“How many dimensions is our Universe?” Bob asked his friends.
Alice confidently replied, “Don’t be silly, it is 3D!”
Bob, intrigued, countered, “But I watched a video on Sci-A channel stating that according to General Relativity, we live in 3+1D, with the extra dimension being time and holding equal significance to the other three spatial dimensions. Charlie, you’re the expert here, what do you think?”
Charlie, a freshly retired physicist, pondered with a heavy sigh. He then plucked one of Alice’s hairs and remarked, “How many dimensions does this hair have?”...he then continued “ If you’re far enough away, you might think it’s effectively one-dimensional. However, under the microscope, it is 3D. Such might be our Universe.”
Alice, looking confused, asked, “Can you explain more?”
Alice confidently replied, “Don’t be silly, it is 3D!”
Bob, intrigued, countered, “But I watched a video on Sci-A channel stating that according to General Relativity, we live in 3+1D, with the extra dimension being time and holding equal significance to the other three spatial dimensions. Charlie, you’re the expert here, what do you think?”
Charlie, a freshly retired physicist, pondered with a heavy sigh. He then plucked one of Alice’s hairs and remarked, “How many dimensions does this hair have?”...he then continued “ If you’re far enough away, you might think it’s effectively one-dimensional. However, under the microscope, it is 3D. Such might be our Universe.”
Alice, looking confused, asked, “Can you explain more?”
Introduction
Introduction
Exploring the state-of-the-art of phenomena in astrophysics, discernible through advanced observatories like KAGRA, Virgo, LIGO, and the soon-to-be-launched LISA ( also known as NGO), hinges on sophisticated models conducted in 3 +1 dimensions. Most of these models assume our universe to possess three spatial dimensions and one temporal dimension inspired by the framework of general relativity, even if they are constructed within modified theories of gravity, or through semi-classical analysis, attempting to incorporate quantum fields on a curved background, neglecting the backreaction of these fields on the spacetime geometry itself.
Our intuitive perception aligns with a three spatial dimensional universe, it is hard to imagine drawing more than three perpendicular lines at a point. This intuition is supported by stability conditions observed in most solar system orbits and the structure of Maxwell equations [1]. However, theories seeking a quantum version of Einstein’s Gravity, even more trying to unify it with other known interactions, aren’t constrained by this intuition and this number of dimensions. String theory, for example, proposes dimensions far beyond four—up to ten or even twenty-six [2], even if there are some exception to that [4]. Similarly, Kaluza-Klein models necessitate 11 dimensions for grand unifications.
Despite these theories addressing longstanding questions about Quantum Gravity, a crucial consideration is how they operate within the energy scales accessible through experiments or observations (currently a TeV). While many models propose higher dimensions, the observable universe remains 3+ 1 D at our energy scale. String theory, for instance, suggests compactifying the extra dimensions at our energy scale, treating them as effective excitations of a 4D universe . This dynamical dimensionality concept is common in Quantum Gravity theories. On the other hand, some evidence hinting at dimensionality reduction at higher energy scales [4].
Before delving into this dimensionality debate, it’s essential to discuss the utility of scale-dependent dimensions from a classical standpoint. In Sec. I, we’ll explore classical motivations for considering dynamic dimensions. Sec. II will detail how various Quantum Gravity theories, such as Causal Dynamical Triangulation (CDT), and Causal Set Theory incorporate this concept. In Sec. III, our discussion will predominantly draw insights from [2,4], forming the basis for our conclusions. Finally, in Sec. IV, we aim to extend these discussions to contemplate the concept of dimension reduction within the Wolfram Model.”
Our intuitive perception aligns with a three spatial dimensional universe, it is hard to imagine drawing more than three perpendicular lines at a point. This intuition is supported by stability conditions observed in most solar system orbits and the structure of Maxwell equations [1]. However, theories seeking a quantum version of Einstein’s Gravity, even more trying to unify it with other known interactions, aren’t constrained by this intuition and this number of dimensions. String theory, for example, proposes dimensions far beyond four—up to ten or even twenty-six [2], even if there are some exception to that [4]. Similarly, Kaluza-Klein models necessitate 11 dimensions for grand unifications.
Despite these theories addressing longstanding questions about Quantum Gravity, a crucial consideration is how they operate within the energy scales accessible through experiments or observations (currently a TeV). While many models propose higher dimensions, the observable universe remains 3+ 1 D at our energy scale. String theory, for instance, suggests compactifying the extra dimensions at our energy scale, treating them as effective excitations of a 4D universe . This dynamical dimensionality concept is common in Quantum Gravity theories. On the other hand, some evidence hinting at dimensionality reduction at higher energy scales [4].
Before delving into this dimensionality debate, it’s essential to discuss the utility of scale-dependent dimensions from a classical standpoint. In Sec. I, we’ll explore classical motivations for considering dynamic dimensions. Sec. II will detail how various Quantum Gravity theories, such as Causal Dynamical Triangulation (CDT), and Causal Set Theory incorporate this concept. In Sec. III, our discussion will predominantly draw insights from [2,4], forming the basis for our conclusions. Finally, in Sec. IV, we aim to extend these discussions to contemplate the concept of dimension reduction within the Wolfram Model.”
I. Dimension Reduction from a Classical POV
I. Dimension Reduction from a Classical POV
Recognizing the advantages of altering manifold dimensionality at different scales, an intriguing perspective emerges. This dynamical dimensionality holds implications for both cosmology and particle physics. For a more in-depth exploration of the arguments presented in this section, we strongly encourage the reader to refer to [2].
Higgs
Higgs
As discussed in [2], lower dimensions exhibit suppressed Ultraviolet divergence in the standard model [5]. For instance, on a Minkowskian background with dimension, n+1, the Higgs field with Yukawa coupling has divergence of for n = 2,3 while Log[Λ] for n=1. The transition from quadratic (linear) divergence to logarithmic divergence as the spatial dimensions shift from three (two) to one is crucial. In that sense, on 1+1 D background, there is no ultraviolet divergence at all, and the theory is super-renormalizable.
Consider the mentioned Lagrangian, we can look the first loop of fermions, gauge bosons and Higgs fields,
n-1
Λ
Consider the mentioned Lagrangian, we can look the first loop of fermions, gauge bosons and Higgs fields,
In[]:=
ToExpression["L_H=D_\\nu \\Phi^{\\dagger} D^\\nu \\Phi-\\mu^2 \\Phi^{\\dagger} \\Phi+\\frac{\\lambda}{2}\\left(\\Phi^{\\dagger} \\Phi\\right)^2-\\sum_f g_f \\Phi \\bar{\\psi}_f \\psi_f",TeXForm]
Out[]=
L
H
∑
f
g
f
ψ
f
ψ
f
2
μ
†
Φ
ν
D
D
ν
†
Φ
1
2
2
Φ
2
()
†
Φ
where Φ is the Higgs field, = λ= , and are the fermion and Higgs mass respectively, while ν is the Higgs field vacuum expectation value. is the covariant derivative for the field.
2
g
f
2
m
f
2
ν
2
m
H
2
2
ν
m
f
m
H
ν
D
Applying the Feynman rules to those digrams one find the cut off momentum ( conversely energy) scale Λ, one obtain the following integrals in three, two and one spatial dimensions respectively.
QCD
QCD
Quantum Chromo Dynamics (QCD) is also super-renormalizable in 2+1 D and 1+1 D, obviating the need for lattice regularization common in 3+1 D. The shift in dimensionality will make QCD coupling constant has a positive dimension, facilitates convergence in perturbative schemes akin to Quantum Electro Dynamics (QED).
ToExpression[" \\mathcal{L}=-\\frac{1}{4} F_{\\mu \\nu}^a F^{a \\mu \\nu}+i \\bar{\\psi} \\gamma^\\mu\\left(\\partial_\\mu+i g A_\\mu^a T^a\\right) \\psi \\\\ ",TeXForm]ToExpression[" F_{\\mu \\nu}^a=\\partial_\\mu A_\\nu^a-\\partial_\\nu A_\\mu^a+g f^{a b c} A_\\mu^b A_\\nu^b",TeXForm]
L-+i(ψ)
1
4
aμν
F
a
F
μν
D
μ
ψ
μ
γ
a
F
μν
∂
μ
a
A
ν
∂
ν
a
A
μ
abc
f
b
A
μ
c
A
ν
Moreover in the same way dimension reduction can cure Ultraviolet divergence at higher energy scales, dimension construction could be used to cure infrared divergence. The reader can find an example illustrated in [2] Sec I. 5.
Cosmology
Cosmology
Illustratively, we can also consider the utility of dimension reduction in a cosmological model. By solving Einstein Field equations in 4+1 D and regarding our universe as a hypersurface in 3+1 D, one can account for the cosmological constant from a geometric point of view. For someone living in our universe, a hypersurface on the 4+1 D, a effective stress-energy tensor will be detectable, this could be used to the recovery of the well-known FRLW metric.
In[]:=
<<OGRe`
OGRe:
OGRe: An Object-Oriented General Relativity Package for Mathematica |
v1.7.0 (2021-09-17) |
• To view the full documentation for the package, type |
• To list all available modules, type |
• To get help on a particular module, type ? followed by the module name. |
• To enable parallelization, type |
• OGRe`Private`UpdateMessage To disable automatic checks for updates at startup, type |
TNewCoordinates["Spherical5D",{t,r,θ,ϕ,ψ}]TNewCoordinates["Spherical4D",{t,r,θ,ϕ}]
Clear[Λ,ψ]
TNewMetric"Universe5D","Spherical5D",{1,0,0,0,0},0,-Exp2t,0,0,0,0,0,-Exp2t,0,0,0,0,0,-Exp2t,0,{0,0,0,0,-1}TNewMetric"FLRW","Spherical4D",{1,0,0,0},0,-Exp2t,0,0,0,0,-Exp2t,0,0,0,0,-Exp2tTNewMetric["FLRWTraditional","Spherical4D",{{1,0,0,0},{0,-,0,0},{0,0,-,0},{0,0,0,-}}]
Λ[ψ]
3
2
r
Λ[ψ]
3
2
r
2
Sin[θ]
Λ[ψ]
3
Λ[ψ]
3
2
r
Λ[ψ]
3
2
r
2
Sin[θ]
Λ[ψ]
3
2
a[t]
2
r
2
a[t]
2
r
2
Sin[θ]
2
a[t]
In that sense we can recognize the flat (k=0) FLRW as a hypersurface of the 4+1 D universe through examining the slice of = . We wrote also the FLRW metric in the convenient way to make it easy to recognize the scale factor a(t)
2
ψ
3
Λ
In[]:=
TShow["Universe5D"]TShow["FLRW"]TShow["FLRWTraditional"]
OGRe:
Universe5D: (t,r,θ,ϕ,ψ) =
g
μν
1 | 0 | 0 | 0 | 0 |
0 | - 2t Λ[ψ] 3 | 0 | 0 | 0 |
0 | 0 | - 2t Λ[ψ] 3 2 r | 0 | 0 |
0 | 0 | 0 | - 2t Λ[ψ] 3 2 r 2 Sin[θ] | 0 |
0 | 0 | 0 | 0 | -1 |
OGRe:
FLRW: (t,r,θ,ϕ) =
g
μν
1 | 0 | 0 | 0 |
0 | - 2t ψ | 0 | 0 |
0 | 0 | - 2t ψ 2 r | 0 |
0 | 0 | 0 | - 2t ψ 2 r 2 Sin[θ] |
II. Dimension Reduction, a trend in Quantum Gravity Theories
II. Dimension Reduction, a trend in Quantum Gravity Theories
Causal Dynamical Triangulations (CDT)
Causal Dynamical Triangulations (CDT)
Once more, this aligns seamlessly with the overarching theme of this report— the reduction of dimensions at higher energy scales implies the potential for more renormalizable theories; wether we are considering quantum gravity theory or QFTs coupled with it. As articulated in [7], “It is natural to conjecture it will provide an effective short-distance cut-off by which the non-perturbative formulation of quantum gravity employed here, causal dynamical triangulations, evades the ultraviolet infinities of perturbative quantum gravity.”
Causal Set Theory
Causal Set Theory
III. Conclusion and Discussion
III. Conclusion and Discussion
At this juncture, the reader might be grappling with numerous questions surrounding the pervasive trend of dimension reduction in aspiring quantum gravity theories. What imparts uniqueness to gravity in 1+1 dimensions? Is the phenomenon could formulated in terms of renormalization? While concrete answers remain elusive in the absence of a comprehensive quantum gravity theory, intriguing speculations that align coherently with theoretical arguments have been proposed [4, 2].
Scale Invariance
Scale Invariance
Let’s delve into the realm of 1+1 dimensional gravity. In this context, the Einstein-Hilbert Action is solely contingent on the topological (global) features of the manifold, the action in that is only the Euler characteristic of the manifold [2]. This implies a lack of local- wether curvature or fluctuations namely Gravitational waves or Gravitons- measures in the theory. Conversely, the gravitational coupling constant G is dimensionless and consequently endowing the theory with scale invariance—insensitivity to energy scales.
In terms of renormalization, 1+1 dimensional gravity exhibits a non-Gaussian UV fixed point, suggesting that reliability as a quantum gravity theory may hinge on the deliberate loss of sensitivity to energy scale, with dimension reduction as a mechanism for that. In the language of quantum field theory, this aligns with the asymptotic safety theory of gravity, although there is currently no empirical evidence supporting General Relativity as such a theory. Moreover, this intriguing scale invariance feature can also be explored within the realm of CDT, where phase transitions reveal scales at which dimension reduction becomes notably pronounced. Post-transition, the theory approaches its continuum limit, with a predisposition towards scale invariance—a coherence that resonates with the renormalization argument presented [4].
In terms of renormalization, 1+1 dimensional gravity exhibits a non-Gaussian UV fixed point, suggesting that reliability as a quantum gravity theory may hinge on the deliberate loss of sensitivity to energy scale, with dimension reduction as a mechanism for that. In the language of quantum field theory, this aligns with the asymptotic safety theory of gravity, although there is currently no empirical evidence supporting General Relativity as such a theory. Moreover, this intriguing scale invariance feature can also be explored within the realm of CDT, where phase transitions reveal scales at which dimension reduction becomes notably pronounced. Post-transition, the theory approaches its continuum limit, with a predisposition towards scale invariance—a coherence that resonates with the renormalization argument presented [4].
Asymptotic Silence
Asymptotic Silence
IV. On the Wolfram Model
IV. On the Wolfram Model
Sketch for Asymptotic Silence in WM
Sketch for Asymptotic Silence in WM
Can we have Bottom-Top approach for WM?
Can we have Bottom-Top approach for WM?
Acknowledgments
Acknowledgments
I extend my sincere gratitude to my mentors, Xerxes D. Arsiwalla and Hatem Elshatlawy, for their invaluable guidance and support during the winter school. Their expertise and mentorship have greatly enriched my learning experience. Additionally, I would like to express my thanks to all the organizers, speakers, and Stephen Wolfram for their dedicated effort and time.
References
References
[1] P. Ehrenfest, KNAW Proc. 20 I (1918) 200.
[2] Stojkovic, D. (2014). “Vanishing dimensions: Review.” arXiv preprint arXiv:1406.2696.
[3] Graña, M., & Triendl, H. (2017). String theory compactifications. SpringerBriefs in Physics, 1–74. https://doi.org/10.1007/978-3-319-54316-1_1
[4] Carlip, S. (2017). “Dimension and dimensional reduction in quantum gravity.” Classical and Quantum Gravity, 34(19), 193001. doi:10.1088/1361-6382/aa8535
[5] Peskin, M. E., & Schroeder, D. V. (1995). An introduction to quantum field theory. Perseus Books.
[6] Loll, R. (2019). “Quantum gravity from causal dynamical triangulations: a review.” Classical and Quantum Gravity, 37(1), 013002. doi:10.1088/1361-6382/ab57c7
[7] Ambjørn, J., Jurkiewicz, J., & Loll, R. (2005). “The Spectral Dimension of the Universe is Scale Dependent.” Physical Review Letters, 95(17), 171301. doi:10.1103/PhysRevLett.95.171301
[8] Abajian, J., & Carlip, S. (2018). “Dimensional reduction in manifoldlike causal sets.” Physical Review D, 97(6), 066007. doi:10.1103/PhysRevD.97.066007
[9] Khatsymovsky, V. M. (2020). “On the discrete version of the black hole solution.” International Journal of Modern Physics A, 35(11n12), 2050058. doi:10.1142/S0217751X2050058X
CITE THIS NOTEBOOK
CITE THIS NOTEBOOK
How many dimensions are there? Dynamical Dimensions & Dimension Reduction
by Fawzi Aly
Wolfram Community, STAFF PICKS, January 11, 2024
https://community.wolfram.com/groups/-/m/t/3101823
by Fawzi Aly
Wolfram Community, STAFF PICKS, January 11, 2024
https://community.wolfram.com/groups/-/m/t/3101823