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Thanks, Ill try to google the "Quantizations"

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Also I'll add my own two cents on all of this - while energy clearly comes in small packets that we call quanta, and spin and charge are also quantized, the vast majority of physical quantities remain continuous values. The energy spectrum of photons is completely continuous, as is momentum, kinetic energy and everything derived from it, and, as far as we know, spacetime coordinate. No one really knows what it would mean to quantize spacetime. Is every possible physical location a point on a discrete grid like pixels on a computer monitor? What would that mean for SPECIAL relativity, let alone General Relativity? These are all unanswered quesitons.

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The usual way quantization is treated is often either fairly heuristic, or arbitrarily axiomatic by starting with a classical model and then quantize it. And the resulting models have been succesful, but nevertheless it's easy to get a feeling of lack of deeper understanding.

I personally don't see clearly how a continuos state space would be distinguishable from a discrete state space. Because any real observer are likely to have a limited observational resolution. I personally see the continuum in most of our models as a mathematical model. Ie. the continuum isn't necessarily physical. Then one might argue that if the continuum is indistinguishable from a discrete model, there are mathematical ghost degrees of freedom that lack physical basis. Therefore i would rather like to replace the description with the equivalence class of models that allows both continous as well as discrete spaces.

But my gut feeling is that I have hard time to appreciate the concept of a physical continuum, from the point of view of information capacity. Sure one can imagine that there may be a continuum and I as a simple observer can only see a part of it, but then it means I can not keep the reference to the larger structure. Any kind of larger theory would violate the information constraint where the theory lives. In this sense, one might imagine time as relative change to be discrete relative to an outside observer. However, how would the observer himself be able to judge wether what he sees is discrete relative to some larger construct that he are unable to imagine?

/Fredrik

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Then the question is still what is a relation in a physical context? Does there need to be a physical basis for the relation too? Or does it make sense to consider mathematical relations with unbounded complexity? I think not. I personally think the measurement problem is part of this. If we have two observers, observing things. Then we claim that there exists a deterministic relation between the observers. Who is observing this relation? And how is objectivity handled?

/Fredrik

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xristy

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I think that Christoph Schiller's paper: http://arxiv.org/abs/gr-qc/9610066" [Broken] is a pedagogically nice treatment of the issue of spacetime quantization - as well as energy and so on.

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Surely no-one understands both (or either for that matter) :-)

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In any event, you can quantize an ordinary relativistic point-particle where the particle's proper time is the thing conjugate to the Hamiltonian. In that case, the spacetime coordinates [tex]x^\mu[/tex] are all operators on a Hilbert space, and all the delicacies of quantum mechanics follow. You can always choose a gauge, however, in which [tex]x^0[/tex], or any other linear combination, is diagonal. In that gauge, [tex]x^0[/tex] behaves simply as an ordinary number, and one is not normally aware of it being an operator.

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nrqed

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Really?!?String theory is one framework in which spacetime emerges as a low-energy, long-distance description of a quantum system. Now, it might not be our universe, but there is a place in which the quantization of spacetime can be understood.

It seems to me that the staring point of any string theory action is a string propagating through a background spacetime. The spacetime is here to start with, it does not emerge from the theory!

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Unfortunately that is a common misconception. One normally starts with a particular background and quantizes around it. But while it appears that there is a background in which the strings move, from the point of view of the string there is merely a scalar field [tex]X^\mu(\sigma)[/tex] propagating on it's 2D worldsheet. "Background" fields like the metric make their appearance as coherent states (due to other strings) on this 2D conformal field theory. There are other fields living on this CFT, namely [tex]\psi^\mu[/tex] as well as various ghosts, so how are they to be interpreted as embedding functions? They are too weird to have such an interpretation, and it is better to think of "classical" string theory as some Riemann surface with a particular conformal field theory living on it.Really?!?

It seems to me that the staring point of any string theory action is a string propagating through a background spacetime. The spacetime is here to start with, it does not emerge from the theory!

From what I understand (for open strings at least), you can start with a CFT with a spacetime type [tex]X^\mu[/tex], and continuously move to another vacuum, by a process known as Tachyon condensation, in which this field disappears (no more spacetime). So spacetime is merely an excitation in the theory, i.e. a "space-filling D-brane". But it is just one state, and there are other states which have to be very bizarre indeed.

If this hasn't convinced you, all known "string theories", or known string vacua, are dual to eachother and to something that has an 11D supergravity low-energy limit. So here you have a number of 10D string theories (states) and they can be mapped to something with 11 dimensions. If the number of dimensions isn't even fundamental, then spacetime itself is on shaky ground =)

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nrqed

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Fascinating. Thanks for those details.Unfortunately that is a common misconception. One normally starts with a particular background and quantizes around it. But while it appears that there is a background in which the strings move, from the point of view of the string there is merely a scalar field [tex]X^\mu(\sigma)[/tex] propagating on it's 2D worldsheet. "Background" fields like the metric make their appearance as coherent states (due to other strings) on this 2D conformal field theory. There are other fields living on this CFT, namely [tex]\psi^\mu[/tex] as well as various ghosts, so how are they to be interpreted as embedding functions? They are too weird to have such an interpretation, and it is better to think of "classical" string theory as some Riemann surface with a particular conformal field theory living on it.

From what I understand (for open strings at least), you can start with a CFT with a spacetime type [tex]X^\mu[/tex], and continuously move to another vacuum, by a process known as Tachyon condensation, in which this field disappears (no more spacetime). So spacetime is merely an excitation in the theory, i.e. a "space-filling D-brane". But it is just one state, and there are other states which have to be very bizarre indeed.

If this hasn't convinced you, all known "string theories", or known string vacua, are dual to eachother and to something that has an 11D supergravity low-energy limit. So here you have a number of 10D string theories (states) and they can be mapped to something with 11 dimensions. If the number of dimensions isn't even fundamental, then spacetime itself is on shaky ground =)

I am always confused because it seems that at some point people switch from a description in terms of a 2D theory to a description in terms of strings/branes moving in some target space and it's never clear to me how/why the transition is made.For example, couldn't one describe

But then, what do they mean when they say that certain excitations on the worldsheet correspond to spacetime fermions? On the worldsheet, there are scalar fields and fermioninic fields. From the 2d worldsheet perspective, what does it mean to have an excitation that corresponds to a worldsheet fermion? The mu are simply scalar fields, they are not coordinates. It sounds as if this would imply that the fermion fields are dependent on the scalar fields from the point of view of the 2D CFT??

And what is a brane in that case if you keep the point of view of the 2D worldsheet. Again, the X\mu are scalar fields, not coordinates.

Thanks again. Sorry for the simple-minded questions

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This picture is known as the RNS model. There is also the Green-Schwarz model, which has manifest spacetime spinors but worldsheet supersymmetry is hard to see, and this model is hard to quantize. They are equivalent, however.

Back to your first question. You can do everything from the worldsheet point of view. Things like string scattering amplitudes can be written completely in terms of the worldsheet CFT. I think you can also describe branes using what's called "boundary states" on the CFT. But this is too microscopic to be able to handle big calculations. So, starting with your CFT, you figure out what the spacetime fields look like and how they behave to whatever order in quantum perturbation theory you like. Then you solve those equations like classical equations, because that is easier to do...

It's a bit like trying to solve fluid dynamics by talking about individual atoms. A gnarly task indeed.

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The simplest answer and therefore the least scientifically rigid is that time is relatively quantised by us, as it involves the orbital period of the sun, the moons phases, the energetic emissions of caesium amongst other things. Not of course the answer you want, but none the less yes it is and might not and might be depending on how you define time. I've been reading too much philosophy I think.

Quantized:

1.to subdivide (as energy) into small but measurable increments.

2.to calculate or express in terms of quantum mechanics

Quantized:

1.to subdivide (as energy) into small but measurable increments.

2.to calculate or express in terms of quantum mechanics

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nrqed

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Thanks again, this is very very interesting stuff.Well the worldsheet field-theory has scalars [tex]X^\mu(\sigma, \tau)[/tex] and fermions [tex]\psi^\mu(\sigma, \tau)[/tex] and ghosts. These fields admit mode expansions in terms of creation and annihilation operators. The 'states' of the string then are found by acting on the ground state [tex]\left|0\right\rangle[/tex] by these creation operators. Note that the creation operators also carry a Lorentz vector index, so creation operators always add 1 to spin.

Before discussiong your other points, let me ask you about what you said above. the index mu is a Lorentz vector index. By this here we mean a Lorentz transformation in the target space, right? But if, for the sake of the discussion, we keep everything from the point of view of the 2D CFT, there is no ambient space so this is a purely internal symmetry, correct? But this symmetry is a Lorentz symmetry so it's kind of strange. From this point of view (of the 2D CFT), this is imposed by hand or does it follow from some consistency condition?

Or maybe I am completely missing the point.

Thanks again

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Anyway, in the sector that the 2D CFT is free (i.e., non interacting), each field living on it will cause an anomaly to appear that (seems to) break conformal invariance. It contributes a number to the trace of the energy momentum tensor (which needs to be zero for conformal invariance to hold). Generally commuting fields add to this number while anticommuting fields subtract from it. So as soon as you add one field to your CFT, you'll need to add another to cancel the anomaly.

I think an arbitrary (but probably not the most general) free CFT will have some # of commuting scalars, some # of anticommuting scalars (ghost), some # of commuting spinors (ghosts again) and some # of anticommuting spinors. These fields are all representations of the 2D poincare group, so that's why we pick them. You can then pick these numbers so that the anomaly vanishes. I believe the solutions are 26 commuting scalars and no anticommuting spinors, 10 commuting scalars and 10 anticommuting spinors, and some really funky solutions which I don't know much about (N = 2 and N = 4 worldsheet supersymmetry, for those who are keeping track).

So in a sense, the Lorentz symmetry is sort of along for the ride. If you write an action

[tex]S = \int\!d^2\sigma\, \sum_i \partial_\alpha \phi^i \partial^\alpha \phi^i[/tex]

then this is the action of some number [tex]n[/tex] of scalar fields. But you get an [tex]SO(n)[/tex] symmetry for free. There are some important subtleties why the metric happens to be [tex]SO(n-1, 1)[/tex] rather than [tex]SO(n)[/tex], and I'm not quite sure I understand these, (it has to do with Weyl invariance), but that's another story.

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nrqed

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Ok. I can see the SO(n) symmetry popping out for free (as long as the scalar field are non-interacting). The issue of getting an SO(n-1,1) invariance instead really intrigues me but we could talk about this later.

Anyway, in the sector that the 2D CFT is free (i.e., non interacting), each field living on it will cause an anomaly to appear that (seems to) break conformal invariance. It contributes a number to the trace of the energy momentum tensor (which needs to be zero for conformal invariance to hold). Generally commuting fields add to this number while anticommuting fields subtract from it. So as soon as you add one field to your CFT, you'll need to add another to cancel the anomaly.

I think an arbitrary (but probably not the most general) free CFT will have some # of commuting scalars, some # of anticommuting scalars (ghost), some # of commuting spinors (ghosts again) and some # of anticommuting spinors. These fields are all representations of the 2D poincare group, so that's why we pick them. You can then pick these numbers so that the anomaly vanishes. I believe the solutions are 26 commuting scalars and no anticommuting spinors, 10 commuting scalars and 10 anticommuting spinors, and some really funky solutions which I don't know much about (N = 2 and N = 4 worldsheet supersymmetry, for those who are keeping track).

So in a sense, the Lorentz symmetry is sort of along for the ride. If you write an action

[tex]S = \int\!d^2\sigma\, \sum_i \partial_\alpha \phi^i \partial^\alpha \phi^i[/tex]

then this is the action of some number [tex]n[/tex] of scalar fields. But you get an [tex]SO(n)[/tex] symmetry for free. There are some important subtleties why the metric happens to be [tex]SO(n-1, 1)[/tex] rather than [tex]SO(n)[/tex], and I'm not quite sure I understand these, (it has to do with Weyl invariance), but that's another story.

But what about the wolrdhseet fermion fields. Is there a simple way to see why they should carry a spacetime index mu? Why would they have to come up in such a representation if we look at things from the point of view of the 2D theory?

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Haelfix

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Of course the calculations are done mostly on the worldsheet (and interactions thereof) but there is nothing stopping you from retransforming out of that picture.

Of course sometimes thats a hard calculation, and afaik people are only able to get explicit closed form answers for special simplified cases (sometimes you can guess the new answer based solely on consistency criteria)

The real issue is what do do when the full background metric g cannot be seperated into weak field approximations (eg curvature is strong). Thats an open and active field of research, and the procedures are involved.

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Well, you could call the index of the fermions something else, like [tex]i[/tex]. The point is that there are a number of them, (for anomaly cancellation purposes), and in the case that there happen to be 10, we might as well call the index [tex]\mu[/tex] as well.Ok. I can see the SO(n) symmetry popping out for free (as long as the scalar field are non-interacting). The issue of getting an SO(n-1,1) invariance instead really intrigues me but we could talk about this later.

But what about the wolrdhseet fermion fields. Is there a simple way to see why they should carry a spacetime index mu? Why would they have to come up in such a representation if we look at things from the point of view of the 2D theory?

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Haelfix

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I find that pretty neat.

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Only one slight issue, he kinda demolishes... all of physics at the Planck scale. How valid is this paper?I think that Christoph Schiller's paper: http://arxiv.org/abs/gr-qc/9610066" [Broken] is a pedagogically nice treatment of the issue of spacetime quantization - as well as energy and so on.

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nrqed

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Yes, I also find that quite awesome. There are so many things that make string theory fascinating that I can't help feeling there must be something there that Nature has used. It's possible that it's not the case and that it will turn out that string theory has no relevance to the real world (or maybe we will never find out) but it is so amazing that it's worth exploring further, IMHO.

I find that pretty neat.

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nrqed

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But to say that it is a spacetime vector index implies a non-trivial relation between these (since they transform under a specific way under "Lorentz transformations of the ambient space") . This is the part that I don't quite understand. Do you see what I mean?Well, you could call the index of the fermions something else, like [tex]i[/tex]. The point is that there are a number of them, (for anomaly cancellation purposes), and in the case that there happen to be 10, we might as well call the index [tex]\mu[/tex] as well.

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