Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

Marcolli, Matilde; Port, Alexander

doi:10.1007/s11786-015-0236-y

Cited by 7 publications

(5 citation statements)

References 24 publications

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“…Another work that associates a Lie algebra with a graph grammar is Marcoli and Port (2015) . In this case, the basis Fock space over which the Lie algebra operators are defined is a space of labeled graphs G , rather than labeled graph grammar rules G in → G out , so it is a different and smaller operator Lie algebra than ours.…”

Section: Discussionmentioning

confidence: 99%

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Mjolsness

2022

Front. Syst. Biol.

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Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, “stochastic parameterized graph grammars,” composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to “dynamical graph grammars,” which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics.

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Section: Discussionmentioning

confidence: 99%

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Mjolsness

2022

Front. Syst. Biol.

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“…In recent years, what one means by language and grammar has vastly generalized beyond traditional forms of written language. Examples include graph grammars in graphical languages [57,83], pregroup grammars in diagrammatic languages involving string diagrams [40,77], visual grammars corresponding to compositions involving visual scenes and actions [96,139], and hypergraph / operator algebras that extend graph grammars [4,136,137], just to name a few. In our case here, the collections of objects in both, Obj P and Obj M are not arbitrary collections.…”

Section: States Of Owmentioning

confidence: 99%

Qualia and the Formal Structure of Meaning

Arsiwalla

2024

Preprint

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This work explores the hypothesis that subjectively attributed meaning constitutes the phenomenal content of conscious experience. That is, phenomenal content is semantic. This form of subjective meaning manifests as an intrinsic and non-representational character of qualia. Empirically, subjective meaning is ubiquitous in conscious experiences. We point to phenomenological studies that lend evidence to support this. Furthermore, this notion of meaning closely relates to what Frege refers to as "sense", in metaphysics and philosophy of language. It also aligns with Peirce's "interpretant", in semiotics. We discuss how Frege's sense can also be extended to the raw feels of consciousness. Sense and reference both play a role in phenomenal experience. Moreover, within the context of the mind-matter relation, we provide a formalization of subjective meaning associated to one's mental representations. Identifying the precise maps between the physical and mental domains, we argue that syntactic and semantic structures transcend language, and are realized within each of these domains. Formally, meaning is a relational attribute, realized via a map that interprets syntactic structures of a formal system within an appropriate semantic space. The image of this map within the mental domain is what is relevant for experience, and thus comprises the phenomenal content of qualia. We conclude with possible implications this may have for experience-based theories of consciousness.

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“…The unexpected connections to computer science by way of combinatorics hint at a vast untapped potential of interdisciplinary collaboration. So far little research has gone into this direction but articles like [23,24], or [25] -where it was shown that Feynman graphs can be interpreted as a type of formal language generated by a theory dependent graph grammar -seem to suggest that there are deep connections between the two fields whose study might benefit both disciplines.…”

Section: A Different Perspective -Dirac Words As Cartier-foata Monoidsmentioning

confidence: 99%

Contraction of Dirac matrices via chord diagrams

Golz¹

2017

Preprint

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Chord diagrams and combinatorics of word algebras are used to model products of Dirac matrices, their traces, and contractions. A simple formula for the result of arbitrary contractions is derived, simplifying and extending an old contraction algorithm due to Kahane. This formula is then used to express the Schwinger parametric integrand of a QED Feynman integral in a much simplified form, with the entire internal tensor structure eliminated. Possible next steps for further simplification, including a specific conjecture, are discussed.

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Graph Grammars, Insertion Lie Algebras, and Quantum Field Theory

Cited by 7 publications

References 24 publications

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics

Qualia and the Formal Structure of Meaning

Contraction of Dirac matrices via chord diagrams

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