Field equations as conservation laws, Frobenius integrability conditions, and a connection with quaternion analyticity
The following represents qualitative picture of field equations of TGD trying to emphasize the physical aspects. What is new is the discussion of the possibility that Frobenius integrability conditions are satisfied and correspond to quaternion analyticity.
- Kähler action is Maxwell action for induced Kähler form and metric expressible in terms of imbedding space coordinates and their gradients. Field equations reduce to those for imbedding space coordinates defining the primary dynamical variables. By GCI only four of them are independent dynamical variables analogous to classical fields.
- The solution of field equations can be interpreted as a section in fiber bundle. In TGD the fiber bundle is just the Cartesian product X4× CD× CP2 of space-time surface X4 and causal diamond CD× CP2. CD is the intersection of future and past directed light-cones having two light-like boundaries, which are cone-like pieces of light-boundary δ M4+/-× CP2. Space-time surface serves as base space and CD× CP2 as fiber. Bundle projection Π is the projection to the factor X4. Section corresponds to the map x→ hk(x) giving imbedding space coordinates as functions of space-time coordinates. Bundle structure is now trivial and rather formal.
By GCI one could also take suitably chosen 4 coordinates of CD× CP2 as space-time coordinates, and identify CD× CP2 as the fiber bundle. The choice of the base space depends on the character of space-time surface. For instance CD, CP2 or M2× S2 (S2 a geodesic sphere of CP2), could define the base space. The bundle projection would be projection from CD× CP2 to the base space. Now the fiber bundle structure can be non-trivial and make sense only in some space-time region with same base space.
- The field equations derived from Kähler action must be satisfied. Even more: one must have a preferred extremal of Kähler action. One poses boundary conditions at the 3-D ends of space-time surfaces and at the light-like boundaries of CD× CP2.
One can fix the values of conserved Noether charges at the ends of CD (total charges are same) and require that the Noether charges associated with a sub-algebra of super-symplectic algebra isomorphic to it and having conformal weights coming as n-ples of those for the entire algebra, vanish. This would realize the effective 2-dimensionality required by SH. One must pose boundary conditions also at the light-like partonic orbits. So called weak form of electric-magnetic duality is at least part of these boundary conditions.
It seems that one must restrict the conformal weights of the entire algebra to be non-negative r≥ 0 and those of subalgebra to be positive: mn>0. The condition that also the commutators of sub-algebra generators with those of the entire algebra give rise to vanishing Noether charges implies that all algebra generators with conformal weight m≥ n vanish so the dynamical algebra becomes effectively finite-dimensional. This condition generalizes to the action of super-symplectic algebra generators to physical states.
M4 time coordinate cannot have vanishing time derivative dm0/dt so that four-momentum is non-vanishing for non-vacuum extremals. For CP2 coordinates time derivatives dsk/dt can vanish and for space-like Minkowski coordinates dmi/dt can be assumed to be non-vanishing if M4 projection is 4-dimensional. For CP2 coordinates dsk/dt=0 implies the vanishing of electric parts of induced gauge fields. The non-vacuum extremals with the largest conformal gauge symmetry (very small n) would correspond to cosmic string solutions for which induced gauge fields have only magnetic parts. As n increases, also electric parts are generated. Situation becomes increasingly dynamical as conformal gauge symmetry is reduced and dynamical conformal symmetry increases.
- The field equations involve besides imbedding space coordinates hk also their partial derivatives up to second order. Induced Kähler form and metric involve first partial derivatives ∂αhk and second fundamental form appearing in field equations involves second order partial derivatives ∂α∂βhk.
Field equations are hydrodynamical, in other worlds represent conservation laws for the Noether currents associated with the isometries of M4× CP2. By GCI there are only 4 independent dynamical variables so that the conservation of m≤ 4 isometry currents is enough if chosen to be independent. The dimension m of the tangent space spanned by the conserved currents can be smaller than 4. For vacuum extremals one has m= 0 and for massless extremals (MEs) m= 1! The conservation of these currents can be also interpreted as an existence of m≤ 4 closed 3-forms defined by the duals of these currents.
- The hydrodynamical picture suggests that in some situations it might be possible to assign to the conserved currents flow lines of currents even globally. They would define m≤ 4 global coordinates for some subset of conserved currents (4+8 for four-momentum and color quantum numbers). Without additional conditions the individual flow lines are well-defined but do not organize to a coherent hydrodynamic flow but are more like orbits of randomly moving gas particles. To achieve global flow the flow lines must satisfy the condition dφA/dxμ= kABJBμ or dφA= kABJB so that one can special of 3-D family of flow lines parallel to kABJB at each point - I have considered this kind of possibility in detail earlier but the treatment is not so general as in the recent case.
Frobenius integrability conditions follow from the condition d2φA=0= dkAB∧ JB+ kABdJB=0 and implies that dJB is in the ideal of exterior algebra generated by the JA appearing in kABJB. If Frobenius conditions are satisfied, the field equations can define coordinates for which the coordinate lines are along the basis elements for a sub-space of at most 4-D space defined by conserved currents. Of course, the possibility that for preferred extremals there exists m≤ 4 conserved currents satisfying integrability conditions is only a conjecture.
It is quite possible to have m<4. For instance for vacuum extremals the currents vanish identically For MEs various currents are parallel and light-like so that only single light-like coordinate can be defined globally as flow lines. For cosmic strings (cartesian products of minimal surfaces X2 in M4 and geodesic spheres S2 in CP2 4 independent currents exist). This is expected to be true also for the deformations of cosmic strings defining magnetic flux tubes.
- Cauchy-Riemann conditions in 2-D situation represent a special case of Frobenius conditions. Now the gradients of real and imaginary parts of complex function w=w(z)= u+iv define two conserved currents by Laplace equations. In TGD isometry currents would be gradients apart from scalar function multipliers and one would have generalization of C-R conditions. In citeallb/prefextremals,twistorstory I have considered the possibility that the generalization of Cauchy-Riemann-Fuerter conditions could define quaternion analyticity - having many non-equivalent variants - as a defining property of preferred extremals. The integrability conditions for the isometry currents would be the natural physical formulation of CRF conditions. Different variants of CRF conditions would correspond to varying number of independent conserved isometry currents.
- This picture allows to consider a generalization of the notion of solution of field equation to that of integral manifold. If the number of independent isometry currents is smaller than 4 (possibly locally) and the integrability conditions hold true, lower-dimensional sub-manifolds of space-time surface define integral manifolds as kind of lower-dimensional effective solutions. Genuinely lower-dimensional solutions would of course have vanishing (g41/2) and vanishing Kähler action.
String world sheets can be regarded as 2-D integral surfaces. Charged (possibly all) weak boson gauge fields vanish at them since otherwise the electromagnetic charge for spinors would not be well-defined. These conditions force string world sheets to be 2-D in the generic case. In special case 4-D space-time region as a whole can satisfy these conditions. Well-definedness of Kähler-Dirac equation demands that the isometry currents of Kähler action flow along these string world sheets so that one has integral manifold. The integrability conditions would allow 2<m≤ n integrable flows outside the string world sheets, and at string world sheets one or two isometry currents would vanish so that the flows would give rise 2-D independent sub-flow.
- The method of characteristics is used to solve hyperbolic partial differential equations by reducing them to ordinary differential equations. The (say 4-D) surface representing the solution in the field space has a foliation using 1-D characteristics. The method is especially simple for linear equations but can work also in the non-linear case. For instance, the expansion of wave front can be described in terms of characteristics representing light rays. It can happen that two characteristics intersect and a singularity results. This gives rise to physical phenomena like caustics and shock waves.
In TGD framework the flow lines for a given isometry current in the case of an integrable flow would be analogous to characteristics, and one could also have purely geometric counterparts of shockwaves and caustics. The light-like orbits of partonic 2-surface at which the signature of the induced metric changes from Minkowskian to Euclidian might be seen as an example about the analog of wave front in induced geometry. These surfaces serve as carriers of fermion lines in generalized Feynman diagrams. Could one see the particle vertices at which the 4-D space-time surfaces intersect along their ends as analogs of intersections of characteristics - kind of caustics? At these 3-surfaces the isometry currents should be continuous although the space-time surface has "edge".
For details see the chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds" of "Quantum physics as infinite-dimensional geometry" or the article Could One Define Dynamical Homotopy Groups in WCW?.
For a summary of earlier postings see Links to the latest progress in TGD.