[Paper Review] Oriented matroid theory and loop quantum gravity in (2+2) and eight dimensions

J. A. Nieto|arXiv (Cornell University)|Mar 24, 2010

Noncommutative and Quantum Gravity Theories2 references7 citations

TL;DR

This paper establishes a novel link between oriented matroid theory and loop quantum gravity (LQG) in (2+2) and eight-dimensional spacetimes by connecting the chirotope concept in oriented matroids to the structure constants of normed division algebras—real numbers, complex numbers, quaternions, and octonions—thereby suggesting that LQG may be consistently formulated in 8D and (2+2) signatures. The key contribution is a duality framework in LQG that generalizes the self-dual curvature formalism using oriented matroid duality, extending Ashtekar's formalism beyond 4D and (1+3) signature.

ABSTRACT

We establish a connection between oriented matroid theory and loop quantum gravity in (2+2) (two time and two space dimensions) and 8-dimensions. We start by observing that supersymmetry implies that the structure constants of the real numbers, complex numbers, quaternions and octonions can be identified with the chirotope concept. This means, among other things, that normed divisions algebras, which are only possible in 1,2, 4 or 8-dimensions, are linked to oriented matroid theory. Therefore, we argue that the possibility for developing loop quantum gravity in 8-dimensions must be taken as important alternative. Moreover, we show that in 4-dimensions, loop quantum gravity theories in the (1+3) or (0+4) signatures are not the only possibilities. In fact, we show that loop quantum gravity associated with the (2+2)-signature may also be an interesting physical structure.

Motivation & Objective

• To explore the viability of loop quantum gravity in (2+2) and eight-dimensional spacetimes beyond the standard (1+3) signature.
• To establish a connection between oriented matroid theory and the self-dual curvature formalism in quantum gravity.
• To show that the chirotope concept in oriented matroids corresponds to structure constants of normed division algebras (1,2,4,8 dimensions), linking algebraic structures to quantum gravity.
• To propose a dual formulation of LQG states using circuit and cocircuit spaces, generalizing the Ashtekar formalism via duality.
• To suggest that the (2+2) signature and 8D spacetime are viable alternatives for quantum gravity, motivated by self-duality and division algebra constraints.

Proposed method

• Using the chirotope concept in oriented matroid theory to model the structure constants of normed division algebras (real, complex, quaternions, octonions), which are only defined in 1,2,4,8 dimensions.
• Applying the generalized dual tensor ${}^*R^{AB}$ in 8D using the $η$-symbol (analogous to the $symbol in 4D), showing self-duality is preserved.
• Extending the self-dual curvature formalism from 4D to (2+2) and 8D spacetimes by defining ${}^+R^{AB} = \frac{1}{2}(R^{AB} + \alpha {}^*R^{AB})$, where $\alpha = \{1,i\}$, and $\alpha$ is replaced by a suitable octonionic structure in 8D.
• Formulating physical states $\Psi_C(A,L)$ and $\Psi_{C^*}(A^*,L^\perp)$ as holonomies over circuits and cocircuits in a graph $G$, with $L$ and $L^\perp$ representing circuit and cocircuit spaces.
• Introducing dual Hamiltonian constraints $\hat{H}^*$ and $\hat{H}_l^*$ acting on dual states $|\Psi^*>$, generalizing the Heisenberg and Schrödinger-like duality in LQG.
• Establishing a duality between the original and dual gravitational fields via $E^*$ and $\omega^*$, with corresponding self-dual curvature ${}^+R^{*AB}$, suggesting a unified dual quantum gravity framework.

Experimental results

Research questions

• RQ1 Can loop quantum gravity be consistently formulated in (2+2) spacetime dimensions, given that self-duality is preserved via the $\eta$-symbol?
• RQ2 Is there a fundamental connection between oriented matroid theory and the structure constants of normed division algebras in quantum gravity?
• RQ3 How does the duality between circuit and cocircuit spaces in oriented matroids generalize the Ashtekar formalism in LQG?
• RQ4 Can the self-dual curvature formalism in 8D spacetime be realized using the octonionic $\eta$-symbol, analogous to the quaternions in 4D?
• RQ5 What is the physical role of dual gauge fields $A^*$ and dual Hamiltonian constraints $\hat{H}^*$ in a quantum gravity framework based on oriented matroid duality?

Key findings

• The chirotope concept in oriented matroid theory is shown to correspond directly to the structure constants of normed division algebras, explaining why only 1, 2, 4, and 8 dimensions support self-dual gravity.
• Self-duality of the curvature 2-form is preserved in 8D spacetime using the $\eta$-symbol, which generalizes the $\epsilon$-symbol from 4D and is linked to octonion algebra.
• The (2+2) signature is identified as a viable alternative to (1+3) and (0+4) in loop quantum gravity, with self-duality achievable via the same formalism.
• A dual formulation of LQG states is proposed: $\Psi_C(A,L)$ for circuits and $\Psi_{C^*}(A^*,L^\perp)$ for cocircuits, with corresponding dual Hamiltonian constraints $\hat{H}^*$ and $\hat{H}_l^*$.
• The duality between $L$ and $L^\perp$ (circuit and cocircuit spaces) satisfies a generalized Farkas property, making the formalism self-dual and symmetric, a key feature of oriented matroids.
• The framework suggests that 8D and (2+2) signatures are not just mathematical curiosities but physically viable candidates for quantum gravity, motivated by division algebra constraints and self-duality.