dorsal/arxiv
View SchemaPath integral formulation and Feynman rules for phylogenetic branching models
| Authors | P. D. Jarvis, J. D. Bashford, J. G. Sumner |
|---|---|
| Categories | |
| ArXiv ID | q-bio/0411047 |
| URL | https://arxiv.org/abs/q-bio/0411047 |
| DOI | 10.1088/0305-4470/38/44/002 |
Abstract
A dynamical picture of phylogenetic evolution is given in terms of Markov models on a state space, comprising joint probability distributions for character types of taxonomic classes. Phylogenetic branching is a process which augments the number of taxa under consideration, and hence the rank of the underlying joint probability state tensor. We point out the combinatorial necessity for a second-quantised, or Fock space setting, incorporating discrete counting labels for taxa and character types, to allow for a description in the number basis. Rate operators describing both time evolution without branching, and also phylogenetic branching events, are identified. A detailed development of these ideas is given, using standard transcriptions from the microscopic formulation of nonequilibrium reaction-diffusion or birth-death processes. These give the relations between stochastic rate matrices, the matrix elements of the corresponding evolution operators representing them, and the integral kernels needed to implement these as path integrals. The `free' theory (without branching) is solved, and the correct trilinear `interaction' terms (representing branching events) are presented. The full model is developed in perturbation theory via the derivation of explicit Feynman rules which establish that the probabilities (pattern frequencies of leaf colourations) arising as matrix elements of the time evolution operator are identical with those computed via the standard analysis. Simple examples (phylogenetic trees with 2 or 3 leaves), are discussed in detail. Further implications for the work are briefly considered including the role of time reparametrisation covariance.
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"abstract": "A dynamical picture of phylogenetic evolution is given in terms of Markov\nmodels on a state space, comprising joint probability distributions for\ncharacter types of taxonomic classes. Phylogenetic branching is a process which\naugments the number of taxa under consideration, and hence the rank of the\nunderlying joint probability state tensor. We point out the combinatorial\nnecessity for a second-quantised, or Fock space setting, incorporating discrete\ncounting labels for taxa and character types, to allow for a description in the\nnumber basis. Rate operators describing both time evolution without branching,\nand also phylogenetic branching events, are identified. A detailed development\nof these ideas is given, using standard transcriptions from the microscopic\nformulation of nonequilibrium reaction-diffusion or birth-death processes.\nThese give the relations between stochastic rate matrices, the matrix elements\nof the corresponding evolution operators representing them, and the integral\nkernels needed to implement these as path integrals. The `free\u0027 theory (without\nbranching) is solved, and the correct trilinear `interaction\u0027 terms\n(representing branching events) are presented. The full model is developed in\nperturbation theory via the derivation of explicit Feynman rules which\nestablish that the probabilities (pattern frequencies of leaf colourations)\narising as matrix elements of the time evolution operator are identical with\nthose computed via the standard analysis. Simple examples (phylogenetic trees\nwith 2 or 3 leaves), are discussed in detail. Further implications for the work\nare briefly considered including the role of time reparametrisation covariance.",
"arxiv_id": "q-bio/0411047",
"authors": [
"P. D. Jarvis",
"J. D. Bashford",
"J. G. Sumner"
],
"categories": [
"q-bio.PE",
"physics.bio-ph",
"q-bio.QM"
],
"doi": "10.1088/0305-4470/38/44/002",
"title": "Path integral formulation and Feynman rules for phylogenetic branching models",
"url": "https://arxiv.org/abs/q-bio/0411047"
},
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