Could the notion of hyper-determinant be useful in TGD framework?

Hyperdeterminants have stimulated interesting discussions in viXra blog and also Kea has talked about them. The notion is new to me but so interesting from TGD point of view that I cannot resist the temptation of making fool of myself by declaring why it looks so interesting. This gives also an excellent opportunity to demonstrate my profound ignorance about the notion;-). Instead of typing all my ignorance in html, I give a link to pdf article Could the notion of hyper-determinant be useful in TGD framework?.

Addition: I decided to glue the response to a comment by Phil Gibbs summarizing my motivations for getting interested in hyper-determinants.

1. Why the equations stating the vanishing of n:th variation of Kähler action are interesting in TGD framework is due to the infinite vacuum degeneracy of Kähler action making possible an infinite hierarchy of criticalities: one can say that TGD Universe is quantum critical. Criticality means a hierarchy of vanishing n:th variations. Phase transitions inside phase transitions inside.... This property is responsible for a lot of new physics and mathematics involved with TGD.

2. The equations for n:th variation of Kähler action formulated in terms of functional derivatives are formally of this form and the existence of solution means vanishing of a generalized hyper-determinant. In standard QFT vanishing n≥3:th variations are not terribly interesting and even their existence is questionable. Vanishing second variations correspond to zero modes and vanishing of Gaussian determinant.

3. n:th variations correspond formally to infinite tensor product with same dimension for all tensor factors and in this case there should be no restrictions on the number of tensor factors. The definition of hyper-determinant in this case is of course highly non-trivial. Already functional (Gaussian) determinants are tricky objects. What makes hyper-determinant so interesting from TGD view point is that it applies to multilinear equations involving homogeneous polynomials. Something between linear and genuinely non-linear and solvable.

What hopes one has for genuine multilinearity, which seems to be almost synonymous to non-locality?

1. In the general case multilinearity requires non-locality and in purely local non-linear field theories there are not must hopes about multilinearity. The field equations for n:th variation should not contain powers of the same imbedding space coordinate or same derivative of it at same point. This is certainly not the case for a typical action principle. If the equations are genuinely multilinear in some basis for the deformations of space-time surface they are solvable and generalized hyper-determinant should tell whether this is the case. Its vanishing would also code for criticality for a higher order phase transition.

2. When one constructs perturbation theory for a functional integral using exponent of Kähler function, one considers Kähler function identified as Kähler action for a preferred extremal. Formally this is a non-local functional of the data about 3-surface but actually reduces to 3-D Chern-Simons Kähler action with constraints characterizing weak form of electric magnetic duality. By effective 2-dimensionality Chern-Simons action is however a non-local functional of data about partonic 2-surface and its tangent space. n:th variation for 3-surface and 4-surface reduce to a non-local function of n:th variation of partonic 2-surface and its tangent space data. This is just what genuine multilinearity means so that multilinearity seems to hold true!

3. This also relates to the local divergences of quantum field theories. They are present just because of higher order purely local couplings. Now they are absent if non-locality implying multilinearity holds true so that the functional integral over partonic 2-surfaces plus tangent space data should be free of infinities. Hence multilinearity might be behind integrability and absence of divergences. Maybe this relates also to the Yangian algebras which are non-local.

For more details see the chapter Does the Modified Dirac Equation Define the Fundamental Action Principle?.