Cliff(+,+,+,+,+)⊕Cliff(-,-,-,-,-) or a tale of warp and weft in the weave structure of spacetime

There is more (signs issues in uncovering the hidden mathematical structure of physics) than meets the (low energy phenomenologist's) eye...
This post is a sort of continuation of a personal comment recently left on P. Woit's Not Even Wrong blog.

One great virtue of the standard Hilbert space formalism of quantum mechanics is that it incorporates in a natural manner the essential “variability” which is the characteristic feature of the Quantum: repeating twice the same experiment will generally give different outcome, only the probability of such outcome is predicted, the various possibilities form the spectrum of a self-adjoint operator in Hilbert space. We have discovered a geometric analogue of the Heisenberg commutation relations [p,q]=ih. The role of the momentum p is played by the Dirac operator. It takes the role of a measuring rod and at an intuitive level it represents the inverse of the line element ds familiar in Riemannian geometry, in which only its square is specified in local coordinates... 

It has been known for quite some time that in order to encode a geometric space one can encode it by the algebra of functions (real or complex) acting in the same Hilbert space as the above line element, in short one is dealing with “spectral triples”. Spectral for obvious reasons and triples because there are three ingredients: the algebra A of functions, the Hilbert space H and the above Dirac operator D. It is easy to explain why the algebra encodes a topological space. This follows because the points of the space are just the characters of the algebra, evaluating a function at a point P ∈ X respects the algebraic operations of sum and product of functions. The fact that one can measure distances between points using the inverse line element D is in the line of the Kantorovich duality in the theory of optimal transport. It takes here a very simple form. Instead of looking for the shortest path from point P to point P' as in Riemannian Geometry, which only can treat path-wise connected spaces, one instead takes the supremum of |f(P) − f(P')| where the function f is only constrained not to vary too fast, and this is expressed by asking that the norm of the commutator [D, f] be ≤ 1. In the usual case where D is the Dirac operator the norm of [D, f] is the supremum of the gradient of f so that the above control of the norm of the commutator [D, f] means that f is a Lipschitz function with constant 1, and one recovers the usual geodesic distance. But a spectral triple has more information than just a topological space and a metric, as can be already guessed from the need of a spin structure to define the Dirac operator (due to Atiyah and Singer in that context) on a Riemannian manifold. This additional information is the needed extra choice involved in taking the square root of the Riemannian ds2 in the operator theoretic framework. The general theory ... naturally introduces decorations for a spectral triple such as a chirality operator γ in the case of even dimension and a charge conjugation operator J which is an antilinear isometry of H fulfilling commutation relations with D and γ which depend upon the dimension only modulo 8. All this has been known for quite some time as well as the natural occurrence of gravity coupled to matter using the spectral action applied to the tensor product A⊗A of the algebra A of functions by a finite dimensional algebra A corresponding to internal structure. In fact it was shown in [4] that one gets pretty close to zooming on the Standard Model of particle physics when running through the list of irreducible spectral triples for which the algebra A is finite dimensional. The algebra that is both conceptual and works for that purpose is 

A=M₂(H)⊕M₄(C) 

where H is the algebra of quaternions and Mk the matrices.

Geometry and the Quantum: Basics

Ali H. Chamseddine, Alain Connes, Viatcheslav Mukhanov

(Submitted on 4 Nov 2014)

... when extracting the very operator algebraic substance of the low energy physics gauge sector...

...what should one beg for in a quest of reconciling gravity with quantum mechanics? In our view such a reconciliation should not only produce gravity but it should also naturally produce the other known forces, and they should appear on the same footing as the gravitational force. This is asking a lot and, in the minds of many, the incorporation of matter in the Lagrangian of gravity has been seen as an unnecessary complication that can be postponed and hidden under the rug for a while. As we shall now explain this is hiding the message of the gauge sector which in its simplest algebraic understanding is encoded by the ... algebra A=M₂(H)⊕M₄(C) . The answer that we discovered is that the package formed of the 4-dimensional geometry together with the above algebra appears from a very simple idea: to encode the analogue of the position variable q in the same way as the Dirac operator encodes the components of the momenta, just using the Feynman slash [of real scalar fields].

Id. 

... in the form of two quanta of geometry...

To be more precise we let Y∈A⊗C_κ be of the Feynman slashed form Y=Y^AΓ_A, and fulfill the equations Y²=κ, Y^*=κY (1) Here κ=±1 and C_κ⊂Ms(C), s=2^n/2, is the real algebra generated by n+1 gamma matrices Γ^A, 1≤a≤n+1 [it is n+1 and not n where Γ_n+1 is up to normalization the product of the n others]  Γ^A∈C_κ, {Γ^A,Γ^B}=2κδ^AB, (Γ^A)^*=κΓ^A 

The one-sided higher analogue of the Heisenberg commutation relations is 

1/n! <Y[D,Y]···[D,Y]> = γ√κ    (n terms [D,Y]) (2) 

where the notation <T> means the normalized trace of T = Tij with respect to the above matrix algebra Ms(C) (1/s times the sum of the s diagonal terms Tii). We shall show below in Theorem 1 that a solution of this equation exists for the spectral triple (A, H, D) associated to a Spin compact Riemannian manifold M (and with the components Y A ∈ A) if and only if the manifold M breaks as the disjoint sum of spheres of unit volume. This breaking into disjoint connected components corresponds to the decomposition of the spectral triple into irreducible components and we view these irreducible pieces as quanta of geometry.

 Id.

... and weaving them according to the real structure of spacetime...

The corresponding picture, with these disjoint quanta of Planck size is of course quite remote from the standard geometry and the next step is to show that connected geometries of arbitrarily large size are obtained by combining the two different kinds of geometric quanta. This is done by refining the one-sided equation (2) using the fundamental ingredient which is the real structure of spectral triples, and is the mathematical incarnation of charge conjugation in physics. It is encoded by an anti-unitary isometry J of the Hilbert space H fulfilling suitable commutation relations with D and γ and having the main property that it sends the algebra A into its commutant as encoded by the order zero condition : [a, JbJ^-1]=0 for any a,b∈A. This commutation relation allows one to view the Hilbert space H as a bimodule over the algebra A by making use of the additional representation a→Ja^*J^-1 . This leads to refine the quantization condition by taking J into account as the two-sided equation 

1/n! <Z [D, Z] · · · [D, Z]>=γ   Z=2EJEJ^-1−1, (3) 

where E is the spectral projection for {1,i}⊂C of the double slash Y=Y⁺⊕Y^-∈C^∞(M,C⁺⊕C^-). More explicitly E=1/2(1+Y⁺)⊕1/2(1+iY^-). It is the classification of finite geometries of [4] which suggested to use the direct sum C⁺⊕C^- of two Clifford algebras and the algebra C^∞(M,C⁺⊕C^-).

Id.

...with its warp and weft 

In the above formulation of the two-sided quantization equation th[is] algebra... appears as a byproduct of the use of the Feynman slash. It is precisely at this point that the connection with our previous work on the noncommutative geometry (NCG) understanding of the Standard Model appears. Indeed...the algebra A=M₂(H)⊕M₄(C) [i]s the right one to obtain the Standard Model coupled to gravity from the spectral action applied to the product space of a 4-manifold M by the finite space encoded by the algebra A. Thus the full algebra is the algebra C^∞(M,A) of A-valued functions on M. Now the remarkable fact is that in dimension 4 one has 

C⁺=M₂(H),  C^-=M₄(C)    (4) 

More precisely, the Clifford algebra Cliff(+,+,+,+,+) is the direct sum of two copies of M₂(H) and thus in an irreducible representation, only one copy of M₂(H) survives and gives the algebra over R generated by the gamma matrices Γ^A. The Clifford algebra Cliff(−,−,−,−,−) is M₄(C) and it also admits two irreducible representations (acting in a complex Hilbert space) according to the linearity or anti-linearity of the way C is acting. In both the algebra over R generated by the gamma matrices ΓA is M₄(C). This fact clearly indicates that one is on the right track and in fact together with the above two-sided equation it unveils the following tentative “particle picture” of gravity coupled with matter, emerging naturally from the quantum world. First we now forget completely about the manifold M that was used above and take as our framework a fixed Hilbert space in which C=C⁺⊕C^- acts, as well as the grading γ, and the anti-unitary J all ful- filling suitable algebraic relations. So far there is no variability but the stage is set. Now one introduces two “variables” D and Y=Y⁺⊕Y^- both selfadjoint operators in Hilbert space. One assumes simple algebraic relations such as the commutation of C and JCJ^-1, of Y and JYJ^-1, the fact that Y^±=ΣY^±_AΓ^A_± with the Y_A commuting with C, and that Y²=1₊⊕(−1)-  and also that the commutator [D,Y] is bounded and its square again commutes with both C_± and the components Γ^A, etc... One also assumes that the eigenvalues of the operator D grow as in dimension 4. One can then write the two-sided quantization equation (3) and show that solutions of this equation give an emergent geometry. The geometric space appears from the joint spectrum of the components Y^±_A . This would a priori yield an 8-dimensional space but the control of the commutators with D allows one to show that it is in fact a subspace of dimension 4 of the product of two 4-spheres. The fundamental fact that the leading term in the Weyl asymptotics of eigenvalues is quantized remains true in this generality due to already developed mathematical results on the Hochschild class of the Chern character in Khomology. Moreover the strong embedding theorem of Whitney shows that there is no a-priori obstruction to view the (Euclidean) space-time manifold as encoded in the 8-dimensional product of two 4-spheres. The action functional only uses the spectrum of D, it is the spectral action which, since its leading term is now quantized, will give gravity coupled to matter from its infinitesimal variation.

Id.