What's the difference between open string field theory and closed string field theory?
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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.Originally posted by meteor
What's the difference between open string field theory and closed string field theory?
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 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.
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> yieldingOriginally posted by sol
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?
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.Originally posted by sol
I apologize if this seems like a stupid question.
Yeah, I just got "pm'ed" by greg about that.Originally posted by selfAdjoint
Well that's a bit harsh
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.Originally posted by selfAdjoint
I define as a single theory that produces the real predictions of GR
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.Originally posted by selfAdjoint
It's not enough to generate some U(1) x SU(2) x SU(3) generic theory.
No, and I wouldn't hold my breath waiting for a good one.Originally posted by selfAdjoint
do you know of an online tutorial for string field theory?
Trying to learn strings on your own is a b*tch no matter how you do it.Originally posted by selfAdjoint
Trying to understand it from research papers is a bummer.
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
I want to say this gently..
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
You say they haven't got GR, but they do have approximations
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.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.
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-invariantOriginally 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.
You haven't come across a reference because it's not true.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.
What do you mean by mystical?Originally posted by sol
You would be scare of my mystical approach then
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
Just know, that I endeavor to understand and do not want to be criticized for my thinking outside of this issue:0)
I'm having trouble decoding this. Maybe you want a general explanation of D-branes?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
Could you be more specific?Originally posted by sol
could you explain SU(2)
This is going to take quite a bit of typing. I'll get to this sometime this week. Sorry for the delay.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.
jeff-Originally posted by jeff
Finally, to move beyond U(1) to non-abelian U(n) gauge symmetry, we need to introduce Chan-Paton factors.
that was awesome.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.
ouch, that was tough. i can t wait to see Baez' reply.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.
I made an edit which bears on the state-operator correspondence: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.
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...Originally posted by lethe
...quick and dirty explanation of what a Chan-Paton factor is?
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.Originally posted by sol
Is it possible to have a generalization put in front of this equative formulation on a simpler level.
Jeff, I detect as much panic in your posts as your peers, you are not alone!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 = - (1/4πα′) ∫ 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)⋅⋅⋅Vn(kn)> ~ ∑g=0,1,2,... ∫g DγabDXμ V1(k1)⋅⋅⋅Vn(kn)e-SG.
Gμν(X) = ημν + εμν(X)
εμν(X) = ∫ d26k εμν(k)eik⋅X
everywhere small compared to ημν. Then
e-SG = e-(Sη + Sε) = e-Sη ∑n=0,1,...(- 4πα′)-n(1/n!) ∫ d26k1⋅⋅⋅d26kn V(k1)⋅⋅⋅V(kn)
V(k) ≡ εμν(k)Vμν(k) ≡ εμν(k) ∫ dμγ γab∂aXμ∂bXνeik⋅X
is a vertex operator coupling strings to fluctuations in the background metric Gμν. 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 εμν picks out the symmetric part of Vμν, 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⋅X to transform properly under spacetime translations Xμ → Xμ + aμ. 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 = k2εμν(k) = 0 ↔ ⇑εμν(X) = ⇑Gμν(X) = 0
kμεμν(k) = 0 ↔ ∂μεμν(X) = ∂μGμν(X) = 0,
εμμ(k) = 0 ↔ εμμ(X) = 0.
In addition to showing that the spin-2 excitations are massless, because the ricci tensor Rμν satisfies
Rμν ∝ ∂μ∂νελλ - 2∂λ∂(μεμ)λ + ⇑εμν + O(ε2),
this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background Gμν satisfy the vacuum einstein equations Rμν = 0.
Because massless states are transversally polarized, V must be invariant under the shift
εμν(k) → εμν(k) + kμξν + kνξμ
by longitudinal polarizations. In terms of the metric, this gauge-invariance
εμν(X) → εμν(X) + kμξν(X) + kνξμ(X)
is an infinitesimal diffeomorphism generated by the vector field ξμ(X) in the approximation where O(ε2) terms are neglected and under which Rμν = 0 is invariant. In fact Rμν = 0 is the only spacetime diffeo-invariant equation that reduces to ⇑Gμν(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-iσ + τ , z* = eiσ + τ
with σ = σ + 2π the periodic coordinate along the string and τ the time coordinate on the world-sheet. We then have
V ∝ εμν∫d2z ∂zXμ(z)∂z*Xν(z*)eik⋅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
∂zXμ(0) ↔ α-1μ , ∂z*Xμ(0) ↔ (α-1μ)* , eik⋅X(0,0) ↔ |0;0;k>
where α-1μ and (α-1μ)* excite left- and right-moving n = 1 modes.
Putting these together gives
V ↔ εμνα-1μ(α-1ν)*|0;0;k>.
Here it is. http://groups.google.com/groups?dq=&hl=en&lr=&ie=UTF-8&threadm=bkdc9k%244qv%241%40glue.ucr.edu&prev=/groups%3Fhl%3Den%26lr%3D%26ie%3DUTF-8%26group%3Dsci.physics.researchOriginally posted by lethe
ouch, that was tough. i can t wait to see Baez' reply.
I see that the reply to my post from jeff dissapeared? unless he re-routed it into a different 'sum-over-history'!Originally posted by Tom
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