Does String Theory have any equations yet? I chose not to....

[physics22]

Back when I was studying at Princeton University, J.A. Wheeler introduced me to Ed Witten while we were having lunch at the Institute for Advanced Study (IAS). I knew Feynman’s and Wheeler’s views on “String Theory,” and I had looked into it a bit as a field of research, but I couldn’t seem to find any actual equations reflecting physical entities. And so I took the opportunity to ask Ed Witten what the actual equations were.

He smiled at first, thought about it, and said there weren’t really any, yet.

“When do you think you might get some I asked?”

He wasn’t smiling after that.

So I was wondering if they had since found any? If the answer is “yes,” please do provide the equation (or equations) as I would love to see the math which I have heard is beuatiful! :) Also please provide the physical entities it refers too. Thanks! :)

 

[Simon Bridge]

The answer is still "not really". But be careful of that "really" qualifier.
For example, there are equations used to describe the strings - and they have equations describing the modes of vibrations of the strings ... the modes correspond, in the theory, to fundamental particles. These fundamental particles are physical entities.
That formulation is available in any non-qualitative intro to string theory.

The "really" part is that it is unclear how we'd go about doing physics with this. But there have been some attempts. See replies to:
http://physics.stackexchange.com/questions/62117/fundamental-equations-of-string-theory

 

[Demystifier]

String theory, of course, contains a lot of equations. What is missing, however, is a small set of fundamental equations, from which all other equations should follow as special cases.

 

[haushofer]

The most fundamental equation of string theory is given by the action. This (Polyakov-)action describes a 2-dimensional conformal field theory which we can solve exactly in a flat background: it's a two-dimensional Lorentz-invariant wave equation. The 'problem' comes when we try to translate this to spacetime-dynamics; then we can only approximate the equations of motions as viewed from a spacetime observer.

 

[rootone]

I'm not an above average math guy.
My opinion of string theory is that is very aesthetic pleasing to those who understand it.
That doesn't mean it describes reality though, it needs to make testable predictions for that.

 

[p-brane]

I'm not an above average math guy.
My opinion of string theory is that is very aesthetic pleasing to those who understand it.
That doesn't mean it describes reality though, it needs to make testable predictions for that.

My opinion of [insert name of theory you don't really understand here but nonetheless don't like] is that [insert name of theory you don't really understand here but nonetheless don't like] is very aesthetically pleasing to those who understand [insert name of theory you don't really understand here but nonetheless don't like]. That doesn't mean [insert name of theory you don't really understand here but nonetheless don't like] describes reality though, [insert name of theory you don't really understand here but nonetheless don't like] needs to make testable predictions for that.

 

[no-ir]

p-brane: Are you saying that making testable predictions is not necessary for a theory of physics? How does such a theory then differ from (mathematically sophisticated) philosophy?

 

[rootone]

My opinion of [insert name of theory you don't really understand here but nonetheless don't like] is that [insert name of theory you don't really understand here but nonetheless don't like] is very aesthetically pleasing to those who understand [insert name of theory you don't really understand here but nonetheless don't like]. That doesn't mean [insert name of theory you don't really understand here but nonetheless don't like] describes reality though, [insert name of theory you don't really understand here but nonetheless don't like] needs to make testable predictions for that.

I didn't offer an opinion as to whether I 'like' string theories, and I do understand the concept though not the deep math.
I have no reason to dislike string theories, I don't consider them to be implausible.
However liking or disliking some idea has no bearing on whether the idea is close to physical truth, only evidence does.

 

[ShayanJ]

I didn't offer an opinion as to whether I 'like' string theories, and I do understand the concept though not the deep math.
I have no reason to dislike string theories, I don't consider them to be implausible.
However liking or disliking some idea has no bearing on whether the idea is close to physical truth, only evidence does.

I don't see the problem with string theory. Its not like we have a lot of experimental data in the QG regime but no evidence for string theory, but that we don't have enough experimental data in the QG regime! Now in this absence of experimental evidence for or against string theory, the only way to decide whether you want to work on it or not, are theoretical evidence and aesthetics. You certainly don't expect people to simply abandon string theory, do you?
Also, the alternatives aren't doing better. So why is the objection only towards string theory?

 

[Urs Schreiber]

If the answer is “yes,” please do provide the equation (or equations) as I would love to see the math which I have heard is beuatiful! :) Also please provide the physical entities it refers too. Thanks! :)

This is answered in string theory FAQ -- What are the equations of string theory? I just went and expanded a bit more there. Have a look there for the following text equipped with hyperlinks for all keywords.

So:

All local field theories in physics are prominently embodied by key equations, their equations of motion. For instance classical gravity (general relativity) is essentially the theory of Einstein's equations, quantum mechanics is governed by the Schrödinger equation, and so forth.

But perturbative string theory is not a local field theory. Instead it is an S-matrix theory (see What is string theory?). Therefore instead of being given by an equation that picks out the physical trajectories, it is given by a formula for how to compute scattering amplitudes. That formula is the string perturbation series: it says that the probability amplitude for $n_{in}$ asymptotic states of strings coming in (into a particle collider experiment, say), scattering, and $n_{out}$ other asymptotic string states emerging (and hitting a detector, say) is a sum over all Riemann surfaces with $(n_{in}, n_{out})$-punctures of the $n$-point functions of the given 2d SCFT that defines the scattering vacuum.

More in detail, a string background is equivalently a choice of 2d SCFT of central charge 15 (a “2-spectral triple”), and in terms of this the formula for the S-matrix element/scattering amplitude for a bunch of asymptotic string states $\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}$ coming in, and a bunch of states $\psi^1_{out}, \cdots, \psi^{n_{out}}_{out}$ coming out is schematically of the form

$$
S_{\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}}
\;=\;
\underset{g \in \mathbb{N}}{\sum}
\lambda^g
\underset{
{moduli \; space \; of}
\atop
{{(n_{in},n_{out}) punctured}
\atop
{{super\; Riemann \; surfaces}
\atop
{{\Sigma^{n_{in}, n_{out}}_g}
\atop
{of\; genus\; g}}}}
}{
\int
}
\left(
SCFT \; Correlator \; over \; \Sigma \; of \; states\;
{\psi^1_{in}, \cdots, \psi^{n_{in}}_{in}, \psi^1_{out}, \cdots, \psi^{n_{out}}_{out}}
\right)
$$

expressing the S-matrix element (scattering amplitude) shown on the left as a formal power series in the string coupling constant with coefficients the integrals over moduli space of super Riemann surfaces of the worldsheet correlators (nn-point functions) for the given incoming and outgoing string states.

With more technical details filled in, this formula reads as follows (for the bosonic string, as found in Polchinski's "String theory", volume 1, equation (5.3.9))

and for the superstring, as found in Polchinski 01, volume 2, equation (12.5.24):

This is the equation (formula) that defines perturbative string theory.

And this is of just the same form as as the Feynman perturbation series in local quantum field theory, the only difference being that the latter is more complicated: there one has to some over Feynman diagrams with labeling for all intermediate particles (virtual particles) and with some arbitrary “cutoff” to make the integrals well defined, whereas here we simply sum over all super Riemann surfaces. The different intermediate virtual particles as well as the renormalization counterterms are all taken care of by the higher string modes, encoded in the worldsheet CFT correlators.

There was a time in the 1960s, when quantum field theorists around Geoffrey Chew proposed that precisely such formulas for S-matrix elements should be exactly what defines a quantum field theory, this and nothing else. The idea was to do away with an explicit concept of spacetime and local interactions, and instead declare that all there is to be said about physics is what is seen by particles that probe the physics by scattering through it. This is an intrisically quantum approach, where there need not be any classical action functional defined in terms of spacetime geometry. Instead, all there is a formula for the outcome of scattering experiments.

Historically, this radical perspective fell out of fashion for a while with the success of QCD and the quark model in its formulation as as local field theory coming from an action functional: Yang-Mills theory.

But fashions come and go, and the original idea of Geoffrey Chew and the S-matrix approach continues to make sense in itself, and it is this form of a physical theory that perturbative string theory is an example of.

Ironically, more recently, the S-matrix-perspective also becomes fashionable again in Yang-Mills theory itself, with people noticing that scattering amplitudes at least in super Yang-Mills theory have good properties that are essentially invisible when expressing them as vast sums of Feynman diagram contributions as obtained from the action functional. For more on this see at amplituhedron.

On the other hand, there is also an analog of the second quantized field-theory-with-equations for string scattering: this is called string field theory, and this again is given by equations of motion. For instance the equations of motion of closed bosonic string field theory are of the form

$$
Q \Psi + \tfrac{1}{2} \psi \star \psi + \tfrac{1}{6} \psi \star \psi \star \psi + \cdots = 0
\,,
$$

where $\Psi$ is the string field, $Q$ is the BRST operator and $\star$ is the string field star product.

The string field $\Psi$ has infinitely many components, one for each excitation mode of the string. Its lowest excitations are the modes that correspond to massless fundamental particles, such as the graviton. Expanding the equations of motion of string field theory in mode expansions (“level expansion”) does reproduce the equations of motions of these fields as a perturbation series around a background solution and together with higher curvature corrections.

 

[Demystifier]

For instance the equations of motion of closed bosonic string field theory are of the form
$$
Q \Psi + \tfrac{1}{2} \psi \star \psi + \tfrac{1}{6} \psi \star \psi \star \psi + \cdots = 0
\,,
$$
where $\Psi$ is the string field, $Q$ is the BRST operator and $\star$ is the string field star product.

What about string-field theory for superstrings?

 

[Urs Schreiber]

What about string-field theory for superstrings?

The equation of motion of closed superstring field theory has the same form, for the appropriate super-versions of the string field $\Psi$, the BRST operator $Q$ and the product $\star$. This is Sen 15, equation (2.22)

And I should have mentioned: the equation of motion for closed bosonic string field theory are due to Zwiebach 92, equation (4.46).

 

[Demystifier]

This is Sen 15, equation (2.22)

Is this a non-perturbative formulation of super-strings? If so, can it say something about other non-perturbative aspects such as compactification, branes, AdS/CFT etc?

 

[Urs Schreiber]

Is this a non-perturbative formulation of super-strings? If so, can it say something about other non-perturbative aspects such as compactification, branes, AdS/CFT etc?

String field theory is meant to be some kind of non-perturbative formulation of string theory, yes.

The string field theory action functional is to the string perturbation series exactly as the action functional of a field theory is to its Feynman perturbation series.

Now, while the full non-perturbative theory is in principle encoded in the action functional (for instance by putting it on a lattice and running Monte-Carlo simulation), in practice it is hard to extract the non-perturbative effects,even if you have the action (that's why the mass gap problem for Yang-Mills theory is a millenium problem). Accordingly, progress on these matters in string field theory goes more slowly still.

But the key success of string field theory is the computation of D-brane decay energy densities. Whenever there is an unstable D-brane configuration in string theory (say the D25 brane in bosonic string theory, or a brane/anti-brane pair in superstring theory) there is a tachyon mode in the corresponding perturbative string theory. Way back, Sen realized that this tachyon must be the perturbative incarnation of the fact that one is looking at the perturbation series around the maximum of a potential, instead of around a minimum (the latter is what one needs to do to get a consistent perturbation theory). He hypothesized that this potential must be that of an unstable D-brane configuration, which tends to decay by rolling down this potential, finding its true energy minimum.

This is famous as "Sen's conjecture".

For the decay of the D25-brane in bosonic string theory, this conjecture was checked numerically using bosonic string field theory. Then Martin Schnabl found an analytic solution. Later Ted Erler did the analogous computation for the decay of brane/anti-brane pairs in supstring theory.

(For references see at nLab:string field theory -- References -- Tachyon dynamics, decaying D-branes and Sen's conjecture)

These proofs of variants of Sen's conjecture are so far the big successes of string field theory as a non-pertubative formulation of string theory. But many other things remain to be understood.

In particular, so far the string field action is always treated as a classical action. It's quantization, that would be the proper "second quantization" of the string. Hence so far string field theory is a means to succinctly encode all the inteaction of the the first quantized string, together with some global information on the relevant potentials, which cannot be seen from the point of view of anyone perturbative expansion.

A complete non-perturbative understanding of string theory starting from the string field action should be possible (certainly could be possible), but is hard, arguably at least as hard as solving problems like the mass-gap problem of Yang-Mills theory.

 

[Demystifier]

String field theory is meant to be some kind of non-perturbative formulation of string theory, yes.

But Eq. (2.22) is an expansion in the powers of $\Psi$, so it is a kind of a perturbative expansion, isn't it?

 

[Urs Schreiber]

But Eq. (2.22) is an expansion in the powers of $\Psi$, so it is a kind of a perturbative expansion, isn't it?

Each of these terms $\langle \underset{n \mathrm{factors}}{\underbrace{\psi \star \cdots \star \psi}}\rangle$ in the action is an interaction term (the star-product encodes the merging of two strings to one, the $n$-fold star product encodes the merging of $n$ strings to one). For instance for the open string, for $n =3$ and expanding in the string field excitations, this contains the interaction vertices of the form $\overline{\phi} A_\mu \Gamma^{\mu} \phi$ encoding a fermion and an anti-forming annihilating to a gauge boson. So the fact that in the string field theory there such vertices for arbitrary $n$ means that there are fundamental interaction vertices of arbitrary many particles in the particle, limit.

That's just what the full action functional needs to contain. Of course it is tempting now to make a formal expansion in these powers. But a priori, the string field action contains them all.

 

[Demystifier]

So the fact that in the string field theory there such vertices for arbitrary n means that there are fundamental interaction vertices of arbitrary many particles in the particle, limit.

Why these vertices need to be fundamental? Isn't it more natural to assume that this is only the effective 1PI action, rather than a fundamental action?

 

[Urs Schreiber]

Why these vertices need to be fundamental? Isn't it more natural to assume that this is only the effective 1PI action, rather than a fundamental action?

Yes, right, that's what it comes out to be. The only fundamental interaction is the $\psi \star \psi$. What I wrote down is the tree level version, while the full 1PI action is more complicated, see for instance equation (4.12) in Jurco-Muenster 13

 

[Demystifier]

The only fundamental interaction is the $\psi \star \psi$.

What is the corresponding fundamental equation of motion?

 

[Urs Schreiber]

What is the corresponding fundamental equation of motion?

Hm, if the action is truncated to that first interaction term, then the equation of motion reduces to $Q \psi = \psi \star \psi$.

That's actually the form of the full equations of motions for string field theory of just the open string.

I might not be saying it completely properly. Maybe the right way to say it is that the closed closed string field theory action only exists at the level of (the analog of) a 1PI action. I'll check with my local SFT experts when I see them next.