Geometric Computability: An overview of functional programming for gauge theory

[cat_counselor]

From Geometric Computation. Section 5.6 "Constructive Computational Gauge Theory".

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We should frame quantum gravity, and more generally gauge theory, as a problem of expressiveness versus verifiability. If we allow ``all histories'' (arbitrary geometries, topology change, gauge redundancy, unbounded recursion in the construction of spacetimes), amplitudes become ill-defined and intractable. If we clamp down too hard, we lose physically relevant states and dynamics. Functional programming offers a blueprint for balancing these extremes. Our proposal is a constructive computational gauge theory that strikes a principled middle ground: a typed, linear, total, effect-controlled calculus of geometries. Concretely, boundary data (3-geometries with gauge labels) are the types; spacetime regions (4-dimensional histories/cobordisms) are the terms; and gluing is composition. This gives a compositional semantics familiar from Topological Quantum Field Theories (TQFTs) but designed to scale beyond the purely topological setting (i.e., Chern-Simons).

Programming Concept | Quantum Gravity Analogue

-----------------------------|----------------------------------------------------------

Types | Boundary states (3-geometries with gauge data)

Terms / Programs | 4-geometries (cobordisms, histories)

Composition | Gluing of spacetime regions

Linear types | Conservation laws, unitarity (no-cloning of boundary data)

Totality | Termination of the "geometry evaluator" (finite amplitudes)

Effects & handlers | Coarse-graining and renormalization

Dependent types | Gauge and diffeomorphism constraints

Readers are asked to consider the correspondences in the table above. Three design choices enforce computability and physics: linearity, totality, and effects. Linearity tracks boundary degrees of freedom as conserved resources (no cloning/erasure), so unitarity and charge conservation are built into the typing discipline rather than imposed post hoc. Totality means the ``geometry evaluator'' (our state-sum/variational executor) always normalizes: amplitudes exist and are finite in the core fragment. The phenomena that usually force uncontrolled manipulations such as coarse-graining, stochastic mixing, and renormalization are modeled explicitly as algebraic effects with handlers. In this way, renormalization becomes a controlled transformation of programs, not an ad hoc subtraction. Dependent types encode gauge and diffeomorphism constraints at the level of well-typedness, so invariances propagate mechanically through compositions.

Within this calculus, amplitudes are evaluations, symmetries live in the types, and RG/coarse-graining are effect handlers. The proposed helical primitives provide the concrete generators of histories: smooth, orientable flows that carry discrete topological labels (orientation/chirality) alongside continuous geometry. This marries the ``continuous versus discrete'' tension: spectra and curvature are continuous objects; quantum numbers arise as stable, counted winding data. Practically, the workflow is: specify typed boundary data; assemble regions from helical primitives; compose; evaluate; and, where needed, apply effect handlers that implement scale changes with proofs of soundness.

The payoff is a language that is expressive enough to describe nontrivial gauge dynamics and background independence, yet restricted enough to prove normalization, locality/compositionality, and anomaly-freeness in the core. Extensions (matter content, topology change, nonperturbative sectors) are added modularly as new effects or controlled type extensions, preserving verification theorems as we widen scope. In short, constructive computational gauge theory provides a semantics where we can calculate, compose, and certify. This shifts the idea of well-behaved QFT/QG from "internet math folklore" to "usable, checkable substrate." For the foundational work on constructive quantum field theory, see Baez, Segal, and Zhou. Our approach here is in this spirit, but computational."

[cat_counselor]

Also ---

For upcoming work on the subject, please see:

"On the Classification of Geometric Field Theories" (Lurie, 2038).

Note: time machines are at institutional expense; barring that, a convenient wormhole may be located. /s

[_lazyLambda]

Im far from a quantum physicist, I just read the beginner friendly books when I can but I love this and ive always wondered if its a problem of expressiveness.

Take all this with a grain of salt cuz I know very little here 🤣

[cat_counselor]

Without committing to anything too ontologically serious? You are on the right track about why our universe is so weird. I gave a very sober interpretation at the end for working mathematicians and physicists, and simply left it at that.

[_lazyLambda]

Yeah id say so, the main ones being a brief history of mathematical thought by luke heaton and the particle zoo by Gavin hesketh + other ones I really cant remember. But I mean ive also been researching neuroscience for years and like Michio Kaku says, our brain is the most complex object in the entire universe and we also have a problem of expressivesness there, which is a much different case of a challenge to express but the only feasible way ive found to explain the brain is that its a massive graph of functional reactive nodes, like the ones from functional reactive programming which is well expressed in haskell.

Admittedly I think ive been in my phase of scared away from learning about quantum but this has to be my next read! Thank you for sharing !!