# Does a smooth world sheet imply strings moving faster than light?

In string theory, one wants to study the interaction of strings throughout time. The interacting strings sweep out a surface in Minkowski space. To write down an action for the string, we need to assume that the world sheet is at least differentiable (usually smooth). But if we assume that strings cannot move faster than light, we get an immediate contradiction. Because if the world sheet would be smooth, then at a spacetime point where strings join, the tangent plane would need to be space-like, which implies that the string moves faster than light.

It is obviously easier to handle string theory defined on smooth surfaces, but the above implies that this would violate fundamental principles in physics. So what is the solution?

(I know that this is not an issue if the metric of the target space is Euclidean. But from a physical point of view, we are interested in Lorentzian metrics. Also, a Wick rotation does not solve the problem, because in the end, the string would still need to move faster than light to sweep out a smooth surface.)

TL;DR: The string is not moving faster than light.

1. A 2D tangent plane $$T_pN$$ of a regular/generic point $$p$$ of the 2D string world sheet $$N:=X(\Sigma) \subseteq M$$ imbedded into a Lorentzian target space $$M$$ has Lorentzian signature, i.e. the induced metric on $$T_pN$$ has split signature 1+1, cf. e.g. Ref. 1.

2. At isolated string splitting-joining interaction points (which are associated with spatial topology change), the manifold is singular, cf. e.g. Ref. 2.

Presumably the singularities should be cured via an appropriate Feynman $$i\epsilon$$ prescription/analytic continuation via Wick rotation from an Euclidean formulation, cf. e.g. Ref. 3.

References:

1. B. Zwiebach, A first course in String Theory, 2nd edition, 2009; section 6.3.

2. L. Freidel, R.G. Leigh & D. Minic, Metastring Theory and Modular Space-time, arXiv:1502.08005; p. 19 + Fig. 1.

3. E. Witten, The Feynman $$i\epsilon$$ in String Theory, arXiv:1307.5124.

• Zwiebach shows that a world-sheet governed by the Nambu-Goto action always admits a timelike tangent vector (or at least a null vector). But then, interactions in closed string theory in Minkowski space should be impossible: whenever we have two closed strings which interact into one string, we get topologically a pair of pants (picture). There clearly exists a point with spacelike tangent plane. Does it imply that most pictures in string theory are false? And interactions in closed string theory don't exist?
– KuSi
Commented Jun 5, 2022 at 16:36
• I updated the answer. Commented Jun 5, 2022 at 17:12