How do the quark momenta in M8 and H relate to each other?
The relationship between quark momenta in M8 and H is not clear. M8-H duality suggests the same momentum and mass spectrum for quarks in M8 and H. However, the mass spectrum of color partial waves for quark spinors for the Dirac operator D(H) is very simple and characterized by 2 integers labeling triality t=1 representations of SU(3) (see this). D(H) does not allow a mass spectrum as algebraic roots of polynomials.
What can one conclude if M8-H duality holds true in a strong sense so that these spectra are identical?
The only possible conclusion seems to be that the propagator in both M8 and H is just the M4 Dirac propagator D(M4) and that the roots of the polynomial P give the spectrum of off-mass-shell masses. Also tachyonic mass squared values are allowed as roots of P. The real on-shell masses would be associated with Galois singlets.
Consider first the nice features of the D(M4) option.
- The integration over momentum space reduces to a finite summation over virtual mass shells defined by the roots of P and one avoids divergences. This tightens the connection with QFTs.
- Massless propagation conforms with the twistorialization since mass one obtains holomorphy in twistor variables.
- Massless quarks are consistent with the QCD picture about quarks.
- The consideration of problems related to right-handed neutrino (see this) led to the proposal that the quark spinor modes in H are annihilated only by the H d'Alembertian but not by the H Dirac operator. The assumption that on mass shell H-spinors are annihilated by the M4 Dirac operator leads to the same outcome.
This allows different M4 chiralities to propagate separately and solves problems related to the notion of right-handed neutrino νR (assumed to be 3-antiquark state and modellable using leptonic spinors in H. This also conforms with the right and left-handed character of the standard model couplings.
- A further argument favoring the D(M4) option comes from the following observation. Suppose that one takes seriously the idea that the situation can be described also by using massless M8 momenta. This implies that for some choices of M4⊂ M8 the momentum is parallel to M4 and therefore massless in 4-D sense. Only the quarks associated with the same M4 can interact. Hence M4 can be always chosen so that the on mass-shell 4-momenta are light-like.
- For some years ago I found that the space-time propagators for points of H connected by a light-like geodesic behave like massless propagators irrespective of mass. CP2 type extermals have a light-like geodesic as an M4 projection. This would suggest that quarks associated with CP2 type extremals effectively propagate as massless particles even if one assumes that they correspond to modes of the full H Dirac operator. This allows us to consider D(H) as an alternative. For this option most quarks would be extremely massive and practically absent from the interior of the space-time surface.
- Since the color group acts as symmetries, one can assume that spinor modes correspond to color partial waves as eigen states of CP2 spinor d'Alembertian D2(CP2). One would get rid of the constraint on masses but the correlation between color and electroweak quantum numbers would be still "wrong".
- If M4 Kähler form is trivial, there would be no need for the tachyonic ground state in p-adic mass calculations to reduce the mass quark squared to zero. On the other hand, this might lead also to the problems since the earlier calculations were sensitive to the negative ground state conformal weight and would not work as such. This conformal weight could be generated by conformal generators with weights h coming as roots of P with a negative real part.
- If leptons are allowed as fundamental fermions, D(H) allows νR as a spinor mode which is covariantly constant in CP2. If leptons are not allowed, one can argue that νR as a 3-quark state can be modeled as a mode of H spinor with Kähler coupling giving correct leptonic charges.
The M4 Kähler structure favored by the twistor lift of TGD (see this) implies that νR with negative mass squared appears as a mode of D(H). This mode allows the construction of tachyonic ground states.
For D(M4) with a M4 Kähler coupling, one obtains for all spinor modes states with both positive and negative mass squared from the JklΣkl term. Physical on-mass- shell states with negative mass squared cannot be allowed. These would however allow to construct tachyonic ground states needed in the p-adic mass calculations. Note that a pair of these two modes gives massless ground state.
For CP2 harmonics, the correlation between color and electroweak quantum numbers is wrong (see this). Therefore the physical quarks cannot correspond to the solutions of D2(H)Ψ=0. The same applies also to the solutions of D(M4)Ψ=0 if one assumes that they belong to irreducible representations of the color group as eigenstates of D(CP2).
How to construct quark states, which are physical in the sense that they are massless and color-electroweak correlation is correct?
- For D(H) option, the reduction of quark masses to zero requires a tachyonic ground state in p-adic mass calculations (see this). Also for D(M4) with M4 Kähler structure ground states can be tachyonic.
Colored operators with non-vanishing conformal weight are required to make all quark states massless color triplets. This is possible only if the ground state is tachyonic, which gives strong supper for M4 Kähler structure.
- This is achieved by the identification of physical quarks as states of super-symplectic representations. Also the generalized Kac-Moody algebra assignable to the light-like partonic orbits or both of these representations can be considered. These representations could correspond to inertial and gravitational representations realized at "objective" embedding space level and "subjective" space-time level.
Supersymplectic generators are characterized by a conformal weight h completely analogous to mass squared. The conformal weights naturally correspond to algebraic integers associated with P. The mass squared values for the Galois singlets are ordinary integers.
- It is plausible that also massless color triplet states of quarks can be constructed as color singlets. From these one can construct hadrons and leptons as color singlets for a larger extension of rationals. This conforms with the earlier picture about conformal confinement. These physical quarks constructed as states of super-symplectic representation, as opposed to modes of the H spinor field, would correspond to the quarks of QCD.
One can argue that Galois confinement allows to construct physical quarks as color triplets for some polynomial Q and also color singlets bound states of these with extended Galois group for a higher polynomial P\circ Q and with larger Galois group as representation of group Gal(P)/Gal(Q) allowing representations of a discrete subgroup of color group.
At the level of M8 octonionic spinors are natural. M8-H duality requires that they are equivalent with H-spinors. The most natural identification of octionic spinors is as bi-spinors, which have octonionic components. Associativity is satisfied if the components are complexified quaternionic so that they have the same number of components as quark spinors in H. The H spinors can be induced to X4⊂ M8 by using M8-H duality. Therefore the M8 and H pictures fuse together.
The quaternionicity condition for the octonionic spinors is essential. Octonionic spinor can be expressed as a complexified octonion, which can be identified as momentum p. It is not an on-mass shell spinor. The momenta allowed in scattering amplitudes belong to mass shells defined by the polynomial P. That octonionic spinor has only quaternionic components conforms with the quaternionicity of X4⊂ M8 eliminating the remaining momentum components and also with the use of D(M4).
Can one allow complex quark masses?
One objection relates to unitarity. Complex energies and mass squared values are not allowed in the standard picture based on unitary time evolution.
- Here several new concepts lend a hand. Galois confinement could solve the problems if one considers only Galois singlets as physical particles. ZEO replaces quantum states with entangled pairs of positive and negative energy states at the boundaries of CD and entanglement coefficients define transition amplitudes. The notion of the unitary time evolution is replaced with the Kähler metric in quark degrees of freedom and its components correspond to transition amplitudes. The analog of the time evolution operator assignable to SSFRs corresponds naturally to a scaling rather than time translation and mass squared operator corresponds to an infinitesimal scaling.
- The complex eigenvalues of mass squared as roots of P be allowed when unitarity at quark level is not required to achieve probability conservation. For complex mass squared values, the entanglement coefficients for quarks would be proportional to mass squared exponents exp(im2λ), λ the scaling parameter analogous to the duration of time evolution. For Galois singlets these exponentials would sum up to imaginary ones so that probability conservation would hold true.
For a summary of earlier postings see Latest progress in TGD.