# Why is it hard to give a lattice definition of string theory?

In Polyakov's book, he explains that one possible way to compute the propagator for a point particle is to compute the lattice sum $\sum_{P_{x,x'}}\exp(-m_0L[P_{x,x'}])$, where the sum goes over all paths between $x$ and $x'$. One then needs to compute this sum and choose the bare mass so that there's a good continuum limit.

Polyakov then goes on to say that this doesn't work for string theory. I skimmed the literature and couldn't find any explanation of this fact. Naively I would think that in order to find the propagator, you could just compute the sum $\sum_{W_{C,C'}}\exp(-T_0 A[W_{C,C'}])$, where the sum is over worldsheets that end on the curves $C$ and $C'$. What goes wrong? Is this just a hard sum to do?

M theory and string theory both have propogators and Feynman diagrams.

The concrete form of the propagators can be easily obtained, since the problem can be mapped onto the problem of finding the propagators for the one-dimensional harmonic oscillator.

See:Becker and Matrix theory as wells as string field theory

Also the SO32 closed string is an E8xE8 even lattice.

• That's not true. M-theory does not have any precise formulation in terms of a perturbative expansion. Also I don't understand in what sense saying that some theory has a good perturbative expansion shows the difficulties in trying to put it on a lattice. – Ramiro Hum-Sah Oct 26 '20 at 0:55
• Can I ask, How do M branes evolve if not a path integral.A propogator is a time evolution operator,not a pertubative method. What we change essentially in M theory is the branes and the integral, making it a volume integral – user220348 Oct 26 '20 at 1:01
• Everything you have written is complete nonsense. Just a bunch of words without any real meaning. – Ramiro Hum-Sah Oct 26 '20 at 1:19
• This answer has been flagged as low-quality and voted for deletion. – joseph h Oct 26 '20 at 2:11
• The question is about obtaining a theory (in this case, string theory) by working on discretized space-time then taking continuum limit en.wikipedia.org/wiki/Lattice_field_theory – Mitchell Porter Oct 26 '20 at 3:00