M-theory 11/11/11

With so many 11's around today, it seems fitting to mention some M-theory related material. A few days ago, Hisham Sati updated his On the geometry of the supermultiplet in M-theory paper which argues that the massless supermultiplet of D=11 supergravity can be generated from the decomposition of reps of the exceptional Lie group F4 and its maximal compact subgroup Spin(9). The dynamical origin of this is proposed to result from Cayley plane bundles over eleven-dimensional spacetime.

The Cayley plane, OP^2, is a projective plane over the octonions and its isometries form the group F4. Lines in OP^2 are 8-spheres and given any two points in OP^2 there is a unique 8-sphere passing through them. Given any three distinct points, if we apply an F4 transformation that fixes one of the points, we get a Spin(9) transformation.

In matrix parlance, F4 is the automorphism group of the algebra of 3x3 Hermitian matrices over the octonions, the exceptional Jordan algebra J(3,O). We can construct OP^2 using the rank one projectors of J(3,O). It can actually be defined as the space of all such rank one projectors. Normalizing the rank one projectors turns them into primitive idempotents, that is, matrices P that satisfy P^2=P which cannot be decomposed as an orthogonal sum of other idempotents. As the identity matrix of J(3,O) is just a 3x3 matrix with ones on the diagonal, its straightforward to see that it decomposes into an orthogonal sum of three primitive idempotents. This is called the capacity and is why J(3,O) is an algebra of degree three.

Going back to the geometry of OP^2, the three distinct points mentioned earlier can be interpreted as three orthogonal primitive idempotents of J(3,O) with orthogonality being a result of these matrices satisfying P1.P2=0 under regular matrix multiplication. To simplify the picture, let's just imagine applying an F4 transformation on the identity matrix where we want to keep one of the diagonal ones fixed. This can be done with a Spin(9) transformation. Since we can fix any of the three diagonal ones of the identity matrix, there are three copies of Spin(9) inside F4 we can use. This freedom of choice we have is what some people refer to as triality.

QM over Split Composition Algebras

Over at viXra log, Philip Gibbs had a nice post on quantum mechanics and non-locality. In traditional quantum mechanics, it is often assumed one is constructing projective spaces over the complex field. However, as John Baez has noted at the n-category cafe, one can always formulate quantum mechanics over the quaternions and octonions as well. In order for octonionic quantum mechanics to be properly formulated, the Jordan formulation must be used in order to define projective spaces. Even then, one is limited to constructing a projective plane in the best case, due to algebraic topological constraints.

Back in my undergrad days, I was interested in studying quantum mechanics over arbitrary division algebras, which inevitably leads to the study of Jordan algebras as normed spaces over the reals. In the octonionic case, first studied by Jordan, Wigner and von Neumann back in the 1920's, one can have an algebra of 3x3 Hermitian operators in the maximal case. This case yields the exceptional Jordan algebra, with its corresponding projective space OP^2, the Cayley-Moufang plane. Even in this somewhat pathological case, it is possible to construct a 27-dimensional normed vector space over the reals. This is done by defining an inner product on the exceptional Jordan algebra, (X,Y)=tr(XoY), which induces a positive definite form, the norm, (X,X)=tr(X^2)=|X|^2. This norm also works for any nxn Jordan algebras over R,C,H. In all these cases, the length of a Hermitian operator is zero if and only if it's the zero vector (zero matrix). This means, in particular there are no rank one operators with zero length, and hence our projective spaces as manifolds, are easily described with the number of charts given by the degree of the Jordan algebra. In quantum mechanics this means we can normalize our rank one operators and the norm squared acquires a nice probabilistic interpretation.

When one attempts to give a similar normed space construction for Jordan algebras over the split composition algebras, it turns out the story isn't so nice. The first property that goes out the window is positive definiteness. So in quantum mechanics over split composition algebras there are a bunch of rank one projectors that have zero length. To this, one may say, "so what?" Well, for one, one can't assign a probabilistic interpretation to pure states described by these vectors. Now one may reply, "so just mod these out and define your projective space accordingly" Sure, we can try to do this but what if the physics actually requires the use of these pathological rank one operators?

Quantum mechanics over split composition algebras has already found use in M-theory compactifications, especially in describing extremal black hole charge vectors. In M-theory on T^5 and T^6, the charge vector spaces are actually Jordan algebras over the split octonions. In the black hole context, rank one operators describe 1/2 BPS states with zero entropy. This can be seen by noting rank one operators are those with zero determinant. So what does the (semi)norm mean in this context? I'm not really sure yet. In a literal sense, it gives the distance squared of an operator from the zero matrix. If one borrows some terminology from D-brane constructions, perhaps the norm can be interpreted as giving a type of tension, proportional to some theoretical mass. This would give an interpretation to the non-trivial charge vectors with zero norm: they describe some type of "massless" 1/2 BPS black holes. The other non-zero norm, rank one charge vectors describe "massive" 1/2 BPS black holes. The spectral decomposition of a full rank 3x3 Hermitian operator in the charge space then says that a 1/8 BPS black hole can be viewed as a bound state of elementary massive 1/2 BPS black holes, in some sense.

Universe as Black Hole Quantum Computer

In my last post I entertained the idea of a possible M-theoretical computronium. By definition, computronium is a programmable substrate which can model virtually any object. So M-theory computronium is a realization of such a hypothetical substrate, using objects from M-theory. Along these lines, using the qudit/qubit correspondence, M-theory computronium would essentially be a substrate of programmable extremal black holes. So is an M-theory universe ultimately just a large-scale black hole quantum computer?

Leading researchers such as Seth Lloyd and David Deutsch have argued that the universe, as a giant quantum mechanical system, is indistinguishable from a big quantum computer. Lloyd has stated [1] that since all elementary particles and their interactions register and process quantum information, the universe is constantly performing quantum computations. Moreover, since elementary particles such as the electron and photon can be mapped to qubits and qutrits, their interactions can be seen as the result of quantum logic operations, i.e., quantum computational gates. Hence, the collection of all such qudit computations is indistinguishable from the universe itself.

The arguments by Lloyd and Deutsch are convincing, but from the perspective of string/M-theory we can go one step further, and note that all elementary particles actually arise from more fundamental higher-dimensional objects. In string theory, the popular explanation was that all elementary bosons and fermions arise from vibrations of a string. However, in 1995, after Edward Witten showed that the five known, consistent superstring theories are related by dualities, and it became clear that there was a deeper theory in 11-dimensions, called M-theory that was behind it all. Such a theory, at low energies becomes the unique D=11 supergravity, and contains (among other things) two-dimensional and five-dimensional branes (M2-branes and M5-branes), but no strings. So, from an M-theory perspective all elementary particles should arise from objects in eleven-dimensions, such as the M2 and M5-branes and their various configurations.

In attempting to recover elementary particles from M-theory one might attempt to study BPS-saturated solutions in supergravity theories, which are believed to survive at the full quantum level and give a glimpse of the true, non-perturbative structure of M-theory. The easiest way to find such BPS solutions is searching for extremal p-brane solitons, either in D=10,11 or in Kaluza-Klein reductions to lower dimensions. The Kaluza-Klein reductions, which include toroidal compactifications of M-theory, are especially quite nice as the procedure preserves all the original supersymmetry of the full D=11 configuration. This means, given a lower dimensional extremal BPS solution, it can be "oxidized" back up into a higher dimensional supergravity solution that preserves the same amount of supersymmetry.

The simplest extremal BPS solutions are those that preserve 1/2 of the original supersymmetry. These are solutions that contain a single charge, carried by a single field strength in supergravity. Such solutions arise in toroidally compactified M-theory down to dimensions D=3,4,5,6, and take the form of 1/2 BPS black holes.

In D=4,5,6 compactifications (M-theory on T^7, T^6 and T^5), Duff et al. noticed [2] that extremal black holes and entangled qubits share the same invariants and algebraic structures. On the M-theory side the invariants give the entropy of the black holes, while also helping to classify the various BPS solutions. On the quantum information side, the invariants help to classify the entanglement classes for qubits and qutrits. In D=3 compactifications, this black hole/qudit correspondence was even used by Levay [3] and Duff [4] to predict 9 entanglement families for four entangled qubits, where in quantum information theory the exact number has yet to be determined and predictions range from 8 to 21 families.

Recently, it has been shown [5] that by defining generalized qubits and qutrits over composition algebras (e.g., the quaternions, octonions and their split forms, etc.), it is possible to directly identify 1/2 BPS black hole solutions in D=5,6 with these generalized qubits and qutrits. This allows one to interpret qubits and qutrits (qudits) with the simplest extremal black hole solutions that contain a single charge and preserve 1/2 supersymmetry. The U-duality groups of the corresponding D=5,6 supergravity theories are then interpreted as transformations of these qudits through stochastic local operations and classical communication (SLOCC). This gives rise to new kinds of SLOCC gates in quantum information theory, which in the case of a non-associative composition algebras, endows qubits with SO(9,1), SO(5,5) symmetry and qutrits with the symmetry of the E6 exceptional Lie group.

It is quite remarkable that E6 can be interpreted as the SLOCC symmetry group of qutrits over non-associative composition algebras. Moreover, from the viewpoint of interpreting the universe as a quantum computer, it's quite desirable to have a quantum computer that processes quantum information with E6 symmetry. This is because in grand unification theories E6 is a possible gauge group which, after symmetry breaking, gives rise to the SU(3)xSU(2)xU(1) gauge group of the standard model of particle physics.

Hence, by studying BPS-saturated solutions in M-theory on T^6 (D=5, N=8 supergravity), and interpreting the simplest 1/2 BPS solutions within quantum information theory, we are inevitably led to the picture of a quantum computational theory containing qutrits with E6 symmetry. [Note: M-theory on T^6 actually has E6(6) non-compact U-duality symmetry, but upon using the full bioctonion algebra for the qutrits, compact E6(C) is recovered.] This quantum computational theory, for all practical purposes, is indistinguishable from an E6 grand unified theory, from which the standard model can be recovered. However, here, the local geometry is inherently nonassociative, as each black hole charge space, being a nonassociative C*-algebra, is associated to a spectral triple.

Ultimately, using the simplest solutions in M-theory that preserve half of their higher-dimensional supersymmetry, we arrive at a picture of the universe as a quantum computer that encodes information in the form of black holes with zero entropy. The logical operations on these black holes, as qudits, transform states within an exceptional projective space, preserving the entropy of the black holes in the process. In this picture, the ten-dimensional Lorentz group SO(9,1) and the D=5 T-duality group SO(5,5) take the form of groups of qubit transformations, which can be embedded inside E6 qutrit transformations. Thus, the dreams of Lloyd and Deutsch might eventually be realized if our universe is described by M-theory. And such a universe is computationally elegant indeed.

M-theory Computronium

A fine post at the Physics and Cake blog got me thinking about computronium and how it might be realized in M-theory. Of course, this is a purely theoretical musing, but nevertheless is worthy of some consideration. For surely any advanced intelligent civilization, who have already solved M-theory will necessarily develop advanced technology that makes use of quantum gravity and its higher dimensional physics. This will especially be the case in the area of computational technology. So, using M-theory, what form might such computational technology take? Is there an M-theoretical computronium? If so, how do we program it?

Surely, the ultimate computronium is the quantum vacuum itself. Along these lines, in a string/M-theory context, by invoking the correspondence between black holes and qubits, one can see hints as to how the vacuum might eventually serve as a computational substrate. See, for example:

L. Borsten, M.J. Duff, A. Marrani, W. Rubens, On the Black-Hole/Qubit Correspondence.

In M-theory, there exist stable non-perturbative states (BPS states) with mass equal to a fraction of the supersymmetry central charge. These states arise from configurations of two and five-dimensional branes, gravitational waves and Taub-NUT-like monopoles. (Note there are no superstrings in M-theory. They arise from compactifications of M-branes in dimensional reduction from D=11 to D=10).

The black hole/qubit correspondence so far has made use of toroidal compactifications of M-theory. That is, one begins with the full 11-dimensions of M-theory and starts to curl up dimensions so that n of them form a higher-dimensional torus (doughnut shape), T^n. This then describes a lower dimensional supergravity theory, in D-n dimensions.

In the D-n dimensional supergravity theory, some BPS states arising from configurations in M-theory behave like microscopic black holes. These black holes are called extremal black holes, as they can be thought of as the ground states of black holes undergoing Hawking radiation. These states have no analog in general relativity, but do exist in supergravity and M-theory which consider quantum effects.

So far what has been found is that in M-theory compactifications down to dimensions D=3,4,5,6, BPS black hole solutions behave like entangled qubits and qutrits. More precisely, the invariants used to classify black holes with different fractions of supersymmetry, end up being the same invariants used to classify entanglement classes of qubits and qutrits. Even more, the black hole mathematical techniques classify qubits and qutrits over not only the real and complex numbers, but over higher dimensional division algebras in four and eight dimensions. So string theory actually predicts new types of qubits and qutrits and classifies their entanglement classes in advance.

Now, in practice, if M-theory is correct, the vacuum should be teeming with such microscopic black holes. They would, in a sense, serve as the qudits of an M-theoretical computronium. Specific types of transformations in M-theory called U-duality transformations, that map between BPS black hole solutions, would then serve as ‘quantum gates’ for these qudits.

Hence, to tell the M-theory vacuum what we would like to do, amounts to the programming of microscopic black holes via U-duality machine code.

Motives, Twistors and Amplitudes

Nima Arkani-Hamed gave a recent talk on 01/26 entitled "Space-Time, Quantum Mechanics and Scattering Amplitudes". He essentially covers all the recent progress in the study of scattering amplitudes in dual twistor variables. He ends with hints at an underlying theory that gives rise to AdS/CFT and QFT, which might be based on the mathematical theory of motives.

For those unfamiliar the theory of motives, the goal within the mathematical community is to define a unified cohomology theory, from which all others (de Rham, Čech, singular, etc.) are special cases. It is interesting that a unified theory of physics would coincide with this platonic goal of mathematicians. Perhaps Edward Witten foresaw such a convergence and the 'M' of M-theory stood for motive all along. Either way, category theorists saw this coming a few years ago.

How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?

— Albert Einstein