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Stream: community: our work

Topic: Causal-net condensation

Xuexing Lu (Aug 04 2025 at 16:01):

Hello, every one. I am here to give a brief introduction to my work on causal-net condensation, which is purely categorical construction about symmetric monoidal categories. You can find the details in https://arxiv.org/abs/2201.08963. And you can find a version at https://www.researchgate.net/publication/386015651_A_category_of_causal-nets.

There are many mathematical and physical ideas behind this work. I list several of them as follows.

Give a purely categorical formulation of Baez construction. In the paper "Spin Network States in Gauge Theory"(https://arxiv.org/abs/gr-qc/9411007), Baez gave an interesting construction of measures on the moduli of gauge fields in terms of spin networks. We view this construction as the first step of giving a background-independent formulation of lattice gauge theory.
Give a mathematical formulation of string-net condensation by Levin and Wen in https://arxiv.org/abs/cond-mat/0404617. We think that Baez construction and string-net condensation are two sides of one thing. This work paves a way to prove a "physics theorem" that "topological order=symmetric monoidal category".
Give a new formation of the graphical calculus for symmetric monoidal categories developed by Joyal and Street in https://www.sciencedirect.com/science/article/pii/000187089190003P. In this work, we interpret symmetric monoidal categories as cosheaves on the category of causal-net, which is similar to view categories as simplicial sets satisfying the Segal condition.
Provide a framework for quantum gravity. We view causal-net condesation as a framework for non-perturbative quantum field theory. The framework of causal-net condensation is parallel to that of factorization homology. Maybe in some sense we can view causal-net condensation as an $E_\infty$ version of factorization homology.
Provide a non-perturbative and background-independent quantum twistor theory. Faraday thought that light was produced by the vibration of lines of force. I think this is the old idea of string-net condensation. What are lines of force for light and other gauge fields? They are strings in string-net condensation and also twistors of Penrose (https://en.wikipedia.org/wiki/Twistor_theory).

John Baez (Aug 04 2025 at 16:51):

This is an interesting paper - thanks for bringing it to our attention here!

I'm curious about your motivation for working, not with all graphs (in the category theorist's sense), but only acyclic ones. By calling them causal networks and mentioning quantum gravity, you seem to be hinting that they describe causality in the physics sense: i.e. that vertices are 'events' and a directed path from a vertex $x$ to a vertex $y$ means that $x$ occurred 'before' $y$ . Every causal network thus has an underlying [[causal set]], and causal sets have been extensively investigated in quantum gravity.

But you also mention spin networks and string nets, and these have mainly been used to describe states of quantum systems in space, not in spacetime - and thus, without any causal structure.

There has been a bit of work on 'spin foams' with a causal structure, but it hasn't really gone that far yet.

Xuexing Lu (Aug 04 2025 at 18:13):

@John Baez Thank you for your interest in it and and for mentioning causal set. From my point of view, causal-net condensation aims at to be an upgraded version of causal-set theory. Causal-set theory is a statistic theory, based on probability theory, while causal-net condensation is based on (multi-)linear algebra and monoidal category theory. I think causal-net condensation combines the advantages of the three theories: causal-set theory, string-net condensation and loop quantum gravity. It is background-independent (a trouble for string-net condensation to remove the backgound lattice) and has the hope to solve the problem of time (which troubled both loop quantum gravity and string-net condensation), just as causal set theory, but has more fundamental mathematical foundation and involves more mathematical structures. As far as I know, the main trouble of causal set theory is the problem of classical limit, that is, emerging the classical theory of general relativity. In this aspect, I think causal-net condensation is more hopeful, because its framework is similar to string-net condensation which is good at emerging classical theory.

In fact, I think that there should be three parallel theories, corresponding to three classes of graphs: causal-nets(acyclic directed graphs), general directed graphs(also called quivers) and undirected graphs. The three theories correspond to, respectively, three classes monoidal categories: symmetric monoidal categories, traced symmetric monoidal categories and self-dual symmetric monoidal categories (I guess there is no standard name for this class of monoidal categories). Spin networks and string nets should be element of the second class of theories.

There has been a bit of work on 'spin foams' with a causal structure, but it hasn't really gone that far yet.

Yes. I have learned and know little about spin foam theory, which is a theory based on loop quantum gravity with the aim to solve the problem of time and to be a dynamical theory of quantum gravity. But I feel that it is more like a Euclidean and perturbative theory than a Minkowski quantum field theory.

John Baez (Aug 05 2025 at 08:08):

There are spin foams based on SO(4) representations and on SO(3,1) representations, so they come in both Euclidean and Minkowskian versions. I would not say they're perturbative. Not much work has been done on causal spin foams, but Markopoulou and Smolin have written papers on spin foams where the set of vertices is equipped with the structure of a causal set.

A few references:

I've only read the first two of these papers, and I can vouch for those. I generally don't trust work on quantum gravity, so I include the other papers just to point out that Lorentzian spin foams are indeed studied.

John Baez (Aug 05 2025 at 08:13):

In general, no work on causal sets or spin foams or loop quantum gravity has convincingly shown that these theories are able to give a correct classical limit; this is apparently a hard problem, and people should work on it more.

Xuexing Lu (Aug 05 2025 at 08:32):

Interesting information. Thanks very much. I haven't read a paper on spin foam for about ten years, though most of the ones I've encountered before were yours. Regarding the classical limit problem, I'm studying optimal transport theory, hoping to combine it with causal net condensation to derive a classical theory.

An optimal transport formulation of the Einstein equations of general relativity https://arxiv.org/abs/1810.13309

Xuexing Lu (Aug 05 2025 at 08:46):

I was greatly inspired by this article.
How many quanta are there in a quantum spacetime?
https://arxiv.org/abs/1404.1750
The slides is here.
How Many Quanta are there in a Quantum Spacetime slides (1).pdf

A crucial structure appearing in spin foam theory, string-net condesation(or, wave function renormalization), tensor network renormalization and causal-net condensation is the construction of coarse-graining.

In the category of causal-nets, coarse-graining is characterized exactly as coequalizer.

Xuexing Lu (Aug 05 2025 at 09:01):

A fundamental result about the category of causal-nets is that there are exactly six type of indecomposable morphisms, which represent exactly the six basic conventions in the graphical caluclus for symmetric monoidal categories, where the first three conventions (subdivision of an edge, adding a vertex, adding an directed edge)were introduced in Baez construction.

I extend Baez construction to include more conventions (coarse-graining two vertices-->tensor product of morphisms, coarse-graining two parallel edges-->tensor product of objects, and contraction an edge--->composition of morphisms), which essentially represent the algebraic operations of symmetric monoidal categories, and are simple coarse-grainings. By this, we can consider general diagrams, not only spin networks whose edge-labelings must be indecomposable objects in symmetric monoidal categories. However, after Kan extension, only spin networks (diagrams labelled by indecomposable objects) are left, representing the real physical degree of freedoms.