Cosmological Duality in Four Time and Four Space Dimensions

We describe a duality transformation in a cosmological model of four time and four space dimensions ((4 + 4)-dimensions). In particular, we show that via the Fourier transform, at the level of the zero-point energy of quantum mechanics and the de Sitter space, a Gaussian distribution in four dimensions leads to a dual Gaussian distribution also in four dimensions, with duality transformation 1 σ σ → , in the standard deviation σ . Moreover, we show that as a consequence of such a duality in σ a duality of the cosmological constant Λ can be obtained. Finally, we comment on the possibility that both the oriented matroid theory as well as the surreal number theory are related to the formalism presented in this work.


Introduction
It is known that the (5 + 5)-dimensional space-time (five time and five space dimensions) is a common signature to both type IIA strings and type IIB strings [1]. In fact, versions of M-theory [2] lead to type IIA and to type IIB string in space-time of signatures (5 + 5). It turns out that by duality transformations string theories of signatures (5 + 5) can be related to other string signatures such as (1 + 9) [3].
Just as the (1 + 3)-dimensional signature can be considered as a reduced world of the de Sitter (1 + 4)-dimensional or anti-de Sitter (2 + 3)-dimensional signatures via the cosmological constant 0 Λ > and 0 Λ < (see Refs [4] [5] and 1 σ σ ↔ of a Gaussian of distribution of 4-space coordinates associated with the de Sitter space (anti-de Sitter) and the vacuum zero-point energy yields to a Gaussian of 4-time coordinates of the same vacuum scenario. This is in fact one of our main contribution and its importance emerges when we notice that only in 4-dimensions such a totally duality symmetry is achieved. In the process, we discover that in 4-dimensions the cosmological constant 0 Λ > associated with the de Sitter space ( 0 Λ < , anti de Sitter space) is dual to the cosmological constant There are at least two frameworks where a (4 + 4)-world has emerged as interesting physical scenario. First, it has been proved [6] that the Dirac equation in (4 + 4)-dimensions admit a Majorana-Weyl physical spinor state with only 8 real components which can be identified with the 4-complex components of the usual electron components. Second, in Ref. [7] it has been shown that a general Kruskal-Szekeres transform, in black-hole physics, implies 8 hidden regions instead of just 4-regions as it is usually believed. It turns out that this 8-regions admit better interpretation in a world of (4 + 4)-dimensions.
Technically, this work is organized as follows. In Section 2, we consider the geodesic of a point particle in the de Sitter (anti-de Sitter) space and show that the classical harmonic oscillator equation in 4-dimensions emerges. In Section 3, we quantize the system obtained in the previous section and focus in the case of zero-point energy showing that this case admits a Gaussian distribution solution. While in Section 4, by the use of the Fourier transform we investigate a duality scenario. Thus combining the zero-point energy and de Sitter vacuum space we discover a duality of the cosmological constant. Finally in Section 5, we express a number of final remarks. In particular, from our analysis of the Gaussian distribution from 4-space dimensions to 4-time dimensions we conclude that the space-time involved may be considered corresponds to a (4 + 4)-dimensional spacetime. Moreover, we briefly comment about the possibility that, for further work, the mathematical structures of matroid theory and surreal number theory may be interesting routes for a connection with our approach.

Geodesic Equation of the de Sitter (Anti-de Sitter) Space
Let us start recalling some geometrical aspects of the de Sitter space (or anti de Sitter space). In particular, it is well known that the Christoffel symbols in the de gue role of energy and momentum, are dimensionless. Thus, the formula (7) leads to the Schrödinger equation This leads to 2 0ˆ.
Of course, in analogy to the usual harmonic oscillator equation, (9) must imply that ε is quantized. Since one is mainly interested in the lowest energy state, here we shall be concerned only with the zero-point "energy". For this purpose for each coordinate in Thus, in the zero-point "energy", for each coordinate in ( ) , , , x y z ξ one must have the equation where is the inverse of the standard deviation. Following similar steps, for the other coordinates , , y z ξ one finally discovers that Here, q is a positive integer number to be determined below. Of course, (15) (or (16)) corresponds to Gaussian distribution in 4-dimensions.

Duality and the Fourier Transform
It turns out that the Fourier transform of (15) or (16) is given by In fact, let us show in some detail that (15) (or (16)) implies (17). For this purpose let be the Fourier transform that sends us from the one dimen- Here the function ( ) 2 e e d , 2 then, completing the squares one finds ( ) Now, introducing the variable it u x σ σ = − so that the differential remains as So, let us generalize the above procedure to any dimension d. One has One can again make the change of variable Performing the integrals in (25) yield In particular, by setting 2 q = one obtains 4 d = and thus (16) in agreement with (17). Note the surprising duality relation between  and *  , namely which only is fully achieved in 4 d = . In this sense one can say that the 4-dimensions associated with *  and and 4-dimensions associated with  are dual to each other. Observe that * *    which is the usual requirement for a duality symmetry. From (14) one knows that , for 1 n = , one obtains that α is of the order of 10 −5 cm 2 which is too large to be identified with any fundamental atomic radius. However, an interesting and attractive possibility is to assume the condition . It is remarkable that there exist a relation between the de Sitter length 0 l and the cosmological constant Λ , namely (see [8] and references therein) Thus, the duality relation (31) can be written as where one has defined Since one is assuming the formula (33) one sees that 2 P β Λ Λ = , with 2 2 2 9 n β α = and hence one has discovered a cosmological constant duality 1 .
So, considering the case 1 n = and 1 α = , since p l is of the order of 10 −33 cm, one finds that P Λ is of the order of 10 66 cm −2 which is, of course, very large, but according to (38) the cosmological constant Λ will be very small of the order 10 −66 cm −2 . It is worth mentioning that the type of duality (38) has been previously described in the context of S-duality for linearized gravity [9].

Final Remarks
In the above discussion, we have focused in the de Sitter space with 0 Λ > , however similar conclusions must be achieved in the case of anti de Sitter space with 0 Λ < (see Refs [4] [5] and references therein). Further, in the usual quantum mechanics, the Fourier transform relate in one to one correspondence the configuration space of q-coordinate with the conjugate momentum p-coordinates. If there are n such q-coordinates there must n p-coordinates and this mean that the total phase space must be (n + n)-dimensional. Following this route of though one can say that the Fourier transform between the 4 x µ -coordinates and the 4 t α -coordinates in our approach must lead inevitably to a (4 + 4)-dimensional space-time. In fact, combining (23), (28) and (29) one may write e e e e d d d d 2 and 2 2 f σ * = π . So, from (39) one may conclude that the space-time involved may be considered to correspond to a (4 + 4)-dimensional. Therefore, a surprising scenario emerges from our formalism: The point is that our result refers to both vacuum de Sitter space associated with the "macroscopic" x µ -world of 4-dimensional space and the zero-point "energy" linked to the lowest possible energy at that quantum "microscopic" t α -world of 4-dimensions, and vice versa. The key result is that these two worlds are related by duality symmetry; a standard deviation σ of a x µ -world is dual to the 1 σ in the t α -world. This means that a x µ -world with thin and tall (small σ ) Gaussian distribution is dual to wide and low (large σ ) Gaussian distribution in the other t α -world and vice versa. Moreover, this is verified by observing that the large cosmological constant P Λ in the quantum world is dual to the small cosmological constant Λ in the classical cosmological world. This seems to explain, in a cosmological context, the smallness of the cosmological constant. Thinking in the possible geometrization of the duality symmetry in (4 + 4)-dimensions space, one is tempted to propose the modified line element The idea that duality emerges as a key concept in vacuum cosmology in 4-dimensions has been explored before in [10]. Here, we have shown that duality also seems to play a fundamental role in vacuum cosmology in (4 + 4)-dimensions, at both macroscopic and microscopic level. So, one wonders whether duality can be the central mathematical notion in quantum gravity. This route of though shall bring us sooner or later to look for a mathematical structure as a frame for the duality concept. In turns out that such a formalism already exist, namely oriented matroid theory (see Refs. [11]- [17] and references therein). This mathematical framework is a generalization of both graphs and matrices and establishes that every oriented matroid has a dual. This means that although every graph corresponds to a matroid, there are matroids which are not graphic. So, this clarify why the complete graphs are not graphic. Thus, oriented matroid theory has entered its key concept: duality. So, one may be interesting for further work to establish the relation between the duality described in this work and oriented matroid theory.
Finally, let us just mention another source of interest for further work. We refer to the surreal number theory. This is a mathematical structure that was discovered by Conway [18] (see also Ref. [19]) and Gonshor [20]. Among its interesting properties of such numbers is that natural, integers and real numbers are contained in the surreal number framework. In Refs. [21] and [22] it was established the main reasons why we believe that such a numbers may be physically interesting. In particular one may be interested in considering surreal number theory in the context of quantum gravity in (4 + 4)-dimensions. Let us clarify this comment. Classical general relativity in (4 + 4)-dimensions can be formulated in R 8 , that is in a continuum 8-dimensional space-time background. So quantizing gravity may be though as the framework, which must imply at mechanism for starting with a continuum differential geometry in 8-dimensions and then discretize such a geometry. But what if we go backwards? That is, we start with a discrete mathematical structure of surreal numbers  and we end up with a continuum 8-dimensional manifold based on R. Consider the set [18] { }