# String worldsheets and lqg two-complexes

• Coin
In summary: is the amplitude of a transition between the two spinfoam......the spinfoam formalism provides a calculational road to transition amplitudes in quantum gravity.
Coin
So this is something that's been bothering me for awhile and I seem to be getting no closer to understanding it on my own. I want to quote two things here; one, an Aaron Bergman post about String Theory at Uncertain Principles; two, Rovelli's LQG textbook, "Quantum Gravity".

http://scienceblogs.com/principles/2007/08/what_is_string_theory.php

What I would mainly like to do... is to answer the much easier question, "What is string perturbation theory?" But before getting to that, let's talk a bit about what perturbation theory is... Feynman discovered an amazing and intuitive way of organizing the calculations involved in perturbation theory. Each term is a sum of graphs where the lines represent different types of particle in the free theory and the vertices correspond to the terms which describe the interaction. These are the famous Feynman diagrams... These diagrams are remarkably useful as it's very easy to picture them as particles interacting with each other...

Feynman diagrams were far from Feynman's only contribution to quantum field theory. He also showed how to take the fields out of free field theories. Instead of considering fields, one considers all possible ways a particle can move in spacetime. Mathematically, this corresponds to a theory that lives on a one dimensional line as opposed to on spacetime. This line is called the 'worldline' of the particle. The important field that lives on this one dimensional line is a map that embeds the line into spacetime (where we'd ordinarily be doing our physics). This values of this map give usual worldline of a particle... Because we're doing quantum field theory on this one dimensional line, we integrate over every possible embedding...

String perturbation theory is a generalization of this final construction. All we do is replace our one dimensional line with a two dimensional surface called the worldsheet. Two dimensions is a lot more than one, so we have some freedom about the type of theory that can live there. This leads to the various types of string theories, ie, the bosonic string and the various superstring theories. Some amazing things begin to happen once you fix your theory, however. For one, you don't have to put in interactions by hand anymore. If you look at every possible two dimensional shape to map into spacetime, you automatically get interactions...

Another consequence is that... the spacetime you map into is forced to obey the Einstein field equations. In other words, string perturbation theory only makes sense when you are doing perturbations around a spacetime that satisfies the equations of gravity... When you look at how the vibrations of the string manifest themselves as particles in spacetime, one of them looks exactly like a graviton. On reasonably general grounds, any sensible theory that contains a graviton pretty much has to be Einstein's theory. So, it seems that a minor miracle has occurred. Solely by generalizing Feynman's description of perturbation theory from one dimensional objects to two dimensional objects, you automatically get gravity

In short, Bergman is saying something that I've seen said elsewhere as well: String theory is really just peturbation theory with 2-D worldsheets instead of 1-D worldlines. In fact as I understand things, if you use worldsheets, you're automatically doing string theory-- if you start with the idea of 2-D worldsheets you can derive string theory backward from the worldsheets.

Meanwhile, this is Rovelli on LQG, page 26:

A road toward the calculation of transition amplitudes in quantum gravity is provided by the spinfoam formalism.

Following Feynman's ideas, we can give W(s,s') a representation as a sum-over-paths. This representation can be obtained in various manners. In particular, it can be intuitively derived from a perturbative expansion, summing over different histories of sequences of actions of H that send s' into s.

A path is then the "world-history" of a graph, with interactions happening at the nodes. This world-history is a two-complex, as in Figure 1.5, namely a collection of faces (the world-histories of the links); faces join at edges (the world-histories of the nodes); in turn, edges join at vertices. A vertex represents an individual action of H... Thus, a two-complex is like a Feynman graph, but with one additional structure. A Feynman graph is composed by vertices and edges, a spinfoam by vertices, edges, and faces... a spinfoam is a Feynman graph of spin networks, or a world-history of spin networks...

In the perturbative expansion of W(s, s'), there is a term associated with each spinfoam σ bounded by s and s'. This term is the amplitude of σ. THe amplitude of a spinfoam turns out to be given by... the product over the vertices v of a vertex amplitude A_v(σ). The vertex amplitude is determined by the matrix element of H between the incoming and the outgoing spin networks and is a function of the labels of the faces and the edges adjacent to the vertex. This is analagous to the amplitude of a conventional Feynman vertex, which is determined by the matrix element of the hamiltonian between the incoming and outgoing states.

So, assuming I understand these two quotes correctly, I can summarize them as:

String theory can be peturbatively described by taking the worldline formalism and generalizing it to the next dimension up; in this case the worldsheets form a two-dimensional surface, and the classical action is described by the surface area of the sheet.

LQG can be peturbatively described by taking the Feynman Diagram formalism and generalizing it to the next dimension up; in this case the diagram graphs form a web of two-dimensional graph faces, and the graph faces give quantum amplitudes.

This seems deeply weird to me. Looking at what are as far as I'm aware two essentially conceptually opposite approaches to quantum gravity based on two very different underlying mathematical objects (strings and spinnets), they both seem to be doing something extremely similar if not identical when it comes down to their strategy for peturbative calculations: they seem, from the descriptions I see, to be taking the Feynman path integral technique and generalizing the mathematical structures by one dimension. However what I'm not sure about is whether these two things really are all that similar, or if they just seem similar to me because I am not informed enough about quantum theory to understand the differences between them. What do I make of this? i feel like I must be misunderstanding something somewhere.

I'm wondering if anyone who's more familiar with these issues could help me understand:

- Do I correctly understand what string theory and lqg are doing with their respective "peturbative theory + 1 dimension" things?

- I'm assuming that generalizing the worldline and generalizing on Feynman diagrams are analagous operations. Although I can find in many places vague assertions the worldline and Feynman diagram are somehow mathematically connected to one another, and I know both are used in calculating path integrals, I don't actually understand what the connection between these two things is. Are these indeed related structures?

- I'm assuming that producing the classical action is somehow relatable to producing the quantum amplitudes. But I don't in fact really understand how (or even whether) these two things are related, or in what form the classical action survives when you move to a quantum theory. I have some idea that in a classical theory you have an action which is minimized, and the quantum path integral ultimately assigns the greatest amplitudes to the paths where the classical action is minimized, but I'm not sure what that says about how action and amplitude are connected.

- Assuming I understand correct that when you calculate path integrals in normal quantum theory, worldlines have Feynman diagrams associated with them and vice versa (and it's not just that I'm falsely assuming these structures to be related because both have Feynman's name associated with them...). What would the worldlines or worldline analogues associated with a "2D Feynman diagram" look like? What would the Feynman diagrams or analogues associated with a "2D worldsheet" look like? Is it possible to directly compare the String Theory worldsheets Bergman describes and the LQG spinfoams Rovelli describes in this manner as mathematical structures, if that makes sense, despite their fairly different purposes?

- Assuming the answer to all above questions is "yes, the thing LQG is doing with 2D Feynman diagrams and the thing string theory is doing with 2d worldsheets are similar"-- does this even mean anything?

Please excuse me if I have garbled any of the concepts above.

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In part of this you are in good company, Coin. John Baez has several slide talks (going back IIRC to 2005) where he pictorially points out the similarity between a string worldsheet (drawn in 3D, as a 2D surface) and a LQG spinfoam.

The two things look pretty much the same except the spinfoam is piecewise linear, composed of polygonal flats. It is after all defined combinatorially as a collection of labeled edges, labeled triangles etc. It doesn't require an embedding into a pre-arranged space.

Visually, in Baez lecture slides, the two objects look very much alike. Both something like a tee shirt joined to a pair of short pants. One smooth and the other a sort of Cubist rendition.

I'm not sure in what way you can say the spinfoam approach is perturbative, however. It is usually described as a nonperturbative approach to quantum geometry (or gravity). The spin networks at the beginning and end of the spinfoam are treated as quantum states of geometry. I'm not sure how one would consider a path integral (a history of geometries) as perturbative. No perturbation series. No peturbation around any pre-established geometric "norm".

It's interesting, as both you and Baez have done, to draw the analogies and point out the similarities (both visual and deeper). It's probably harder to elucidate the differences.
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EDIT
BTW this is an interesting thread for sure! I just posted this initial response and already I'm thinking of other things to say. Maybe I'll wait and come back to this later. One point to make is that in the spinfoam approach there is, in a certain sense, no spacetime. There is only geometry. Matter is supposed to be defined on the quantum states of geometry rather than on a continuum. Matter fields live on the states and history of geometry as additional labels (besides the spin labels). This is apparently quite challenging to realize. Laurent Freidel is one to watch in this area (treating matter on spinfoam histories) and also Etera Livine seems to have a paper in preparation with some other people. What they are attempting is definitely possible in 3D but one doesn't know if it is possible in 4D. If something like Feynman diagrams comes out, it would be as a flat limit---a gravity-less limit---judging from the experience in 3D. I'll let this sit and come back later, Coin. Good post!

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Coin said:
- Assuming the answer to all above questions is "yes, the thing LQG is doing with 2D Feynman diagrams and the thing string theory is doing with 2d worldsheets are similar"-- does this even mean anything?

I think there are many ways to see approximate dualities between the different research programs. My personal take on the questions you address would be to look at the foundations here.

Clearly the actions principle, together with feynmanns path intergrals represents some sort of statistics, but with a different logic than classical statistics. I think the deeper understanding of the relation between the classical logic here and the quantum logic is not understood. At least I haven't understood it, and I havent' seen anyone else that seems to either. We have models, that's all, but I think these are great questions. Neither the loop program nor the string program addresses these questions. This is why I think the connections here seem to remain fascinating mathematical observations. My quess is that a deeper understanding of the physics of the similarity will require stepping outside of the linea of reasoning advocated by the two approaches. But I am personally convinced that the foundations of the quantum formalism here, quantum actions and path intergrals is still awaiting to be understood, and I think this understanding might well be more "model independent". I think the very logic implicit here is not understood in the deeper sense. I think one problem is that some people dismiss such quests as "philsophical toyery" without physical contact. It's a bit mysterical though that this is not a symmetric situation, you don't equallty often hear people (except when talking about string theory) addressing these question think of these elaborations as "mathematical toyery" without physical contact. Mathematical toyery seems more accepted than philosophical toyery for some odd reason. I think we need both kinds of toyery :)

But I noticed there is one talk by Florian Conrady from Loops 07 on what you address

Spin foams, gauge–string duality and renormalization
"In the first part of the talk, we present a recent result on 3d SU(2) lattice Yang-Mills theory, showing that it can be cast in the form of an exact string representation. The derivation starts from the spin foam representation of the lattice gauge theory. We demonstrate that every spin foam can be equivalently described as a worldsheet of strings on a framing of the lattice. Using this correspondence, the expectation value of a Wilson loop is translated into a sum over worldsheets that are bounded by strings along a framing of the loop. In the second part of the talk, we take the worldsheet picture as a motivation to discuss a possible approach to renormalization in SU(2) lattice Yang-Mills theory. The Yang-Mills theory is cast in the form of a lattice BF Yang-Mills theory with both the connection A and a 2-form B as variables. Then, a block spin renormlization is proposed where the block variables are a connection and a B-field on a coarser lattice. We suggest a perturbative scheme for the integration over the UV variables."
-- http://www.matmor.unam.mx/eventos/loops07/cont_abs.html

/Fredrik

Marcus and Fra, thanks so much for the fascinating responses! The Conrady talk is especially interesting, they actually claim to prove the existence of a "Bijection between worldsheets and spin foams" in this paper:

We prove that every [spin foam representation] can be equivalently described as a self--avoiding worldsheet of strings on a framing of the lattice.

And elaborates:

To some extent, the concept of spinfoams already embodies the idea of an exact gauge–string duality: spinfoams can be considered as branched surfaces that are world sheets of fluxlines... Due to the branching and the labelling with representations, these surfaces are not worldsheets as in string theory, however. The new element of this paper is the following: we show that in 3 dimensions spinfoams of SU(2) can be decomposed
into worldsheets that do not branch and carry no representation label. They can be regarded as worldsheets of strings in the fundamental representation.

I'm not sure I understand the "world sheets of fluxlines" point, but that's kind of exciting! It still seems a little confusing though as to what this all "means". As Marcus points out in spinfoams there is "no spacetime", or rather spacetime is in some sense a thing that emerges from the structure of the spinfoam. But worldsheets are supposed to provide a description of strings propagating over time, right? Where does this time come from? And strings propagating are supposed to be dual to particles, right..?... what do these particles mean? Are these questions even sensible? (Maybe I am just overthinking this; the Conrady paper, which treats the spinfoam-corresponding worldsheets as just mathematical formalisms, ignores these types of questions altogether and gets right down to using the duality to calculate things... :) ) I will try to listen to the talk later and see if it helps any at building an intuition.

## 1. What are string worldsheets?

String worldsheets are two-dimensional surfaces in spacetime that are used to describe the behavior of fundamental particles, such as strings. They represent the paths that these particles take through spacetime and are a key concept in string theory.

## 2. What is LQG (Loop Quantum Gravity)?

LQG is a theoretical framework that attempts to combine the principles of quantum mechanics and general relativity to describe the behavior of gravity at the smallest scales. It views spacetime as a network of interconnected loops and uses mathematical tools such as spin networks and spin foams to describe the dynamics of spacetime.

## 3. What is the relationship between string worldsheets and LQG two-complexes?

String worldsheets and LQG two-complexes are both mathematical structures used to describe the behavior of particles and spacetime at the quantum level. They are complementary approaches, with string worldsheets focusing on the behavior of fundamental particles and LQG two-complexes focusing on the structure of spacetime itself.

## 4. What are two-complexes in LQG?

In LQG, two-complexes are two-dimensional surfaces made up of interconnected edges and vertices. They are used to describe the topology of spacetime and the interactions between different regions of spacetime. These two-complexes play a crucial role in the mathematical formulation of LQG.

## 5. How are string worldsheets and LQG two-complexes being used in modern physics?

String worldsheets and LQG two-complexes are both currently being studied and explored in the field of theoretical physics. They are being used to gain a better understanding of the fundamental building blocks of the universe and to potentially unify our understanding of gravity with the other fundamental forces of nature.

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