In the early days of Loop Quantum Gravity there were people who expressed a hope that ideas from LQG could be applied to string theory in order to provide the background independent formulation that people were looking for. It was a natural idea because we knew that string theory lacked a formulation that was non-perturbative and diffeomorphism invariant. LQG is constructed in a way that respects the general relativistic structure of spacetime but it fails to provide the right classical limit with gravity. The failing of one theory were the strengths of the other so perhaps a mergence of the two would solve everything
It seemed like there were good reasons to believe that LQG and string theory could be brought together because they both had origins in loop models of gauge theory. In three dimensional gravity string theory and LQG could be viewed as different views of the same model. This just had to be extended to higher dimensions.
Lee Smolin, one of the founders of LQG, responded by learning string theory and trying to reformulate it in LQG terms, but string theorists were not impressed. I don’t think you can generalise the point of view of every string theorists but the open secret passed around was that LQG was a wrong lead. Physicists working on LQG were actually proud of the fact that it worked only in four dimensions and that it worked without including matter, (except that it didn’t actually work). To string theorists these ideas were simply out of date. There were even mad people who thought that the loops in LQG could be he same thing as strings and that was plain crazy. Every string theorist knows that strings can pass cleanly through themselves and each other without restriction while knotted loops can’t.
Over time the split between LQG and string theory grew stronger. It was aided by the politcal need to divide research into separate disciplines for the purposes of funding. With most people working on string theory, LQG had to be characterised as an alternative approach to be funded separately. Smolin abandoned his attempt to apply LQG methods to string theory and instead wrote his infamous book “The Trouble with Physics: The Rise of String Theory, the Fall of a Science and What Comes Next” . His followers went with him and the wedge between LQG and string theory moved further in.
So what is the reality? Could LQG tell us anything about string theory? Well first of all the fact that LQG has not worked on its own does not mean that it can’t be applied to string theory. Perhaps it can only work when the right matter fields are included in the right way. The knotty mathematics of LQG is related to quantum groups which also make an appearance in the conformal field theories used by string theorists so perhaps the two approaches are not so distant. But what of the argument that strings are not like knots because they pass through without hindrance? Actually this is not how things really are. When a string passes through itself it may pass unhindered or it may interact with itself at the crossing point causing it to split into two separate strings. The resulting interaction looks just like the Skein relations used in the mathematics of knot invariants. Other relations in spin networks related to associativity can be connected to crossing symmetry of the string theory S-matrix. In fact the list of connections between LQG and string theory is endless once you start looking for them.
At this year’s string conference David Gross renewed the call to find out what string theory is and how it should be formulated correctly. So why doesn’t someone just apply LQG quantisation to string theory and solve the problem? Gross and other string theorists are quick to dismiss LQG, so does that stop other smart people trying? I think the real problem is that there is a genuine mathematical gap between LQG and string theory that is hard to bridge. LQG relies on a special property of 4-dimensional spacetime that allows gravity to be reformulated in a way that succumbs to quantization. The same trick cannot be applied to the higher dimensional supersymmetric spacetimes in String Theory.
Yet there is still hope. There are hints of relationships between supersymmetry in 3,4,6 and 10 dimensions and the division algebras R, C, H and O. Perhaps if the complex based quantum mechanics of LQG could be replaced with quaternionic or octonionic quantum mechanics it would work in 6 and 10 dimensions. Who knows?
A string theorist recently said in his blog that you have to look at the equations and discover the solution rather than inventing new theories. In string theory this approach has been a very successful way of discovering cross-dimensional dualities and other unexpected phenmena like holography. As they are fond of saying around here “you couldn’t make it up”. But LQG was also ”discovered” by quantising gravity in a particular way. Like string theory it was not invented, so it also has a special power to show us new things.
For the moment everybody is stuck. I think the situation is stalled because of a lack of vision that blinds people to the mathematical relationships they need to be thinking about. Perhaps something observed in the LHC will provide a new light that shows the way forward, or perhaps the inspiration will come from someone else who steps back and sees the big picture. Community attitudes may fog our view but it will only require one person with vision to see the way. Let’s hope we dont have to wait too long.
