An improved Lorentzian noncommutative (non)associative formalism (en route to a physical spectral standard model)?

Real structure + fundamental symmetry = real even Lorentzian spectral triple
...we present a Lorentzian version of the noncommutative geometry of the Standard Model... 
We describe the main aspects of a Lorentzian spectral triple, slightly changing the notation to make it more compatible with the physics literature. A Krein space is a Hilbert space H equipped with a self-adjoint unitary operator J=J=J-1 called a fundamental symmetry... 
By putting together the works of Strohmaier [25] and Paschke and Sitarz [26] we can propose the following definition. A real even Lorentzian spectral triple consists of
  • (i) a ∗-algebra A; 
  • (ii) a Krein space (H,J ) where every a ∈ A is represented by a bounded operator π(a) such that π(a ∗ ) = π(a) × and [J , π(a)]=0; 
  • (iii) a Krein-anti-self-adjoint operator D such that [D, π(a)] is bounded for all a ∈ A; 
  • (iv) a self-adjoint unitary operator γ that commutes with A and anticommutes with D and J
  • (v) an antilinear unitary operator J such that J
    =εI, JD=ε'DJ, Jγ=ε′′γJ and JJJJ, where (ε, ε′, ε'′) depends on the KO-dimension as for the usual spectral triples [1].
Note that the interplay between fundamental symmetry, real structure and chirality was also discussed in studies of topological insulators [27]. 
As compared to other definitions of a Lorentzian spectral triple [25,28,29] we choose D to be Krein-anti-selfadjoint because the standard Dirac operator is so and we do not need to modify the ε-table giving  (ε, ε′, ε'′) as a function of the KO-dimension...
(Submitted on 15 Apr 2015 (v1), last revised 5 Jun 2015 (this version, v2))

A coda to the defence of the  (−, +, +, +) metric signature for spacetime

We consider a four-dimensional smooth Lorentzian spin manifold M and we choose the metric signature (−, +, +, +) with p = 3 positive signs and q = 1 negative sign because:
  • (i) it corresponds to the spectral triple of a spin manifold with KO-dimension p− q = 2, as advocated by Barrett [31]; 
  • (ii) it was argued that this is the only signature where a neutrinoless double beta decay can be correctly described [32].

Since the following discussion will be local, we can choose the γ-matrices to satisfy {γµν}=gµν, where gµν is diagonal with diagonal elements (−1,+1,+1,+1) [33]. We define the helicity operator γM5=−iγ0γ1γ2γthe Dirac operator DM=−iγµµ and the fundamental symmetry JM=β=iγ0, which is used in the calculation of expectation values: <ψb|T|ψ> with |ψb>=β|ψ>. The product <ψb|ψ′> i is called a Krein product. The antilinear map is JM=ζK, where ζ=γ5γ2 and K is the complex conjugate operator. The operator JM is different from the physical charge conjugation operator γK=−βγ2βK [33]. In fact there are two possible charge conjugation operators corresponding to two different ε-tables [20] JM is the first one and the physical charge conjugation is the second one. It can be checked that all the axioms of a real even Lorentzian spectral triple hold with these definitions... 

4D KO-dim 2  Lorentzian ⊗ finite KO-dim 6 non-Lorentzian  = 4D KO-dim 0 real even Lorentzian?
To describe the Standard Model with Lorentzian metric, we make the tensor product of the Lorentzian spectral triple of M and the finite spectral triple AF of section IV C. The tensor product of pseudo-Riemannian spectral triples was investigated by van den Dungen [29]. The grading of the tensor product is γ=γ5⊗γF, its Dirac operator D=DM⊗Id2+γ5DM, its charge conjugation is J=JMγFJF because the KO-dimensions of the first and second spectral triples are 2 and 6 [19, 20] and its fundamental symmetry is J=JM⊗I. The finite spectral triple cannot be Lorentzian because this would not be compatible with the anticommutation of γ and J. Moreover, using the finite spectral triple of section IV C provides the correct fermionic Lagrangian in the Lorentzian metric [29]. It can be checked that, with this definition, the tensor product of spectral triples is indeed a real even Lorentzian spectral triple of KO-dimension zero and this solves the fermion multiplicity problem [31]. The order-zero and -one conditions hold by construction. Moreover, the calculation of section IV A can be repeated to show that the order-two condition holds iff {[DF, a], [DF, b◦]}=0 up to the junk. Since this was already proved, the Lorentzian spectral triple of the Standard Model satisfies [also] the order-two condition... 
Chamseddine and Connes based their derivation of the Standard Model on a bimodule over an algebra A. Boyle and Farnsworth proposed to use a bimodule over the universal differential algebra Ω which is physically more satisfactory because it contains (up to the junk) the gauge fields, the field intensities, the curvature and the Lagrangian densities. But their approach was not compatible with the manifold part of the Standard Model. To take into account the differential graded structure of ΩD, we built a differential graded bimodule that takes the junk into account. The grading transforms the Boyle and Farnsworth condition on the commutator [π(δa), π(δb)◦]=0 into a condition on the anticommutator {π(δa), π(δb)◦}∈K, which is now satisfied for the full Lorentzian Standard Model and not only for its finite part. 
Our differential graded bimodule retains some of the advantages of the Boyle and Farnsworth approach: (i) it unifies the conditions of order zero and one and the condition of massless photon into a single bimodule condition; (ii) it can be adapted to non-associative or Lie algebras. 
Now we intend to investigate the symmetries of this approach by using the morphisms defined by Eilenberg [11]. It will be interesting to compare these symmetries with the ones found by Farnsworth and Boyle [34].


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