Massline and other recentresults

Of CDT Quantum Gravity

Németh Dániel, Heidelberg 2019.09.9-13.

Regge CalculusLattice regularization via triangulation

Volume is just the sum of the simplices

Curvature is related to the deficit angle:

𝜖 = 2𝜋 −

𝑛

Θ𝑛

𝑇

𝐴𝑇𝜖𝑇 = ∫ 𝑑4𝑔 𝑔𝑅 𝑔 = 𝑆𝑅

Path integral formalism

𝑆𝐸𝐻 =1

16𝜋 𝐺∫ 𝑑4𝑥 −𝑔 𝑅 − 2Λ

𝑆𝑅 𝜅, 𝜆, 𝑇𝜶 =

𝜎𝑑−2∈𝑇𝑑−2

(2𝜅𝜖𝜎𝑑−2𝜈𝜎𝑑−2 − 𝜆𝜈(𝜎𝑑−2))

𝑍 = ∫ 𝐷 𝑔 𝑒𝑖𝑆𝐸𝐻 𝑔 → 𝑊𝑖𝑐𝑘 + 𝑅𝑒𝑔𝑔𝑒 →

𝑇

𝑒−𝑆𝑅[𝑇]

Simplices in 2+1 DEvery slice is a 2 dimensional triangulation

Simplices in 3+1 DEvery slice is a 3 dimensional triangulation

• Every slice is a closed 3D triangulation

• Integer time: {4,1}

• Fractional time: 3,2 → 2,3 → {1,4}

• Slab: adjacent 4 fractional timeslices

Topology and geometry of a configuration

4D - Action

𝑆𝐸𝐻 = −𝐾0𝑁0 + 𝐾4𝑁4 + Δ(𝑁4(4,1)

− 6𝑁0)

𝐾0~1

𝐺0→ 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝐾4~𝛬0𝐺0

→ 𝐶𝑜𝑠𝑚𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Δ ~ 𝛼 → 𝐴𝑠𝑖𝑚𝑚𝑒𝑡𝑟𝑦 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟

Vo

lum

ePr

ofi

le

For fixed volume: 𝐾4(K0, Δ) plane presents the phase structure

Cb

B

C

A

Massline

• Action contribution: 𝑆𝑚 = 𝑚 ⋅ 𝑙, 𝑍 = σ𝑇 𝑒−𝑆𝑅 − 𝑆𝑚

• Semi-non-backward:

• 4,1 → 1,4

• 1,4 → 2,3 → 3,2 → 4,1

• Self-avoiding - (technical, significant difference in 2D in

Hausdorff/spectral dimension)

Curvature

• Radial volume distribution:

< 𝑉 𝑟 >𝑡=1

𝑛𝑡

𝑜∈𝑇𝑡

𝑉(𝑟, 𝑜)

• Order of a spatial link 𝑙 is 𝑜(𝑙)

• Def(𝑙) = 2𝜋 − 𝑜(𝑙) ⋅ arccos(1

3)

• 𝐶 𝑞 = σ𝑙∈𝑞𝐷𝑒𝑓(𝑙)

𝑜(𝑙)→ 𝐶 𝑞 ≈ σ𝑙∈𝑞

2

𝑜(𝑙)

Structure

• Eccentricity: 𝜖 𝑣 = max𝑢

𝑑(𝑣, 𝑢)

• Radius: 𝑟 = min𝑣

𝜖(𝑣)

• Diameter: 𝑑 = max𝑣

𝜖(𝑣)

• Central point: where r = 𝜖(𝑣)

• Peripheral point: where d = 𝜖(𝑣)

• Central region: set of 𝑁3with themin(𝜖 𝑣 )

Eccentricity distributionNumber of 𝑁3 with given eccentricity

Centrum and outgrows

Geometric structure of the sphereAdjecent ringparts are connected components

Geometric structure of the torusCentrum is large but there are many outgrows

Comparison of torus and sphereCaused by the size of the minimal configuration

Radial volume distribution of sphere and torus

Measured around a line and around a random point

Difference of the maximas of volume profile

Between the line and the random point

Average curvature for various massesMeasured around a line and around a random point

Curvature around the line and at random

In the function of mass and distance

Length of the lineThe line can go backwards until a {4,1} simplex

Line length as a function of mass𝑙 𝑚 ≈ 40 ⋅ 𝑒−1⋅𝑚

Next steps

Quantumline

Two or more lines

Introduction of massline as a sourceof a scalar/vector field

Summary

• Introduction of a point particle

•Critical behavior was observed

•Better understanding of thegeometric structure

•Dependence of the length on themass was determined

Thank you foryour

attention!