Mathematical Physics


Could the Notion of Hyperdeterminant be Useful in TGD Framework?

Authors: Matti Pitkänen

The vanishing of ordinary determinant tells that a group of linear equations possesses non-trivial solutions. Hyperdeterminant generalizes this notion to a situation in which one has homogenous multilinear equations. The notion has applications to the description of quantum entanglement and has stimulated interest in physics blogs. Hyperdeterminant applies to hyper-matrices with n matrix indices defined for an n-fold tensor power of vector space - or more generally - for a tensor product of vector spaces with varying dimensions. Hyper determinant is an n-linear function of the arguments in the tensor factors with the property that all partial derivatives of the hyper determinant vanish at the point, which corresponds to a non-trivial solution of the equation. A simple example is potential function of n arguments linear in each argument.

Why the notion of hyperdeterminant- or rather its infinite-dimensional generalization- might be interesting in TGD framework relates to the quantum criticality of TGD stating that TGD Universe involves a fractal hierarchy of criticalities: phase transitions inside phase transitions inside... At classical level the lowest order criticality means that the extremal of Kähler action possesses non-trivial second variations for which the action is not affected. The system is critical. In QFT context one speaks about zero modes. The vanishing of the so called Gaussian (of functional) determinant associated with second variations is the condition for the existence of critical deformations. In QFT context this situation corresponds to the presence of zero modes.

The simplest physical model for a critical system is cusp catastrophe defined by a potential function V(x) which is fourth order polynomial. At the edges of cusp two extrema of potential function stable and unstable extrema co-incide and the rank of the matrix defined by the potential function vanishes. This means vanishing of its determinant. At the tip of the cusp the also the third derivative vanishes of potential function vanishes. This situation is however not describable in terms of hyperdeterminant since it is genuinely non-linear rather than only multilinear.

In a complete analogy, one can consider also the vanishing of n:th variations in TGD framework as higher order criticality so that the vanishing of hyperdeterminant might serve as a criterion for the higher order critical point and occurrence of phase transition. Why multilinearity might replace non-linearity in TGD framework could be due to the non-locality. Multilinearty with respect to imbedding space-coordinates at different space-time points would imply also the vanishing of the standard local divergences of quantum field theory known to be absent in TGD framework on basis of very general arguments. In this article an attempt to concretize this idea is made. The challenge is highly non-trivial since in finite measurement resolution one must work with infinite-dimensional system.

Comments: 4 pages.

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Submission history

[v1] 1 Nov 2011

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