduction

2 Cuboid spin foams





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.


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.


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?


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


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!


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