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".(adsbygoogle = window.adsbygoogle || []).push({});

http://scienceblogs.com/principles/2007/08/what_is_string_theory.php [Broken]

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. 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

Meanwhile, this is Rovelli on LQG, page 26:

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

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 seemsdeeplyweird 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: theyseem, 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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# String worldsheets and lqg two-complexes

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