Monday, September 24, 2018

Atiyah's proposed proof of the Riemann Hypothesis


After much anticipation, M. Atiyah unveils his proposed proof of the Riemann hypothesis.  It depends on an interesting function called the Todd function.  His approach of course still must be properly evaluated and time will reveal if all assumptions are sensible.

Update: the Riemann hyptothesis remains unproven. Atiyah will release a more detailed treatment, but many remain skeptical.

Wednesday, March 14, 2018

Hawking's Legacy

On this 14th day of March, 2018, the world honors Stephen Hawking as an extraordinary scientist, teacher and futurist.  From A Brief History of Time to The Universe in a Nutshell, it is clear Hawking's deepest fascination has been the quest for a unified theory of physics.  In particular, Hawking was anticipating the completion of M-theory.

When can we expect the completion of M-theory?  Is it really an 11-dimensional completion of 11D supergravity and 10D superstring theory?  Or is it much more?  Ed Witten once mused, "String theory is 21st century physics that fell accidentally into the 20th century."  So the joke continues, in that 22nd century mathematics is needed to solve M-theory.  Might M stand for motive?  Time will tell.


Wednesday, November 22, 2017

Exceptional Periodicity

As was hinted at in a previous post, it is possible to view the exceptional Lie algebras as the tip of an infinite algebraic spectrum.  This novel concept, we coined Exceptional Periodicity (EP), is now available for download on the arXiv: arXiv:1711.07881 [hep-th].

This EP structure was inspired by certain Yang-Mills-like gradings of the exceptional Lie algebras, as well as higher dimensional spin groups, used in approaches to unification.  It differs from the conventional infinite dimensional generalizations of e8 in that the Jacobi identity is not in general obeyed by these higher algebras, yet do retain structure similar to lattice vertex algebras.  Moreover, building on the "Magic Star" projection of e8, each of these higher algebras can be projected to higher Magic Stars, that generalize that of e8.  At the six inner vertices of the star, the cubic Jordan algebras are generalized to a cubic ternary algebra, first envisioned by Vinberg, dubbed T-algebras.  Such T-algebras are reminiscent of spin factors and Peirce decompositions of cubic Jordan algebras.

The e8 Magic Star thus encodes the exceptional Jordan algebra on its six star vertices, which exhibits triality, from its off-diagonal 8D components.  These higher stars do not retain this triality, as the bosonic off-diagonal parts do not grow as fast as the spinor part, which grows exponentially.

So what can be done with these higher EP Magic Stars?  The T-algebras appear to encode a rich matter sector, that generalize the 16, 32, 64 and 128 spinors found in the exceptional Lie algebras.  Such higher stars can be used to design higher mathematical universes, in a periodic, algebraic fashion.  More details will be given in a series of upcoming papers.  Stay tuned.

Thursday, November 16, 2017

A Kind of Magic

It is a wondrous result that one can construct the Freudenthal Magic Square from the normed division algebras.  Using tensored division algebras to generate the Magic Square has also proven to be useful in the double copy procedure for Yang-Mills theories and gravity, resulting in the Magic Pyramid.  However, Landsberg and Manivel found midway between e7 and e8 there should be a non-reductive Lie algebra e7(1/2), which is related to the 'sextonions', a six-dimensional algebra midway between the quaternions and octonions.  This implies a generalization of the Freudenthal Magic Square and Magic Pyramid.  Marrani and Borsten took this analysis to its logical completion in A Kind of Magic, by filling in the "missing gaps" with the 3-dimensional "ternions" and the sextonions, and even studying their U-duality counterparts.  On the M-theory side, one can then ask, what type of compactification leads to such non-reductive Lie algebraic symmetries?  Or even better, are these symmetries hinting at something larger, beyond 11-dimensions? 

Saturday, November 04, 2017

The Mathematical Universe Hypothesis

Over at Backreaction, Max Tegmark's Mathematical Universe Hypothesis (MUH) was evaluated.  What is the MUH exactly?

The Mathematical Universe Hypothesis (MUH): Our external physical reality is a mathematical structure.
 What kind of structure could this be?  We have general relativity and quantum field theory, and these are based on (pseudo) Riemannian geometry and Lie algebraic fiber bundle theory, respectively.  Tegmark has asserted the MUH implies a so-called "Theory of Everything" (TOE) will be a purely mathematical theory.  This seems reasonable, and the devil is always in the details.  So what are the details?

One can approach the problem by uniting general relativity with quantum field theory, and this is usually dubbed quantum gravity.  Being that general relativity does not inherently contain, say, SU(3) symmetry, which is central to our understanding of quarks and gluons and baryonic matter in general, it is logical to seek a mathematical formalism that does include it, and attempt to derive general relativity at large scales.

Some remark that general relativity is akin to a hydrodynamic theory which cannot discern individual H2O molecules and their bonding properties.  In the case of spacetime, the "molecules" would analogously be gravitons.  Garrett Lisi's E8 model takes the Lie algebraic structure as an axiom and studies a unified model in which spacetime and the standard model arise from gradings of the largest exceptional Lie algebra.  Alain Connes refines fiber bundle theory with noncommutative geometry and takes a certain C*-algebra that acts on finite points as encoding the standard model over a 4D base space.  In string theory, there are D-brane models that encode standard model structure in worldvolumes, and this is also an example of noncommutative geometry.  Once the graviton is identified, it is possible, such as in string theory, to show large curvature in general relativity is equivalent to a coherent state of gravitons.

Surely progress is being made in our search for a TOE.  And the closer we get, Max Tegmark's hypothesis seems much more likely.  However, the MUH does not give the mathematical structure we seek.  To find such a structure, researchers must journey past the boundaries of known mathematics and physics.  And no single road will dominate all searches, and one must be versed in a myriad of approaches to arrive at the coveted TOE.

Wednesday, September 27, 2017

The Magic Star

In arXiv:1112.1258 [math-ph], P. Truini gave an elegant construction of the exceptional algebras, via a star-like projection under an A2 (su(3)).  For the F4, E6, E7, E8 cases, the 6 star vertices contain root vectors that transform as a Jordan algebra of degree three (containing 3x3 hermitian elements over the division algebras A=R, C, H, O).  In the center is the reduced structure group of the Jordan algebra.

In the maximal case, the Jordan algebra is the exceptional one, first mentioned in the famous quantum mechanical classification paper by Jordan, Wigner and von Neumann (1934).  This "E8 Magic Star" construction makes the symmetries of extremal black holes in D=4 N=8 SUGRA manifest, as one can take vertical slices and notice the 5-grading 1+56+(E7+1)+56+1 appears naturally.  Another application of the star would be to identify 27-dimensional Jordan algebraic star vertex as the matrix degrees of freedom in a "bosonic M-theory" Chern-Simons theory and notice the reduced structure group E6 is the full invariance group of its cubic form.  P. Truini and A. Marrani also proposed an emergent model based on this star projection.  (Note: there is also a spectacular way to extend this star beyond E8, coming in a forth-coming paper with the author and P. Truini and A. Marrani. Stay tuned!)

In Garrett Lisi's elementary particle explorer, the E8 Magic Star is visualized as:

One can view it by moving the V direction under the E6 Coxeter view mode with these settings:

Saturday, April 01, 2017

Generalizing Einstein's Spacetime

Now that we're in the year 2017, it is a great time to return to Einstein's view of spacetime.  One of his more memorable quotes on the subject is:

"People before me believed that if all the matter in the universe were removed, only space and time would exist. My theory proves that space and time would disappear along with matter."

In our current struggles to formulate quantum gravity, any fruitful approach to the problem must invariably include matter.  Hence, to speak of a purely spacetime-centered formulation of quantum gravity is doomed to failure.

Looking at the problem abstractly, let us return to the mathematics.  General relativity is a theory based on (pseudo) Riemannian geometry, which involves smooth (infinitely differentiable) manifolds with a metric.  We have come so far since the 19th century, in our mathematical view of manifolds.  The 19th century physicists and mathematicians did not have quantum theory in mind while developing their algebraic and geometric structures.  It took the early 20th century to introduce the quantum behavior of matter.

With quantum matter in hand, we can revisit Einstein's view of spacetime.  Indeed, Einstein rejected the probabilistic nature of quantum theory and this prejudice has remained for more than 100 years, for many physicists and mathematicians.  Yet with over 100 years of experimental verification, it is clear quantum theory is fundamental to our understanding of microscopic reality.

This begs the question: how does one build geometry, in the sense of Einstein's matter-driven spacetime, while also incorporating the quantum nature of matter?

We must approach this question with 20th century mathematics.  Many physicists are not aware that a revolution took place in mathematics, initiated by luminaries such as Emmy Noether and John von Neumann.  What Emmy suggested in the 1920's was to clarify the notion of generic point in an algebraic variety by using the following recipe:

1)  Start with the coordinate ring of an algebraic variety (the ring of polynomial functions defined over the variety)

2)  The maximal ideals of this ring will correspond to ordinary points of the variety (given suitable conditions are met)

3) The non-maximal prime ideals will correspond to the various generic points, for each subvariety and by taking all prime ideals, one recovers the whole collection of ordinary and generic points.

Emmy did not pursue this idea, but mathematicians such as Krull, Weil, Zariski, Serre, Chevalley, Nagata, Martineau and Grothendieck did follow the prescription and generalized to very general ring spectra.  In light of this, it is thus clear how Connes' noncommutative geometry is the more mature manifestation of Emmy Noether's vision.  The keyword here is generalized schemes.  What is a scheme? 

Definition:  A scheme is a topological space together with commutative rings for all its open sets, arising by gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets.

To generalize this to noncommutative rings, we declare:

Definition:  A noncommutative scheme is a topological space together with noncommutative rings for all its open sets, arising by gluing together spectra (spaces of prime ideals) of noncommutative rings along their open subsets.

To recover noncommutative geometry, and its spectral triples, we merely take the noncommutative ring to be a noncommutative C*-algebra.  The relevant topology is usually taken to be the Zariski topology.

Now, what does this mathematical machinery mean for physics and the study of spacetime?  The key is to view quantum theory in its abstract form.  We recall that matter is described by states in a Hilbert space and an algebra of observables acts on these states.  This algebra of observables is a noncommutative C*-algebra!

So let's consider X as our topological space of possible states of some physical system and the elements of C(X) (our noncommutative ring) are the observables for this system.  The value of an observable at a point in X is the result of our observation (which we perceive in the form of its spectrum, or eigenvalues).  The Zariski topology captures all the semidecidable properties one can decide using observations in C(X).  For example, an element in C(X) could give position as a semidecidable property, which is decided by computing the position to finite precision.

Since quantum theory has been shown to be built on noncommutative C*-algebras, it is natural to take a noncommutative scheme as our building block for spacetime.  The matter of the standard model comes in representations of the standard model gauge group SU(3) x SU(2) x U(1).  Connes has formulated a noncommutative model that incorporates the standard model symmetry by using a carefully selected C*-algebra for this spectral triple.  In light of grand unified theory, which must ultimately be re-cast in the noncommutative framework, one can consider much more general C*-algebras than that of Connes.  Also, one must keep in mind Minkowski space must be emergent in this choice of noncommutative scheme and its accompanied C*-algebra.  One might also suspect more general algebraic structures might become relevant at the big bang.  The search goes on.