Tuesday, December 13, 2011

Unofficial Higgs combined plots are in

As promised, Philip Gibbs has produced combined plots for the Higgs mass, which includes data from LHC, Tevatron and LEP. Notice that nice peak centered at 124-125 GeV!

If this is a light E6 GUT Higgs, we'd expect to see a light isosinglet quark such as the D quark at the LHC very soon. We should even be able to predict its mass (~>250 GeV). A Z' boson would also be nice. For more info, see slides here and here. On to 2012!

Higgs rumors were correct

The official results are in, at least for the LHC's 2011 data, and it appears the rumors were quite accurate. See TRF and QDS for further details. To quote CMS member Dorigo, who stated there is now "Firm Evidence" with the current data,

So the summary is that ATLAS has a 3.6-sigma significance at 126 GeV, by combining their three most sensitive channels; CMS has a 2.4-sigma significance at 124 GeV, by combining all the meaningful search channels -even less sensitive ones.

After an ATLAS and CMS combined plot is produced, there might very well be over 4 sigma evidence for a light Higgs. I'm certain Philip Gibbs is working on a combined plot at this very moment. There is still a Christmas present to be delivered! Stay tuned.

Monday, December 12, 2011

Why a light Higgs is cool

As to why many theorists are excited over news of a possible light Higgs boson with 125 GeV mass, here's a memorable excerpt from a September 2011 interview with Clerk Maxwell Professor of Theoretical Physics and former CERN staff member John Ellis:

In the first scenario (114-135 GeV), we could be looking at a Standard Model Higgs boson. This range has been refined experimentally: recent LHC results presented in Mumbai excluded the Standard Model Higgs from about 135 GeV to about 500 GeV, while LEP had previously excluded it up to 114GeV. That leaves a narrow low-mass range of about 20 GeV where it could lie. But if found in this range, the Standard Model theory would still be incomplete; the present electroweak vacuum would be unstable for such a light Higgs in the Standard Model, so we would have to come up with new physics to stabilise it.

Higgs Candidate Events

While there has been no official announcement on the possible Higgs mass from CERN, there are some nice images available on possible candidate events where the Higgs might have appeared, as mentioned at TRF.

Candidate events in the CMS Standard Model Higgs Search using 2010 and 2011 data
(Click images to enlarge)

A typical candidate event including two high-energy photons whose energy (depicted by red towers) is measured in the CMS electromagnetic calorimeter. The yellow lines are the measured tracks of other particles produced in the collision.

A typical candidate event including two high-energy photons whose energy (depicted by red towers) is measured in the CMS electromagnetic calorimeter. The yellow lines are the measured tracks of other particles produced in the collision. The pale blue volume shows the CMS crystal calorimeter barrel.

Real CMS proton-proton collision events in which 4 high energy electrons (green lines and red towers) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

Real CMS proton-proton collision events in which 4 high energy electrons (green lines and red towers) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

Real CMS proton-proton collision events in which 4 high energy muons (red lines) are observed. The event shows characteristics expected from the decay of a Higgs boson but is also consistent with background Standard Model physics processes.

All images and descriptions copyrighted property of © 2011 CERN and used for educational purposes.

Friday, December 09, 2011

Higgs rumors at 124.6 GeV

December 13 comes ever closer and the rumors about the Higgs mass get more detailed. Lubos Motl has commented on a recent post at QDS by Tommaso Dorigo in which he seems to hint at a possible Higgs mass from diphoton Higgs decay channels

- gamma: a gamma-ray is a photon, i.e. a quantum of light. A very energetic one, to be sure: a gamma ray is such only if it carries significantly more energy than a x-ray, so above a Mega-electron-Volt or so. The gammas we will be hearing about are those directly coming from a Higgs boson decay, and these have an energy of 62.3 GeV, equivalent to the kinetic energy of a mosquito traveling at 9 centimeters per second.

Here, the Higgs mass 124.6 GeV = 62.3 GeV x 2, from a process that can be written as H -> gamma gamma - where the Higgs decays to two high energy photons. Of course, Tommaso admits
I teased my most gullible readers with a (wrong) covert give-away of the Higgs mass ...
Either way, it is fun to speculate when the actual announcement is only a few days away. So let's see how close this 124.6 GeV is to the official (statistical) CMS value on Monday.

Friday, December 02, 2011

Higgs rumors at 125 GeV

As we all await CERN's official CMS and ATLAS results for the 2011 Higgs hunt, rumors about its mass have surfaced at notable blogs such as Philip Gibbs' viXra log, Peter Woit's Not Even Wrong and Tommaso Dorigo's Quantum Diaries Survivor. As mentioned by "Alex" in the viXra comment section,
Today rumour is: Higgs at 125 Gev around 2-3 sigma…
Such a rumor, if true, would not only indicate evidence for the existence of the Higgs boson, but is evidence for a light Higgs boson (115-135 GeV), which popular models such as E6 GUTs and M-theory on G2-manifolds predict. Of course, 2-3 sigma evidence isn't really conclusive but it does favor physics beyond the Standard Model. These are exciting times and by December 12 and 13 we'll all get to see if the rumors are true. Moreover, Philip Gibbs has also promised everyone a combined CMS and ATLAS plot once the data is released. How's that for an early Christmas present?

Update: Over at Lubos Motl's TRF blog, a commenter "azerty13" said he received the following email from CERN Director General Rolf Heuer:
Dear colleagues,

I would like to invite you to a seminar in the main auditorium on 13 December at 14:00, at which the ATLAS and CMS experiments will present the status of their searches for the Standard Model Higgs boson. These results will be based on the analysis of considerably more data than those presented at the Summer conferences, sufficient to make significant progress in the search for the Higgs boson, but not enough to make any conclusive statement on the existence or non-existence of the Higgs. The seminar will also be webcast.

Rolf Heuer

Such an email, if genuine, definitely supports the 2-3 sigma portion of the 125 GeV Higgs mass rumor. Stay tuned.

Update: As mentioned at viXra log, the latest incarnation of the rumor at Woit's blog gives 3.5 sigma in ATLAS and 2.5 sigma in CMS which amounts to about 4.3 sigma combined for the 10/fb. Keep in mind 5 sigma evidence is what is required at this stage of the Higgs hunting game.

Friday, November 11, 2011

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.

Thursday, November 03, 2011

Creation of Matter and E7

Recently, Ferrara and Kallosh posted a paper on arxiv entitled Creation of Matter in the Universe and Groups of Type E7. The abstract is as follows:

We relate the mechanism of matter creation in the universe after inflation to a simple and universal mathematical property of extended N > 1 supergravities and related compactifications of superstring theory. We show that in all such models, the inflaton field may decay into vector fields due to a nonminimal scalar-vector coupling. This coupling is compulsory for all scalars except N=2 hyperscalars. The proof is based on the fact that all extended supergravities described by symmetric coset spaces G/H have duality groups G of type E7, with exception of U(p,n) models. For N=2 we prove separately that special geometry requires a non-minimal scalar-vector coupling. Upon truncation to N=1 supergravity, extended models generically preserve the non-minimal scalar-vector coupling, with exception of U(p,n) models and hyperscalars. For some string theory/supergravity inflationary models, this coupling provides the only way to complete the process of creation of matter in the early universe.

As mentioned in the article, one can consider d=4, N=8 supergravity arising from M-theory on T^7, with duality group G=E7(7) acting on a 56-dimensional Freudenthal triple system (FTS) over the split-octonions. Another type arises from the N=2 magic supergravity based on the FTS over the octonions, with duality group G=E7(-25). At an algebraic level, E7(7) and E7(-25) are unified in the complexified duality group G=E7(C) acting on the FTS over the bioctonions. There is, however, no corresponding E7(C) supergravity theory at this time.

Friday, September 09, 2011

Dilogarithm Motives in Physics

Click Here for Talk

About this episode

Spencer Bloch

"Dilogarithm Motives Arising in Physics"

Talk given at "Algebraic Geometry, K-theory, and Motives" (a conference dedicated to Andrei Suslin's 60th birthday) St. Petersburg, Russia June 25-29, 2010.

In this very clear talk, Bloch explains how one is often writing down interesting periods on n-dimensional projective space when doing high energy physics. One can start with a hypersurface defined by the vanishing of a certain configuration polynomial F of degree d. F determines the hypersurface X, and X determines the motive.

Tuesday, May 17, 2011

Harmony of Scattering Amplitudes

The KITP program The Harmony of Scattering Amplitudes is still underway and there have been many wonderful talks on the geometry of scattering amplitudes in twistor space. The twistor approach allows one to look at scattering processes in a more algebraic geometrical fashion where as Ed Witten noted (arXiv:hep-th/0312171), one should focus on holomorphic curves.

At the most basic level, one is interested in degree one genus zero curves. In the complex case, such curves are copies of CP^1, 2-spheres. Witten argued that n-particle MHV amplitudes with two particles of negative helicity and n-2 with positive helicity localize on such degree one genus zero curves. This is the special case of Witten's more general conjecture that the twistor version of the n particle scattering amplitude is nonzero only if the points are supported on an algebraic curve in twistor space of degree d=q-1+l (where q is number of negative helicity particles and l is the number of loops). So for example, the tree level ++--- amplitude is nonzero on a curve of degree d=2-1+0=1, a degree one genus zero curve, a 2-sphere, as expected (see diagram above).

More recently, there is a more combinatorial way to view the MHV (and N^kMHV) amplitudes. This approach allows one to use associahedra, bubble diagrams and chorded polygons, for example. Below is a chorded polygon for the ++--- amplitude, and up to rotation and CPT transformation, is the only one contributing to the amplitude. For the --+++ amplitude there is another such polygon, so for the n=5 MHV amplitudes only (2(n-3))!/(n-3)!(n-2)!=2 total chorded polygons contribute.

In such chorded polygons the chords physically correspond to twistor fields exchanged between degree one genus zero instantons. So given an n-point N^kMHV amplitude, one can draw many different chorded polygons (given by Catalan number C_{n-2}), but those which contribute are those that have no internal twistor field triangles, and describe configurations where each genus zero curve in the process has at least two points with different helicities. Below is a diagram for a non-contributing chorded polygon for the n=8 NNMHV amplitude (note the "illegal" internal twistor field triangle).

As Nima Arkani-Hamed has noted, the twistor approach to scattering amplitudes is revealing a deeper mathematical unity that Feynman diagrams obscure. The mathematics so far involves the Riemann moduli space of surfaces of genus g with n marked points, Gromov-Witten invariants, symplectic geometry, quantum cohomology and motives.

Tuesday, April 05, 2011

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.

Thursday, March 31, 2011

FBI can't crack code. Can you?

For those interested in a cryptographical challenge, the Feds are asking for help on the cracking of a code written by Ricky McCormick, 41, three days before he was found dead on June 30th, 1999 in a St. Louis field.

"We are really good at what we do," said Dan Olson, the chief of the FBI's Cryptanalysis and Racketeering Records Unit. "But we could use some help with this one."

Read more on the challenge at MSNBC.

Is N=8 Supergravity Finite?

In a recent paper arXiv:1103.4115 [hep-th], Renata Kallosh shows that E7(7) U-duality predicts the all-loop UV finiteness of perturbative N=8 supergravity.

Tuesday, March 08, 2011

Automorphic Instanton Partition Functions on Calabi-Yau Threefolds

Persson posted a nice paper recently, exploring the relationships between Calabi-Yau threefolds, U-duality groups and automorphic instanton partition functions. The paper discusses recent attempts at describing the moduli space in type IIA/B string theory on X×S^1, or the hypermultiplet moduli space in type IIB/A on X. It is well known that in certain classes of N=2 supergravities (e.g. magic supergravities) one can use automorphic techniques to constrain quantum corrections. Given a D=3 U-duality group G_3(Z), BPS-degeneracies are recovered from the Fourier coefficients of its related automorphic forms. From this stems the conjecture that the instanton partition function in N=2 supergravity on R^3×S^1 should correspond to an automorphic representation in the quaternionic discrete series of G_3.


Abstract: We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that whenever the low-energy theory in D=3 exhibits an arithmetic "U-duality" symmetry G(Z) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.

Monday, March 07, 2011

Ferrara: Black Holes and Supergravity

Part I: The Attractor Mechanism

Part II: N=8 Supergravity and Black Hole Charge Orbits

November 10, 2010
Lectures given for 25th Anniversary of ICTP's Dirac Medal, ICTP, Trieste, Italy.

Wednesday, February 09, 2011

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.

Monday, February 07, 2011

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.

Sunday, January 30, 2011

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