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Difference between open string field theory and closed string field theory

  1. Sep 8, 2003 #1
    What's the difference between open string field theory and closed string field theory?
     
    Last edited by a moderator: Feb 5, 2013
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  3. Sep 9, 2003 #2

    selfAdjoint

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    Re: differences

    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.
     
  4. Sep 10, 2003 #3

    jeff

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    Re: Re: differences

    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.
     
  5. Sep 10, 2003 #4

    jeff

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    Re: U(1)=Photon

    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

    αi-1|0;k>.

    This is analogous to the simple harmonic oscillator one studies in introductory courses in quantum mechanics. The subscript "-1" of αi-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 αi-1 takes values i = 1,...,24 and indicates that αi-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

    α′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.

    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.
     
    Last edited: Sep 10, 2003
  6. Sep 10, 2003 #5

    selfAdjoint

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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.
     
  7. Sep 10, 2003 #6

    jeff

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

    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.




    Yeah, but you've gotta 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.

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

    Trying to learn strings on your own is a b*tch no matter how you do it.
     
    Last edited: Sep 10, 2003
  8. Sep 10, 2003 #7

    selfAdjoint

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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.
     
  9. Sep 10, 2003 #8

    jeff

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    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.

    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?

    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 hμν 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±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 θμν, i.e. ∂μθμν = 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 hμν has been impossible until strings, which as I said require it.

    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

    α′Rμν + 2α′∇μ∇νΦ - (α′/4)HμλωHνλω = 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.

    You haven't come across a reference because it's not true.
     
    Last edited: Sep 10, 2003
  10. Sep 10, 2003 #9

    jeff

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    Re: Encapsulating Geometrical Interpetation

    What do you mean by mystical?

    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.

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

    Could you be more specific?
     
  11. Sep 11, 2003 #10

    jeff

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    Re: Probabilty Distributions

    By this do you mean that you're building your own theory or that you're simply trying to learn some physics?
     
  12. Sep 11, 2003 #11

    selfAdjoint

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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¿ŒÞƒÊރ˃³ - (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.
     
  13. Sep 15, 2003 #12

    jeff

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    This is going to take quite a bit of typing. I'll get to this sometime this week. Sorry for the delay.
     
  14. Sep 15, 2003 #13

    selfAdjoint

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


    selfAdjoint
     
  15. Sep 19, 2003 #14

    jeff

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    helicity-2 string excitations are gravitons

    selfAdjoint,

    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 = - (1/4πα′) ∫ dμγγabGμν(X)∂aXμ∂bXν

    in which the basic fields Xμ of the theory embed the world-sheet with metric γab and measure dμγ in a background spacetime with metric Gμν. 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 γab → eφγab 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 Xμ 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>.
     
    Last edited: Sep 21, 2003
  16. Sep 19, 2003 #15
    Re: Re: U(1)=Photon

    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:

    that was awesome.
     
  17. Sep 19, 2003 #16
    i couldn t find this thread. do you remember the title?
     
  18. Sep 19, 2003 #17

    selfAdjoint

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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.
     
  19. Sep 19, 2003 #18
    ouch, that was tough. i can t wait to see Baez' reply.
     
  20. Sep 19, 2003 #19

    jeff

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    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.
     
    Last edited: Sep 20, 2003
  21. Sep 24, 2003 #20

    jeff

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    chan-paton factors

    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.
     
    Last edited: Sep 25, 2003
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