# References and Citations

## How to Cite This Library

If you use this library in your research or project, please cite it using the information provided in our [CITATION.cff](CITATION.cff) file. This file contains
structured citation metadata that can be automatically processed by GitHub and other platforms.

**Quick citation:**

```text
Adam Getchell. 2025. causal-triangulations: A Causal Dynamical Triangulation library for quantum gravity research.
GitHub. https://github.com/acgetchell/causal-triangulations
```

For BibTeX, APA, or other citation formats, please refer to the [CITATION.cff](CITATION.cff) file or use GitHub's "Cite this repository" feature.

## AI-Assisted Development Tools

- **ChatGPT**: OpenAI. <https://openai.com/chatgpt>.
- **Claude**: Anthropic. <https://www.anthropic.com/claude>.
- **CodeRabbit**: CodeRabbit AI, Inc. <https://coderabbit.ai/>.
- **Codex**: OpenAI. <https://openai.com/codex/>.
- **WARP**: Warp Dev, Inc. <https://www.warp.dev/>.

All AI-generated output was reviewed and/or edited by the maintainer.
No generated content was used without human oversight.

## Computational Geometry and Delaunay Triangulations

This library delegates Delaunay construction, validation, periodic triangulation support, and local geometric edit primitives to the `delaunay` crate. Refer to
that crate and its bibliography for detailed computational-geometry citations:

- Getchell, Adam. "delaunay: A d-dimensional Delaunay triangulation library." Zenodo, 2026. DOI:
  [10.5281/zenodo.16931097](https://doi.org/10.5281/zenodo.16931097). Repository: <https://github.com/acgetchell/delaunay>

## Discrete Approaches to General Relativity

### Lattice Approaches to Quantum Gravity

- Sorkin, R. D. "Causal sets: discrete gravity." In _Lectures on Quantum Gravity_, edited by A. Gomberoff and D. Marolf, 305-327. Boston: Springer, 2005. arXiv:
  [gr-qc/0309009](https://arxiv.org/abs/gr-qc/0309009)

- Williams, R. M., and Tuckey, P. A. "Regge calculus: a brief review and bibliography." _Classical and Quantum Gravity_ 9, no. 5 (1992): 1409. DOI:
  [10.1088/0264-9381/9/5/021](https://doi.org/10.1088/0264-9381/9/5/021)

### Spin Foam Models and Loop Quantum Gravity

- Perez, A. "The Spin Foam Approach to Quantum Gravity." _Living Reviews in Relativity_ 16, no. 1 (2013): 3. DOI:
  [10.12942/lrr-2013-3](https://doi.org/10.12942/lrr-2013-3). arXiv: [1205.2019](https://arxiv.org/abs/1205.2019)

- Rovelli, C. "Quantum Gravity." Cambridge: Cambridge University Press, 2004. DOI: [10.1017/CBO9780511755804](https://doi.org/10.1017/CBO9780511755804)

## Foundational Causal Dynamical Triangulations Theory

This section contains the seminal papers and foundational work that established CDT as a non-perturbative approach to quantum gravity.

### Comprehensive Reviews

- Ambjørn, J., Görlich, A., Jurkiewicz, J., and Loll, R. "Nonperturbative Quantum Gravity." _Physics Reports_ 519, no. 4-5 (2012): 127-210. DOI:
  [10.1016/j.physrep.2012.03.007](https://doi.org/10.1016/j.physrep.2012.03.007). arXiv: [1203.3591](https://arxiv.org/abs/1203.3591)

- Loll, R. "Quantum Gravity from Causal Dynamical Triangulations: A Review." _Classical and Quantum Gravity_ 37, no. 1 (2020): 013002. DOI:
  [10.1088/1361-6382/ab57c7](https://doi.org/10.1088/1361-6382/ab57c7). arXiv: [1905.08669](https://arxiv.org/abs/1905.08669)

### Original CDT Framework

- Ambjørn, J., Jurkiewicz, J., and Loll, R. "Dynamically Triangulating Lorentzian Quantum Gravity." _Nuclear Physics B_ 610, no. 1-2 (2001): 347-382. DOI:
  [10.1016/S0550-3213(01)00297-8](https://doi.org/10.1016/S0550-3213(01)00297-8). arXiv: [hep-th/0105267](https://arxiv.org/abs/hep-th/0105267)

- Ambjørn, J., Jurkiewicz, J., and Loll, R. "Emergence of a 4D World from Causal Quantum Gravity." _Physical Review Letters_ 93, no. 13 (2004): 131301. DOI:
  [10.1103/PhysRevLett.93.131301](https://doi.org/10.1103/PhysRevLett.93.131301). arXiv: [hep-th/0404156](https://arxiv.org/abs/hep-th/0404156)

### Volume Profiles, Topology, and Dimensional Observables

- Ambjørn, J., Budd, T., and Watabiki, Y. "Scale-dependent Hausdorff dimensions in 2d gravity." _Physics Letters B_ 736 (2014): 339-343. DOI:
  [10.1016/j.physletb.2014.07.047](https://doi.org/10.1016/j.physletb.2014.07.047). arXiv: [1406.6251](https://arxiv.org/abs/1406.6251)

- Ambjørn, J., Gizbert-Studnicki, J., Görlich, A., Jurkiewicz, J., and Németh, D. "The phase structure of causal dynamical triangulations with toroidal
  spatial topology." _Journal of High Energy Physics_ 2018, no. 6 (2018): 111. DOI:
  [10.1007/JHEP06(2018)111](https://doi.org/10.1007/JHEP06(2018)111). arXiv: [1802.10434](https://arxiv.org/abs/1802.10434)

- Ambjørn, J., Görlich, A. T., Jurkiewicz, J., Loll, R., Gizbert-Studnicki, J., and Trześniewski, T. "The semiclassical limit of causal dynamical
  triangulations." _Nuclear Physics B_ 849, no. 1 (2011): 144-165. DOI:
  [10.1016/j.nuclphysb.2011.03.019](https://doi.org/10.1016/j.nuclphysb.2011.03.019). arXiv: [1102.3929](https://arxiv.org/abs/1102.3929)

- Ambjørn, J., Jurkiewicz, J., and Loll, R. "Reconstructing the Universe." _Physical Review D_ 72, no. 6 (2005): 064014. DOI:
  [10.1103/PhysRevD.72.064014](https://doi.org/10.1103/PhysRevD.72.064014). arXiv: [hep-th/0505154](https://arxiv.org/abs/hep-th/0505154)

- Ambjørn, J., Jurkiewicz, J., and Loll, R. "The Spectral Dimension of the Universe is Scale Dependent." _Physical Review Letters_ 95, no. 17 (2005): 171301.
  DOI: [10.1103/PhysRevLett.95.171301](https://doi.org/10.1103/PhysRevLett.95.171301). arXiv: [hep-th/0505113](https://arxiv.org/abs/hep-th/0505113)

- van der Duin, J., Loll, R., Schiffer, M., and Silva, A. "Quantum gravity and effective topology." _The European Physical Journal C_ 86, no. 2 (2026): 102.
  DOI: [10.1140/epjc/s10052-026-15322-x](https://doi.org/10.1140/epjc/s10052-026-15322-x)

## Monte Carlo Methods in Quantum Gravity

These references provide the algorithmic foundations for Monte Carlo simulations in discrete quantum gravity.

### CDT Simulation Implementation and Ensembles

- Brunekreef, J., Görlich, A., and Loll, R. "Simulating CDT quantum gravity." _Computer Physics Communications_ 300 (2024): 109170. DOI:
  [10.1016/j.cpc.2024.109170](https://doi.org/10.1016/j.cpc.2024.109170). arXiv: [2310.16744](https://arxiv.org/abs/2310.16744)

- Israel, N. S., and Lindner, J. F. "Quantum gravity on a laptop: 1+1 Dimensional Causal Dynamical Triangulation simulation." _Results in Physics_ 2 (2012):
  164-169. DOI: [10.1016/j.rinp.2012.10.001](https://doi.org/10.1016/j.rinp.2012.10.001)

- Getchell, Adam. "CDT-plusplus: Causal Dynamical Triangulations in C++." Repository: <https://github.com/acgetchell/CDT-plusplus>. A Zenodo DOI will be added
  after a formal release.

### Ergodic Moves and Alexander Moves

- Alexander, J. W. "The combinatorial theory of complexes." _Annals of Mathematics_ 31, no. 2 (1930): 292-320. DOI:
  [10.2307/1968099](https://doi.org/10.2307/1968099)

- Pachner, U. "P.L. homeomorphic manifolds are equivalent by elementary shellings." _European Journal of Combinatorics_ 12, no. 2 (1991): 129-145. DOI:
  [10.1016/S0195-6698(13)80080-7](https://doi.org/10.1016/S0195-6698(13)80080-7)

### Generic MCMC Framework

This library delegates generic Markov-chain stepping, delayed-commit proposal APIs, log-space transition handling, and future reusable MCMC algorithm
improvements to the `markov-chain-monte-carlo` crate. Refer to that crate and its bibliography for detailed generic MCMC and statistical-physics citations:

- Getchell, Adam. "markov-chain-monte-carlo: A composable MCMC framework for Rust." Zenodo, 2026. DOI:
  [10.5281/zenodo.20033111](https://doi.org/10.5281/zenodo.20033111). Repository: <https://github.com/acgetchell/markov-chain-monte-carlo>

### Metropolis-Hastings Algorithm in CDT

- Ambjørn, J., and Loll, R. "Non-perturbative Lorentzian quantum gravity, causality and topology change." _Nuclear Physics B_ 536, no. 1-2 (1998): 407-434.
  DOI: [10.1016/S0550-3213(98)00692-0](https://doi.org/10.1016/S0550-3213(98)00692-0). arXiv: [hep-th/9805108](https://arxiv.org/abs/hep-th/9805108)

- Hastings, W. K. "Monte Carlo sampling methods using Markov chains and their applications." _Biometrika_ 57, no. 1 (1970): 97-109. DOI:
  [10.1093/biomet/57.1.97](https://doi.org/10.1093/biomet/57.1.97)

### 1+1 CDT Coupling Calibration

- The default non-volume-fixed 1+1 CDT action constants use the critical 2D CDT cosmological coupling `lambda_c = ln 2` from the exactly solved model described
  by Ambjørn and Loll (1998) and reviewed by Ambjørn, Görlich, Jurkiewicz, and Loll (2012). Because this crate's action writes the cosmological term as
  `lambda_edge N1` while the standard 2D CDT convention weights triangle volume `N2`, the default toroidal edge-count value is
  `lambda_edge = (2 / 3) ln 2`, using the closed 1+1 relation `N1 = 3 N2 / 2`.

### Regge Calculus and Discrete Action

- Hamber, H. W. "Quantum Gravitation: The Feynman Path Integral Approach." Berlin: Springer-Verlag, 2009. DOI:
  [10.1007/978-3-540-85293-3](https://doi.org/10.1007/978-3-540-85293-3)

- Regge, T. "General Relativity without coordinates." _Il Nuovo Cimento_ 19, no. 3 (1961): 558-571. DOI:
  [10.1007/BF02733251](https://doi.org/10.1007/BF02733251)

## Numerical Methods and Performance

### Formal Verification in Scientific Computing

- Boldo, S., and Melquiond, G. "Computer Arithmetic and Formal Proofs." London: ISTE Press, 2017. ISBN: 978-1-78548-254-3

- Muller, J.-M., et al. "Handbook of Floating-Point Arithmetic." 2nd ed. Basel: Birkhäuser, 2018. DOI:
  [10.1007/978-3-319-76526-6](https://doi.org/10.1007/978-3-319-76526-6)

### High-Performance Scientific Computing

- Golub, G. H., and Van Loan, C. F. "Matrix Computations." 4th ed. Baltimore: Johns Hopkins University Press, 2013. ISBN: 978-1-4214-0794-4

- Press, W. H., et al. "Numerical Recipes: The Art of Scientific Computing." 3rd ed. Cambridge: Cambridge University Press, 2007. ISBN: 978-0-521-88068-8
