The purpose of this post is to explain a recent discovery that we did with my two physicists collaborators Ali Chamseddine and Slava Mukhanov. We wrote a long paper Geometry and the Quantum: Basics which we put on the arXiv, but somehow I feel the urge to explain the result in non-technical terms.
The subject is the notion of particle in Quantum Gravity. In particle physics there is a well accepted notion of particle which is the same as that of irreducible representation of the Poincaré group. It is thus natural to expect that the notion of particle in Quantum Gravity will involve irreducible representations in Hilbert space, and the question is "of what?".
What we have found is a candidate answer which is a degree 4 analogue of the Heisenberg canonical commutation relation [p,q]=ih. The degree 4 is related to the dimension of space-time. The role of the operator p is now played by the Dirac operator D. The role of q is played by the Feynman slash of real fields, so that one applies the same recipe to spatial variables as one does to momentum variables. The equation is then of the form E(Z[D,Z]^4)=\gamma where \gamma is the chirality and where the E of an operator is its projection on the commutant of the gamma matrices used to define the Feynman slash.
Our main results then are that:
1) Every spin 4-manifold M (smooth compact connected)
appears as an irreducible representation of our two-sided equation.
2) The algebra generated by the slashed fields is the algebra of functions on M
with values in A=M_2(H)\oplus M_4(C), which is exactly the slightly noncommutative
algebra needed to produce gravity coupled to the Standard Model minimally
extended to an asymptotically free theory.
3) The only constraint on the Riemannian metric of the 4-manifold is that its volume
is quantized, which means that it is an integer (larger than 4) in Planck units.
The result 1) is a consequence of deep results in immersion theory going back to the work of Smale, and also to geometric results on the construction of 4-manifolds as ramified covers of the 4-sphere, where the optimal result is a result of Iori and Piergallini asserting that one can always assume that the ramification occurs over smooth surfaces and with 5 layers in the ramified cover. The dimension 4 appears as the critical dimension because finding a given manifold as an irreducible representation requires finding two maps to the sphere such that their singular sets do not intersect. In dimension n the singular sets can have (as a virtue of complex analysis) dimension as low as n-2 (but no less) and thus a general position argument works if (n-2)+(n-2) is less than n, while n=4 is the critical value.
The result 2) is a consequence of the classification of Clifford algebras. When working in dimension 4, the sphere lives in five dimensional Euclidean space and to write its equation as the sum of squares of the five coordinates one needs 5 gamma matrices. The two Clifford algebras Cliff(+,+,+,+,+) and Cliff(-,-,-,-,-) are respectively M_2(H)+ M_2(H) and M_4(C). Thus taking an irreducible representation of each of them yields respectively M_2(H) and M_4(C).
The result 3) comes from the index formula in noncommutative geometry. One shows that the degree 4 equation implies that the volume of the manifold (which is defined as the leading term of the Weyl asymptotics of the eigenvalues of the Dirac operator) is the sum of two Fredholm indices and is thus an integer. It relies heavily on the cyclic cohomology index formula and the determination of the Hochschild class of the Chern character.
The great advantage of 3) is that, since the volume is quantized, the huge cosmological term which dominates the spectral action is now quantized and no longer interferes with the equations of motion which as a result of our many years collaboration with Ali Chamseddine gives back the Einstein equations coupled with the Standard Model.
The big plus of 2) is that we finally understand the meaning of the strange choice of algebras that seems to be privileged by nature: it is the simplest way of replacing a number of coordinates by a single operator. Moreover as the result of our collaboration with Walter van Suijlekom, we found that the slight extension of the SM to a Pati-Salam model given by the algebra M_2(H)\oplus M_4(C) greatly improves things from the mathematical standpoint while moreover making the model asymptotically free! (see Beyond the spectral standard model, emergence of Pati-Salam unification.)
To get a mental picture of the meaning of 1), I will try an image which came gradually while we were
working on the problem of realizing all spin 4-manifolds with arbitrarily large quantized volume as a
solution to the equation.
"The Euclidean space-time history unfolds to macroscopic dimension from the product of two 4-spheres of Planckian volume as a butterfly unfolds from its chrysalis."