Path 4: Causal Dynamical Triangulations (CDT)

Build spacetime like a quantum Lego: add up simple blocks with causality and see a universe emerge.

Inventory

Rationale: This approach is a modern implementation of the gravitational path integral (Feynman’s sum-over-histories) using a discrete approximation of spacetime. In CDT, one approximates spacetime by gluing together elementary building blocks (simplices, like four-dimensional triangles) in all possible ways, but with a crucial rule: preserve a causal structure (no silly configurations where time loops back on itself). By doing Monte Carlo simulations of this summation, CDT has shown remarkable results: from microscopic randomness, a 4-dimensional spacetime with an extended, quasi-smooth structure emerges. Notably, in simulations the “average universe” that forms looks like a de Sitter universe (our universe with a cosmological constant) without putting that in by hand. This suggests that quantum gravity, when done right, naturally produces an expanding universe. CDT thus provides a concrete, computable model of quantum gravity that respects key principles (like causality and unitarity) and can recover large-scale classical behavior from Planck-scale quantum fluctuations – a significant achievement.

Prerequisite Themes: Path integrals and lattice methods (CDT is analogous to lattice QCD but for gravity), basic differential geometry (simplices approximate manifolds), and statistical mechanics (the Monte Carlo and phase transition language). One should also be comfortable with general relativity concepts like spacetime manifolds and the concept of “summing over geometries.”

Dependencies: CDT is related to Path 3 (Asymptotic Safety) in that the continuum limit of the lattice can reveal fixed points (indeed, the emergence of proper 4D geometry in CDT is tied to being near a critical point). It shares lineage with older Path 10 (Euclidean Quantum Gravity and Regge calculus) – Regge calculus was a precursor using fixed lattice triangulations, and Euclidean (imaginary-time) path integrals were pioneered by Hawking; CDT refines those by restoring a distinction between time and space (Lorentzian signature) and letting the lattice dynamically grow. CDT’s success in generating a cosmos complements Path 2 (LQG): both suggest spacetime discreteness, but CDT works in a covariant (summing histories) way while LQG is canonical – interestingly, some results like discrete spectra might coincide if both are right. Currently, CDT does not include matter fields (or at least, adding them is non-trivial), so it may eventually depend on integrating techniques from lattice gauge theory to incorporate matter.

Signs of Progress: A major next step is to show that as the simulation’s lattice spacing is taken to zero (and volume to infinity), it yields quantitative agreement with general relativity at large scales – so far, the evidence is qualitative (dimension ~4, de Sitter geometry). Detecting different “phases” of quantum spacetime and identifying a second-order phase transition (needed for a continuum limit) is ongoing work; success there would firmly establish CDT as a viable theory of QG. Another sign would be if CDT could compute something like the spectral properties of the Laplacian on quantum spacetime and match semi-analytical results, or if it could reproduce black hole-like configurations in a simulation. From an observational viewpoint, if CDT can incorporate simple matter and make even statistical predictions (like probability of certain spatial topologies in the early universe), that could one day be compared to cosmological data. In summary, progress is measured by how well CDT can show classical gravity (and perhaps simple quantum corrections) emerging from the sum over causal triangulations, and by exploring whether different “microscopic rules” in the model yield physically sensible alternate outcomes (e.g. could changing a parameter produce something like a phase of “crumpled” spacetime vs. “extended” spacetime – understanding such phases is underway).

Base Camp 4.1: Discrete Geometry and Regge Calculus

Scope: Learn how to approximate a smooth spacetime by a triangulation (simplicial decomposition) where edges, triangles, etc., encode distances. Regge calculus is the formalism for general relativity on a piecewise-flat lattice of simplices: understand Regge’s equation (analog of Einstein eq) and the concept of deficit angle as measure of curvature at a bone (triangle in 3D, hinge in 4D).

Stepping Stones: (1) In 2D, triangulate a surface with equilateral triangles – see that specifying the number of triangles meeting around a vertex fixes curvature via deficit angle $2\pi - n \theta$ (with $\theta$ internal angle of triangle). (2) Write the Regge action in 4D: $S_{\rm Regge} = \frac{1}{16\pi G}\sum_{\text{hinges}} A_h \delta_h + \Lambda \sum_{\text{simplices}} V_{\text{simplex}}$ (where $\delta_h$ is deficit angle at a hinge and $A_h$ its area). Understand conceptually why this mimics $\int R$ (curvature concentrated on hinges). (3) Derive Regge equations by varying edge lengths: one gets that deficit angles satisfy equations analogous to $G_{\mu\nu}=0$. (Though heavy to solve, conceptual: if no matter, the net deficit tends to zero if lengths can adjust, corresponding to flatness in absence of curvature sources). (4) Recognize limitations: solving Regge equations is hard, but it’s a well-defined discrete classical GR. In quantum context, one sums over all triangulations weighted by $\exp(-S_{\rm Regge}/\hbar)$.

Base Camp 4.2: Path Integral and Euclidean Dynamical Triangulations (EDT)

Scope: Understand the idea of summing over geometries by summing over triangulations. First, consider Euclidean signature (Wick-rotated) for simplicity: define the partition function $Z = \sum_{\text{triangulations }T} \frac{1}{C_T} e^{-S_{\rm Regge}(T)}$, where $C_T$ is symmetry factor. See how in practice one sums by Monte Carlo sampling triangulations. EDT (Euclidean DT) tried this but found the “universe” either crumpled or split into baby universes (phase issues).

Stepping Stones: (1) Formulate the path integral discretely: fix number of simplices or use $\Lambda$ to weight them (in Regge action $\Lambda$ term roughly $\Lambda V$ encourages large volume). (2) Discuss the need for an ensemble of different topologies or fix topology (CDT fixes topology to $S^1 \times S^3$ typically). (3) See what went wrong in EDT: the two observed phases – one “crumpled” (high $G$ regime: effectively, gravity is too attractive, geometry collapses to a high curvature lump) and one “branched polymer” (low $G$: behaves like a tree of baby universes, not 4D smooth). There was no continuum-like extended phase. (4) Recognize that EDT did not enforce a notion of causality/time slicing and allowed “spatial topology change” spontaneously, which might cause those phase pathologies.

Base Camp 4.3: Causal Dynamical Triangulations Formulation

Scope: Learn how CDT differs: it uses a global time slicing (each slice is a spatial triangulation of a fixed topology) and only “causal” connections (simplices connect such that time always moves forward, no arbitrary identification that break Lorentzian causal structure). The path integral is taken over causal triangulations with a proper time foliation.

Stepping Stones: (1) Define the building blocks of CDT in 4D: 4-simplices of types that straddle adjacent time slices (e.g. “(4,1)” simplex with 4 vertices at time $t$ and 1 at $t+1$, or “(3,2)” with 3 at $t$, 2 at $t+1$). These are like “tent” pieces connecting slices. (2) Write the Regge action for these simplices, noting that because of fixed edge lengths (usually take all time-like edges of one length, spatial of another), the action simplifies to something like $S = -\kappa_2 N_2 + \kappa_4 N_4$ (where $N_2$ number of triangles (2-faces) and $N_4$ number of 4-simplices) plus potential $\alpha (N_{41}-2N_{32})$ term distinguishing (4,1) vs (3,2) simplices. In practice, $\kappa$ parameters relate to $G$ and $\Lambda$. (3) Understand how one performs Wick rotation in CDT: because of the fixed foliation, can analytically continue the time-like edge length squared from + to - (or vice versa) giving a real weight $e^{-S}$ from the Lorentzian amplitude $e^{iS_{\rm Lorentz}}$. Remarkably, this is possible because of no causality-violating configurations. (4) Emphasize differences: no baby universe branching (except possibly at disconnected phases, but by design topology fixed); no graphs that break causality; so the hope is this yields a physically sensible large-scale limit. (5) Summarize: the Monte Carlo sim will vary the arrangement of (4,1) and (3,2) simplices across slices, subject to gluing rules and fixed total time extension.

Base Camp 4.4: Monte Carlo Simulation and Observables

Scope: Get familiar with how CDT simulations are done: for each set of bare couplings ($\kappa_0, \Delta$ etc.), generate ensembles of triangulations (with computer algorithms doing local moves like flipping simplices) and measure observables like average spatial volume at each time slice (the “volume profile”), spectral dimension, etc.

Stepping Stones: (1) Understand the concept of finite-size scaling: near critical points (continuum limit) one expects certain observables to scale with system size (number of simplices) in specific ways that can be extrapolated. (2) Key observable: the volume profile $N_3(t)$ vs $t$. CDT found that in the extended phase, $N_3(t)$ matches the shape of a Euclidean de Sitter (4-sphere) with fluctuations. Essentially, the universe dynamically forms a large 4D “blob” of spacetime with a Gaussian profile in time, rather than crumpling or splitting. This is evidence of a semi-classical universe emerging. (3) Spectral dimension $d_s$: measure by random walks on the triangulation – they found $d_s \approx 4$ at large scales but running to $\approx 2$ at small scales (so a dynamical dimensional reduction). (4) Also mention the phase diagram: CDT in addition to the physically interesting “de Sitter” phase, has at least two other phases (an “A” phase with effectively no extended structure, and a “B” phase with highly oscillatory spatial volume – recently identified as a “condensed” phase). The interesting phase is where $G$ and $\Lambda$ are such that an extended quasi-classical spacetime appears (call it “C” phase). (5) Steps: each Monte Carlo step modifies the triangulation slightly (moves like breaking a (4,1) simplex into (3,2)+(3,2), etc.), which change $N_4$ and $N_2$ counts accordingly, then Metropolis accept/reject according to action difference. After equilibration, sample configurations and calculate observables. (6) This yields expectation values like $\langle N_3(t)\rangle$ which can be fitted to continuum form $N_3(t) \sim \cos^3(t/\tau)$ like a 4-sphere.

Base Camp 4.5: Continuum Limit and Effective Theories

Scope: Discuss how one takes the continuum limit: ideally fine-tune to a critical point where correlation length (in lattice units) $\to \infty$. Identify what in CDT is analogous to a second-order phase transition for continuum gravity. Also what effective action has been extracted (e.g. does the universe follow Einstein’s equations at large scales? There are studies measuring effective action via fluctuations).

Stepping Stones: (1) Indications of criticality: the transition between phase C and another might be second-order (recent evidence suggests the B-C transition might be second-order). If so, that’s the point to take lattice spacing to 0 while sending number of simplices to infinity, achieving continuum limit. (2) Renormalization in CDT: one tries to observe how bare parameters $\kappa_0,\Delta$ should be tuned with volume to maintain consistent physics – some attempts show running of effective coupling with system size consistent with asymptotic safety’s predictions (there is some interplay with FRG results). (3) Effective action measurement: by analyzing shape fluctuations of the volume profile, Ambjørn et al. extracted an effective minisuperspace action which was consistent with the continuum Einstein-Hilbert action plus a small correction. Specifically, they fitted the dynamics of $N_3(t)$ to an action $S \sim \int dt [\frac{(\partial_t N_3)^2}{N_3} + \ldots + \mu N_3 - \lambda N_3^{1/3}]$ etc., finding it matches de Sitter with cosmological constant term. (4) If time permits, mention ongoing work: exploring coupling to matter in CDT, and inhomogeneous initial conditions, etc. But main point: continuum semi-classical behavior seen, next step to verify approach to classical equations fully and examine UV limit (maybe connect to asymptotic safety continuum couplings).

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