Monday, February 24, 2014

Can string field theory be a full definition of string theory?

Michio Kaku is 67 today. He's a well-known popularizer of science who often likes to talk about crazy futurist concepts, promote physics using shows and statements that look like excessive if not cheap commercials, and spread excessive fear about things that are not really worrisome. But I still think that many criticisms against him have been unfair and he has done lots of useful work.

In science, he's known primarily for a pair of 1974 papers written with Keiji Kikkawa that established string field theory. More precisely, it was the light-cone gauge bosonic string field theory. People like Green and Schwarz would develop and exploit the superstring version of the formalism in the early 1980s. Witten would initiate a different, covariant (i.e. with manifest Lorentz symmetry) "cubic" string field theory as well as the 1992 boundary string field theory.

I've discussed the status of string field theory in the past and reported various exciting results, like Martin Schnabl's explicit vacuum solution of CSFT proving Sen's conjectures. But let me add a few words.

The straightforward, "constructive" quantum field theory are defined as the quantization of a classical action that depends on fields \(\Phi_i(x^\mu)\). The interactions are pretty much inevitably local which means that only products \(\Phi_i(x^\mu)\Phi_j(x^\mu)\cdots\) of fields at the same spacetime point may be added. The addition of terms that are products of fields at generic, different spacetime points would violate locality – unless there would be a nonlocal field redefinition that would turn the action into a local construct once again.

A natural strategy to define string theory is to phrase it as a "generalized quantum field theory" with infinitely many component fields \(\Phi_i(x^\mu)\) i.e. a string field theory. In fact, beginners tend to think that this approach is "inevitable" or "the only possible one". Well, they are wrong. In string theory, people prefer to compute the scattering amplitudes and other physically relevant observables "directly", and a quantum field theory-like formalism isn't necessarily an intermediate step in the calculation. In fact, a "first-quantized" approach – see e.g. Witten's introduction to string theory – is the standard one. One directly constructs the thickened, stringy counterparts of the Feynman diagrams and he never talks about spacetime fields.

However, string field theory is a possible way to define string theory and at least with certain disclaimers, it works and may be used to calculate the same scattering amplitudes etc. as in other approaches to string theory. The basic degrees of freedom in string field theory are functionals\[


\] where \(\sigma\) parameterizes a compact set (either a line interval for open strings or a circle for closed strings) and for each function \(x^\mu(\sigma)\) i.e. for each possible shape of a string in the target spacetime, one has one value of the field. This string field theory ultimately has to be quantized so these "values" of the string fields become operators (well, operator-valued functional distributions).

In the Hilbert space of a single string, one may also write the wave functional \(\Psi[x^\mu(\sigma)]\) for the ground state of the string as well as all the excited states in the stringy Hagedorn tower. These wave functionals form a basis of the relevant Hilbert space and the (classical or quantum) string field \(\Phi[x^\mu(\sigma)]\) may be expanded in this basis. One obtains countably many fields that still depend on the "center of mass coordinates" \(x_0^\mu\) and if one truncates the theory to the low-lying, light excitations of the string, the resulting action will look like the usual actions in quantum field theory (Yang-Mills theory or general relativity coupled to additional matter fields).

At the same moment, all the interactions in quantum field theory (think about all the cubic, quartic, and other vertices in Feynman diagrams of Einstein's or Yang-Mills or another theory that emerges at low energies) may be derived from some unified interactions in string field theory. Just like the local quantum field theories required the fields in a product to be evaluated at the same spacetime point, string field theory only allows e.g. the cubic splitting/joining vertex above for shapes of the string that overlaps so that the longest string spans the same 8-shaped curve in the target spacetime as the two "smaller" strings combined.

The light-cone gauge string field theory is the first form of string theory I embraced at the technical which was a reason why it was relatively straightforward to see that perturbative string theory emerges as a limit of matrix string theory. In the light cone gauge, the coordinate \(x^+\sim x^0+x^9\) is used as a "time" (the slices of the spacetime are null), the other null coordinate \(x^-\sim x^0-x^9\) is dual to the momentum component \(p^+\) which is identified with the length of the string in the \(\sigma\)-space, and the string fields only depends on the remaining purely spatial 8 transverse coordinates \(x^i(\sigma)\).

There are no unphysical states, no good (Faddeev-Popov) ghosts, and no bad ghosts (states of negative norm) anywhere. So the unitarity manifestly holds. However, the Lorentz symmetry isn't quite manifest due to the special treatment of the null directions. Nevertheless, the Lorentz and (super)Poincaré symmetry may be proven to hold with the straightforward definition of the (super)Poincaré generators if the supercommutators are approximated by the Poisson brackets. If the supercommutators are calculated exactly, some extra work is needed and we find out that the algebra still closes as long as \(D=26\) for bosonic string theory or \(D=10\) for superstring theory etc.

Witten's cubic string field theory works differently – and it only works well for open strings. The string fields generalize matrices with entries \(M^i{}_j\) except that the indices \(i,j\) don't take values just in a finite set. Instead, these two "indices" are replaced by "huge indices" that take values in all possible shapes of an open half-string in the target spacetime. So the information carried by these "indices" \(i,j\) tells you exactly where the left half-string and the right half-string are stretched in space – i.e. where the whole string is located. The advantage of representing the string's shape using two half-strings is that we may multiply the string fields by the matrix multiplication. The matrix multiplication is therefore realized as a "star-product" which is nothing else than an infinite-dimensional version of a "star-product" in field theories defined in noncommutative geometries and the whole string field theory may be defined using the action\[

S(\Psi) = \tfrac{1}{2} \langle \Psi |Q_B |\Psi \rangle + \tfrac{1}{3} \langle \Psi,\Psi,\Psi \rangle

\] where both the quadratic and cubic terms in the string field \(\ket{\Psi}\) are defined using the star-product. The cubic term is giving all the desired interactions. If you forget about the cubic term, you only get the quadratic action whose equations of motion are effectively saying that the string field is \(Q_B\)-closed (BRST-closed) while the variation of the string field \(\ket{\Psi}\) by a \(Q_B\)-exact expression \(Q_B\ket{\Lambda}\) is a gauge symmetry of the string field theory. In this Witten's cubic string field theory, the string fields depends on all \(D=10\) or \(D=26\) coordinates \(x^\mu(\sigma\) as well as the antighosts such as \(c(\sigma),\tilde c(\sigma)\) (I won't discuss the more technically annoying superstring case here).

In the previous paragraph, I encouraged you to think about the simplified action in which the cubic term is dropped; we get quite manifestly the free string theory of a single string (which may be multiply created because there are creation/annihilation operators hiding in the string field) in the usual modern covariant i.e. BRST formalism. The BRST variations in the single-string Hilbert space are reinterpreted as "ordinary" gauge transformations in the string field theory.

But it's also possible to drop the quadratic term and derive the whole string field theory using a purely cubic action, essentially\[

S = \frac 13 \Psi * \Psi * \Psi.

\] At least superficially, it looks really great. The equations of motion are simply \(\Psi*\Psi=0\). The \(Q_B\)-based quadratic term is obtained by giving a string field vev to the field \(\Psi\), i.e. rewriting \(\Psi = \Psi_0 + \Phi\). The vev \(\Psi_0\) is something like \(Q_B*1\), i.e. the right BRST operator acting on an "identity functional". At least formally, the purely cubic action is then equivalent to the "quadratic plus cubic" action. It's known as the background-independent string field theory because the allowed backgrounds (encoded in the form of the operator \(Q_B\)) emerge as solutions to the classical equations of motion \(\Psi_0*\Psi_0=0\) that is imposed on the vev. Note that in a certain approximation, the equation \(\Psi_0*\Psi_0=0\) carries the same condition as the nilpotency \(Q^2=0\) – so finding classical solutions to the purely cubic string field theory is more or less equivalent to the search for "stringy" BRST-operators \(Q_B\) that are nilpotent.

Because of the success of quantum field theory, most people (?) would tend to expect that a string field theory-based definition of string theory may be the complete one, one that clarifies all uncertainties about the non-perturbative behavior of string theory and many other things. More than 15 years ago, however, it was becoming clear that string field theory is just another way to "generate" more or less the same formalism of perturbative string theory. Non-perturbative aspects of string theory were as hidden in string field theory as they were hidden in other, "more direct" ways to calculate physical observables in string theory.

Well, this comment needs a clarification. String field theory is great for analyzing D-brane-like solutions, lower-dimensional D-branes as solitons in (tachyon-including) fields that live in a higher-dimensional D-brane, and so on. And because D-branes are non-perturbative objects, you could say that string field theory "sees into" non-perturbative string dynamics. However, D-branes are not equivalent to the "full non-perturbative dynamics" of string theory. Instead, D-brane terms and states are "transperturbative", like the ordinal numbers e.g. \(\omega+5\) if you wish, and they are accessible through different formalisms than string field theory, too.

While useful and extremely clear for the understanding of tachyon condensation on D-branes, string field theories couldn't really be used to prove the S-duality or other aspects of the strongly coupled limit of some string vacua. They just don't seem to "know" anything that goes beyond other perturbative approaches to string theory.

In fact, there were always many reasons to expect that string field theory just wouldn't be "much more complete" than what people already knew. The proofs of the equivalence of the string field theory diagrams to other stringy calculations are very analogous to the proof of the equivalence of the Schrödinger, Heisenberg, and Feynman pictures (i.e. path-integral formalism) in quantum mechanics. And we know very well that this equivalence is complete so it isn't conceivable that something possible to calculate in one picture would be utterly inaccessible by another picture.

Moreover, the problems of cubic string field theory with the closed strings suggested that string field theory is really no better than regular effective quantum field theories when it comes to really difficult problems such as those in quantum gravity (recall that gravity emerges from closed strings in string theory). Open string fields may live on a pre-existing gravitational (or closed string) backgrounds and their background-dependence isn't too different from the coupling of regular matter fields to a background geometry in a GR-like theory coupled to these matter fields.

Finally, the progress in black-hole-information sciences made it clear that the black hole backgrounds must really break the locality in some subtle yet inevitable way – because the information must be able to get out of the black hole interior. This seems impossible with the string field theoretical attitude where you would expect the background geometry to be one of the closed-string fields – which implies that a string field theoretical description of a black hole should preserve the locality and it should therefore destroy the information in the black hole interior, too. That's fundamentally bad.

So it seems that the non-perturbative, exponentially tiny effects that are needed for the information to escape are intrinsically incompatible with any string field theory. In other words, it apparently follows that every string field theory has to be inaccurate at the perturbative level.

However, this pessimistic conclusion is going too far. After all, we have a perturbatively exact definition of string theory that reduces to light-cone-gauge string field theory in the perturbative limit: our beloved matrix string theory. The exact Kaku-Kikkawa/Green-Schwarz light-cone-gauge perturbative string field theory emerges from matrix string theory if we integrate the off-diagonal elements of the matrices out. This allows to reinterpret the eigenvalues of the matrices as strands of independent strings, and the small effects of the off-diagonal modes of the matrix only allow the strings to split and join (at the same spacetime point).

One is led to believe that a more accurate treatment of the off-diagonal modes in matrix string theory should be enough to see that the information may escape from the black hole and why. What is problematic is that we must use the light-cone gauge i.e. we must Fourier-expand the black hole geometry in terms of modes with well-defined values of \(p^+\): the locality in the null coordinate \(x^-\) (pretending to be spacelike) isn't manifest (no covariant matrix string theory is known as of today). So we can't quite insert the full black hole geometry and study what happens.

But if this problem is overcome, it should be possible to see in what sense the off-diagonal modes of the matrix string theory's matrices and the \(U(N)\) gauge symmetry of this theory is enough to prove the black hole complementarity – the refusal of the black hole interior to be quite independent from the exterior. I have often tried to understand how this small violation of locality may result from matrix string theory – which is a non-perturbatively exact refinement of light-cone gauge string field theory. I have never succeeded and I cannot be sure that the success is possible but I still believe that there should be a way to derive such a conclusion from matrix string theory.

The existence of string fields seems to be a "perturbative approximation", much like the existence and fundamental character of strings in any other formalism of perturbative string theory. At a general coupling, the dynamics simply doesn't seem to be "just about strings". Nevertheless, it's still conceivable that even the exact degrees of freedom may be expanded in the string fields, and some particular features of the non-perturbative dynamics (like the off-diagonal modes and non-Abelian gauge symmetry in matrix string theory) could be enough to gain a "local perspective" to the issues such as black hole complementarity.

In my opinion, these are important questions and I wonder whether some people on this blue, not green planet are spending their time with similar questions at all.

This text is too technical, the expected number of readers is low, and I won't be proofreading it.


  1. "The existence of string fields seems to be a "perturbative approximation" Perturbative treatments cannot evolve emergent observables. There's your problem, emergence of unassumed symmetries impacting BRST invariance. Here is one solution to an intermnable pursuit of that which has no demonstrated empirical relevance,

  2. What is the relation between String field theory and Higher spin theory?

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  5. Dear Lubos,

    I liked reading your comments about SFT!

    I think that history may paint a worse picture of SFT than maybe it deserves.

    First there was a lot of mental capital devoted to developing closed SFT in the late 80's which didn't really seem to pay off. Partly the resulting formalism was too complicated for practical computations, and partly there was a gradual realization that closed string dynamics is not "field theoretic." The nonperturbative corrections are too large, and that there is no closed string vertex which "automatically" generates the genus expansion.

    Second there was the duality revolution. This revealed that the strong coupling limit of string theory is either another weakly coupled string theory, or it was something more exotic which would reduce to 11D sugra. In the former case SFT is superfluous since the string is weakly coupled, and in the second case SFT is inappropriate since we don't have a string theory. Simultaneously, the duality revolution brings the realization that string theory has D-branes and other membrane-like objects which appear to be as fundamental as the string.

    Regarding the first point, I might submit that closed string fields are not the right degrees of freedom for a nonperturbative definition of string theory (though some might advocate other interpretations). Nevertheless, despite the resemblance, there may be really a fundamental difference between closed SFT and open SFT in this regard. For the open string, the nonperturbative corrections are not too large; there is a fundamental vertex which "automatically" generates the perturbation theory; there are actions which, though challenging to study, can be used to make explicit calculations, and have slowly delivered almost all we could hope to learn about tree level string theory.

    Regarding the second point, dualities seem to be somewhat of a strange argument against SFT. After all, AdS/CFT duality does not eliminate the utility of having an action for N=4 super Yang-Mills. It might be that strings are not more fundamental than other objects. But it also might be that string theory has several nonperturbative formulations which utilize different weakly coupled degrees of freedom, and some of these nonperturbative formulations might be string field theories.

    What do you think of all this?

    I'm not sure I follow why you think that SFT would violate unitary BH evaporation. SFT is, after all, nonlocal. But honestly I don't see SFT having anything concrete to say about BHs in the forseeable future.

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  8. Dear Ted, thanks for your visit and comments! I agree with everything, including the promises, and I would be convinced that it was written in the blog post above as well but you wrote it more clearly and more concisely.

    Your longer essay of this kind - about any HEP/maths topic, not just SFT - would be welcome.

  9. Other than connections between massless higher spins and string theory in general (which are currently being explored, mostly in the holographic context), SFT is interesting to some higher spin experts since it gives an action for an infinite number of interacting higher spin fields. Since massless higher spins don't have a first quantized description, you need a field theory formalism which would probably resemble SFT. But to be honest, other than free Lagrangians, I don't think SFT has been of much concrete help for massless higher spins. This is probably because we don't know how to covariantly quantize a string in AdS, and in flat space there are some serious doubts as to whether interacting massless higher spins exist.

  10. I like this short overview of the merits of string field theory, as far as I can say :-)

    And the last paragraphs about the importance of the off-diagonal elements of the matrices of matrix string theory being important to understand the coupling between the black hole interior and exterior in the context of complementarity would make a nice answer to one of my old Physics SE questions

    Ok, the title seems a bit clumsy when looking at it again ...

    Michio Kaku is capable of talking about funny things at times :-), but this does not take the fact that he is a capable cool physicist away. The anti theoretical-physics trolls who feel entitled to badly and personally insult him (including name calling etc) are just assholes that should be .... you know :-(0)

  11. Thanks, interesting post, and I will really need to read up more on String Field Theory too. I had bookmarked a source on SFT a long time ago, and have totally forgotten to access it. This post helps remind me about it, too.

  12. Dear Lubos, what is the analogous of creation and annihilation operators in usual string theory? Are they useful in this framework? Thanks

  13. Right, they're elementary degrees of freedom in string field theory.

    In light-cone SFT, it's particularly clear. There is a functional Psi[x^i(sigma)] that contains an annihilation operator for every function (string) x^i(sigma), and the Hermitian conjugate Psi^dagger, with the obvious Dirac-delta-functional-proportional commutator.

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