# String vacua and particles?

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I've been doing a little bit of reading on string theory, and the very large number of string vacua that are possible (i.e., perhaps 10^500 or more). One thing that is not clear to me is exactly what constitutes a 'vacuum' in string theory. In QFT theory, the vacuum is defined as the state with no physical particles (i.e., the ground state)... however I'm not sure, and I haven't been able to clearly find a source, which says that the same is true for string vacua?

I do understand that the choice of string vacuum ultimately determines what particles may exist, etc.

I would surmise that to call anything a 'vacuum,' it would have to be devoid of particles (excitations above the ground state), but not 'empty' (e.g., filled with fields, etc., just as in QFT). Is this true for string vacua?

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Perhaps I can expand on the above briefly - quoting from Brian Greene's pop-sci "Elegant Universe", he mentions "virtual string pairs" that are created by temporarily borrowing energy from the vacuum. As discussed in the context of QFT by @A. Neumaier in his excellent Insight articles, this seems just as impossible. (The virtual string pairs are discussed in the context of the perturbative approach.)

Any comments on this as related to the posting above? I would assume Greene is just appealing to figurative language...

Let's start with the "vacua" of string theory that are defined in terms of the scattering of strings on some flat space-time - either completely uncompactified space-time in the maximum dimension, or some space-time which is a product of a lower-dimensional Minkowski space and some flat compact space.

In both cases, you can define a perturbative S-matrix by figuring out possible asymptotic states of a single string, and then defining the analogue of the usual Dyson-Feynman expansion of the infinite-time S-matrix, but with a sum over string histories rather than ordinary Feynman diagrams.

Now the question is, what is the relation of all these S-matrix theories to each other? Do they represent perturbations around different minima of the same fundamental theory, so that one could actually tunnel from one background geometry to another, or are they truly disjoint possibilities?

So far as I know, there is nothing like a clear general answer to this question, which would require a clear formulation of the underlying theory, and knowledge of whether the "vacuum" in question is stable or not. Since these perturbative S-matrices are defined in a bottom-up way, they can be missing contributions from new objects like branes, and one also has to ask whether the perturbation expansion converges or not.

These same questions are even more acute for the post-2003 landscape of flux vacua, which were constructed to yield de Sitter space rather than Minkowski space, and which are generally thought to be metastable. If they are metastable, then they really have to be embedded in a bigger deeper formulation of string theory, but that is lacking.

Perhaps I have just overlooked the papers where all these questions have already been answered, but I don't think so. The evidence of this is Tom Banks's long string of papers presenting a heterodox view, according to which (if I understand him correctly) string theory in de Sitter space is stable, and each possible value of the cosmological constant defines a separate sector of string theory.

I believe that if this was definitely wrong, Banks's papers would have been refuted by now. But instead, what I see is just a widespread presumption (which Susskind in particular has championed) that there exists some cosmological framework for string theory, akin to chaotic eternal inflation, in which all the metastable vacua really are just local minima in some multiverse potential.

This is only a presumption because, to my knowledge, no implementation of this framework at the level of strings exists. Instead we have all sorts of heuristic toy models implemented using field theory (and some weird speculations out of Stanford about FRW/CFT duality and a p-adic multiverse). But I see profound uncertainty about how to even describe tunneling between vacua in string theory (e.g. what sort of instanton describes the tunneling process), let alone what the "potential on the landscape" is.

If we retreat to some simpler questions, then there is something that looks more like knowledge. E.g. in geometric backgrounds which preserve a high degree of supersymmetry, there may be a lot of scalars and the supersymmetry tells you that there is no potential preferring one set of vevs over another. But with lower degrees of supersymmetry, there should be a potential (which implies time evolution of the geometry, and that is largely an unsolved problem).

Or there's Brian Greene's own work showing that the moduli space of one Calabi-Yau space overlaps with that of another, so that string theories on those two backgrounds really should be regarded as sectors of the same theory. There are huge webs of such overlaps between Calabi-Yaus. I don't know offhand if the resulting combined theories all fall into the case of flat unified moduli space, though I think if there are duality webs for N=1 vacua there should be a big potential which would govern time evolution from one vacuum state to another.

That may all be confusing so I will try to sum up. There are many quantum theories of the form "string theory X on background Y", that can be defined with varying degrees of rigor and calculability. The deeper question is then, do all these quantum theories link up, and if so, how do they link up? I usually suppose that string theory on the various AdS backgrounds represent disjoint theories, or disjoint sectors of string theory; and that string theory on various dS backgrounds ought to be embedded in some larger quantum theories, in which the "dS vacua" really are minima of a shared cosmological potential; and for string theory in flat space, I am agnostic about the extent to which we are dealing with separate quantum theories, or with connected sectors of unified quantum theories.

P.S. When Greene talks about virtual string pairs, it's just like talking about virtual particles. Virtual particles are intermediate objects which appear "in the middle" of a path integral, but not in the in state or the out state. These virtual strings would be the same - handles that bud from the worldsheet and then get reabsorbed, and so on.

Spinnor and asimov42
@mitchell porter Thanks! That's exceedingly helpful and a great summary!

P.S. When Greene talks about virtual string pairs, it's just like talking about virtual particles. Virtual particles are intermediate objects which appear "in the middle" of a path integral, but not in the in state or the out state. These virtual strings would be the same - handles that bud from the worldsheet and then get reabsorbed, and so on.
Right - I'd understood the virtual string pairs as being intermediate. My thought, going back to @A. Neumaier's articles about virtual particles, and other sources, is that the virtual pairs are intermediate quantities in perturbative calculations, which when taken together (infinite series) define the S-matrix. But Greene's assertion that energy is 'borrowed' from the vacuum somehow by Heisenberg's uncertainty principle is merely pop-sci jargon - in QFT this does not occur (i.e., "particles popping in and out of existence" all over the place, since there's no way to borrow energy from the ground state afaik, and the theory doesn't have creation/annihilation operators for off-shell particles anyway). My assumption is that that same is true in string theory?

But Greene's assertion that energy is 'borrowed' from the vacuum somehow by Heisenberg's uncertainty principle is merely pop-sci jargon - in QFT this does not occur [...] My assumption is that that same is true in string theory?
I won't try to adjudicate Greene vs Neumaier, at least not without seeing Greene's own words, but I will say that you may as well think of vacuum fluctuations in string theory, as working in much the same way as they do in field theory, but also with some extra space-time uncertainty due to quantum gravity (metric fluctuations). All the QFT phenomena like vacuum polarization, Casimir effect, Schwinger pair production also occur in string theory, but with extra stringy details.

But there is no time-energy uncertainty principle... Greene's own words (in the book) state that string/anti-string pairs can "borrow energy from the universe... so long as they annihilate one another with sufficient haste." If we're looking to QFT, then this is not what happens (see @A. Neumaier)

Basically, is a virtual string a term in a calculation, or (unlike in QFT), something else?

Urs Schreiber
Gold Member
To amplify a point that may be missed in what has been said above:

As far as we restrict to perturbative string theory then we know precisely what a vacuum state is: It the choice of worldseet 2d SCFT.

This is by comparison with perturbative QFT: The information which tuns formal products of field observables such as ##T\mathbf{\Phi}^a(x)\mathbf{\Phi}^b(y)## into n-point functions is the choice of vacuum state, which allows us to evaluate this product to the propagator ##\langle T\mathbf{\Phi}^a(x)\mathbf{\Phi}^b(y) \rangle##.

Now in perturbative string theory, by definition, the information that turns specification of in/out states into actual scattering amplitudes, that's the worldsheet 2dSCFT with which the string perturbation series is computed.

Now, it's hard to build full 2dSCFTs. Therefore the usual approach is to write down a classical 2d susy sigma-model and quantize perturbatively. To lowest order that sigma model will force a solution to the corresponding supergravity equations of motion, to higher orders it will impose a sequence of additional effects and constraints.

That notorious "landcape of string theory vacua" is the result of rough approximation to what these consistent sigma-model backgrounds could be.

(string theory FAQ: What is a string vacuum?, deserves to be expanded, but not tonight,)

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dextercioby
Thanks @Urs Schreiber. Any thoughts on the nature of virtual strings?

Urs Schreiber
Gold Member
I would surmise that to call anything a 'vacuum,' it would have to be devoid of particles
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One wants to be a bit more sophisticated here in order for the concept to apply also to curved backgrounds, where global particle number need not well defined

In perturbative QFT the vacuum states, ##\langle ...\rangle##, or their closest analogs on curved spacetimes, are characterized by the properties of the 2-point functions which they induce by

$$\Delta^{a b}_H(x,y) = \langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y) \rangle$$

The condition is that these 2-point functions be Hadamard distributions. With these, one may show, causal perturbation theory works and hence the S-matrix (the Feynman perturbation series ) may be constructed by renormalization.

Now for perturbative string theory the reasoning is the other way around, because perturbative string theory by definition is an "S-matrix theory", meaning that it is defined right away by a formula that produces something that looks like a renormalized S-matrix (namely the string perturbation series). So now a vacuum for string theory is whatever makes the string perturbation series be defined on given asymptotic scattring states And that information is precisely that of a 2d superconfromal field theory of central charge 15 defined on all genera (because the string perturbation series by definition is the sum over the correlators of such a 2d SCFT over genera).

dextercioby
Going further - the only reference I can find to "virtual string pairs" being produced by borrowing energy from the vacuum is Greene's book (Elegant Universe). I'd really like to get a better handle on this - in several places the loops (sum of string histoires) are referred to a quantum corrections, but without any resort to 'borrowing energy from the vacuum'.

Is there any reason to this of the string case as different from the QFT case? (as I noted above - calculations in a series only) Even if someone could point to a reference it would be very helpful.

haushofer
I see no reason why string theory would differ from other QFT's concerning the ontological status of virtual particles/strings/... I would forget about the "borrowing energy from the vacuum" interpretation. Personally I've never seen a satisfying explanation of it. It merely serves to sell people a feeling of understanding in a very cheap way.

Urs Schreiber
Thanks @haushofer! Exactly my reasoning (based, as I noted, on @A. Neumaier's articles here, and whole set of other sources that don't even mention virtual processes in this way).

I'm certainly happy to entertain other explanations... if someone wants to chime in.

haushofer
Maybe this analogy adds a bit to your understanding. We can express the number 1 as the following infinite series:

##1 = \sum_{n=1}^{\infty} \frac{1}{2^n} \ \ \ \ \ \ \ \ (1)##

This is just a reformulation of a familiar number as an infinite series. Now, some people take this too literally. They say that if you want to walk 1 meter, this series tells you you have to take an infinite amount of steps! How could that be? Well, the sane person would tell you that this is because you choose to divide the 1 meter in this case into an infinite amount of steps. And infinity is not a concept we encounter in every day life; it's just a mathematical concept. How about taking just one step?

In QFT's we encounter amplitudes to calculate probabilities which we cannot calculate exactly (in most cases). The strategy is to rewrite it as an infinite series, and approximate by calculating just the first few terms and leaving out the rest. These intermediate terms in your series show themselves as (or:contain) "virtual particles". They are just a remnant of being too stupid to calculate stuff in an exact way. If we could do it, every "virtual particle" would disappear, just as every term in our series (1) would disappear if we rewrite it as simply 1.

Urs Schreiber
@haushofer Right, I'm familiar with virtual particles as propagators in Feynman diagrams. My question, which you answered above, was whether virtual particles somehow have a different ontological status in string theory, because, e.g., the vacuum is defined differently. A number of folks here have a good grasp of of string backgrounds, etc., - I would love to hear their comments on this (if any) in addition to your own (which I appreciate).

Urs Schreiber
Gold Member
@haushofer Right, I'm familiar with virtual particles as propagators in Feynman diagrams. My question, which you answered above, was whether virtual particles somehow have a different ontological status in string theory, because, e.g., the vacuum is defined differently. A number of folks here have a good grasp of of string backgrounds, etc., - I would love to hear their comments on this (if any) in addition to your own (which I appreciate).
I wasn't sure what you were after. But if you are happy with virtual particles as "being" whatever corresponds to the edges in a Feynman diagram, then the answer is right in front of you: By definition, perturbative string theory is a modification of the Feynman diagram rules, where Feynman graphs with their vertices and edges are replaced by surfaces, which you may roughly think of a being sued together from cylinders/strips (replacing the edges) and 3-holed spheres/3-punctured disks (replacing the vertices) and to be thought of as string worldsheets. So if a vitual particle before was whatever corresponds to an edge in a Feynman diagram, so a virtual string is now what happens along such a cylinder.

Only that (and this is part of what makes string theory good and interesting) there is no invariant decompositing of surfaces into cyliners/strips and 3-holed spheres/3-punctured disks, so in a sense the virtual thing concept for strings gets more virtual, still.

So in a sense all that is stringy about string theory rests on "virtual strings". The reason that people don't speak this way very much is that it's not particularly contentful. Already the "virtual particle" imagery in QFT is not particularly contentful.

The real point to take away from all this is this: Perturbative string theory by definition is a modification of the Feynman rule for computing the S-matrix. That means perturbative string theory is an S-matrix theory from the get go. There is a priori no spacetime in which one could count "string number", instead its all defined just by generalized Feynman diagrams, hence by "virtual" stuff, if you wish.

asimov42
@Urs Schreiber Thanks! Yes, I believe this is exactly what I was after. My main question was whether virtual strings had the same status as virtual particles in @A. Neumaier's FAQ ... appearing as terms in the computation of scattering amplitudes, but calculation devices rather 'physical entities', per se. Indeed, I agree with the use of the word 'imagery' with respect to virtual particles in QFT.

@Urs Schreiber Is there a reason to be unhappy with virtual particles being edges in Feynman diagrams? My understanding was that this is in fact the definition... Quoting @A. Neumaier,

"Virtual particles are defined as (intuitive imagery for) internal lines in a Feynman diagram (Peskin/Schroeder, p.5, or Zeidler, QFT I Basics in mathematics and physics, p.844). They are frequently used by professionals to illustrate processes in quantum field theory, and as a very useful shorthand language for complicated multivariate integrals over internal (real, but off-shell) momenta."

Reference https://www.physicsforums.com/insights/physics-virtual-particles/

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A. Neumaier
Virtual strings (and the assocoated borrowing stuff) are a loose - and if taken too serious misleading - terminology for objects arising in formal perturbation theory. The latter objects make sense, and the talk is just there to bring it alive in a nonexisting virtual reality for nonexperts.

@A. Neumaier are you referring to terms in integrals, as in my quote (from you) above, or something else? When you say "object," are you referring to a mathematical representation? (i.e., doesn't the string perturbative expansion follow the same form, essentially, as the QFT perturbative expansion?)

Urs Schreiber
Gold Member
@Urs Schreiber Is there a reason to be unhappy with virtual particles being edges in Feynman diagrams? My understanding was that this is in fact the definition... Quoting @A. Neumaier,
Yes, I just meant if you are in fact after this technical definition, instead of the poetry of "borrowing energy from the vacuum", then it's easy to see what the stringy analog is.

asimov42
Thanks @Urs Schreiber - I'm assuming when @A. Neumaier refers to "objects" above, he's referring to terms in the perturbative expansion (i.e., virtual strings)?

Just want to make sure I'm getting my definitions correct...

Hi all - one more thing for my own clarity:

At present, if one uses the perturbative approach to calculate the S-matrix, incorporating the higher order (virtual string) processes, the solution diverges, is this correct? Presumably (hopefully!) then, a non-perturbative approach would suppress contributions from virtual processes (loops), leading to a finite result for string amplitudes. Is this essentially correct?

Or is there some other known method to ensure the amplitudes are not infinite (thinking renormalization QFT, but that doesn't apply here or course)?

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And one last last thing: in string Feynman diagrams, edges are replaced by cylinders, which join, and then split (the result of an interaction). I presume one should not take this as being imagery for the real interaction ... that is, if I do an electron-electron scattering experiment, would I expect the 'electron' strings to actually join and then split? (e.g., why would the strings ever come that 'close'... although I realize we're dealing with QM). Standard Feynman diagram are not representative of interactions in this way, so I just wanted to be sure...

Final question (perhaps I should move these last three to a new thread): if one were to develop a non-perturbative approach to string theory, could one then dispense with virtual strings (as part of virtual processes), just as in non-perturbative versions of QFT?

My difficulty is that I'm not clear on the role that perturbative calculations with virtual strings have on the outcome of real string scattering experiments. Here, there is no renormalization involved - you still have a perturbative calculation... but the situation appears different from QFT, and I'm not sure that it is. That is, you have a calculation involving a 'virtual' worldsheet - purely a calculation device?

Thanks again everyone - if someone has a pointer to a clarifying article/text, that would be great too. As @Urs Schreiber mentioned previously, virtual strings don't seem to be a particularly contentful topic, so it's hard to find resources that make things clear...

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A. Neumaier