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Locally Causal CDT and Matter Coupling vs Foundational CDT Program

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Causal Dynamical Triangulation· within family

	Locally Causal CDT and Matter Coupling Frontier	Foundational CDT Program Frontier
Proposed	2013	1998
Key figures	Steven Jordan, Renate Loll, Ben Ruijl	Jan Ambjorn, Renate Loll
In one sentence	Jordan and Loll demonstrated in 2013 that the causality constraint that makes CDT work can be enforced locally rather than globally, removing the preferred-foliation construction that is the family's central foundational concern. The Locally Causal Dynamical Triangulation extension is the active 2020-2026 research frontier of the program, alongside matter-coupled CDT and the continuum-limit search.	Ambjorn and Loll proposed in 1998 that the gravitational path integral can be defined nonperturbatively by summing over discrete Lorentzian geometries built from simplices, with one structural rule: every building block must agree on which direction is the past and which is the future. The causal-foliation constraint is what rescued lattice quantum gravity from the pathological geometries that defeated earlier Euclidean approaches.
Predictions	The causality constraint that makes the CDT path integral well-defined can be enforced locally rather than globally, recovering the original construction as a special case while permitting a wider class of triangulated geometries. The local construction is computationally much more expensive than the global one; most LCDT results to date are in two dimensions, with three-dimensional investigations active and four-dimensional results still out of reach at the precision needed for full phase-structure mapping. Matter-coupled CDT modifies the phase diagram in computable ways, with the modification dependent on the matter content (scalar, gauge, fermion) and the matter-matter and matter-gravity couplings. The pattern of modification is a direct probe of the gravitational degrees of freedom in the presence of matter and is the most direct CDT-side contact with Asymptotic Safety's matter-coupled fixed-point calculations.	The gravitational path integral can be defined nonperturbatively on a lattice without picking a background metric, by summing over all causal triangulations weighted by the Regge action. Imposing a global causal ordering on the simplicial geometries suppresses the crumpled and branched-polymer geometries that dominate the Euclidean Dynamical Triangulation path integral, allowing a smooth four-dimensional phase to exist. The continuum limit of the lattice theory, if it exists, should approach a true renormalized quantum field theory of gravity, possibly connected to the ultraviolet fixed point of the Asymptotic Safety program.
Where it breaks	Most of the post-2013 LCDT literature lives in two dimensions because the local-causality construction is computationally much more expensive than the global one. Four-dimensional LCDT, where the program's most important questions live, remains computationally limited. Critics argue that the LCDT response to the preferred-foliation concern is real in principle but not yet realized in the dimension that matters. Matter coupling at present requires simplified matter sectors (single scalar fields, abelian gauge fields, fermions in restricted regimes) because of the same computational cost. Realistic Standard Model matter content coupled to CDT is beyond current resources and may require substantial algorithmic innovation rather than just more compute time. The continuum-limit search has identified candidate second-order phase transitions but has not closed the rigorous argument that the CDT lattice theory defines a continuum quantum field theory of gravity. Without this the LCDT and matter-coupling extensions remain extensions of a lattice construction whose long-distance status is not yet settled. The cross-program contact with Asymptotic Safety is qualitative at present rather than quantitative. CDT and Asymptotic Safety see similar dimensional reduction at short scales and similar features of matter-coupling, but a direct numerical match between the discrete and continuum approaches is still work in progress.	The causal-foliation construction picks a global time direction at the microscopic level. Critics argue this breaks full background independence and is at odds with the diffeomorphism invariance that any quantum theory of gravity should preserve. The Locally Causal Dynamical Triangulation extension is the program's response, but most published LCDT work is in lower dimensions because the local-causality construction is computationally much more expensive than the global one. The continuum limit of the lattice theory has not been rigorously established. Without it, the program's results are statements about finite-size simulations rather than about a true continuum quantum theory of gravity. Recent work on phase transitions in the C-to-bifurcation region is encouraging but does not yet close the argument. The bare action contains only the Einstein-Hilbert term and a [[cosmological constant]]. Whether the results survive the inclusion of higher-derivative operators or matter sectors is the subject of the program's ongoing extensions.
Key unresolved problem	The too-slow-to-compute problem: the upgraded version that drops any built-in time direction, locally causal CDT, is so demanding to simulate that no one can yet run it in the full four dimensions where the theory must finally be tested.	The zoom-out problem: the results come from a finite grid of building blocks, and no one has proven that shrinking the blocks to nothing, the continuum limit, yields a genuine quantum theory of gravity rather than a grid artifact.
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Locally Causal CDT and Matter Coupling

2013 · Frontier

Foundational CDT Program

1998 · Frontier

Proposed

2013

1998

Key figures

Steven Jordan, Renate Loll, Ben Ruijl

Jan Ambjorn, Renate Loll

In one sentence

Jordan and Loll demonstrated in 2013 that the causality constraint that makes CDT work can be enforced locally rather than globally, removing the preferred-foliation construction that is the family's central foundational concern. The Locally Causal Dynamical Triangulation extension is the active 2020-2026 research frontier of the program, alongside matter-coupled CDT and the continuum-limit search.

Ambjorn and Loll proposed in 1998 that the gravitational path integral can be defined nonperturbatively by summing over discrete Lorentzian geometries built from simplices, with one structural rule: every building block must agree on which direction is the past and which is the future. The causal-foliation constraint is what rescued lattice quantum gravity from the pathological geometries that defeated earlier Euclidean approaches.

Predictions

The causality constraint that makes the CDT path integral well-defined can be enforced locally rather than globally, recovering the original construction as a special case while permitting a wider class of triangulated geometries.
The local construction is computationally much more expensive than the global one; most LCDT results to date are in two dimensions, with three-dimensional investigations active and four-dimensional results still out of reach at the precision needed for full phase-structure mapping.
Matter-coupled CDT modifies the phase diagram in computable ways, with the modification dependent on the matter content (scalar, gauge, fermion) and the matter-matter and matter-gravity couplings. The pattern of modification is a direct probe of the gravitational degrees of freedom in the presence of matter and is the most direct CDT-side contact with Asymptotic Safety's matter-coupled fixed-point calculations.

The gravitational path integral can be defined nonperturbatively on a lattice without picking a background metric, by summing over all causal triangulations weighted by the Regge action.
Imposing a global causal ordering on the simplicial geometries suppresses the crumpled and branched-polymer geometries that dominate the Euclidean Dynamical Triangulation path integral, allowing a smooth four-dimensional phase to exist.
The continuum limit of the lattice theory, if it exists, should approach a true renormalized quantum field theory of gravity, possibly connected to the ultraviolet fixed point of the Asymptotic Safety program.

Where it breaks

Most of the post-2013 LCDT literature lives in two dimensions because the local-causality construction is computationally much more expensive than the global one. Four-dimensional LCDT, where the program's most important questions live, remains computationally limited. Critics argue that the LCDT response to the preferred-foliation concern is real in principle but not yet realized in the dimension that matters.
Matter coupling at present requires simplified matter sectors (single scalar fields, abelian gauge fields, fermions in restricted regimes) because of the same computational cost. Realistic Standard Model matter content coupled to CDT is beyond current resources and may require substantial algorithmic innovation rather than just more compute time.
The continuum-limit search has identified candidate second-order phase transitions but has not closed the rigorous argument that the CDT lattice theory defines a continuum quantum field theory of gravity. Without this the LCDT and matter-coupling extensions remain extensions of a lattice construction whose long-distance status is not yet settled.
The cross-program contact with Asymptotic Safety is qualitative at present rather than quantitative. CDT and Asymptotic Safety see similar dimensional reduction at short scales and similar features of matter-coupling, but a direct numerical match between the discrete and continuum approaches is still work in progress.

The causal-foliation construction picks a global time direction at the microscopic level. Critics argue this breaks full background independence and is at odds with the diffeomorphism invariance that any quantum theory of gravity should preserve. The Locally Causal Dynamical Triangulation extension is the program's response, but most published LCDT work is in lower dimensions because the local-causality construction is computationally much more expensive than the global one.
The continuum limit of the lattice theory has not been rigorously established. Without it, the program's results are statements about finite-size simulations rather than about a true continuum quantum theory of gravity. Recent work on phase transitions in the C-to-bifurcation region is encouraging but does not yet close the argument.
The bare action contains only the Einstein-Hilbert term and a [[cosmological constant]]. Whether the results survive the inclusion of higher-derivative operators or matter sectors is the subject of the program's ongoing extensions.

Key unresolved problem

The too-slow-to-compute problem: the upgraded version that drops any built-in time direction, locally causal CDT, is so demanding to simulate that no one can yet run it in the full four dimensions where the theory must finally be tested.

The zoom-out problem: the results come from a finite grid of building blocks, and no one has proven that shrinking the blocks to nothing, the continuum limit, yields a genuine quantum theory of gravity rather than a grid artifact.

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