can there be such an ST with BI?
I wouldn't be surprised. Stringy physics has so many varieties and tunable features that I wouldn't be surprised at anything they built with it.
are you familiar to any such theories?
Just after I posted, Urs (Urs Scheiber) posted here about the IKKT model which defines it's own spacetime dynamically. That's at least close to background independence. I tried to read the paper he linked to, but it's too specialized for me. Nevertheless this is an example, and the theory Urs developed to link it to LQG is very interesting. Look at his thread.
The more prominent candidate for a non-perturbative formulation of string theory is the BFFS matrix model, which came before the IKKT idea. It is obtained from 10D super Yang Mills theory with U(N), N>>1 gauge group by dimensionally reducing not to a single point but to a 1-d subspace, the time axis. (T-dualizing this time axis switches between BFSS and IKKT.) One thus obtains an "ordinary" (supersymmetric) quantum mechanical Hamiltonian theory, where however coordinates are replaced by large matrices.
The BFSS matrix model has several intepretations: The one inherited from the YM theory is that of a quantum mechanics of N D0 brains (pointlike brains) between which ground state strings stretch. The matrices in the model can be interpreted as the coordinates of the D0 branes and the non-commutativity of these coordinates comes, heuristically, from this non-local interconnection.
The second interpretation is maybe more surprising: One can start quantizing the supermembrane, the +1dim analogue of the string qhich lives in 11 dimensions. It turns out that the Hamiltonian of this membrane can be regularized by replacing coordinates by matrices. Fiddling around with this the Hamiltonian can be shown to be that of the BFSS matrix model. Blocks of block diagonal matrices represent blobs of membrane which interact via the non-diagonal elements.
In any case, one can show that many elements and amplitudes of string theory can be reproduced from the matrix model. Because the matrix model is non-perturbatively formulated it was conjectured to be _the_ non-perturbative formulation for a given asymptotical defined spacetime.
This is not precisely "background independent", but it is very interesting.
Urs, many thanks for that backgrounder. That's more information about matrix theory than I've ever seen before outside a technical paper in the arXiv. It will repay much studying.
Ah!..I see something here, could I ask why the diagonal matrices seem to be DYNAMICAL?..and if the standard blocks are really static?
I did some work on why the standard block 'matrix' have non-connecting evolutions and always have 'isolated' systems. But diagonal blocks have an evolution that has to be Dynamical and interacting movements, I found some very interesting things, but as the link shows my knowledge is really basic, though it was a revelation to me? forgive the crudness of my page:http://homepage.ntlworld.com/paul.valletta/PRIME GRIDS.htm
But I spent little time putting it together, my computer webpage skill is really basic.
So what I need to ask, is it that all matrices that are non-diagonal, relate to Static solutions?..and all diagonal matrices relate to systems that are Dynamical in some way?
I ask this because I recall something to do with the Riemmianian zeta function being the central function fo all dynamic systems that are not in isolation?
If I am understanding the question correctly, if calculations are made only on the string, then there is no information included about a background potential off the string.
But perhaps the Feynman path integral formulation, where every conceiveable path is considered is a calculation that does include every conceivable point in space.
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