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{{Short description|Hypothetical approach to quantum gravity with emergent spacetime}}
{{Beyond the Standard Model|expanded=]}} {{no footnotes|date=April 2020}}{{Beyond the Standard Model|expanded=]}}


'''Causal dynamical triangulation''' (abbreviated as '''CDT''') theorized by ], ] and ], and popularized by ] and ], is an approach to ] that like ] is ]. '''Causal dynamical triangulation''' ('''CDT'''), theorized by ], ] and Jerzy Jurkiewicz, is an approach to ] that, like ], is ].


This means that it does not assume any pre-existing arena (dimensional space), but rather attempts to show how the ] fabric itself evolves. This means that it does not assume any pre-existing arena (dimensional space) but, rather, attempts to show how the ] fabric itself evolves.


There is evidence <ref> There is evidence <ref>
{{cite journal | last=Loll | first=Renate| year =2019| title = Quantum gravity from causal dynamical triangulations: a review | journal = Classical and Quantum Gravity | volume = 37 | issue = 1 | page = 013002 | doi = 10.1088/1361-6382/ab57c7| arxiv= 1905.08669 }}</ref> {{cite journal | last=Loll | first=Renate| year =2019| title = Quantum gravity from causal dynamical triangulations: a review | journal = Classical and Quantum Gravity | volume = 37 | issue = 1 | page = 013002 | doi = 10.1088/1361-6382/ab57c7| arxiv= 1905.08669 | s2cid=160009859}}</ref>
that at large scales CDT approximates the familiar 4-dimensional spacetime, but shows spacetime to be 2-dimensional near the ], and reveals a ] structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called ], and with other recent theoretical work. that, at large scales, CDT approximates the familiar 4-dimensional spacetime but shows spacetime to be 2-dimensional near the ], and reveals a ] structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called ], and with other recent theoretical work.


== Introduction == == Introduction ==
Near the ], the structure of ] itself is supposed to be constantly changing due to ]s and topological fluctuations. CDT theory uses a ] process which varies ] and follows ] rules, to map out how this can evolve into dimensional spaces similar to that of our universe. Near the ], the structure of ] itself is supposed to be constantly changing due to ]s and topological fluctuations. CDT theory uses a ] process which varies ] and follows ] rules, to map out how this can evolve into dimensional spaces similar to that of our universe.


The results of researchers suggest that this is a good way to model the ]{{citation needed|date=March 2013}}, and describe its evolution. Using a structure called a ], it divides spacetime into tiny triangular sections. A simplex is the multidimensional analogue of a ] ; a 3-simplex is usually called a ], while the 4-simplex, which is the basic building block in this theory, is also known as the ]. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes, where previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few. The results of researchers suggest that this is a good way to model the ]{{citation needed|date=March 2013}}, and describe its evolution. Using a structure called a ], it divides spacetime into tiny triangular sections. A simplex is the multidimensional analogue of a ] ; a 3-simplex is usually called a ], while the 4-simplex, which is the basic building block in this theory, is also known as the ]. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes. Whereas previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few, CDT avoids this problem by allowing only those configurations in which the timelines of all joined edges of simplices agree.

CDT avoids this problem by allowing only those configurations in which the timelines of all joined edges of simplices agree.


== Derivation == == Derivation ==
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CDT is a modification of quantum ] where spacetime is discretized by approximating it with a piecewise linear ] in a process called ]. In this process, a ''d''-dimensional spacetime is considered as formed by space slices that are labeled by a discrete time variable ''t''. Each space slice is approximated by a ] composed by regular (''d'' − 1)-dimensional simplices and the connection between these slices is made by a piecewise linear manifold of ''d''-simplices. In place of a smooth manifold there is a network of triangulation nodes, where space is locally flat (within each simplex) but globally curved, as with the individual faces and the overall surface of a ]. The line segments which make up each triangle can represent either a space-like or time-like extent, depending on whether they lie on a given time slice, or connect a vertex at time ''t'' with one at time ''t'' + 1. The crucial development is that the network of simplices is constrained to evolve in a way that preserves ]. This allows a ] to be calculated ]ly, by summation of all possible (allowed) configurations of the simplices, and correspondingly, of all possible spatial geometries. CDT is a modification of quantum ] where spacetime is discretized by approximating it with a piecewise linear ] in a process called ]. In this process, a ''d''-dimensional spacetime is considered as formed by space slices that are labeled by a discrete time variable ''t''. Each space slice is approximated by a ] composed by regular (''d'' − 1)-dimensional simplices and the connection between these slices is made by a piecewise linear manifold of ''d''-simplices. In place of a smooth manifold there is a network of triangulation nodes, where space is locally flat (within each simplex) but globally curved, as with the individual faces and the overall surface of a ]. The line segments which make up each triangle can represent either a space-like or time-like extent, depending on whether they lie on a given time slice, or connect a vertex at time ''t'' with one at time ''t'' + 1. The crucial development is that the network of simplices is constrained to evolve in a way that preserves ]. This allows a ] to be calculated ]ly, by summation of all possible (allowed) configurations of the simplices, and correspondingly, of all possible spatial geometries.


Simply put, each individual simplex is like a building block of spacetime, but the edges that have a time arrow must agree in direction, wherever the edges are joined. This rule preserves causality, a feature missing from previous "triangulation" theories. When simplexes are joined in this way, the complex evolves in an orderly{{how|date=March 2013}} fashion, and eventually creates the observed framework of dimensions. CDT builds upon the earlier work of ], ], and ], but by introducing the causality constraint as a fundamental rule (influencing the process from the very start), Loll, Ambjørn, and Jurkiewicz created something different. Simply put, each individual simplex is like a building block of spacetime, but the edges that have a time arrow must agree in direction, wherever the edges are joined. This rule preserves causality, a feature missing from previous "triangulation" theories. When simplexes are joined in this way, the complex evolves in an orderly{{how|date=March 2013}} fashion, and eventually creates the observed framework of dimensions. CDT builds upon the earlier work of ], ], and ], but by introducing the causality constraint as a fundamental rule (influencing the process from the very start), Loll, Ambjørn, and Jurkiewicz created something different.

== Advantages and disadvantages ==

CDT derives the observed nature and properties of spacetime from a small set of assumptions, without adjusting factors. The idea of deriving what is observed from first principles is very attractive to physicists.{{citation needed|date=December 2015}} CDT models the character of spacetime both in the ultra-microscopic realm near the Planck scale, and at the scale of the cosmos, so CDT may provide insights into the nature of reality.{{citation needed|date=December 2015}}

Evaluation of the observable implications of CDT relies heavily on ] by computer. Some{{who|date=March 2013}} feel that this makes CDT an inelegant quantum gravity theory. Also, it has been argued{{according to whom|date=March 2013}} that discrete time-slicing may not accurately reproduce all possible modes of a dynamical system. However, research by ] and ]{{citation needed|date=January 2012}} demonstrates that the cause for those concerns may be limited{{how|date=March 2013}}. Therefore, many physicists still regard this line of reasoning as promising{{citation needed|date=January 2012}}.


== Related theories == == Related theories ==


CDT has some similarities with ], especially with its ] formulations. For example, the Lorentzian ] is essentially a non-perturbative prescription for computing path integrals, just like CDT. There are important differences, however. Spin foam formulations of quantum gravity use different degrees of freedom and different Lagrangians. For example, in CDT, the distance, or "the interval", between any two points in a given triangulation can be calculated exactly (triangulations are eigenstates of the distance operator). This is not true for spin foams or loop quantum gravity in general. CDT has some similarities with ], especially with its ] formulations. For example, the Lorentzian ] is essentially a non-perturbative prescription for computing path integrals, just like CDT. There are important differences, however. Spin foam formulations of quantum gravity use different degrees of freedom and different Lagrangians. For example, in CDT, the distance, or "the interval", between any two points in a given triangulation can be calculated exactly (triangulations are eigenstates of the distance operator). This is not true for spin foams or loop quantum gravity in general. Moreover, in spin foams the discreteness is thought to be fundamental, while in CDT it is viewed as a regularization of the path integral, to be removed by the ].


Another approach to quantum gravity that is closely related to causal dynamical triangulation is called ]. Both CDT and causal sets attempt to model the spacetime with a discrete causal structure. The main difference between the two is that the causal set approach is relatively general, whereas CDT assumes a more specific relationship between the lattice of spacetime events and geometry. Consequently, the Lagrangian of CDT is constrained by the initial assumptions to the extent that it can be written down explicitly and analyzed (see, for example, , page 5), whereas there is more freedom in how one might write down an action for causal-set theory. Another approach to quantum gravity that is closely related to causal dynamical triangulation is called ]. Both CDT and causal sets attempt to model the spacetime with a discrete causal structure. The main difference between the two is that the causal set approach is relatively general, whereas CDT assumes a more specific relationship between the lattice of spacetime events and geometry. Consequently, the Lagrangian of CDT is constrained by the initial assumptions to the extent that it can be written down explicitly and analyzed (see, for example, , page 5), whereas there is more freedom in how one might write down an action for causal-set theory.

In the continuum limit, CDT is probably related to some version of ]. In fact, both theories rely on a foliation of spacetime, and thus they can be expected to lie in the same universality class. In 1+1 dimensions they have actually been shown to be the same theory,<ref>{{cite journal |last1=Ambjørn |first1=J. |last2=Glaser |first2=L. |last3=Sato |first3=Y. |last4=Watabiki |first4=Y. |date=2013 |title=2d CDT is 2d Hořava–Lifshitz quantum gravity |journal=Physics Lettetters B |volume=722 |issue=1–3 |pages=172–175 |doi=10.1016/j.physletb.2013.04.006 |arxiv =1302.6359 |bibcode=2013PhLB..722..172A |s2cid=85444972 }} </ref> while in higher dimensions there are only some hints, as understanding the continuum limit of CDT remains a difficult task.


== See also == == See also ==
Line 51: Line 46:


== References == == References ==
;Notes
{{no footnotes|date=April 2020}}
{{reflist}} {{reflist}}
;Bibliography
* * {{Webarchive|url=https://web.archive.org/web/20070311061337/http://www.phys.lsu.edu/mog/mog19/node12.html |date=2007-03-11 }}
* ], ], and ] - , ], July 2008 * ], ], and ] – , ], July 2008
* Alpert, Mark "The Triangular Universe" Scientific American page 24, February 2007 * Alpert, Mark "The Triangular Universe" Scientific American page 24, February 2007
* Ambjørn, J.; Jurkiewicz, J.; Loll, R. - * Ambjørn, J.; Jurkiewicz, J.; Loll, R. –
* Loll, R.; Ambjørn, J.; Jurkiewicz, J. - - a less technical recent overview * Loll, R.; Ambjørn, J.; Jurkiewicz, J. – – a less technical recent overview
* Loll, R.; Ambjørn, J.; Jurkiewicz, J. - - a technically detailed overview * Loll, R.; Ambjørn, J.; Jurkiewicz, J. – – a technically detailed overview
* Markopoulou, Fotini; Smolin, Lee - - shows that varying the time-slice gives similar results * Markopoulou, Fotini; Smolin, Lee – – shows that varying the time-slice gives similar results
* Loll, R – A review from May 2019, focusing on results that were recent at that time


Early papers on the subject: Early papers on the subject:
* R. Loll, ''Discrete Lorentzian Quantum Gravity'', 21 Nov 2000 * R. Loll, ''Discrete Lorentzian Quantum Gravity'', 21 Nov 2000
* J Ambjørn, A. Dasgupta, J. Jurkiewicz, and R. Loll, ''A Lorentzian cure for Euclidean troubles'', v1 14 Jan 2002 * J Ambjørn, A. Dasgupta, J. Jurkiewicz, and R. Loll, ''A Lorentzian cure for Euclidean troubles'', v1 14 Jan 2002
*


== External links == == External links ==
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* *
* *
⚫ * from Renate Loll's homepage
*
⚫ * as broadcast by ]
*
⚫ * from Renate Loll's homepage
⚫ * as broadcast by ]


{{Theories of gravitation}} {{Theories of gravitation}}

Latest revision as of 17:58, 21 February 2024

Hypothetical approach to quantum gravity with emergent spacetime
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Causal dynamical triangulation (CDT), theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent.

This means that it does not assume any pre-existing arena (dimensional space) but, rather, attempts to show how the spacetime fabric itself evolves.

There is evidence that, at large scales, CDT approximates the familiar 4-dimensional spacetime but shows spacetime to be 2-dimensional near the Planck scale, and reveals a fractal structure on slices of constant time. These interesting results agree with the findings of Lauscher and Reuter, who use an approach called Quantum Einstein Gravity, and with other recent theoretical work.

Introduction

Near the Planck scale, the structure of spacetime itself is supposed to be constantly changing due to quantum fluctuations and topological fluctuations. CDT theory uses a triangulation process which varies dynamically and follows deterministic rules, to map out how this can evolve into dimensional spaces similar to that of our universe.

The results of researchers suggest that this is a good way to model the early universe, and describe its evolution. Using a structure called a simplex, it divides spacetime into tiny triangular sections. A simplex is the multidimensional analogue of a triangle ; a 3-simplex is usually called a tetrahedron, while the 4-simplex, which is the basic building block in this theory, is also known as the pentachoron. Each simplex is geometrically flat, but simplices can be "glued" together in a variety of ways to create curved spacetimes. Whereas previous attempts at triangulation of quantum spaces have produced jumbled universes with far too many dimensions, or minimal universes with too few, CDT avoids this problem by allowing only those configurations in which the timelines of all joined edges of simplices agree.

Derivation

CDT is a modification of quantum Regge calculus where spacetime is discretized by approximating it with a piecewise linear manifold in a process called triangulation. In this process, a d-dimensional spacetime is considered as formed by space slices that are labeled by a discrete time variable t. Each space slice is approximated by a simplicial manifold composed by regular (d − 1)-dimensional simplices and the connection between these slices is made by a piecewise linear manifold of d-simplices. In place of a smooth manifold there is a network of triangulation nodes, where space is locally flat (within each simplex) but globally curved, as with the individual faces and the overall surface of a geodesic dome. The line segments which make up each triangle can represent either a space-like or time-like extent, depending on whether they lie on a given time slice, or connect a vertex at time t with one at time t + 1. The crucial development is that the network of simplices is constrained to evolve in a way that preserves causality. This allows a path integral to be calculated non-perturbatively, by summation of all possible (allowed) configurations of the simplices, and correspondingly, of all possible spatial geometries.

Simply put, each individual simplex is like a building block of spacetime, but the edges that have a time arrow must agree in direction, wherever the edges are joined. This rule preserves causality, a feature missing from previous "triangulation" theories. When simplexes are joined in this way, the complex evolves in an orderly fashion, and eventually creates the observed framework of dimensions. CDT builds upon the earlier work of Barrett, Crane, and Baez, but by introducing the causality constraint as a fundamental rule (influencing the process from the very start), Loll, Ambjørn, and Jurkiewicz created something different.

Related theories

CDT has some similarities with loop quantum gravity, especially with its spin foam formulations. For example, the Lorentzian Barrett–Crane model is essentially a non-perturbative prescription for computing path integrals, just like CDT. There are important differences, however. Spin foam formulations of quantum gravity use different degrees of freedom and different Lagrangians. For example, in CDT, the distance, or "the interval", between any two points in a given triangulation can be calculated exactly (triangulations are eigenstates of the distance operator). This is not true for spin foams or loop quantum gravity in general. Moreover, in spin foams the discreteness is thought to be fundamental, while in CDT it is viewed as a regularization of the path integral, to be removed by the continuum limit.

Another approach to quantum gravity that is closely related to causal dynamical triangulation is called causal sets. Both CDT and causal sets attempt to model the spacetime with a discrete causal structure. The main difference between the two is that the causal set approach is relatively general, whereas CDT assumes a more specific relationship between the lattice of spacetime events and geometry. Consequently, the Lagrangian of CDT is constrained by the initial assumptions to the extent that it can be written down explicitly and analyzed (see, for example, hep-th/0505154, page 5), whereas there is more freedom in how one might write down an action for causal-set theory.

In the continuum limit, CDT is probably related to some version of Hořava–Lifshitz gravity. In fact, both theories rely on a foliation of spacetime, and thus they can be expected to lie in the same universality class. In 1+1 dimensions they have actually been shown to be the same theory, while in higher dimensions there are only some hints, as understanding the continuum limit of CDT remains a difficult task.

See also

References

Notes
  1. Loll, Renate (2019). "Quantum gravity from causal dynamical triangulations: a review". Classical and Quantum Gravity. 37 (1): 013002. arXiv:1905.08669. doi:10.1088/1361-6382/ab57c7. S2CID 160009859.
  2. Ambjørn, J.; Glaser, L.; Sato, Y.; Watabiki, Y. (2013). "2d CDT is 2d Hořava–Lifshitz quantum gravity". Physics Lettetters B. 722 (1–3): 172–175. arXiv:1302.6359. Bibcode:2013PhLB..722..172A. doi:10.1016/j.physletb.2013.04.006. S2CID 85444972.
Bibliography

Early papers on the subject:

  • R. Loll, Discrete Lorentzian Quantum Gravity, arXiv:hep-th/0011194v1 21 Nov 2000
  • J Ambjørn, A. Dasgupta, J. Jurkiewicz, and R. Loll, A Lorentzian cure for Euclidean troubles, arXiv:hep-th/0201104 v1 14 Jan 2002

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