Last night (18th April) I attended a talk given by Prof. Alain Connes as part of the The Learned Society of Wales Frontiers lectrures. The talk was entitled “The spectral point of view on geometry and physics”.
The talk was very interesting and not too technical. Prof. Connes outlined his philosophy that the physical word should be described by spectral data. This idea really leads to the notion of noncommutative geometry, something Prof. Connes is well-know for.
Connes philosophy comes from many facts of physics. For example, our knowledge of the shape of the Universe comes from spectra data, the red shift and the CMBR. The meter is defined in a natural way in terms of the wavelength of the krypton-86 emission.
The key idea
The question that Connes really tackled was can we understand geometry spectrally? Connes was motivated by the Atiyah-Singer index theorem, which gives topological data about a space from analytical data about operators on that space.
The fundamental idea is that one can restate Riemannian geometry in terms of the spectra of the Dirac operator on that geometry. The topological (smooth) structure of the manifold is recovered from the algebra of (smooth) functions and the metric structure from the spectra of the Dirac operator.
One can then understand a smooth manifold with a Riemannian metric in terms of an algebra of functions and an operator acting on them.
Note that we do not need the notion of points in this spectal description. In fact, this reformulation of Rienamnnian geometry allows one to define metrics on non-commutative spaces, which are really just algebras.
Applications
One of the main hopes of Connes reformulation of Riemannian geometry is that, via non-commutative geometry, one can classically unite the standard model of particle physics with general relativity in a geometric way. In doing so, it may be possible to construct a unified theory, but Connes is not at that stage.
As it stands, Connes theory does not quite match the standard model and there is also the problem of Lorentzian signature metrics. Not having positve definite metrics almost always makes details of the mathematics tricky.
This should not distract from the fact that Connes is a pioneer of non-commutative geometry and mathematically his work is very important.
The talk itself
Connes is a good and entertaining speaker. If you get chance to listen to him, you should take it.
http://en.wikipedia.org/wiki/Symmetric_space
Riemannian geometry
How do non-commutative geometries generate symmetric spaces? Soibelman with Kontsevich explaining mirror symmetry through homological algebra and non-commutative geometry is unclear to me. “Mirror symmetry is residual commutative geometry.” OK, then “non-commutative geometry of mirror symmetry is the geometry of deformed Calabi-Yau manifolds.” But Calabi-Yau manifolds are mirror symmetric, yes? They manually inserted the answer.
arxiv:hep-th/0504084, Non-Commutative Geometry and Twisted Conformal Symmetry “…conformal invariance need not be viewed as incompatible with non-commutative geometry; the non-commutativity of the coordinates appears as a consequence of the twisting…” Adding machine tape is flat and 2-D. Twist along the long axis. There was no intrinsic twist to the tape before you twisted it.
If non-commutative treatments work to unite GR and the standard model, assumptions of mirror-symmetry toward mass are incomplete. GR should bobble angular momentum. The standard model is a parity violation pigsty. You’ve accomplished the fifth labor of Hercules.
Phys. Rev. Lett. 108, 165502 (2012) “Chiral Surfaces Self-Assembling in One-Component Systems with Isotropic Interactions”
http://prl.aps.org/abstract/PRL/v108/i16/e165502
http://physics.aps.org/articles/v5/45
Spontaneous emergence of resolved chirality, as opposed to a racemic mixture, is a big banger. If gravitation is non-commutative algebra, chiral outcomes are deeply favored. This is important.
Assumptions of rigorous vacuum mirror symmetries toward fermionic mass, as opposed to massless boson photons, have no empirical validation and suffer presumptive falsifications (symmetry breakings). Test for trace vacuum chiral background (left foot) with enantiomorphic atomic mass distributions (opposite shoes): space groups P3(1)21 versus P3(2)21 alpha-quartz single crystal test masses in a geometric parity Eotvos experiment. Everything exactly cancels except the orthogonal geometries. If quartz is too New Age, use P3(1) vs. P3(2) gamma-glycine single crystal test masses.