Causality in SFT - Ted Erler

The papers of David Gross and his students often turn out to be important, as the events in Stockholm will indicate tomorrow. But it is not the only reason why Ted Erler was invited to speak at our Duality Seminar about his recent paper with David Gross - a paper dealing with causality in string (field) theory.

http://arxiv.org/abs/hep-th/0406199

Imagine that you work with the cubic string field theory. Three strings interact, and if you express the particular three strings in terms of fields localized at their centers of mass, you obtain a non-local interaction for your quantum field theory with infinitely many fields, of course. It's simply because the three centers of mass of your three interacting strings connected to the cubic vertex have no reason to coincide. In fact, it will also be non-local in time, not only in space, and such a non-locality in time (all derivatives of each field appear in the action) has disastrous consequences - the unbounded Hamiltonian is one of them.




OK, the center of mass is a bad definition of the position of your string. What about the midpoint? The three interacting strings share the midpoint, and therefore if you express the components of the string field as fields localized at the midpoint, the interaction will suddenly become local. However, there will be a different problem: all finite excitations of the string attached to a given midpoint will carry an infinite energy (squared mass). That's not hard to see - let me use my explanation why it's so: for a regular finite-energy string, all points on the string oscillate away from the center of mass in spacetime into distances that are infinite; the squared averaged distance is logarithmically divergent. Now, if you impose the condition that the midpoint sits at a fixed point, it's equivalent to give the rest of the string an infinite kinetic energy.

The center of mass gives you non-locality for the interactions of the component fields; the midpoint gives you local interactions, but they are interactions of singular fields. Ted and David offer a compromise: they express almost all coordinates as the center of mass coordinates, but one of the coordinates is shifted and expressed in terms of the midpoint coordinate. The singularity coming from the translation to the midpoint won't appear if the single coordinate treated separately - the direction in which we decide to use the midpoint coordinate as opposed to the center of mass coordinate - is light-like. It's because the extra singular energy that appeared previously was proportional to the square of the interval defining the translation of the reference point.

Once you shift the X^+ coordinate of your component string fields to denote X^+ at the midpoint, you will at least obtain interactions that are local in this single coordinate - a coordinate that you may call the light-like time. It seems to be the only choice how can you define "time" in open string theory in such a way that dynamics is local in time.

(For closed strings and quantum gravity there may be additional problems, such as the general covariance and the required non-locality to avoid Hawking's information loss, but they're not discussed in this article at all.)

Actually, it's not just locality in this light-cone time that you can derive from this hybrid formalism; Ted argued that causality holds much more strongly. Although string theory is a theory of non-local objects, the violation of locality always seems to be sufficiently mild so that all the usual good consequences of locality and causality in quantum field theory continue to hold.