Tuesday, October 23, 2018

String (M-) Theory Pros and Cons















Over at backreaction, a video was made discussing the pros and cons of string theory.  As stated in the video, string theory was first proposed (in the late 60's) as a theory to describe the nuclear force.  String theory didn't fare so well as such, but was regarded as promising by luminaries such as Nobel prize winner Murray Gell-Mann, who offered John Schwarz and his colleagues funding to keep researching the theory at Caltech, Pasadena, CA.  With time, string theory became better suited as a unified theory of all forces and particles and is studied at all major universities around the world.

Note: Schwarz is still at Caltech, and holds promise for the future of string theory, and its modern incarnation M-theory.

Let's start with the pros of string theory:

  • It extended the study of quantum field theory beyond the mere study of point particles.  This brought conformal transformations to the forefront of scattering amplitude calculations.
  • The theory of vertex operator algebras, now a prominent branch of mathematics, came from the original "dual resonance model", the first version of string theory.
  • Closed strings naturally give a graviton, which along with open strings, can provide a means to unify gravity with the other fundamental forces.
  • Compactifications of D=10 string theory on Calabi-Yau spaces brought a deeper understanding in mathematics, giving a way to relate such spaces via mirror symmetry.
  • The mysterious Monster group, the largest of the sporadic groups, was found to act as the symmetry of a certain vertex operator algebra for string theory on a 24-torus.
  • M-theory as a matrix theory and D-branes with their noncommutative coordinates, have helped give a deeper understanding of noncommutative geometry, which is a branch of mathematics that asserts that operator algebras can be used to generate the quantum analog of smooth manifolds.
  • M-theory was shown in the late 90's to reduce to 11-dimensional supergravity, a theory Stephen Hawking once considered the most exciting candidate for a unified theory.
As far as cons:

  •  There are vastly many ways to reduce to four spacetime dimensions from D=10 (or D=11in the case of M-theory).  The space of possible compactifications is called the landscape.
  • String theory natural enjoys supersymmetry, and one must break supersymmetry to recover the standard model of particle physics we observe at accelerators.
  • It has proven experimentally difficult to verify string theory, as most of the structure of models is to be found at extremely high energies.
  • After the introduction of M-theory by Edward Witten, the 11-dimensional model has yet to take a final form.  The deep structure of M-theory remains a mystery.
 The mystery of M-theory is actually a blessing in disguise, as many have remarked it will take new mathematics to understand the theory.  Time will tell what surprises M-theory, and quantum gravity in general, have in store.

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.


Friday, May 18, 2018

First testible prediction of string theory?
















I. Bena et al. released a paper (arXiv:1805.06463) "Holographic dual of hot Polchinski-Strassler quark-gluon plasma" a few days ago in which the supergravity dual of hot quark-gluon plasma is studied and a particular phase transition occurs with the 5-sphere horizon topology.  This feature could be checked, they claim, by computing the theory on a lattice (4D super-QCD) and giving the first practical prediction of string theory and AdS/CFT.

Note: the manuscript is dedicated to J. Polchinski.


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.