# Difference between open string field theory and closed string field theory

meteor
What's the difference between open string field theory and closed string field theory?

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## Answers and Replies

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Originally posted by meteor
What's the difference between open string field theory and closed string field theory?

Absolutely none. Open strings can become closd by joining their ends. Closed strings can become open by breaking at some point. Strings can join to make things like figure eights and more, and they can part just as easily. The constant mutation of the strings in all kinds of topologies is the string theory equivalent of the interaction of particles.

Afterthought. It occurs to me that some people may not know what open and closed strings are. An open striing is analogous to a simple length of ordinary strin with two end points. A closed string is a loop - a rubber band provides the most familiar example. No end points.

Originally posted by selfAdjoint
Absolutely none. Open strings can become closd by joining their ends. Closed strings can become open by breaking at some point. Strings can join to make things like figure eights and more, and they can part just as easily. The constant mutation of the strings in all kinds of topologies is the string theory equivalent of the interaction of particles.

meteor was asking about string field theory, not ordinary ST. Purely closed string field theories can in fact be formulated. The case for purely open string field theories is a bit trickier because although no closed string fields appear explicitly in open string field theories their amplitudes are included, except that surfaces without boundary to which open strings cannot couple are not generated.

Originally posted by sol
U(1)=photon

If photon starts out as a one dimensional string(?) can we consider this longitudal, and be used as a determination of the nergy contained in the photons length?

Consider the state |0;k> of a bosonic string (remember that photons are bosons since they have integer spin) with momentum k, but with no internal excitations, i.e. it's not vibrating. The photon is obtained by exciting one internal degree of freedom by applying a creation operator to our ground state |0;k> yielding

&alpha;i-1|0;k>.

This is analogous to the simple harmonic oscillator one studies in introductory courses in quantum mechanics. The subscript "-1" of &alpha;i-1 indicates that the excitation vibrates at a certain "frequency". We say that an n = 1 mode has been excited/created, with the minus sign in front of the 1 indicating that the mode is being created rather than destroyed. The superscript "i" on &alpha;i-1 takes values i = 1,...,24 and indicates that &alpha;i-1|0;k> transforms under the 24-dimensional rotational subgroup SO(24) of the 26-dimensional lorentz group SO(25,1). (More generally, in D-dimensional spacetime, massive and massless particles - like the photon in the latter case - form representations of SO(D-1) and SO(D-2) respectively).

The mass of a bosonic string state (in 26-dimensional spacetime) is given by

&alpha;&prime;m2 = N - 1

in which N counts the number of modes that have been excited. In our case, just a single mode, the n = 1 mode, has been excited. Thus m = 0 for the photon and the only contribution to it's energy comes from it's momentum k, as usual for massless particles.

Now, by the word "longitudinal" I'm guessing you meant along the length of the string. However, this word is used to describe the polarization of massive particles. In our case the particle is massless and is thus transversly polarized meaning, as already mentioned that i = 1,...,24. For a massive vector boson we have that i = 1,...,25 so that massive particles have an extra polarization.

Finally, to move beyond U(1) to non-abelian U(n) gauge symmetry, we need to introduce Chan-Paton factors.

Originally posted by sol
I apologize if this seems like a stupid question.

What I find stupid are the people here who've taken up LQG as a religion and criticize ST without knowing the first damn thing about it. They don't understand that LQG is popular only in these forums, the reason being that string theory is simply too difficult for them to really wrap their minds around. These people try to justify their deference towards LQG in other terms, but really, they're full of sh*t, especially when they criticize strings. High energy theorists simply don't pay much attention to LQG. Just compare the tens of thousands of papers published in string theory to the hundreds in LQG. Also, what little they do know about ST comes not from string theorists, but from LQG people with an agenda to push and who intentionally misrepresent ST in the process. On the other hand, LQG is simple and interesting, so to the extent that it creates interest in learning physics - and there's plenty of neat physics in LQG - I wouldn't discourage people from studying LQG.

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Well that's a bit harsh (but see Lubos Motl's takedown of LQG on yesterday's s.p.r).

I followed the superstring 101 and 102 online seminars (I guess you could call them) given through the www.superstringtheory.com site and worked through about half of Polchinski's book in that context. I am not against stringy solutions to the problem of gravity, but I do say that so far they haven't made closure, which I define as a single theory that produces the real predictions of GR and the real predictions of the standard model. It's not enough to generate some U(1) x SU(2) x SU(3) generic theory.

BTW do you know of an online tutorial for string field theory? Trying to understand it from research papers is a bummer.

Originally posted by selfAdjoint
Well that's a bit harsh

Yeah, I just got "pm'ed" by greg about that.

Originally posted by selfAdjoint
I define as a single theory that produces the real predictions of GR

ST contains GR, and thus for example predicts the existence of spacetime, but whether or not LQG contains GR - indeed, whether it has a classical limit at all - is unknown. It's very hard to believe the classical structure of GR would remain a valid basis for quantization all the way down to the Planck length and that obtaining a correct QGT requires that spacetime be butchered - cleaving it into a spatial and "temporal" parts - in order to quantize. In explicitly spacetime covariant theories like ST, time isn't the problem it is in LQG. LQG people say that GR teaches that any QGT must be relational, but I think it also suggests that time and space should be treated on an equal footing.

There's another level to this that I don't think is appreciated. It's not only whether a prospective QGT contains GR, it's also how GR got there. LQG was specifically "contrived" - and that is the right word - to be a quantum theory of GR, so if GR actually ever shows up in LQG, how surprised should we be? After all, that was the whole point. Yet GR is still nowhere to be found in LQG and options are running out (the fact that they've gone to an algebraic approach signals this. Btw, I got your message. The basic reason that the algebraic method is being explored in LQG is that it's representation theory will allow the kinematical sector of phase space, i.e. solutions of the gauss and diffeomorphism constraints that don't satisfy the hamiltonian constraint, to be studied in a more systematic way. I think I posted this somewhere.)

This and all other attempts to find a self-consistent or reasonable theory containing the graviton failed, that is up until it was realized strings contain gravitons. But it's not just that, it's that it's impossible to formulate a consistent string theory without the graviton showing up, even if you begin with flat spacetime! Remarkable! Also, the way GR shows up seems equally miraculous: It appears as a constraint needed to ensure consistancy of the world-sheet theory! Wow, 2D conformal invariance requires spacetime diffeomorphism invariance? Holy sh*t! Thanks for letting us in on that little secret lord. The psychological impact of this on researchers in this field was and remains immense.

Originally posted by selfAdjoint
It's not enough to generate some U(1) x SU(2) x SU(3) generic theory.

Yeah, but you've got to start somewhere and have evidence and arguments - which string theorists do - that it need not lead to a dead end. String theory - M-theory - is innoculated against sudden death by it's incredible depth and robustness: It's understood that we're still only in the early stages of understanding a theory of which we've only had small glimpses of so that there's every reason to believe that the solutions to current limitations of ST will eventually be found. By comparison, LQG is extremely narrow and the resulting fragility of it makes virtually any little problem a cause for immediate concern, and the big one's a cause for the wide-spread skepticism with which theorists view LQG.

Originally posted by selfAdjoint
do you know of an online tutorial for string field theory?

No, and I wouldn't hold my breath waiting for a good one.

Originally posted by selfAdjoint
Trying to understand it from research papers is a bummer.

Trying to learn strings on your own is a b*tch no matter how you do it.

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I want to say this gently, because I don't consider myself a fanatic for either side, but it does seem to me that you apply a different standard of success to strings than to QGR. You say they haven't got GR, but they do have approximations, and you say strings contain GR which is false. Strings contain the graviton which is a spin 2 boson and by prior theory a spin 2 boson will couple to matter like the einstein tensor. But that is not GR. GR is a generally covariant, background independent theory, and string theory is not. The point of background independence isn't just a barb, it's the nub of the whole contention.

People say string field theory, in at least some of its forms, is background free, but I haven't come upon a real refer3nce to this.

Originally posted by selfAdjoint
I want to say this gently..

There's no need to be gentle as long as you're talking about physics. I'm not gentle with anyone when it comes to physics, unless they turn out to be psychotic or something.

Originally posted by selfAdjoint
You say they haven't got GR, but they do have approximations

For example? The approximations they're looking for are semiclassical ones, but they've yet to find any. Maybe you're referring to something in particular?

Originally posted by selfAdjoint
Strings contain the graviton which is a spin 2 boson and by prior theory a spin 2 boson will couple to matter like the einstein tensor.

Massless spin-2, or more simply, helicity-2 particles are not necessarily gravitons. In order to incorporate the usual inverse-square law of gravitational interactions we need to introduce a field h&mu;&nu; that transforms as a symmetric tensor up to gauge transformations of the sort associated in general relativity with diffeomorphisms. Thus in order to construct a theory of massless particles of helicity&plusmn;2 that incorporates long-range interactions, it is necessary for it to have a symmetry something like general covariance. This is achieved by coupling the field to a conserved current &theta;&mu;&nu;, i.e. &part;&mu;&theta;&mu;&nu; = 0. The only such current is the energy-momentum tensor, aside from possible total derivative terms that don't effect the long-range behaviour of the force produced. Incorporating in a quantum theory a helicity-2 particle something like h&mu;&nu; has been impossible until strings, which as I said require it.

Originally posted by selfAdjoint
GR is a generally covariant, background independent theory, and string theory is not. The point of background independence isn't just a barb, it's the nub of the whole contention. and you say strings contain GR which is false.

In the sense that the entire LQG program is predicated on these ideas, yes it is. But string theorists don't do LQG. Consider the following equation which appears in the bosonic theory as a consistency requirement on the world-sheet theory, namely that it be weyl-invariant

&alpha;&prime;R&mu;&nu; + 2&alpha;&prime;&nabla;&mu;&nabla;&nu;&Phi; - (&alpha;&prime;/4)H&mu;&lambda;&omega;H&nu;&lambda;&omega; = 0.

This is einstein's equation with sources the antisymmetric tensor field and dilaton that arise in string theory. This is a good example of what string theorists mean when they say that strings contain GR. What it means is that strings can propagate consistently only in a background that satisfies appropriate field equations (this bothers me more than anything else in ST). It also means the helicity-2 excitation that occurs in ST is in fact the graviton, and this is what string people mean when they say that strings are a genuine QGT. LQG people know this, and now so do you. On the other hand, whether LQG contains the graviton and hence whether it's a QGT is anybody's guess.

Also, as you know, the full supersymmetric theory reduces in various low energy limits to a variety of supergravity theories.

Originally posted by selfAdjoint
People say string field theory, in at least some of its forms, is background free, but I haven't come upon a real refer3nce to this.

You haven't come across a reference because it's not true.

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Originally posted by sol
You would be scare of my mystical approach then

What do you mean by mystical?

Originally posted by sol
Just know, that I endeavor to understand and do not want to be criticized for my thinking outside of this issue:0)

Understood. My rant wasn't directed at anyone in particular, and certainly not you since I've had no exchanges with you up till now. I'm sorry if I bothered you.

Originally posted by sol
Leading from the length( amplitude?) the transverse understanding of information is understood in the brane? Does this contradict what you have said

I'm having trouble decoding this. Maybe you want a general explanation of D-branes?

Originally posted by sol
could you explain SU(2)

Could you be more specific?

Originally posted by sol
In the undertanding I am developing...

By this do you mean that you're building your own theory or that you're simply trying to learn some physics?

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Jeff, can you expand some detail on how this:

Consider the following equation which appears in the bosonic theory as a consistency requirement on the world-sheet theory, namely that it be weyl-invariant

ƒ¿ŒRƒÊƒË + 2ƒ¿ŒÞƒÊÞƒËƒ³ - (ƒ¿Œ/4)HƒÊƒÉƒÖHƒËƒÉƒÖ = 0.

This is einstein's equation with sources the antisymmetric tensor field and dilaton that arise in string theory. This is a good example of what string theorists mean when they say that strings contain GR. What it means is that strings can propagate consistently only in a background that satisfies appropriate field equations (this bothers me more than anything else in ST).

constrains the background? (Sorry about what copy did to the notation). In general I have problems understanding how definitions on the worldsheet work out in terms of background propagation.

Originally posted by selfAdjoint
Jeff, can you expand some detail on how this:

Consider the following equation which appears in the bosonic theory as a consistency requirement on the world-sheet theory, namely that it be weyl-invariant

ƒ¿?ŒRƒÊƒË + 2ƒ¿?Œ?ÞƒÊ?ÞƒËƒ³ - (ƒ¿?Œ/4)HƒÊƒÉƒÖHƒËƒÉƒÖ = 0.

This is einstein's equation with sources the antisymmetric tensor field and dilaton that arise in string theory. This is a good example of what string theorists mean when they say that strings contain GR. What it means is that strings can propagate consistently only in a background that satisfies appropriate field equations (this bothers me more than anything else in ST).

constrains the background? (Sorry about what copy did to the notation). In general I have problems understanding how definitions on the worldsheet work out in terms of background propagation.

This is going to take quite a bit of typing. I'll get to this sometime this week. Sorry for the delay.

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Thanks for your consideration, Jeff. I really appreciate the effort you put in.

helicity-2 string excitations are gravitons

This is the most direct approach I could come up with:

Transition amplitudes in ST are defined in a 1st quantized formalism based on the world-sheet action

SG = - (1/4&pi;&alpha;&prime;) &int; d&mu;&gamma;&gamma;abG&mu;&nu;(X)&part;aX&mu;&part;bX&nu;

in which the basic fields X&mu; of the theory embed the world-sheet with metric &gamma;ab and measure d&mu;&gamma; in a background spacetime with metric G&mu;&nu;. Recall that in QFT the tree level feynman diagram for an interaction consists of a vertex where legs representing incoming and outgoing particles meet. Analogously, for closed strings we have a sphere with punctures to which are glued the ends of "world-tubes" representing incoming or outgoing strings. The invariance, known as weyl-invariance, of SG under rescalings &gamma;ab &rarr; e&phi;&gamma;ab of the world-sheet metric allows the projection (continuous deformation) of world-tubes onto the punctures, effectively sealing each one by insertion of a point sitting at which is a vertex operator defined in terms of X&mu; and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as the state-operator correspondence, an example of which is given at the end of this post. Higher order interactions are obtained as compact oriented boundaryless surfaces of genus g with a vertex operator insertion Vi(ki) for each incoming/outgoing closed string of momentum ki. Hence, amplitudes for n external string states have the form of a sum of path-integrals with insertions

<V1(k1)&sdot;&sdot;&sdot;Vn(kn)> ~ &sum;g=0,1,2,... &int;g D&gamma;abDX&mu; V1(k1)&sdot;&sdot;&sdot;Vn(kn)e-SG.

Now, take

G&mu;&nu;(X) = &eta;&mu;&nu; + &epsilon;&mu;&nu;(X)

with

&epsilon;&mu;&nu;(X) = &int; d26k &epsilon;&mu;&nu;(k)eik&sdot;X

everywhere small compared to &eta;&mu;&nu;. Then

e-SG = e-(S&eta; + S&epsilon;) = e-S&eta; &sum;n=0,1,...(- 4&pi;&alpha;&prime;)-n(1/n!) &int; d26k1&sdot;&sdot;&sdot;d26kn V(k1)&sdot;&sdot;&sdot;V(kn)

in which

V(k) &equiv; &epsilon;&mu;&nu;(k)V&mu;&nu;(k) &equiv; &epsilon;&mu;&nu;(k) &int; d&mu;&gamma; &gamma;ab&part;aX&mu;&part;bX&nu;eik&sdot;X

is a vertex operator coupling strings to fluctuations in the background metric G&mu;&nu;. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point. Next, observe that &epsilon;&mu;&nu; picks out the symmetric part of V&mu;&nu;, so V is the vertex operator of a spin-2 state. Also, since the state-operator correspondence (see the example at the end of this post) requires that vertex operators transform like the string state vectors they represent, they must include the factor eik&sdot;X to transform properly under spacetime translations X&mu; &rarr; X&mu; + a&mu;. Now, any insertion must respect the local weyl symmetry of the theory. In particular, demanding that V be weyl-invariant requires (see polchinski I Chap 3.6)

k2 = k2&epsilon;&mu;&nu;(k) = 0 &harr; &uArr;&epsilon;&mu;&nu;(X) = &uArr;G&mu;&nu;(X) = 0

k&mu;&epsilon;&mu;&nu;(k) = 0 &harr; &part;&mu;&epsilon;&mu;&nu;(X) = &part;&mu;G&mu;&nu;(X) = 0,

&epsilon;&mu;&mu;(k) = 0 &harr; &epsilon;&mu;&mu;(X) = 0.

In addition to showing that the spin-2 excitations are massless, because the ricci tensor R&mu;&nu; satisfies

R&mu;&nu; &prop; &part;&mu;&part;&nu;&epsilon;&lambda;&lambda; - 2&part;&lambda;&part;(&mu;&epsilon;&mu;)&lambda; + &uArr;&epsilon;&mu;&nu; + O(&epsilon;2),

this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G&mu;&nu; satisfy the vacuum einstein equations R&mu;&nu; = 0.

Because massless states are transversally polarized, V must be invariant under the shift

&epsilon;&mu;&nu;(k) &rarr; &epsilon;&mu;&nu;(k) + k&mu;&xi;&nu; + k&nu;&xi;&mu;

by longitudinal polarizations. In terms of the metric, this gauge-invariance

&epsilon;&mu;&nu;(X) &rarr; &epsilon;&mu;&nu;(X) + k&mu;&xi;&nu;(X) + k&nu;&xi;&mu;(X)

is an infinitesimal diffeomorphism generated by the vector field &xi;&mu;(X) in the approximation where O(&epsilon;2) terms are neglected and under which R&mu;&nu; = 0 is invariant. In fact R&mu;&nu; = 0 is the only spacetime diffeo-invariant equation that reduces to &uArr;G&mu;&nu;(X) = 0 in the linearized limit.

In sum, weyl-invariance requires spin-2 excitations be massless and couple in a gauge-invariant way, that is, it requires the general covariance of GR, justifying the interpretation of helicity-2 excitations as gravitons.

State-operator correspondence for the graviton vertex operator:

Define world-sheet coordinates

z = e-i&sigma; + &tau; , z* = ei&sigma; + &tau;

with &sigma; = &sigma; + 2&pi; the periodic coordinate along the string and &tau; the time coordinate on the world-sheet. We then have

V &prop; &epsilon;&mu;&nu;&int;d2z &part;zX&mu;(z)&part;z*X&nu;(z*)eik&sdot;X(z,z*)

in which we've taken the world-sheet metric in "conformal gauge" so that it effectively drops out. Then (up to proportionality) the state-operator correspondence is

&part;zX&mu;(0) &harr; &alpha;-1&mu; , &part;z*X&mu;(0) &harr; (&alpha;-1&mu;)* , eik&sdot;X(0,0) &harr; |0;0;k>

where &alpha;-1&mu; and (&alpha;-1&mu;)* excite left- and right-moving n = 1 modes.

Putting these together gives

V &harr; &epsilon;&mu;&nu;&alpha;-1&mu;(&alpha;-1&nu;)*|0;0;k>.

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lethe

Originally posted by jeff

Finally, to move beyond U(1) to non-abelian U(n) gauge symmetry, we need to introduce Chan-Paton factors.

jeff-

sorry we re so demanding, but i wonder if i could get a quick and dirty explanation of what a Chan-Paton factor is? where does it come from?

plus:

Originally posted by jeff
There's no need to be gentle as long as you're talking about physics. I'm not gentle with anyone when it comes to physics, unless they turn out to be psychotic or something.

that was awesome.

lethe
Originally posted by selfAdjoint
(but see Lubos Motl's takedown of LQG on yesterday's s.p.r)

i couldn t find this thread. do you remember the title?

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Jeff - thanks much for the explanation. You know I actually understood it? But I'm going to print it and study it some more because it makes some of the things (operator-product correspondence for one) a lot clearer.

Lethe - Motl made the comments in a discussion of John Baez's latest "This Week's Finds" which was posted at sci.physics.reaearch the day my post appeared. Do an in-site search on Motl and you should be able to find it.

lethe
Originally posted by selfAdjoint
Lethe - Motl made the comments in a discussion of John Baez's latest "This Week's Finds" which was posted at sci.physics.reaearch the day my post appeared. Do an in-site search on Motl and you should be able to find it.

ouch, that was tough. i can t wait to see Baez' reply.

Originally posted by selfAdjoint
...I'm going to print it and study it some more because it makes some of the things (operator-product correspondence for one) a lot clearer.

I made an edit which bears on the state-operator correspondence:

I originally posted

"Next, observe that V&mu;&nu; has two spacetime indices in which it's symmetric so that the excitation is spin-2 under lorentz transformations."

What I meant to say was that only V&mu;&nu;'s symmetric part contributes to V.

Thus the edit...

"Next, observe that &epsilon;&mu;&nu; picks out the symmetric part of V&mu;&nu;, so V is the vertex operator of a spin-2 state."

I also illustrate at the end of the post the state-operator correspondence for the graviton.

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chan-paton factors

Originally posted by lethe
...quick and dirty explanation of what a Chan-Paton factor is?

The quick and dirty explanation is that chan-paton factors relate invariant mass and orientation of open strings to global symmetries in the world-sheet theory that are promoted to local symmetries in spacetime corresponding in the low energy limit to yang-mills interactions of particles. In more detail...

In quantum systems it's natural to assign to distinguished points nondynamical internal degrees of freedom giving rise to global symmetries, i.e. conserved charges that don't contribute to the hamiltonian or ruin any pre-existing symmetry.

In string theory we can charge the endpoints of open strings with chan-paton degrees of freedom. Although these charges may interact with external gauge fields, during interactions of several open strings (open strings interact at their endpoints), they flow only along the world-sheet boundaries swept out by the endpoints and so are conserved. Since chan-paton charges have trivial world-sheet dynamics - that is, world-sheet interactions can't change them - consistency requires that only identically charged endpoints may interact.

If the chan-paton degrees of freedom in the 2D quantum conformal field theory governing the physics of open strings are to be useful, they must produce in the low energy limit the familiar yang-mills interactions of particles as they appear in the arena of QFT. This requires that the chan-paton state of one endpoint of each open string lie in the fundamental representation N and the other in it's complex conjugate &Ntilde; of some group G(N) which must be a compact semisimple lie group up to the possible inclusion of additional U(1) factors.

Let |m&ntilde;> with m,&ntilde; = 1,...,N&sup2; denote a complete set of possible but not necessarily physically realized states of endpoint pairs. The part |a> that is physical depends on the mass level of states and orientation of strings (explained below) and may be related to |m&ntilde;> by

(1) |a> = &sum;m&ntilde; |m&ntilde;>(&lambda;a)m&ntilde;

in which the NxN matrices &lambda;a are chan-paton factors. The QCFT transition amplitudes will include a factor Tr(&lambda;a1&lambda;a2&sdot;&sdot;&sdot;) containing one chan-paton factor for each point of interaction on the world-sheet. Consistent with the above, the conformal energy-momentum tensor has no dependence on the chan-paton factors, and conformal and poincare invariance are automatic since the chan-paton factors carry only internal indices and are invariant (i.e. they're matrices whose elements are constants).

Looking at just the bosonic sector, unitarity requires that G(N) = U(N), SO(N), or USp(N) where in this last case N is even. Since for U(N), N is complex and so not equal to &Ntilde;, the two ends of a string are different so these strings must be oriented. On the other hand, the fundamental reps of the other two are real so that there's no real difference between endpoints. Hence these strings must be treated as unoriented which requires their quantum wave functions &psi; satisfy &psi;(&sigma;) = &psi;(&pi; - &sigma;) in which &sigma; runs from 0 to &pi; along the string.

The chan-paton states of any oriented open string fill out the adjoint representation N&otimes;&Ntilde; of U(N), so all of the |m&ntilde;> are realized. In this case the chan-paton factors are the N&sup2; NxN hermitian representation matrices (&lambda;a)m&ntilde; of U(N) with a = 1,...,N&sup2; and normalization Tr(&lambda;a&lambda;b) = &delta;ab. On the world sheet the chan-paton factors transform under the global U(N) symmetry as &lambda;a &rarr; &lambda;&prime;a = U&lambda;aU&dagger; for U &isin; U(N). But consider the massless vector states &alpha;-1&mu;|k;a> obtained by exciting a single n = 1 vibrational mode of an open string having momentum k. It's an odd mass level state because it's the internal modes of vibration that determine a string's invariant mass, and only one - an odd number - has been excited. Using the state-operator correspondence (a brief explanation of which is provided at the end of this post)

&alpha;-1&mu; &harr; &part;X&mu; , |k;a> &harr; &lambda;aeik&sdot;X

beween states and vertex operators on the world-sheet yields the correspondence

&alpha;-1&mu;|k;a> &harr; Va&mu;(X) &equiv; &lambda;a&part;X&mu;eik&sdot;X.

Hence &alpha;-1&mu;|k;a>, since it's a massless vector, may be interpreted as a gauge boson corresponding to a local U(N) symmetry in spacetime since we can choose a different U(N) rotation U(X) at each spacetime point X&mu; under which Va&mu;(X) &rarr; V&prime;a&mu;(X) = U(X)Va&mu;(X)U&dagger;(X) = U(X)&lambda;aU&dagger;(X)&part;X&mu;eik&sdot;X. The needed gauge-invariant expressions in the spacetime action giving the QFT amplitudes to which the QCFT amplitudes must reduce in the low energy limit are produced precisely by the chan-paton factors. This shouldn't be suprising since massless vector mesons in consistent interacting theories always transform in the adjoint of the gauge group. However, this promotion of global symmetries on the world-sheet to local symmetries in spacetime is a generic feature of string theory, and so holds also in the unoriented case, to which we now briefly turn.

As just mentioned, the massless vector states of unoriented open strings are, like their oriented counterparts, gauge bosons and so - along with all even level states, as it turns out - are in the adjoint representations of SO(N) or USp(N), these being the antisymmetric and symmetric parts of N&otimes;N respectively. Then for these states, the SO(N) chan-paton factors are NxN antisymmetric matrices which thus pick out the antisymmetric part of |m&ntilde;> in the sense of equation (1) so that a = 1,..,N(N - 1)/2. Similarly, for USp(N) the chan-paton factors are symmetric so that a = 1,...,N(N + 1)/2. At odd mass levels the SO(N) and USp(N) chan-paton factors are instead symmetric and antisymmetric respectively.

In the supersymmetric case, it turns out that only unoriented open strings with gauge group SO(32) are allowed. For closed strings there's another mechanism that gives rise to gauge bosons, and it allows other groups.

The state-operator correspondence:

String theory transition amplitudes are defined in a 1st quantized formalism based on the world-sheet action

SG = - (1/4&pi;&alpha;&prime;) &int; d&mu;&gamma;&gamma;abG&mu;&nu;(X)&part;aX&mu;&part;bX&nu;

in which the basic fields X&mu; of the theory embed the world-sheet with metric &gamma;ab and measure d&mu;&gamma; in a background spacetime with metric G&mu;&nu;. Recall that in QFT the tree level feynman diagram for an interaction consists of a vertex where legs representing incoming and outgoing particles meet. Analogously, for closed strings we have a sphere with punctures to which are glued the ends of "world-tubes" representing incoming or outgoing strings. The invariance, known as weyl-invariance, of SG under rescalings &gamma;ab &rarr; e&phi;&gamma;ab of the world-sheet metric allows the projection (conformal deformation) of world-tubes onto the punctures, effectively sealing each one by insertion of a point sitting at which is a vertex operator defined in terms of X&mu; and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as the state-operator correspondence.

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Originally posted by sol
Is it possible to have a generalization put in front of this equative formulation on a simpler level.

My point of departure is the view that if a theory contains no gravitons - the quantum of the gravitational field - it's not a quantum gravity theory. For example, it's unknown whether LQG contains gravitons (most believe it doesn't), so it shouldn't be advertised as a QGT, which around here it usually is.

On the other hand, string theory does contain gravitons, and is, from the above point of view, our only known QGT (though of course it's much more).

Now, selfAdjoint remarked that any theory having a massless spin-2 particle contains the graviton because that's what a graviton is, a massless a spin-2 particle, but just because string theory contains a massless spin-2 particle doesn't mean it contains GR. The point of my post was that this statement is untrue: A massless spin-2 particle isn't necessarily a graviton, and a theory containing a graviton in fact must imply GR. I then explained the remarkable way gravitons - and hence GR itself - is contained in string theory. In fact, it's impossible to formulate a consistent theory of interacting strings without the graviton appearing in the particle spectrum.

ranyart

Originally posted by jeff

This is the most direct approach I could come up with:

Transition amplitudes in ST are defined in a 1st quantized formalism based on the world-sheet action

SG = - (1/4&pi;&alpha;&prime;) &int; d&mu;&gamma;&gamma;abG&mu;&nu;(X)&part;aX&mu;&part;bX&nu;

in which the basic fields X&mu; of the theory embed the world-sheet with metric &gamma;ab and measure d&mu;&gamma; in a background spacetime with metric G&mu;&nu;. Recall that in QFT the tree level feynman diagram for an interaction consists of a vertex where legs representing incoming and outgoing particles meet. Analogously, for closed strings we have a sphere with punctures to which are glued the ends of "world-tubes" representing incoming or outgoing strings. The invariance, known as weyl-invariance, of SG under rescalings &gamma;ab &rarr; e&phi;&gamma;ab of the world-sheet metric allows the projection (continuous deformation) of world-tubes onto the punctures, effectively sealing each one by insertion of a point sitting at which is a vertex operator defined in terms of X&mu; and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as the state-operator correspondence, an example of which is given at the end of this post. Higher order interactions are obtained as compact oriented boundaryless surfaces of genus g with a vertex operator insertion Vi(ki) for each incoming/outgoing closed string of momentum ki. Hence, amplitudes for n external string states have the form of a sum of path-integrals with insertions

<V1(k1)&sdot;&sdot;&sdot;Vn(kn)> ~ &sum;g=0,1,2,... &int;g D&gamma;abDX&mu; V1(k1)&sdot;&sdot;&sdot;Vn(kn)e-SG.

Now, take

G&mu;&nu;(X) = &eta;&mu;&nu; + &epsilon;&mu;&nu;(X)

with

&epsilon;&mu;&nu;(X) = &int; d26k &epsilon;&mu;&nu;(k)eik&sdot;X

everywhere small compared to &eta;&mu;&nu;. Then

e-SG = e-(S&eta; + S&epsilon;) = e-S&eta; &sum;n=0,1,...(- 4&pi;&alpha;&prime;)-n(1/n!) &int; d26k1&sdot;&sdot;&sdot;d26kn V(k1)&sdot;&sdot;&sdot;V(kn)

in which

V(k) &equiv; &epsilon;&mu;&nu;(k)V&mu;&nu;(k) &equiv; &epsilon;&mu;&nu;(k) &int; d&mu;&gamma; &gamma;ab&part;aX&mu;&part;bX&nu;eik&sdot;X

is a vertex operator coupling strings to fluctuations in the background metric G&mu;&nu;. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point. Next, observe that &epsilon;&mu;&nu; picks out the symmetric part of V&mu;&nu;, so V is the vertex operator of a spin-2 state. Also, since the state-operator correspondence (see the example at the end of this post) requires that vertex operators transform like the string state vectors they represent, they must include the factor eik&sdot;X to transform properly under spacetime translations X&mu; &rarr; X&mu; + a&mu;. Now, any insertion must respect the local weyl symmetry of the theory. In particular, demanding that V be weyl-invariant requires (see polchinski I Chap 3.6)

k2 = k2&epsilon;&mu;&nu;(k) = 0 &harr; &uArr;&epsilon;&mu;&nu;(X) = &uArr;G&mu;&nu;(X) = 0

k&mu;&epsilon;&mu;&nu;(k) = 0 &harr; &part;&mu;&epsilon;&mu;&nu;(X) = &part;&mu;G&mu;&nu;(X) = 0,

&epsilon;&mu;&mu;(k) = 0 &harr; &epsilon;&mu;&mu;(X) = 0.

In addition to showing that the spin-2 excitations are massless, because the ricci tensor R&mu;&nu; satisfies

R&mu;&nu; &prop; &part;&mu;&part;&nu;&epsilon;&lambda;&lambda; - 2&part;&lambda;&part;(&mu;&epsilon;&mu;)&lambda; + &uArr;&epsilon;&mu;&nu; + O(&epsilon;2),

this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G&mu;&nu; satisfy the vacuum einstein equations R&mu;&nu; = 0.

Because massless states are transversally polarized, V must be invariant under the shift

&epsilon;&mu;&nu;(k) &rarr; &epsilon;&mu;&nu;(k) + k&mu;&xi;&nu; + k&nu;&xi;&mu;

by longitudinal polarizations. In terms of the metric, this gauge-invariance

&epsilon;&mu;&nu;(X) &rarr; &epsilon;&mu;&nu;(X) + k&mu;&xi;&nu;(X) + k&nu;&xi;&mu;(X)

is an infinitesimal diffeomorphism generated by the vector field &xi;&mu;(X) in the approximation where O(&epsilon;2) terms are neglected and under which R&mu;&nu; = 0 is invariant. In fact R&mu;&nu; = 0 is the only spacetime diffeo-invariant equation that reduces to &uArr;G&mu;&nu;(X) = 0 in the linearized limit.

In sum, weyl-invariance requires spin-2 excitations be massless and couple in a gauge-invariant way, that is, it requires the general covariance of GR, justifying the interpretation of helicity-2 excitations as gravitons.

State-operator correspondence for the graviton vertex operator:

Define world-sheet coordinates

z = e-i&sigma; + &tau; , z* = ei&sigma; + &tau;

with &sigma; = &sigma; + 2&pi; the periodic coordinate along the string and &tau; the time coordinate on the world-sheet. We then have

V &prop; &epsilon;&mu;&nu;&int;d2z &part;zX&mu;(z)&part;z*X&nu;(z*)eik&sdot;X(z,z*)

in which we've taken the world-sheet metric in "conformal gauge" so that it effectively drops out. Then (up to proportionality) the state-operator correspondence is

&part;zX&mu;(0) &harr; &alpha;-1&mu; , &part;z*X&mu;(0) &harr; (&alpha;-1&mu;)* , eik&sdot;X(0,0) &harr; |0;0;k>

where &alpha;-1&mu; and (&alpha;-1&mu;)* excite left- and right-moving n = 1 modes.

Putting these together gives

V &harr; &epsilon;&mu;&nu;&alpha;-1&mu;(&alpha;-1&nu;)*|0;0;k>.

Jeff, I detect as much panic in your posts as your peers, you are not alone!

Here is a link to a paper that has PANIC written (not witten!)all through it:http://uk.arxiv.org/abs/hep-th/0308055

As Aristotle said..urhmm I am off!

ranyart

Originally posted by ranyart
Jeff, I detect as much panic in your posts as your peers, you are not alone!

Here is a link to a paper that has PANIC written (not witten!)all through it:http://uk.arxiv.org/abs/hep-th/0308055

As Aristotle said..urhmm I am off!

Or for the unitiated:QUOTE;We argue that this may not be a real problem, given the large range of available fluxes and background geometries in string theory.

The interpretation of this Quote from the linked abstract?[?] [?]

ranyart

Originally posted by Tom
PANIC is an international conference on particles and nuclei.

I see that the reply to my post from jeff dissapeared? unless he re-routed it into a different 'sum-over-history'!

Anyway, I do not think Jeff got the hidden variable contained in the post? It actually relates to the 'Stanford crew', and I must say I agreed totally with his acclamation of the said Theorists, I for one have examined a vast number of their pre-print papers for some while, and I am in awae at their persistence in the devolpment of Inflaton field Evolution theory, if that's a correct term?

Question, what causes Strings? Nature or Theorists?

Originally posted by ranyart
I see that the reply to my post from jeff dissapeared? unless he re-routed it into a different 'sum-over-history'!

Anyway, I do not think Jeff got the hidden variable contained in the post? It actually relates to the 'Stanford crew', and I must say I agreed totally with his acclamation of the said Theorists, I for one have examined a vast number of their pre-print papers for some while, and I am in awae at their persistence in the devolpment of Inflaton field Evolution theory, if that's a correct term?

Question, what causes Strings? Nature or Theorists?

After Tom pointed out the panic thing I removed my moronic post. But thanks for the sentiments and sorry about that.

ranyart

Originally posted by jeff
After Tom pointed out the panic thing I removed my moronic post. But thanks for the sentiments and sorry about that.

No problem.

I mearly wanted to point out that Linde, is moving into the 'side' of the string camp, but I for one think his work has a great distance to cover and go (I believe he knows this), but if he gets results, then there are not many who can deny his and 'et al', team deserve credit.

My own take is on how we treat Time, and Length are the underlaying problem area's..these are two totally incompatable arena's.

Photons are themselves Dimension-less, they do not acknowledge The Past, Present or Future, Time for photons does not exist, yet we throw these dimensionless quantities around and percieve some sort of Measurement from its action?

It is no wonder that the measurement of the Cosmological Constant Varies so?..me I put Alpha IN THE Past, K IN THE Present ,and Omega IN THE Future, but in a developed Cyclonic structure, as was suggested by Einstein and others.

The Past is Static, Present Dynamic and The future is Static, the Equations (einsteins field) can be motioned into the past, and a static solution will be reached, the static of singularity. Interestingly, if one proceeds into the Future, one also reaches the Static solution??..a Big singular crunch!

The perception is that certain models retrace such fields, and end up in a past or forward dimensional field, that has hidden variables, then they are given a name, such as extra- large-dimensions..or 5-d space (see the link with higher variables?) The background dependence is now 'out-of-bounds', it is in a 'not-present-tense'.

It an automatic perception for Man to think of yesterday, or tomorrow, but the reality is that to measure any entity in a future or past zone, we have take our 'presentday-static' tools with us, and its quite easy to see that the geometry of a , let's say a fixed metre length (rod), will be contracted in a Future field, if the future is in Expansion.

And if one takes the same rod back into the past, it will be of a greater length, because the past is in effect smaller, by Expansion model, than the present. This is a Lorentz action, and Einstein clearly thought this through, borne out in his later years of working in isolation, except for the work done with Nathen Rosen and Boris Podolsky.

For this Paradox, he explained some delacate motions with Rosen and Podolsky.

The result of which is a Three Dimensional Exchange, the EPR!...imagine these three kings..throwing such a rod back and fore to each other in a mental exchange that is Unequalled in all of Perceptive Human thought, some would say the EPR is one of the most evasive and yet most true accounts of Reality.

Originally posted by ranyart
No problem

I understood this part.

ranyart

Originally posted by jeff
I understood this part.

Originally posted by jeff
I understood this part.

Out of choice no doubt.

Moronic Statements are inherently genetic, so I guess your not to blame. But I do admire your self-evaluation, on this basis only!

Memory operation is not based on copy and paste!

It may be that your capacity to relay rational thinking beween external states(experience) and cortex operators(internal thinking)Relys on the world-sheet that is the correspondence of deluded Humans, whose world sheets do not retraceback to your own neurons?

Expecting the whole of Human thought to comply with a line of thinking that is perhaps non-communicative at best, and confined at worst, is a personal issue that you have to come to terms with?

For Humans memory relates to past events, thinking relates to current events and String speculation does not corrospond to either. Where is Memory when one is not Memorizing?..Where does Thinking go when one is Memorizing?

It is correspondence of reality that allows one to communicate and operate in either realm, if you have no incline of a basic reality other than copy and paste(by default this is where String Theory has its foundation based upon) , then you can only fool some of the people some of the time!

ranyart
Originally posted by jeff
This is going to take quite a bit of typing. I'll get to this sometime this week. Sorry for the delay.

Starting with
In general I have problems understanding how definitions on the worldsheet work out in terms of background propagation.

Then using
V(k) ß ƒÃƒÊƒË(k)VƒÊƒË(k) ß ƒÃƒÊƒË(k) ç dƒÊƒÁ ƒÁabÝaXƒÊÝbXƒËeik⋅X

is a vertex operator coupling strings to fluctuations in the background metric GƒÊƒË. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point.

Uhmm[?] care to insert this into Present Spacetime? what you mean(and if one goes back to the original paper of Jiangping Hu and Shou-Cheng Zhang) one cannot explicitly transform hydrodynamical modes from within a Boundery to the external boundery, it can only go one-way.

If one inserts a dynamical background that is part of a worldsheet that is not based in 3-dimensional PRESENT spacetime, then effectively you are folding/unfolding space/past/future inwardly, which is exactly what Hu and Zhang mentions in their Paper, I quote;Since the dimension of total configuration space is higher than the dimension of base space, this theory bares similarities to Kaluza-Klein Theory, but with two important differences. First the total confiruration space is topologically non-trivial fiber bundle. Second the iso-spin space does not have a small radius. This leads to the "embarrassment of riches" problem. In order to solve this problem we need to find a mechanism where higher iso-spin states obtain mass gaps dynamically, through interactions. This way the low energy degrees of freedom would scale correctly with the dimension of base space. end
.

Then:The underlying mathematical structure of the current approach is the noncommutative
geometry [18] de ned by Eq. (2). Unlike previous approaches [19], this relation
treats all four Euclidean dimensions on equal footing. If we interpret X4 as energy, which is
dual to time, this quantization rule seem to connect space, time, spin and the fundamental
length unit l0 in an uni ed fashion. In the lowest SO(5) level, there is no ordinary non-relativistic kinetic energy. All the single particle states are representations of this algebra.
The non-trivial features identi ed in this work all have their roots in this algebra.

Now why is it that all string theories have problems with background propergation?[?]

Originally posted by ranyart
Starting with
In general I have problems understanding how definitions on the worldsheet work out in terms of background propagation.

Then using
V(k) ?ß ƒÃƒÊƒË(k)VƒÊƒË(k) ?ß ƒÃƒÊƒË(k) ?ç dƒÊƒÁ ƒÁab?ÝaXƒÊ?ÝbXƒËeik⋅X

is a vertex operator coupling strings to fluctuations in the background metric GƒÊƒË. Note that like all vertex operators, V is an integral over the world-sheet since it can be inserted at any point.

Uhmm[?] care to insert this into Present Spacetime? what you mean(and if one goes back to the original paper of Jiangping Hu and Shou-Cheng Zhang) one cannot explicitly transform hydrodynamical modes from within a Boundery to the external boundery, it can only go one-way.

If one inserts a dynamical background that is part of a worldsheet that is not based in 3-dimensional PRESENT spacetime, then effectively you are folding/unfolding space/past/future inwardly, which is exactly what Hu and Zhang mentions in their Paper, I quote;Since the dimension of total configuration space is higher than the dimension of base space, this theory bares similarities to Kaluza-Klein Theory, but with two important differences. First the total confiruration space is topologically non-trivial fiber bundle. Second the iso-spin space does not have a small radius. This leads to the "embarrassment of riches" problem. In order to solve this problem we need to find a mechanism where higher iso-spin states obtain mass gaps dynamically, through interactions. This way the low energy degrees of freedom would scale correctly with the dimension of base space. end
.

Then:The underlying mathematical structure of the current approach is the noncommutative
geometry [18] de ned by Eq. (2). Unlike previous approaches [19], this relation
treats all four Euclidean dimensions on equal footing. If we interpret X4 as energy, which is
dual to time, this quantization rule seem to connect space, time, spin and the fundamental
length unit l0 in an uni ed fashion. In the lowest SO(5) level, there is no ordinary non-relativistic kinetic energy. All the single particle states are representations of this algebra.
The non-trivial features identi ed in this work all have their roots in this algebra.

Now why is it that all string theories have problems with background propergation?[?]

I don't doubt your sincerity, but help me out here. Could you begin again by raising just one very specific issue of concern to you in a brief - very very brief - clear and uncomplicated way?

ranyart
Originally posted by jeff
I don't doubt your sincerity, but help me out here. Could you begin again by raising just one very specific issue of concern to you in a brief - very very brief - clear and uncomplicated way?

A Note On The Chern-Simons
And Kodama Wavefunctions

Edward Witten
Institute For Advanced Study, Princeton NJ 08540 USA
Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction.
It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes.
We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one
does not fill the fermi sea. Finally, we give a sort of explanation of “why” this state exists.
Similar properties can be expected for the analogous Kodama wavefunction of gravity.

Ed Witten:In the nonabelian case, and e are not invariant under homotopically non-trivial gauge
transformations. We ignore this. Along with the unnormalizability, lack of CPT invariance, etc., and additional properties that we will see below, this is one more reason that the Chern-Simons
state is formal and does not really correspond to a sensible physical theory.

arXiv:gr-qc/0306083 v1

Originally posted by ranyart
A Note On The Chern-Simons
And Kodama Wavefunctions

Edward Witten
Institute For Advanced Study, Princeton NJ 08540 USA
Yang-Mills theory in four dimensions formally admits an exact Chern-Simons wavefunction.
It is an eigenfunction of the quantum Hamiltonian with zero energy. It is known to be unphysical for a variety of reasons, but it is still interesting to understand what it describes.
We show that in expanding around this state, positive helicity gauge bosons have positive energy and negative helicity ones have negative energy. We also show that the Chern-Simons state is the supersymmetric partner of the naive fermion vacuum in which one
does not fill the fermi sea. Finally, we give a sort of explanation of “why” this state exists.
Similar properties can be expected for the analogous Kodama wavefunction of gravity.

Ed Witten:In the nonabelian case, and e are not invariant under homotopically non-trivial gauge
transformations. We ignore this. Along with the unnormalizability, lack of CPT invariance, etc., and additional properties that we will see below, this is one more reason that the Chern-Simons
state is formal and does not really correspond to a sensible physical theory.

arXiv:gr-qc/0306083 v1

When I invited you to raise a specific issue, I meant one that is germaine to this thread. There isn't even a question in your post, it's just more terminology that you don't understand. In this last respect, it's spamming.

lethe

Originally posted by jeff

In quantum systems it's natural to assign to distinguished points nondynamical internal degrees of freedom giving rise to global symmetries, i.e. conserved charges that don't contribute to the hamiltonian or ruin any pre-existing symmetry.
yeah, that sounds like what CVJ says in his book. but that s the first part that i didn t follow. why is it natural to assign nondynamical degrees of freedom to the endpoints of a string? what does the phrase "nondynamical degrees of freedom" even mean? can you give me an example of a more pedestrian quantum theory where we do this? does anything like this ever happen in QED?

In string theory we can charge the endpoints of open strings with chan-paton degrees of freedom. Although these charges may interact with external gauge fields, during interactions of several open strings (open strings interact at their endpoints), they flow only along the world-sheet boundaries swept out by the endpoints and so are conserved. Since chan-paton charges have trivial world-sheet dynamics - that is, world-sheet interactions can't change them - consistency requires that only identically charged endpoints may interact.
they have trivial world sheet dynamics because they don t appear in the lagrangian?

what does it mean? i usually associate conserved charges with some symmetry of the lagrangian, although i guess there are other kinds of charges like topological charges. so where did these conserved charges come from? this all seems very opaque to me.

Originally posted by jeff

If the chan-paton degrees of freedom in the 2D quantum conformal field theory governing the physics of open strings are to be useful, they must produce in the low energy limit the familiar yang-mills interactions of particles as they appear in the arena of QFT.

i guess i can imagine that inserting some charges by hand that lead to Yang-Mills would be useful, but right now, this seems highly artificial to me, i guess because i can t understand where these degrees of freedom came from.

if the goal is to get a yang-mills field, why not just postulate that one of the vector fields in the spectrum carries some nonabelian charge?

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