How quaternicity of space-time could be consistent with Hermitian/Hamilton-Jacobi structure?

The recent progress in the understanding of preferred extremals of Kähler action suggests also an interesting connection to the number theoretic vision about field equations. In particular, it might be possible to understand how one can have Hermitian/Harmilton-Jacobi structure simultaneously with quaternionic structure and how quaternionic structure is possible for the Minkowskian signature of the induced metric.

One can imagine two manners of introducing octonionic and quaternionic structures. The first one is based on the introduction of octonionic representation of gamma matrices and second on the notion of octonion real-analycity.

1. If quaternionic structure is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions represented in terms of gamma matrices. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure in Minkowskian regions or to introduce what I have called hyper-quaternionic structure by imbedding the space-time surface to a sub-space $M^8$ of complexified octonions. This is admittedly artificial.
2. The octonionic representation effectively replaces SO(7,1) as tangent space group with G₂ and means selection of preferred M²⊂ M⁴ having interpretation complex plane of octonionic space. A more general condition is that the tangent space of space-time surface at each point contains preferred sub-space M²(x)⊂ M⁴ forming an integrable distribution. The same condition is involved with the definition of Hamilton-Jacobi structure. What puts bells ringing is that the modified Dirac equation for the octonionic representation of gamma matrices allows the conservation of electromagnetic charge in the proposed sense a observed for years ago. One can ask whether the conditions on the charged part of energy momentum tensor could relate to the reduction of SO(7,1) to G₂.
3. Octonionic gamma matrices appear also in the proposal stating that space-time surfaces are quaternionic in the sense that tangent space of the space-time surface is quaternionic in the sense that induced octonionic gamma matrices generate a quaternionic sub-space at a given point of space-time time. Besides this the already mentioned additional condition stating that the tangent space contains preferred sub-space M²⊂ M⁴ or integrable distribution of this kind of sub-spaces is required. It must be emphasized that induced rather than modified gamma matrices are natural in these conditions.

1. Could the quaternionicity of the space-time surface together with a preferred distribution of tangent planes M²(x)⊂ M⁴ or E²(x)⊂ CP₂ be equivalent with the reduction of the field equations to the analogs of minimal surface equations stating that certain components of the induced metric in complex/Hamilton-Jacobi coordinates vanish in turn guaranteeing that field equations reduce to algebraic identifies following from the fact that energy momentum tensor and second fundamental form have no common components? This should be the case if one requires that the two solution ansätze are equivalent.
2. Can the conditions for the modified Dirac equation select complex of co-complex 2-sub-manifold of space-time surface identified as quaternionic or co-quaternionic 4-surface? Could the conditions stating the vanishing of charged energy momentum currents state that the spinor fields are localized to complex or co-complex (hyper-complex or co-hypercomplex) 2-surfaces?

1. Could these two structures be associated with energy momentum tensor and metric respectively? Or perhaps vice versa? Anti-commutators of the modified gamma matrices define an effective metric as

   G_αβ=T_αμT^μ_β .

   This effective metric should have a deep physical and mathematical meaning but this meaning has remained a mystery. Note that iT takes the role of Kähler form for Kähler metric in this expression with imaginary unit plus symmetry replacing antisymmetry. The neutral and charged parts of T would be analogous to quaternionic imaginary units and real part but one would have sum over the squares of all of them rather than only single square as for Kähler metric.

2. Could G be assigned with the quaternionic structure and induced metric to the Hermitian/Hamilton-Jacobi structures? Could the neutral and charged components of the energy momentum tensor somehow correspond to quaternionic units?

1. If quaternionic structrures is defined in terms of the octonionic representation of the imbedding space gamma matrices, there seems to be no obvious problems since one considers automatically complexification of quaternions.
2. For the approach based on the notion of quaternion real analyticity, one is forced to use Wick rotation to define the quaternionic structure or to introduce hyper-quaternionic structure by imbedding the space-time surface to a sub-space M⁸ of complexified octonions. This is admittedly artificial.

Could one pose the additional requirement that the signature of the effective metric G defined by the modified gamma marices (and to be distinguished from Einstein tensor) is Euclidian in the sense that all four eigenvalues of this tensor would have same sign.

1. For the induced metric the projections of gamma matrices are given by

   Γ_α=Γ^ae_aα , e_aα= e_ak∂_α h^k .

   For the modified gamma matrices their analogs would be given by

   Γ_α=Γ^a E_aα , E_aα= e_aμT^μ_α .

   Projection would be followed by multiplication with energy momentum tensor. One cannot induce G from any metric defined in the imbedding space but the notion of tangent space quaternionicity is well-defined.

2. What quaternionic structure for G could mean? One can imagine several options.
   1. For the ordinary complex structure metric has vanishing diagonal components and the infinitesimal line element ds²=g_zz*dzdz*. Could this formula generalize to

      ds²=g_QQ*dQdQ*?

      The generalization would be a direct generalization of conformal invariance to 4-D context stating that 4-metric is quaternion-conformally equivalent to flat metric. This would give additional strong condition on energy momentum tensor:

      G= T_αμT^μ_β= T²δ_αβ .

      The proportionality to Euclidian metric means in Minkowskian realm that the G is of form G= T²(2u_αu_β -g_αβ. Here u is time-like vector field satisfying u^αu_α=1 and having interpretation as local four-velocity (in Robertson-Walker cosmology similar situation is encountered). The eigen value problem in the form G^α_βx^β= λ x^α makes sense and eigenvectors would be u with eigenvalue λ=T² and three vectors orthogonal to u with eigenvalue -T². This requires integrable flow defined by u and defining a preferred time coordinate. In number theoretic vision this kind of time coordinate is introduced and corresponds to the direction assignable to the octonionic real unit. Note that the vanishing of charged projections of the energy momentum tensor does not imply a reduction of the rank of T so that this option might work.

   2. Quaternionicity could mean also the structure of hyper-Kähler manifold. Metric and Kähler form for Kähler manifold are generalized to metric representing quaternion real unit and three covariantly constant Kähler forms I_i obeying the multiplication rules for quaternions. The necessary condition is that the holonomy group equals to SU(2) identifiable as automorphism group of quaternions. One can also define quaternionic structure: there would exist three antisymmetric tensors, whose squares give the negative of the metric. CP₂ allows quaternionic structure in this sense and only one of these forms is covariantly constant.

      Could space-time surface allow Hyper-Kähler or quaternionic structure somehow induced from that of CP₂? This does not work for G. G is quadratic in energy momentum tensor and therefore involves four power of J rather than being square of projection of J or two other quaternionic imaginary units of CP₂. One can of course ask whether the induced quaternionic units could obey the multiplication of quaternionic units and have same square given by the projection of CP₂ metric. In this case CP₂ metric would define the effective metric and would be indeed Euclidian. For the ansatz for preferred extremals with Minkowskian signature CP₂ projection is at most 3-dimensional but also in this case the imaginary units might allow a realization as projections.