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.


Friday, January 27, 2017

Triality Simulation














It is quite popular in the media these days to think of the universe as a simulation.  But what kind of simulation could this be?  First off, what is the difference between something that is "real" and a simulation?  The revelation would be that there is no difference between what we consider real and what we behold as simulated.

Indeed, billions of dollars could be used to prove we are in a simulation. However, that venture must begin with a theoretical investigation.  And theoretical physics is the correct domain to begin this venture, by discovering the unified theory.

So what is a unified theory?  Is it a theory of our observable universe?  Or can we do much better and discover the theory behind all possible simulated universes?  The answer should be clear.