Re: superstrings on the lattice: OCQ as NCG

On 30 Nov 2003, Urs Schreiber wrote:

> As this application might indicate: Couldn't a lattice formulation of
> quantum strings play a role in a non-perturbative investigation of the
> theory, just like field theories are non-perturbatively studied on the
> lattice?

I apologize for having misunderstood the relation between the algebras,
Dirac operators, and the mathematical language involving the triples on
one side, lattices on the other side, and noncommutative geometry (which I
suppose NCG stands for) on the third side. I hope that someone else will
get it.

Are you working within some existing (string) theory; or do you want to
define a new theory; or do you claim that an old theory behaves in a
different way than everyone else thinks? I feel that this basic question
should always have a clear answer, otherwise it is hard to see more than a
collection of confusing mathematical formulae.

If you ask whether people considered latticization of strings, the answer
is, of course, yes. Many times, a long time ago just like recently, and in
several cases we can show that the latticization of the string is an
important and exact part of stringy physics. Matrix string theory is some
sort of discretization, but the string is still continuous.

A much more typical example of string bits - that is also based on DLCQ,
in a sense - is the BMN limit of N=4 d=4 super-Yang-Mills theory. Well,
only the spatial dimension of the string is latticized

        http://arxiv.org/abs/hep-th/0202021

We don't have any working examples in string theory where time can be
discretized as well. Continuous time remains a necessary condition for the
existence of the Hilbert space and similar objects and tools.
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