About the definition of Hamilton-Jacobi structure

I have talked in previous postings a lot about Hamilton-Jacobi structure without bothering to write detailed definitions. In the following I discuss the notion in more detail. Thanks for Hamed who asked for more detailed explanation.

Hermitian and hyper-Hermitian structures

The starting point is the observation that besides the complex numbers forming a number field there are hyper-complex numbers. Imaginary unit i is replaced with e satisfying e²=1. One obtains an algebra but not a number field since the norm is Minkowskian norm x²-y², which vanishes at light-cone x=y so that light-like hypercomplex numbers x+/- e) do not have inverse. One has "almost" number field.

Hyper-complex numbers appear naturally in 2-D Minkowski space since the solutions of a massless field equation can be written as f=g(u=t-ex)+h(v=t+ex) whith e²=1 realized by putting e=1. Therefore Wick rotation relates sums of holomorphic and antiholomorphic functions to sums of hyper-holomorphic and anti-hyper-holomorphic functions. Note that u and v are hyper-complex conjugates of each other.

Complex n-dimensional spaces allow Hermitian structure. This means that the metric has in complex coordinates (z₁,....,z_n) the form in which the matrix elements of metric are nonvanishing only between z_i and complex conjugate of z_j. In 2-D case one obtains just ds²=g_zz*dzdz*. Note that in this case metric is conformally flat since line element is proportional to the line element ds²=dzdz* of plane. This form is always possible locally. For complex n-D case one obtains ds²=g_ij*dzⁱdz^j*. g_ij*=(g_ji*)* guaranteing the reality of ds². In 2-D case this condition gives g_zz*= (g_z*z)*.

How could one generalize this line element to hyper-complex n-dimensional case? In 2-D case Minkowski space M² one has ds²= g_uvdudv, g_uv=1. The obvious generalization would be the replacement ds²=g_uivjduⁱdv^j. Also now the analogs of reality conditions must hold with respect to uⁱ↔ vⁱ.

Hamilton-Jacobi structure

Consider next the path leading to Hamilton-Jacobi structure.

4-D Minkowski space M⁴=M²× E² is Cartesian product of hyper-complex M² with complex plane E², and one has ds²= dudv+ dzdz* in standard Minkowski coordinates. One can also consider more general integrable decompositions of M⁴ for which the tangent space TM⁴=M⁴ at each point is decomposed to M²(x)× E²(x). The physical analogy would be a position dependent decomposition of the degrees of freedom of massless particle to longitudinal ones (M²(x): light-like momentum is in this plane) and transversal ones (E²(x): polarization vector is in this plane). Cylindrical and spherical variants of Minkowski coordinates define two examples of this kind of coordinates (it is perhaps a good exercize to think what kind of decomposition of tangent space is in question in these examples). An interesting mathematical problem highly relevant for TGD is to identify all possible decompositions of this kind for empty Minkowski space.

The integrability of the decomposition means that the planes M²(x) are tangent planes for 2-D surfaces of M⁴ analogous to Euclidian string world sheet. This gives slicing of M⁴ to Minkowskian string world sheets parametrized by euclidian string world sheets. The question is whether the sheets are stringy in a strong sense: that is minimal surfaces. This is not the case: for spherical coordinates the Euclidian string world sheets would be spheres which are not minimal surfaces. For cylindrical and spherical coordinates hower M²(x) integrate to plane M² which is minimal surface.

Integrability means in the case of M²(x) the existence of light-like vector field J whose flow lines define a global coordinate. Its existence implies also the existence of its conjugate and together these vector fields give rise to M²(x) at each point. This means that one has J= Ψ∇ Φ: Φ indeed defines the global coordinate along flow lines. In the case of M² either the coordinate u or v would be the coordinate in question. This kind of flows are called Beltrami flows. Obviously the same holds for the transversal planes E².

One can generalize this metric to the case of general 4-D space with Minkowski signature of metric. At least the elements g_uv and g_zz* are non-vanishing and can depend on both u,v and z,z*. They must satisfy the reality conditions g_zz*= (g_zz*)* and g_uv= (g_vu)* where complex conjugation in the argument involves also u↔ v besides z↔ z*.

The question is whether the components g_uz, g_vz, and their complex conjugates are non-vanishing if they satisfy some conditions. They can. The direct generalization from complex 2-D space would be that one treats u and v as complex conjugates and therefore requires a direct generalization of the hermiticity condition

g_uz= (g_vz*)*, g_vz= (g_uz*)* .

This would give complete symmetry with the complex 2-D (4-D in real sense) spaces. This would allow the algebraic continuation of hermitian structures to Hamilton-Jacobi structures by just replacing i with e for some complex coordinates.

For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equation or the article with the same title.