# How is causality encoded in string theory?

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How is causality encoded in string theory?

Causality in quantum field theory is embodies in (i) Poincare invariance of the Wightman functions, (ii) local commutation rules for the fields. What is the equivalent of that in string theory?

In view of the answer by Urs Schreiber, a more precise version of my question would be: How does string theory ensure that the singularities of the S-matrix can only occur in ways that don't allow the future to influence the past?

edited Jan 13, 2017

only very loosely related....

Ok, can you give some more details why you think causality needs more (special or different) attention in string theory than in point-particle QFT for example?

This question is a bit short; it is not obvious to me what you are up to ...

@Dilatant,

This idea that string theory might not have equations as other theories do is as weird as it is robustly haunting the chat groups. See  String theory FAQ -- What are the equations of string theory?    for explanation of the basics.

There are certainly points to be careful with and critical about string theory. But the idea that the problem with string theory could be a childish failure at an elementary level should occur, even to people without any technical understanding of it, to be implausible.

The irony with all the silly criticism of string theory ("Has no equations!", "Makes no predictions!") is that it distracts attention from the true issues that deserve discussion. The problem is of course that these true issues require some minimum of sophistication to be appeciated.

@dilaton: I added a bit of detail; don't know how to be more specific.

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Here I'll collect references that treat the question being asked in more detail. I might expand this later when I actually have some time for this:

There is

• Emil Martinec, Strings and Causality (pdf) in L. Baulieu, V.  Dotsenko, V. Kazakov, P. Windey (eds.) Quantum Field Theory and String Theory , NATO ASI Series B: Physics Vol. 328 (1995)

wich is a short commented analysis of the string 2-point function as compared to the particle 2-point function and in view of how causality is read off. In the course martinec referes to string fiels, somewhat colloquially.

A much more in depth analysis is in

From the introduction:

Perhaps then it comes as a surprise that critical string theory produces an analytic S-matrix consistent with macroscopic causality. In absence of any other known theoretical mechanism which might explain this, despite appearances one is lead to believe that string interactions must be, in some sense, local.

and

We find that string theory avoids problems with nonlocality in a surprising way. In particular, we find that the Witten vertex is “local enough” to allow for a nonsingular description of the theory which is completely local along a single null direction.

and

unlike lightcone string field theory, it is clear that cubic string field theory at least has a local limit where all spacetime coordinates are taken to the midpoint. We investigate this limit with a careful choice of regulator and show that at any stage the theory is nonsingular but arbitrarily close to being local and manifestly causal. We believe that the existence of this limit, though singular, must account for the macroscopic causality of the string S-matrix. Thus, string theory is local enough to avoid the inconsistencies of a theory which is acausal and nonlocal in time, but is nonlocal enough to make string theory different from quantum field theory

Then they comment on Martinec's account above, and other's, by saying:

To motivate our particular perspective, it seems appropriate to discuss earlier attempts to understand the role of locality, causality and time in string theory, and explain why we feel these approaches do not adequately address the problems just raised.

answered Jan 13, 2017 by (6,025 points)

Yes, the Erler-Gross paper is of the kind I was looking for. I'll study it, thanks!

As far as I understand, they avoid the main problem (in my opinion), namely causality in the closed string sector, which generalizes GR, in contrast to the open one, generalizing YM. Open strings are defined on a fixed background, so the fact that OSFT is causal doesn't seem surprising.

Closed strings are also defined on a fixed background.

@UrsSchreiber Actually, SFTs are defined initially with no reference to any background at all, it appears dynamically. But in the case of OSFT there are known exact solutions on fixed background (say, Witten's construction of CS theory), with no backreaction of strings (and branes) on geometry. As far as I understand, it is the consequence of absence of gravity modes in OSFT. I have never encountered the exact solution of CSFT, or discussion of causality in CSFT. If you have, could you please give a reference?

No SFTs are all defined with respect to a background, it is encoded in the BRST operator Q that fixes a background (S)CFT. What does happen is that one may argue that their results are independent und suitable shifts of this background, but that's another matter.

@UrsSchreiber BRST operator in SFT is the same as in the first-quantized description of the theory, so it doesn't fix the background in the same way it is not fixed in usual ST mod some constraints following from its nilpotency. Background dynamics is governed by GR, which is a subsector of a closed string sector, so one can't fix the background when the closed strings are involved. The best one can do is to find a solution for the background at (usually) the tree level in $\alpha^{\prime}$, and then to consider certain perturbative processes on it.

No, the worldsheet CFT that enters the definition of the SFT action, that is the background, for instance some sigma-model coming from some geometric background. There is no background-free formulation of string (field) theory at the moment. (All these is are arguments for background *independence*, saying that the solutions of the SFT action computed for one background match these for a suitably shifted background.) That's really right there at the definition of SFT, and I don't think I'll have much to re-iterate this point.

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Whightman axioms and locality of nets of observables are concepts that apply to spacetime local field theory. There is also another axiomatization of (perturbative) field theory, namely the S-matrix approach.

Historically, this was suggested in the 1960s as a fundamental axiomatization of QFT. Then the success of the quark model in QCD made spacetime local field theory become the popular paradigm that it is today. But in fact these days people in QFT are being attracted back to the S-matrix approach (this is what the "amplituhedron" story in super Yang-Mills theory is about).

In any case, the key fact to be aware of is that perturbative string theory (which, as opposed to its would-be non-perturbative version is a mathematically well-defined concept), is an example of an S-matrix theory, not of a spacetime local field theory. For entry points to this fact see for instance the string theory FAQ right at the beginning (also search the text for further occurences of "S-matrix").

So your question really is: How is causality encoded in S-matrix theories?

I am not in position to give an exhaustive reply to this question, but a) broadly causality is encoded in the analycity of the S-matrix and its singularity stucture  and b) there is an extensive literature on this.

For what it's worth, already the Wikipedia page on S-matrix theory informs about analycity prinicples of the S-matrix and that they encode causality (see the paragraph Basic principles):

These  principles [analycity of the S-matrix] were to replace the notion of microscopic causality in field theory,

answered Jan 13, 2017 by (6,025 points)

Causality conditions: the singularities of the S-matrix can only occur in ways that don't allow the future to influence the past

Thus a more precise version of my question would be: How does string theory ensure that the singularities of the S-matrix can only occur in ways that don't allow the future to influence the past?

You say that there is an extensive literature on this. I know approximately everything about singularities of S-matrices in quantum field theory. Could you please give some pointers that relate more directly to the string theory side of it?

True, I should give better references. Some basics and further pointers are recorded in section 2.3 of

Moore's section 2.3 is about the analyticity of amplitudes, but the properties given are unrelated to establishing causality in the sense of my first comment. The word causal is not even mentioned in the paper, so it is difficult to find out which references there might shed light on the question.

Right, sorry. You know, with the keywords "analytic" and "string S-matrix" you should be able to track down the relevant references, I don't have them handy here, and don't quite have the leisure to dig into it. But I opened another reply box here with some more references. I'll try to expand that box later when there is time.

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