### 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.

- 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.
- 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.
- 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.

- 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.
- 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!
- 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.

## 6 Comments:

Matti, it's good to see you looking at hyperdeterminants. Despite the fact that they are still unfamiliar to a lot of people, they are actually quite verstile so they could easily be part of TGD.

The expressions you have given are invariants and are sometimes called hyperdeterminants, but they are not the discriminants for the linear equations that we are most interested in. In fact when n is odd your expression is zero. For n even they are interesting and have a multiplicative property that generalises the multiplicative property of ordinary matrix determinants. Cayley was interested in these too but what we know as "Cayley's hyperdeterminant" is a quartic expression while those are quadratic.

Phil,

thank you for a comment. Hyper-determinants are really interesting. I am talking about hyper-determinant as they are defined in Wikipedia article and on basis of the not so voluminous knowledge gained during one day. Do note take me too seriously!;-). I do not even know general formulas in the general case.

I try to summarize the main points of argument suggesting that hyper-determinants are indeed very interesting in TGD framework.

a) 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.

b) 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.

c) 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 almost synonymous to non-locality?

a) 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.

b) When one constructs perturbation theory for a functional integral using exponent of K\"ahler 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-SimonsKähler 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.

c) This also relates to the local divergences of quantum field theories. They are present just because of higher order purely local couplings. They are absent if non-locality implying multilinearity holds true. Hence multilinearity might be behind integrability and absence of divergences. Maybe this relates also to the Yangian algebras which are non-local.

This is how it looks like at this moment.

I think this is the right track... but it is only the beginning and a crack in the door into a more general and yes topological physics of which speculations here look rather conservative to me.

But we can take the linearity of it all too far from the foundations as there can be other forms of expression by nature of determinants. It all depends on how far we are willing to stray from our traditions that are way to restricted.

Some mathematical idea or proof say of Poinclaire or Riemann's zeta at 1/2 may only describe our limited world view in a multiverse.

How far can we use such a spatial notation or matrix say to describe nucleons and so on? Can we say that such matrices cannot exceed octonians? What sort of determinants moreover would be needed to describe curves in planes if multilnear in paths- similar to recent ideas of "topological insulation" in the spin of electrons as room temperature?

The PeSla pesla.blogspot.com

The proposed application of the notion of generalized hyper-matrices and multilinearity (in generic case non-linearity) appears only for deformations of the preferred extremals , which are extremely non-linear objects as such. Multilinearity in this case serves as a signature for absence of infinities and for exact solvability too. Vanishing of hyper-determinant would be a a signature for criticality against n:th order phase transition.

The number of tensor factors is infinite in the generic case. The number of field variables is 8 (reducing to 4 by general coordinate invariance) and being very naive one could say that something analogous to infinite tensor product of 8x8 or 4x4 matrices over space-time points is in question.

The infinite-D hyper-determinants would be rather intricate objects as compared to their finite-D variants. Generalizations of Gaussian determinants of quantum field theories. One should be able to define them in practical manner.

Second application of hyper-determinants is to the description of quantum entanglement. For pure states the matrix describing entanglement between two systems has minimum rank for pure state and thus vanishing determinant. Hyper-matrix and hyper-determinant come into play when one speaks about entanglement between n quantum systems. If I have understood correctly, the vanishing of hyper-determinant means that the state is not maximally non-pure.

For the called hyper-finite factor defined by second quantized induced spinor fields one has very formally infinite tensor product of 8-D H-spinor space. The hyper-determinant would characterize entanglement of fermionic modes. Could quantum classical correspondence mean that the vanishing of n-particle hyper-determinant for fermionic entanglement has as a space-time correlate n:th order criticality. Probably not.

Good news.

http://physicsworld.com/cws/article/news/43474

Fractal patterns are ubiquitous in nature from the shape of a galaxy to the structure of a snowflake. They may also lie behind the mysterious phenomenon of high-temperature superconductivity, high-temperature superconductors do not appear to create the pairs of electrons needed for zero-resistance conductivity via vibrations of the crystal lattice. suggested that high-temperature superconductivity may be linked to the distribution of oxygen ions in the layers between the copper oxide. They say it might be the formation of some of these ions into rows, or "stripes", that is responsible

What they found was that this intensity followed a power-law distribution, in other words that the superconductor was made up of a small number of very high-ordered regions and larger numbers of disordered regions. This, they say, is the hallmark of a scale-free distribution, which is typical of a fractal pattern – with the oxygen stripes forming a similar structure on all scales up to 400 µm.

http://zone-reflex.blogspot.com/2010/09/walking-molecules_18.html

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