Physicists may appreciate the concept of (the) real structure (of spacetime)

From sign to signature
This post is a follow-up to the last one. It deals with a general problem raised by Peter Woit in his last post namely most physicists don’t appreciate the mathematical concept of “real structure”.

... what about the choice of sign which is often involved when one is taking a square root and hence should play a role here since our line element ds is based on Dirac’s square root for the Laplacian. Now this choice of sign, which amounts in the case of ordinary geometry to the choice of a spin structure which is the substitute for the choice of an orientation, turns out to be deeply related to the very notion of “manifold”. While we have a good recipe for constructing manifolds by gluing charts together, it is far less obvious to understand in a conceptual manner the global properties which characterize the spaces underlying manifolds. For instance the same homotopy type can underly quite different manifolds which are not homeomorphic to each other and are distinguished by the fundamental invariant given by the Pontrjagin classes. In first approximation, neglecting subtleties coming from the role of the fundamental group and also of the special properties of dimension 4, the choice of a manifold in a given homotopy type is only possible if a strong form of Poincaré duality holds (cf. [21]). The usual Poincaré duality uses the fundamental class in ordinary homology to yield an isomorphism between homology and cohomology. The more refined form of Poincaré duality that is involved in the choice of a manifold in a given homotopy type involves a finer homology theory called KO-homology. The operator theoretic realization of cycles in KO-homology was pioneered by Atiyah and Singer as a byproduct of their Index Theorem and has reached a definitive form in the work of Kasparov. The theory is periodic with period 8 and one may wonder how a number modulo 8 can appear in its formulation. This manifests itself in the form of a table of signs 

which governs the commutation relations between two simple decorations of spectral triples (with the second only existing in the even dimensional case) so that they yield a KO-homology class:

(1) An antilinear isometry J of H with J²=ε and JD=ε' DJ.

(2) A Z/2-grading γ of H, such that Jγ=ε''γJ. 

The three signs (ε,ε',ε'')∈{±1}³ detect the dimension modulo 8 of the cycle in KO-homology, as ruled by the above table. These decorations have a deep meaning both from the mathematical point of view – where the main underlying idea is that of a manifold – as well as from the physics point of view. In the physics terminology the operator J is the charge conjugation operator, and the grading γ is the chirality

ON THE FINE STRUCTURE OF SPACETIME ALAIN CONNES

The hidden geometrical structure of the standard model of particle physics was discovered by Connes using non-commutative geometry[C]. His model suffers from two defects from the physical point of view: firstly that the spacetime metric is Euclidean, and secondly that each particle appears four times, not once[LMMS, GIS]. The purpose of this paper is to give the analogous geometrical framework for the standard model with Lorentzian signature which also, at the same time, solves the particle quadrupling problem. This model allows the introduction of neutrino masses using the see-saw mechanism. 

Connes’ formulation of non-commutative geometry is a real spectral triple [CR]. This consists of a Hilbert space H, an algebra A of bounded operators in H, and the Dirac operator D, a self-adjoint operator in H, together with a chirality operator γ in H and an antilinear map J on H called the real structure. The chirality satisfies γ=γ^* and γ²=1 and the idea is that its two eigenvalues label the left- and right- handed particles, left-handed particles having eigenvalue 1, right-handed -1. This data satisfies a number of axioms, some of which depend on an integer parameter mod 8, the signature of the geometry, σ... 

The parameter σ was previously identified by Connes as the dimension of the spectral triple. This makes sense for Euclidean geometries, where, by analogy with a Riemannian metric, signature and dimension are the same thing. However for a Lorentzian metric, signature and dimension are different. 

The geometry of the Euclidean standard model is a product of a real spectral triple (H_M, A_M, D_M, γ_M, J_M) for the space-time manifold M with a finite-dimensional real spectral triple (H_F ,A_F ,D_F , γ_F ,J_F) for a non-commutative ‘internal space’. The Hilbert space HF is constructed with basis the 45 elementary fermions, counting left- and right-handed particles separately, and another 45 basis vectors for the antiparticles of the same fermions. Since the space H_M is the Hilbert space of Dirac spinors on M, the particle quadrupling phenomenon is apparent. The total space of fermions is H=H_M⊗H_F. A given elementary fermion, for example a left-handed electron eL∈ H_F  , determines a subspace of wavefunctions φ=ψ⊗e_L+ψ′⊗e^-_L ∈ H, where e^-_L=J_Fe_L is the independent basis vector corresponding to the antiparticle, and ψ and ψ ′ are arbitrary Dirac spinors. In the standard model, e_L should only have a Weyl spinor, so the degrees of freedom are overcounted four-fold. One would like to reduce the space H by imposing the relations [BK] Jφ=φ and γφ=φ where J = J_M⊗J_F and γ=γ_M⊗ γ_F are the real structure and chirality for the total space H. However this does not make sense; their consistency would require J²=1 and Jγ = γJ which hold only if the total signature is equal to zero. The total signature for the Euclidean model is unfortunately four... 

To construct the Lorentzian signature model it is necessary to replace the spectral triple for the manifold by the analogous structure for a Lorentz signature space-time. The guiding idea for this paper is that the structure of the internal space is determined by the requirement that it should be possible to eliminate the quadrupling of the fermions by imposing the additional requirement that the physical fields are simultaneously eigenvectors of J and γ. The result, explained below, is that the internal space is thus determined to be a real spectral triple of signature six. Since it corresponds to a 0-dimensional manifold (having a finite-dimensional Hilbert space), one can see that it is also in some sense Lorentzian.

A Lorentzian version of the non-commutative geometry of the standard model of particle physics

John W. Barrett

(Submitted on 31 Aug 2006 (v1), last revised 6 Nov 2006 (this version, v2))

From i to γ  Dirac to Majorana

We proceed to defining (charge) conjugation, in both the Lorentzian and the euclidean framework, and its interrelation with γ_2m+1 for D=2m. This will lead us to the concept of KO-dimension which provides another mod 8 characteristic of the Clifford algebras. (It has been used in the noncommutative geometry approach to the standard model... 

It was Majorana [M37] who discovered (nine years after Dirac wrote his equation) that there exists a real representation, spanned by ψ_M of Cl(3, 1), for which 

                                      ψ^C_M = ψ_M ⇔ ψ_R = ψ^*_Lc  ( ψ_L= ψ^*_Rc^-1). (4.7) 

(Dirac equation was designed to describe the electron - a charged particle, different from its antiparticle. Majorana thought of applying his “real spinors” for the description of the neutrino, then only a hypothetical neutral particle - predicted in a letter by Pauli and named by Fermi) 

... the first peculiarity of a Majorana mass term is that it would be a purely quantum effect with no classical counterpart, in contrast to a naive understanding of the “correspondence principle”. An even more drastic departure from the conventional wisdom is displayed by the fact that the reality condition (4.7) ... is not invariant under phase transformation (ψ → e ^iαψ)... In particular, a Majorana neutrino coincides with its antiparticle implying a violation of the lepton number conservation, a consequence that may be detected in a neutrinoless double beta decay (see [BP], [B10] [R11] and references therin) and may be also in a process of left-right symmetry restoration that can be probed at the Large Hadron Collider ([TV], [St]). 

The discovery of neutrino oscillations is a strong indication of the existence of positive neutrino masses (for a recent review by a living classic of the theory and for further references - see [B10]). The most popular theory of neutrino masses, involving a mixture of Majorana and Dirac neutrinos, is based on the so called “seesaw mechanism”, which we proceed to sketch (cf. [K09] for a recent review with an eye towards applications to cosmological dark matter and containing a bibliography of 275 entries)...

Clifford Algebras and Spinors

Ivan Todorov

(Submitted on 16 Jun 2011 (v1), last revised 27 Oct 2011 (this version, v2))

//last edition 9 June 2015