2D Gravity with Torsion, Oriented Matroids and 2+2 Dimensions

Abstract

We find a link between oriented matroid theory and 2d gravity with torsion. Our considerations may be useful in the context of noncommutative phase space in a target spacetime of signature (2+2) and in a possible theory of gravity ramification.

Dirac equation in four time and four space dimensions

“…[20]- [25]) one may expect that there must exist a description of the Dirac equation in terms of qubit notion. In turn, (4 + 4)-dimensions have an interesting connection with qubits and chirotopes [26]- [31]. So, we believe that eventually the (4 + 4)-dimensional Dirac equation may shed some light on the superqubit notion.…”

“…and follow similar procedure inserting the gamma matrices γ 30...0 and γ 10...0 at the beginning and at the end respectively and in all the other terms one inserts a 2-index (or ε) in left level of the previous signature one may prove that in (4 + 4)-signature one obtains the result (26). In fact, in this case (26) One physical reason to be interested in these developments emerges from the observation that massless fermions in (4+4)-dimension can lead to massive one in (1 + 3)-dimension. In order to clarify this observation let us write the massless Dirac equation ( 1) as…”

“…In Refs [26] and [27], in analogy with the curved metric g µν = e i µ e j ν η ij , where η ij is a flat metric and e i µ is a zweibeins, the three-index curved metric was proposed g µνλ = e i µ e j ν e k λ η ijk .…”

“…Therefore, what we have shown is that the basis of M(2, R) as given in ( 9) admits an oriented matroid interpretation (see Refs. [26]- [31] and references therein).…”

Phirotopes, super p-branes and qubit theory

“…[3]- [10] and references therein). The key concept to realize these developments is the so called chirotope notion which provides with one of the possible axiomatization for oriented matroid theory (see Ref.…”