![Page 1: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/1.jpg)
Massline and other recentresults
Of CDT Quantum Gravity
Németh Dániel, Heidelberg 2019.09.9-13.
![Page 2: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/2.jpg)
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
![Page 3: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/3.jpg)
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]
![Page 4: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/4.jpg)
Simplices in 2+1 DEvery slice is a 2 dimensional triangulation
![Page 5: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/5.jpg)
Simplices in 3+1 DEvery slice is a 3 dimensional triangulation
![Page 6: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/6.jpg)
• 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
![Page 7: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/7.jpg)
4D - Action
𝑆𝐸𝐻 = −𝐾0𝑁0 + 𝐾4𝑁4 + Δ(𝑁4(4,1)
− 6𝑁0)
𝐾0~1
𝐺0→ 𝐺𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝐾4~𝛬0𝐺0
→ 𝐶𝑜𝑠𝑚𝑜𝑙𝑜𝑔𝑖𝑐𝑎𝑙 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Δ ~ 𝛼 → 𝐴𝑠𝑖𝑚𝑚𝑒𝑡𝑟𝑦 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟
![Page 8: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/8.jpg)
Vo
lum
ePr
ofi
le
For fixed volume: 𝐾4(K0, Δ) plane presents the phase structure
Cb
B
C
A
![Page 9: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/9.jpg)
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)
![Page 10: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/10.jpg)
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)
![Page 11: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/11.jpg)
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 )
![Page 12: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/12.jpg)
Eccentricity distributionNumber of 𝑁3 with given eccentricity
![Page 13: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/13.jpg)
Centrum and outgrows
![Page 14: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/14.jpg)
Geometric structure of the sphereAdjecent ringparts are connected components
![Page 15: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/15.jpg)
Geometric structure of the torusCentrum is large but there are many outgrows
![Page 16: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/16.jpg)
Comparison of torus and sphereCaused by the size of the minimal configuration
![Page 17: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/17.jpg)
Radial volume distribution of sphere and torus
Measured around a line and around a random point
![Page 18: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/18.jpg)
Difference of the maximas of volume profile
Between the line and the random point
![Page 19: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/19.jpg)
Average curvature for various massesMeasured around a line and around a random point
![Page 20: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/20.jpg)
Curvature around the line and at random
In the function of mass and distance
![Page 21: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/21.jpg)
Length of the lineThe line can go backwards until a {4,1} simplex
![Page 22: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/22.jpg)
Line length as a function of mass𝑙 𝑚 ≈ 40 ⋅ 𝑒−1⋅𝑚
![Page 23: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/23.jpg)
Next steps
Quantumline
Two or more lines
Introduction of massline as a sourceof a scalar/vector field
![Page 24: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/24.jpg)
Summary
• Introduction of a point particle
•Critical behavior was observed
•Better understanding of thegeometric structure
•Dependence of the length on themass was determined
![Page 25: Massline and other recent results - Heidelberg Universityeichhorn/CDT_heidelberg.pdf · Massline and other recent results Of CDT Quantum Gravity Németh Dániel, Heidelberg 2019.09.9-13](https://reader033.vdocuments.pub/reader033/viewer/2022060208/5f0406867e708231d40bf2b5/html5/thumbnails/25.jpg)
Thank you foryour
attention!