Correspondence between dark energy quantum cosmology and Maxwell equations

A Friedmann-Robertson-Walker cosmology with dark energy can be modelled using a quintessence field. That system is equivalent to a relativistic particle moving on a two-dimensional conformal spacetime. The quantized version of that theory can lead to a Supersymmetric Majorana quantum cosmology. The purpose of this work is to show that such quantum cosmological model correspond to the Maxwell equations for electromagnetic waves propagating in a medium with specific values for its permittivity and permeability. The exact form of those media parameters are calculated, implying that a Majorana quantum cosmology can be studied in an analogue optical system.


I. INTRODUCTION
The aim of this work is to show that there exists a correspondence between the seemingly different physical models of a cosmological model using dark energy and Maxwell equations. The link between these two formalisms appears when one considers the quantum version of the cosmological model [1] using the Breit prescription [2] for spin particles.
The representation of dark energy using a quintessence field with a potential, allows us to describe the cosmological dynamics in fashion which is analogous to the description of the dynamics of a relativistic particle. Therefore, this theory may be quantized by using a Klein-Gordon scheme, giving rise to the Wheeler-DeWitt equation [3][4][5][6][7][8][9][10][11][12][13]. However, using the Breit prescription [2], the same model can be quantized as a spinorial theory. This procedure yields a Majorana version for the Quantum cosmology which happens to be supersymmetric [1].
In addition of the cosmological implications of such theory, the aim of this article is to show its direct correspondence with the description of propagating electromagnetic fields in a medium using Maxwell equations. We can identify the permittivity and permeability of the medium with parameters of the quantum cosmological model. As we show, this implies that the Supersymmetric Majorana quantum cosmology can be studied in an analogue system using either normal materials or negativeindex metamaterials (NIMs) [14,15]. * Electronic address: felipe.asenjo@uai.cl † Electronic address: sergio.hojman@uai.cl

II. QUANTUM COSMOLOGY WITH DARK ENERGY
Consider an isotropic and homogeneous Friedmann-Robertson-Walker (FRW) spacetime with a line element [16] where a(t) is the scale factor, and the curvature constant k = ±1, 0. The evolution of a FRW cosmology with cosmological constant Λ, interacting with a quintessence (massless scalar) field φ(x β ) characterized by a potential V(φ), can be found by using Einstein equations [1] (8πG/c 4 = 1, where G is the gravitational constant and c is the speed of light) where we introduced V (φ) = V(φ) − Λ. Also, the Klein-Gordon equation for the quintessence field is The above system describes the evolution of a Friedmann-Robertson-Walker-Quintessence (FRWQ) Universe. In Ref. [1] was shown that the Lagrangian gives rise to all three equations (2), (3) and (4), with variables ξ = ln(2 √ 6a 3/2 /3) and θ = 3φ/(2 √ 6) [hereξ = dξ/dλ andθ = dθ/dλ, with λ is an arbitrary parameter]. We have introduced the general potential This Lagrangian (5) shows that the FRWQ cosmology evolves as a relativistic particle moving in two dimensional spacetime under the influence of potentialV . It can be proved that the Lagrangian (5) gives rise to the FRWQ equations by recalling that the Jacobi-Maupertuis and Fermat principles [17] yield identical equations of motion in classical mechanics and geometrical (ray) optics except for the fact that Fermat principle also produces a constraint equation. Notice that Lagrangian (5) can be written as the one for a relativistic particle in a two-dimensional conformally flat spacetime with the metric g µν = Ω 2 η µν (where η µν is the flat spacetime metric) and the conformal factor Thus, the FRWQ system is equivalent to a relativistic particle moving in a two-dimensional conformally flat spacetime, where the quintessence field plays the role of an effective time.
The above results allow to find the quantum version of the FRWQ cosmology, through quantization of the Larangian (5). Restricting ourselves toV > 0, it is possible to find the the quantum Hamiltonian operator [1] wherep is the momentum operator andπ = −i∂ ξ . This Hamiltonian is found from its classical counterpart H = √ g 00 1 + π 2 /Ω 2 , where π is the canonical momentum, and √ g 00 = Ω. Therefore, the quantum equation that describes the quantization of the FRWQ system is where Ψ is the wavefunction for the quantum FRWQ cosmology. The quantum equation (9) emerges because of the direct correspondence between the FRW geometry and the quintessence scalar field at a Fermat-like Lagrangian level. To find a solution, we need a quantization procedure to solve the square-root of the Hamiltonian operator (8). It is custumary to solve the squareroot using a spinless particle approach to get a Klein-Gordon equation [18], which gives origin to the Wheeler-DeWitt Super-Hamiltonian formalism. However, notice that there is no restriction for the quantization scheme we used. In principle, we can solve the square-root using matrices, obtaining the quantization of a relativistic particle which leads to the Dirac equation. This procedure was introduced by Breit [2], that shows that there is a correspondence between the Dirac and the relativistic pointlike particle Hamiltonians. Breit's interpretation [2] identifies the Dirac matrices as π/H → α, and 1 − (π/H) 2 → β. These identifications are consistent with the postulates of Dirac electron's theory [2,19]. By the Breit's prescription, the Hamiltonian operator (8) becomes [1] where α and β are the two-dimensional flat spacetime Dirac matrices, as the effective curvature is already taken into account in Ω (with = 1). Moreover, now the wavefunction Ψ [in Eq. (9)] is a two-dimensional spinor.
With this Hamiltonian, and defining the wavefunction Φ = √ ΩΨ, we finally find from Eq. (9) the spinor quantum equation [1] iγ 0 ∂Φ ∂θ where γ 0 = β and γ 1 = γ 0 α. The above equation corresponds to a Quantum Cosmology theory for the FRWQ system, modelling now the Universe as a spin particle in a two-dimensional conformally flat spacetime [1]. In order to obtain real wavefunctions Φ, the matrices in Eq. (11) should correspond to the two-dimensional Majorana representation In this form, Eq. (11) becomes a set of supersymmetric equations of quantum mechanics [1,[20][21][22][23][24], which can only be obtained in the Majorana picture. The implications of this system were thoroughly studied in Ref. [1], showing that Eq. (11) with matrices (12) represents a Supersymmetric Majorana quantum cosmology.

III. CORRESPONDENCE TO MAXWELL EQUATIONS
The Majorana Quantum FRWQ system (11) models the Universe in a supersymmetric way that can bring new insights in quantum cosmology. However, in this paper, we left the consequence of this theory for future works, as we are interested in a particular feature of the model. We focus the correspondence between the theory (11) and electromagnetism.
In order to make this correspondence manifest, let us consider Maxwell equations in a medium with permittivity ǫ and permeability µ in the absence of charges [25] ∂D ∂t = ∇ × H , with the electric field E, the magnetic field B, the displacement field D = ǫ E, and the magnetization field H = B/µ. Furthermore, ∇ · D = 0 and ∇ · B = 0. In order to show the correspondence with the Quantum Cosmology model, we assume that the permittivity and the permeability are not constant. Also, let us consider a two-dimensional spacetime system (one temporal and one spatial dimension), with spatial variations in, let us say, theê z -direction. We choose transverse fields B(t, z) = B(t, z)ê x , and D(t, z) = D(t, z)ê y , such that B ·ê z = 0 = D ·ê z , and B · D = 0. Besides, the time-independent permittivity and permeability have the same spatial dependence, ǫ = ǫ(z) and µ = µ(z). Then, Maxwell equations (13) acquires the form By introducing the spinor we can re-write system (14) in a simple way as with γ 0 given in (12), and the matrices with µ ′ = ∂µ/∂z and ǫ ′ = ∂ǫ/∂z. The spinor form (16) of Maxwell equations is general in a two-dimensional spacetime. For any general permittivity and permeability, Eq. (16) does not coincide with the Quantum cosmology equation (11). For example, for vacuum, Γ 2 = 0.
However, we can show that there exists a regime in which Maxwell equations and Supersymmetric Majorana Quantum Cosmology coincide. Let us consider the following form for the permittivity and permeability for a constant ζ, and in terms of a function λ to be determined. We focus our attention in the regime when |λ| ≪ 1. The permeability and permittivity of this material are almost equal, µ/ǫ ≈ 1 − 2λ. With the permittivity and permeability given in (18), the matrices (17) becomes with the matrix γ 1 given in (12), and the two-dimensional unit matrix 1. If the variation scales of B and D are much larger than the variation scale of λ, then Maxwell equation (16) can be written in an approximated form as The correspondence between Maxwell equations (20) and the Supersymmetric Quantum Majorana Cosmologies equation (11) is evident using the re-definitions This last equation implies that this correspondence is only valid in the regime Ω 3 e −4ξ ≪ |V (θ)|.
The results (21) establish the complete correspondence between a Supersymmetric Majorana Quantum cosmological model and Maxwel equations. Materials satisfying (18) must have almost equal permeability and permittivity, and they can be both positive ζ > 0 or both negative ζ < 0. In the former case, we are describing a normal material with such a property. In case that both permeability and permittivity are negative, we are describing a NIM [14,15]. The refraction index is where the positive sign describes a normal material, and the negative refractive index represents a NIM in which a wave propagates backwards.
The above results imply that quantum cosmology theories can be optically tested by using normal materials or NIMs, such that its permittivity, permeability and refraction index have the same spatial dependence that the quintessence and scale factor. For example, for a spatially flat cosmology, the Maxwell equations and the Supersymmetric Majorana quantum cosmology coincide for λ(ξ) = 3V /32 exp(2ξ), withV = −V > 0, and refraction index n(ξ) ≈ ±|ζ|[1 − 3V exp(4ξ)/64].
Our proposal is in the same spirit than similar ones for analogue optical systems for quantum cosmologies [26,27], and for gravity in general (see for example Refs. [28][29][30][31]). However, our result establishes an optical analogue for a new kind of spinor quantized cosmological model. Also, the proposed analogue optical media that corresponds to the quantum cosmology is timeindependent but space-dependent, which is the opposite to previous attempts [26]. All the above makes of this Majorana Supersymetric quantum cosmological model a system worth to be studied in an analogue fashion using Maxwell equations in the appropriate media.