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The 9/11 flip

Clifford Johnson at asks what is the origin of the term "9/11 flip" in theoretical physics - a concept that is much older than the 9/11 attacks. He writes down the correct answer but he also adds a lot of incorrect speculations and misinterpretations, together with David who has made a comment.

Let me clarify the situation

The fact is that the 9/11 flip was only found in the context of Matrix theory and it couldn't have been found earlier as I will explain. That's why the paper by Dijkgraaf, Verlinde, and Verlinde (DVV) from March 1997 is the first paper that uses the term "9/11 flip" while my earlier paper from January 1997 where matrix string theory was actually discovered - even though some slow physicists such as Clifford Johnson haven't yet noticed ;-) - was the first paper where the technique, as opposed to the term, was used.

Figure 1: children's version of screwing strings, as named by your humble correspondent, currently known as matrix strings

I chose to compactify the "X1" coordinate as opposed to DVV's "X9" which is why what I used was actually "the 1-11 flip in screwing string theory" rather than "9-11 flip in matrix string theory". ;-) Note that the 9/11 flip was discovered more than four years before the 9/11 attacks.

Why I am so certain that you won't find anything about the 9/11 flip before our papers, mine and DVV? Well, it's because the method itself couldn't be used earlier - i.e. before the Christmas Eve 1996 when I realized these things in Pilsen. The 9/11 flip is inherently linked to the discrete light cone quantization. It is very important that the circle "X-" or "X10" (or "X11") plays a very different role in the construction of the light-cone-quantized theory than "X9" (or "X1" using my conventions) does.

On the other hand, in all pre-matrix-theory papers such as the paper by Hull and Townsend on U-dualities (a paper mentioned by David as a potential candidate), all circular coordinates always play the same role. The permutation of "X9" and "X11" is thus a trivial "Z2" subgroup of the U-duality group (a large diffeomorphism, in fact) that doesn't deserve any special discussion and that is completely manifest in all pre-matrix-theory pictures or M-theory. Everything changes in Matrix theory which is why you need Matrix theory to talk about the 9/11 flip.

David from seems confused (is it David Gross?), too.

Discrete light-cone quantization

In order to explain what the term means in Matrix theory, we must first explain how you derive the model. Fine.

Return to 1996. Take M-theory in 11 dimensions. It is very mysterious as the name "M" suggests. But you want to write down a complete and well-defined Hamiltonian for this theory anyway. Can you do it? You bet. Follow Seiberg and Sen and compactify the light-like coordinate "X- = X0 + X10" on a circle so large that the identification becomes physically inconsequential (14 billion light years, if you wish). Because light-like compactifications may be subtle, deform the identification rule so that the identified points are space-like separated although the separation is "almost" null in the coordinate space.

Nevertheless, it is space-like anyway so you may apply the Lorentz symmetry of M-theory to change this almost null separation to a standard and very short space-like separation (the proper length of an almost null separation is very small). That makes it clear that physics of M-theory in 11 dimensions is equivalent to some limiting physics of M-theory compactified on a very short circle. The latter is known to be nothing else than type IIA string theory. The amount of momentum in the "X-" direction is translated to the number N of D0-branes in the resulting type IIA string theory - and this number must be sent to infinity if you want to describe a finite-momentum sector (with "P+ = N/R" kept fixed) in decompactified M-theory.

If you look at the limits that you need to get the original theory, you will see that the corresponding limiting physics in the type IIA string theory that is equivalent to M-theory in 11 dimensions is nothing else than the matrix quantum mechanics of massless open string modes ending on D0-branes, i.e. the BFSS matrix model: it is the dimensional reduction of the maximally supersymmetric U(N) gauge theory to 0+1 dimensions.

Note that the circle "X-" or "X10" (or "X11") must always be compactified and it is highly boosted in order to be able to find a matrix description of anything at all.

By the procedure above, we derived the exact Hamiltonian for M-theory - the whole superselection sector of the string-theoretical Hilbert space that approaches flat 11-dimensional spacetime at infinity. Can we find the exact Hamiltonian e.g. for type IIA string theory itself? Yes, we can. But we need to compactify one more circle besides "X-". It must be one of the nine normal transverse directions - I chose "X1" and DVV chose "X9". One must carefully distinguish the transverse "X1" or "X9" direction from the "systemic" almost-null direction "X-" or "X10" (or "X11"). There are two different type IIA string theories that you may "see" inside the compactified M-theory. The relation between them is as miraculous as the ability to use Matrix theory to study M-theory in general and jumping in between these two mental images is referred to as the 9/11 flip.

It sounds simple once we know how it works but it wasn't quite trivial to understand what's going on. The 9/11 flip was, for example, the main obstacle that prevented Edward Witten to understand matrix string theory for some time. ;-)


If you don't believe me, feel free to find a paper before DVV that uses the term "9/11 flip" or a paper before mine that uses the relation between two different type IIA string theories inside the same M-theory to achieve anything non-trivial. I predict that you will fail.

And that's the memo.

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