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

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

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selfAdjoint

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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.

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jeff

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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

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jeff

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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

α

This is analogous to the simple harmonic oscillator one studies in introductory courses in quantum mechanics. The subscript "-1" of α

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

α′m

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

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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selfAdjoint

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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.

- #6

jeff

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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

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

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 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

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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selfAdjoint

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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.

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jeff

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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

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

α′

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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jeff

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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?

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jeff

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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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selfAdjoint

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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.

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jeff

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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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selfAdjoint

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

selfAdjoint

selfAdjoint

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jeff

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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

S

in which the basic fields X

<V

Now, take

G

with

ε

everywhere small compared to η

e

in which

V(k) ≡ ε

is a vertex operator coupling strings to fluctuations in the background metric G

k

k

ε

In addition to showing that the spin-2 excitations are massless, because the ricci tensor R

R

this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G

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

ε

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

ε

is an infinitesimal diffeomorphism generated by the vector field ξ

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.

Define world-sheet coordinates

z = e

with σ = σ + 2π the periodic coordinate along the string and τ the time coordinate on the world-sheet. We then have

V ∝ ε

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

∂

where α

Putting these together gives

V ↔ ε

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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.

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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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selfAdjoint

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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.

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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.

- #19

jeff

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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

What I meant to say was that only V

Thus the edit...

"Next, observe that ε

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

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- #20

jeff

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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

In string theory we can charge the endpoints of open strings with

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

Let |mñ> with m,ñ = 1,...,N² denote a complete set of

(1) |a> = ∑

in which the NxN matrices λ

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),

The chan-paton states of any oriented open string fill out the adjoint representation

α

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

α

Hence α

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

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.

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

S

in which the basic fields X

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- #21

jeff

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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.

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Originally posted by jeff

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

S_{G}= - (1/4πα′) ∫ dμ_{γ}γ^{ab}G_{μν}(X)∂_{a}X^{μ}∂_{b}X^{ν}

in which the basic fields X^{μ}of the theory embed the world-sheet with metric γ^{ab}and measure dμ_{γ}in a background spacetime with metric G_{μν}. 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 asweyl-invariance, of S_{G}under rescalings γ^{ab}→ e^{φ}γ^{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 avertex operatordefined in terms of X^{μ}and it's world-sheet derivatives and carrying the quantum numbers of the original incoming/outgoing string state vector: This is known as thestate-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 V_{i}(k_{i}) for each incoming/outgoing closed string of momentum k_{i}. Hence, amplitudes for n external string states have the form of a sum of path-integrals with insertions

<V_{1}(k_{1})⋅⋅⋅V_{n}(k_{n})> ~ ∑_{g=0,1,2,...}∫_{g}Dγ_{ab}DX^{μ}V_{1}(k_{1})⋅⋅⋅V_{n}(k_{n})e^{-SG}.

Now, take

G_{μν}(X) = η_{μν}+ ε_{μν}(X)

with

ε_{μν}(X) = ∫ d^{26}k ε_{μν}(k)e^{ik⋅X}

everywhere small compared to η_{μν}. Then

e^{-SG}= e^{-(Sη + Sε)}= e^{-Sη}∑_{n=0,1,...}(- 4πα′)^{-n}(1/n!) ∫ d^{26}k_{1}⋅⋅⋅d^{26}k_{n}V(k_{1})⋅⋅⋅V(k_{n})

in which

V(k) ≡ ε_{μν}(k)V^{μν}(k) ≡ ε_{μν}(k) ∫ dμ_{γ}γ^{ab}∂_{a}X^{μ}∂_{b}X^{ν}e^{ik⋅X}

is a vertex operator coupling strings to fluctuations in the background metric G_{μν}. 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 ε_{μν}picks out the symmetric part of V^{μν}, 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 e^{ik⋅X}to transform properly under spacetime translations X^{μ}→ X^{μ}+ a^{μ}. Now, any insertion must respect thelocalweyl symmetry of the theory. In particular, demanding that V be weyl-invariant requires (see polchinski I Chap 3.6)

k^{2}= k^{2}ε_{μν}(k) = 0 ↔ ⇑ε_{μν}(X) = ⇑G_{μν}(X) = 0

k^{μ}ε_{μν}(k) = 0 ↔ ∂^{μ}ε_{μν}(X) = ∂^{μ}G_{μν}(X) = 0,

ε^{μ}_{μ}(k) = 0 ↔ ε^{μ}_{μ}(X) = 0.

In addition to showing that the spin-2 excitations are massless, because the ricci tensor R_{μν}satisfies

R_{μν}∝ ∂_{μ}∂_{ν}ε_{λ}^{λ}- 2∂_{λ}∂_{(μ}ε_{μ)}^{λ}+ ⇑ε_{μν}+ O(ε^{2}),

this also shows that to leading order in metric fluctuations, weyl-invariance in the pure helicity-2 theory requires that the background G_{μν}satisfy the vacuum einstein equations R_{μν}= 0.

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

ε_{μν}(k) → ε_{μν}(k) + k_{μ}ξ_{ν}+ k_{ν}ξ_{μ}

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

ε_{μν}(X) → ε_{μν}(X) + k_{μ}ξ_{ν}(X) + k_{ν}ξ_{μ}(X)

is an infinitesimal diffeomorphism generated by the vector field ξ^{μ}(X) in the approximation where O(ε^{2}) terms are neglected and under which R_{μν}= 0 is invariant. In fact R_{μν}= 0 is the only spacetime diffeo-invariant equation that reduces to ⇑G_{μν}(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* = e^{iσ + τ}

with σ = σ + 2π the periodic coordinate along the string and τ the time coordinate on the world-sheet. We then have

V ∝ ε_{μν}∫d^{2}z ∂_{z}X^{μ}(z)∂_{z*}X^{ν}(z*)e^{ik⋅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

∂_{z}X^{μ}(0) ↔ α_{-1}^{μ}, ∂_{z*}X^{μ}(0) ↔ (α_{-1}^{μ})* , e^{ik⋅X(0,0)}↔ |0;0;k>

where α_{-1}^{μ}and (α_{-1}^{μ})* excite left- and right-moving n = 1 modes.

Putting these together gives

V ↔ ε_{μν}α_{-1}^{μ}(α_{-1}^{ν})*|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 Im off!

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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 Im 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?[?] [?]

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selfAdjoint

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Originally posted by lethe

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

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.research

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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 thats a correct term?

Question, what causes Strings? Nature or Theorists?

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