The following argument suggests that the effective metric defined by the anti-commutators of the modified gamma matrices is effectively one- or two-dimensional. Effective one-dimensionality would conform with the observation that the solutions of the modified Dirac equations can be localized to one-dimensional world lines in accordance with the vision that finite measurement resolution implies discretization reducing partonic many-particle states to quantum superpositions of braids. This localization to 1-D curves occurs always at the 3-D orbits of the partonic 2-surfaces.
- One can multiply both sides of this equation with j^{Ak} and sum over the index A labeling isometry currents for translations of M^{4} and SU(3) currents for CP_{2}. The tensor quantity
∑_{A} j^{Ak}j^{Al} is invariant under isometries and must therefore satisfy
∑_{A }η_{AB}j^{Ak}j^{Al}= h^{kl} ,
where η_{AB} denotes the flat tangent space metric of H. In M^{4} degrees of freedom this statement becomes obvious by using linear Minkowski coordinates.
In the case of CP_{2} one can first consider the simpler case S^{2}=CP_{1}= SU(2)/U(1). The coset space property implies in standard complex coordinate transforming linearly under U(1) that only the the isometry currents belonging to the complement of U(1) in the sum contribute at the origin and the identity holds true at the origin and by the symmetric space property everywhere. Identity can be verified also directly in standard spherical coordinates. The argument generalizes to the case of CP_{2}=SU(3)/U(2) in an obvious manner.
- In the most general case one obtains
T^{α k}_{1} =∑_{A}Ψ_{1}^{A} j^{Ak} × (∇Φ_{1})^{α} == f^{k}_{1} (∇Φ_{1})^{α} ,
T^{α k}_{2} =∑_{A}Ψ_{1}^{A} j^{Ak} × (∇Φ_{2})^{α} ≡ f^{k}_{2} (∇Φ_{2})^{α} .
Here i=1 refers to M^{4} part of energy momentum tensor and i=2 to its CP_{2} part.
- The effective metric given by the anti-commutator of the modified gamma matrices is in turn is given by
G^{α β} = m_{kl}f^{k}_{1}f^{l}_{1} (∇Φ_{1})^{α}(∇Φ_{1})^{β} +s_{kl} f^{k}_{2} f^{l}_{2} (∇Φ_{2})^{α}(∇Φ_{2})^{β} .
The covariant form of the effective metric is effectively 1-dimensional for Φ_{1}=Φ_{2} in the sense that the only non-vanishing component of the covariant metric G_{α β} is diagonal component along the coordinate line defined by Φ≡ Φ_{1}=Φ_{2}. Also the contravariant metric is effectively 1-dimensional since the index raising does not affect the rank of the tensor but depends on the other space-time coordinates. This would correspond to an effective reduction to a dynamics of point-like particles for given selection of braid points. For Φ_{1}≠ Φ_{2} the metric is effectively 2-dimensional and would correspond to stringy dynamics.
For background see the chapter