Oriented matroid theory and loop quantum gravity in (2+2) and eight dimensions

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.

SL(2,R)-geometric phase space and (2+2)-dimensions

“…We already know that this signature arises in a number of physical scenarious, including in a background for N = 2 strings [17]- [18] (see also Refs [19]- [21]), Yang-Mills in Atiyah Singer background [22] (see also Refs. [23] for the importance of the 2 + 2 signature in mathematics), Majorana-Weyl spinor [24]- [25] and more recently in loop quantum gravity in terms of oriented matroid theory [26] (see also references [27]- [29]). But one wonders whether the 2 + 2 signature can be linked to quantum gravity itself in 1 + 3 dimensions.…”

Towards a Canonical Gravity in Two Time and Two Space Dimensions

“…This is because in such case there is not a particular reason for considering the splitting of (31) in terms of only one time coordinate (see Ref. [9]) instead of two times coordinates. Specifically, in this case one has the following splitting of (31):…”

“…Finally, it is worth mentioning that one of the main motivations in Ref. [9] was the idea of establishing a connection between Ashtekar formalism in diag(−1, −1, +1 + 1) signature and oriented matroid matroid theory [22] (see also Ref. [23]- [24] and references therein).…”

“…In fact, it might be 1+3 or 0+4, as it is often considered in most of the current developments of loop quantum gravity, but it might also be 2+2 as has been shown in Ref. [9], where canonical gravity of the splitting type (1)+(1+2) was developed. It is worth mentioning that a canonical approach with a splitting of the type (1+1)+(2) has already be considered (see [10]- [12]and references therein).…”

Alternative self-dual gravity in eight dimensions

“…Higher dimensional Ashtekar formalism has shown to be a promising proposal [1][2][3][4][5] (see also Ref. [6][7][8][9][10]). The main physical reason for this is that it allows a possible connection with string theory [11]; an alternative approach for quantum gravity.…”