Is the effective metric defined by modified gamma matrices effectively one- or two-dimensional?

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.

The argument is based on the following assumptions.

1. The modified gamma matrices for Kähler action are contractions of the canonical momentum densities T^α_k with the gamma matrices of H.

2. The strongest assumption is that the isometry currents

   J^Aα =T^α _kj^Ak

   for the preferred extremals of Kähler action are of form

   J^{A α}= Ψ^A (∇Φ)^α

   with a common function Φ guaranteeing that the flow lines of the currents integrate to coordinate lines of single global coordinate variables (Beltrami property). Index raising is carried out by using the ordinary induced metric.

3. A weaker assumption is that one has two functions Φ₁ and Φ₂ assignable to the isometry currents of M⁴ and CP₂ respectively.:

   J^{A α}₁ = Ψ₁^A (∇Φ₁)^α ,

   J^{A α}₂ = Ψ₂^A (∇Φ₂)^α .

   The two functions Φ₁ and Φ₂ could define dual light-like curves spanning string world sheet. In this case one would have effective 2-dimensionality and decomposition to string world sheets (for the concrete realization see this). Isometry invariance does not allow more that two independent scalar functions Φ_i.

Consider now the argument.

1. One can multiply both sides of this equation with j^Ak and sum over the index A labeling isometry currents for translations of M⁴ and SU(3) currents for CP₂. The tensor quantity ∑_A j^Akj^Al is invariant under isometries and must therefore satisfy

   ∑_A η_ABj^Akj^Al= h^kl ,

   where η_AB denotes the flat tangent space metric of H. In M⁴ degrees of freedom this statement becomes obvious by using linear Minkowski coordinates.

   In the case of CP₂ one can first consider the simpler case S²=CP₁= 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 CP₂=SU(3)/U(2) in an obvious manner.

2. In the most general case one obtains

   T^{α k}₁ =∑_AΨ₁^A j^Ak × (∇Φ₁)^α == f^k₁ (∇Φ₁)^α ,

   T^{α k}₂ =∑_AΨ₁^A j^Ak × (∇Φ₂)^α ≡ f^k₂ (∇Φ₂)^α .

   Here i=1 refers to M⁴ part of energy momentum tensor and i=2 to its CP₂ part.

3. The effective metric given by the anti-commutator of the modified gamma matrices is in turn is given by

   G^{α β} = m_klf^k₁f^l₁ (∇Φ₁)^α(∇Φ₁)^β +s_kl f^k₂ f^l₂ (∇Φ₂)^α(∇Φ₂)^β .

   The covariant form of the effective metric is effectively 1-dimensional for Φ₁=Φ₂ 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≡ Φ₁=Φ₂. 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 Φ₁≠ Φ₂ the metric is effectively 2-dimensional and would correspond to stringy dynamics.

For background see the chapter Does the Modified Dirac Equation Define the Fundamental Variational Principle of "TGD: Physics as Infinite-Dimensional Geometry".