Journal Club: Witten NPB186 (1981) p 412-428

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arivero
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Main Question or Discussion Point

This thread is to comment on the article
"Search for a realistic Kaluza Klein Theory", by Ed. Witten.

If someone has got a scanned copy of the preprint PRINT-81-0056 (the article is under copyright, I am afraid) please post it. But my intention is to copy excertps I am interested on discussing, and I want to invite you to do the same.

Mi main interest lies in pages 419-421
 
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nrqed
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This thread is to comment on the article
"Search for a realistic Kaluza Klein Theory", by Ed. Witten.

If someone has got a scanned copy of the preprint PRINT-81-0056 (the article is under copyright, I am afraid) please post it. But my intention is to copy excertps I am interested on discussing, and I want to invite you to do the same.

Mi main interest lies in pages 419-421
It would be great to discuss that article!
I have a copy because I have access to a copy of the book "The World in Eleven Dimensions" which is a collection fo reprints and contain that paper.
 
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This thread is to comment on the article
"Search for a realistic Kaluza Klein Theory", by Ed. Witten.

If someone has got a scanned copy of the preprint PRINT-81-0056 (the article is under copyright, I am afraid) please post it. But my intention is to copy excertps I am interested on discussing, and I want to invite you to do the same.

Mi main interest lies in pages 419-421
The paper can be found on pages 29-44 of the Google books preview of http://books.google.com/books?hl=en...s=GYV0NeEP82&sig=ZklB2pdjkfOT-xLHn6a3e1jTTRc" By M. J. Duff.
 
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arivero
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The paper can be found on pages 29-44 of the Google books preview of http://books.google.com/books?hl=en...s=GYV0NeEP82&sig=ZklB2pdjkfOT-xLHn6a3e1jTTRc" By M. J. Duff.
Great!

A scanned/copyfree pdf could be better, but lacking KEK, Google books is a good alternative. In fact my first contact with this paper (only two years ago, during a Saturday-Sunday gatecrashing of the library at Trinity College) was via this book.

To start somewhere, let me address the first point that jumped to my attention then. In page 37 (420) Witten explains a Dirac version of Klein argument for mass in extra dimensions. Then he uses it to argue for the need of zero modes of the Dirac operator in the compact space, but the method seems of more general usage. In fact it resembles very much to Connes-Lott model, where mass comes from a non commutative "extra" (but discrete, so D=4+0) dimension.

Does anybody remembers the reference of the original argument of Klein? Witten does not refer to it, but to a work of a L. Palla in 1978.

For a different starting point, more advanced if you prefer: why is this paper (Witten's) usually referred as a "non-go theorem"? It seems to show that 11D supergrav is the way to go.
 
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nrqed
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Great!

A scanned/copyfree pdf could be better, but lacking KEK, Google books is a good alternative. In fact my first contact with this paper (only two years ago, during a Saturday-Sunday gatecrashing of the library at Trinity College) was via this book.

To start somewhere, let me address the first point that jumped to my attention then. In page 37 (420) Witten explains a Dirac version of Klein argument for mass in extra dimensions. Then he uses it to argue for the need of zero modes of the Dirac operator in the compact space, but the method seems of more general usage. In fact it resembles very much to Connes-Lott model, where mass comes from a non commutative "extra" (but discrete, so D=4+0) dimension.

Does anybody remembers the reference of the original argument of Klein? Witten does not refer to it, but to a work of a L. Palla in 1978.

For a different starting point, more advanced if you prefer: why is this paper (Witten's) usually referred as a "non-go theorem"? It seems to show that 11D supergrav is the way to go.

I am out of my element here so I won't be able to contribute much. ANd I haven't read the paper in along time but I seem to recall that the conclusion was that there was no way to obtain a chiral theory from compactification a la Kaluza-Klein. That's what I remember Witten concluding at the end of the paper but I may be wrong. This is what left me with the impression that the paper was essentially killing the whole approach.
 
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nrqed
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Great!

A scanned/copyfree pdf could be better, but lacking KEK, Google books is a good alternative. In fact my first contact with this paper (only two years ago, during a Saturday-Sunday gatecrashing of the library at Trinity College) was via this book.

To start somewhere, let me address the first point that jumped to my attention then. In page 37 (420) Witten explains a Dirac version of Klein argument for mass in extra dimensions. Then he uses it to argue for the need of zero modes of the Dirac operator in the compact space, but the method seems of more general usage. In fact it resembles very much to Connes-Lott model, where mass comes from a non commutative "extra" (but discrete, so D=4+0) dimension.
.
I am not sure if I understand your question so I may be missing the point completely. But his argument is simply that if psi is an eigenstate of [tex] D^{int} [/tex], then

[tex] D^{int} \psi = \lambda \psi [/tex]

and this is simply a mass term in the four-dimensional space. That's all there is to it.But I probably misunderstand your question.
 
  • #7
arivero
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I am not sure if I understand your question so I may be missing the point completely. But his argument is simply that if psi is an eigenstate of [tex] D^{int} [/tex], then

[tex] D^{int} \psi = \lambda \psi [/tex]

and this is simply a mass term in the four-dimensional space. That's all there is to it.But I probably misunderstand your question.
Not really a question, but that really atracted my attention, perhaps because I had never thought in mass as coming from extra dimensions, but also because he was using the Dirac operator for it. That was news, then.

Of course, any post-1905 crackpot can think: "inertia comes because we walk with a kind of angle into an extra dimension". And indeed, if one considers
[tex]E^2 - \sum^n_1 p^2_i= m_0^2[/tex]
and then postulates than some extra dimensions have a constrain
[tex]\sum^n_4 p^2_i = K_0^2[/tex]
the our particle has a fourdimensional mass such that
[tex]M_0^2 \equiv E^2 - \sum^3_1 p^2_i = m_0^2 + K_0^2[/tex]
And we can postulate that all the particles are really massless [tex]m_0=0[/tex] and the mass comes from the term [tex]K_0[/tex]. This was the historical origin, I heard, of Klein-Gordon equation, where the argument can be done rigurously -or so-.

But it seems that nobody had though of an "square root version" of this argument until Witten recalls it in this paper.

And even funnier: he recalls it to argue that [tex]K_0=0[/tex]
 
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Haelfix
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I didn't see the link, but I assume this is the paper where Witten shows that extended N=8 supergravity is only consistent in D>=11 as smaller dimensions necessarily are not big enough to encompass the standard model gauge group and their mirrors upon supersymmetry breaking.

The other paper of the time (I forget the author), uniquely fixes D = 11, since higher dimensional Sugra theories contain terms with spin>2 (which are excluded by the Witten-Weinberg theorem).

Actually Wittens paper (while interesting and historically important) is flawed in a number of ways. Nowdays people can bypass most of the nogo assumptions b/c the compactification techniques are much more elaborate than the simple ones considered at the time, and their are mechanisms in place to generate chiral spectrums (dbranes, orbifolding, etc etc) in smaller dimensions.
 
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arivero---

People have worked out in detail the mode expansions associated with particles of different statistics---see the Les Houches lectures of Tony Gherghetta where he does it in the case of the Randall-Sundrum model, for example.
 
  • #10
arivero
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I didn't see the link, but I assume this is the paper where Witten shows that extended N=8 supergravity is only consistent in D>=11 as smaller dimensions necessarily are not big enough to encompass the standard model gauge group and their mirrors upon supersymmetry breaking.
Not exactly. I think that it is because they because smaller dimensions are not big enough to encompass the gauge group, stop. Perhaps I missed where susy or mirrors enter in the argument.

The other paper of the time (I forget the author), uniquely fixes D = 11, since higher dimensional Sugra theories contain terms with spin>2 (which are excluded by the Witten-Weinberg theorem).
The argument about spin>2 being forbidden and thus D=11 is explicitly told in this paper too. Not sure why it is not stressed beyond a single parragraph.


Actually Wittens paper (while interesting and historically important) is flawed in a number of ways. Nowdays people can bypass most of the nogo assumptions b/c the compactification techniques are much more elaborate than the simple ones considered at the time, and their are mechanisms in place to generate chiral spectrums (dbranes, orbifolding, etc etc) in smaller dimensions.
It seems not a flaw; Witten does not build the paper as a non.go theorem as e.g. in Coleman-Mandula paper. He simply argues, in the second part of the paper, that chiral fermions need some more ingenous approach than naive use of the zero modes of the Rarita Schwinger operator on seven dimensional manifolds. How posterior literature has grown this into a No-Go theorem "essentially killing the whole approach" is a mistery to me. I would better think that while this paper was being done they were already thinking on alternatives.

(appart from this issue, have you heard of specific, technical flaws of the paper?)
 
  • #11
arivero
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Hmm, thinking again about this. Does it imply that tachions are not so serious problem IF the operator in the extra dimensions does not have zero modes?

Of course, any post-1905 crackpot can think: "inertia comes because we walk with a kind of angle into an extra dimension". And indeed, if one considers
[tex]E^2 - \sum^n_1 p^2_i= m_0^2[/tex]
and then postulates than some extra dimensions have a constrain
[tex]\sum^n_4 p^2_i = K_0^2[/tex]
the our particle has a fourdimensional mass such that
[tex]M_0^2 \equiv E^2 - \sum^3_1 p^2_i = m_0^2 + K_0^2[/tex]
 
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A scanned/copyfree pdf could be better...
I actually have that M.J.Duff selection as a PDF - can someone recommend a good site to upload it to? It is 47MB.
 
  • #13
arivero
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another fascinating remark of this paper is between pages 37 and 38, when he discusses the Rarita Schwinger field:
Although this field has spin 3/2 form the point of view of eleven dimensions, the components of [itex]\psi_\mu[/itex] with [itex]5 \le \mu \le 11[/itex] are spin one half fields from the point of view of ordinary four dimensional physics. For [itex]\mu \ge 5[/itex], [itex]\mu[/itex] would be observed as an internal symmetry index, not a space time index; it carries spin zero. Although the components [itex]\psi_\mu[/itex] with [itex]\mu= 1 \dots 4[/itex] are spin 3/2 fields in the four dimensional sense, the components with [itex]\mu= 5 \dots 11[/itex] are spin one half fields. So zero-mode solutions of the spin 3/2 wave equation in the extra dimensions would be observed as massless spin 1/2 fermions in four dimensions
He does not discuss what happens with spin 2 or spin 1 fields; bet it is trivial for him, but if it is trivial for someone here, I'd like to read some expanded explanation.
 

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