Correspondence between dark energy quantum cosmology and Maxwell equations
aa r X i v : . [ g r- q c ] A ug Correspondence between dark energy quantum cosmology and Maxwell equations
Felipe A. Asenjo ∗ and Sergio A. Hojman
2, 3, 4, † Facultad de Ingenier´ıa y Ciencias, Universidad Adolfo Ib´a˜nez, Santiago 7941169, Chile. Departamento de Ciencias, Facultad de Artes Liberales, Departamento de F´ısica, Facultad de Ciencias, Universidad de Chile, Santiago, Chile. Centro de Recursos Educativos Avanzados, CREA, Santiago, Chile.
A Friedmann-Robertson-Walker cosmology with dark energy can be modelled using a quintessencefield. That system is equivalent to a relativistic particle moving on a two-dimensional conformalspacetime. When the quintessence behaves as a free massless scalar field in a Universe with cos-mological constant, the quantized version of that theory can lead to a Supersymmetric Majoranaquantum cosmology. The purpose of this work is to show that such quantum cosmological model cor-respond to the Maxwell equations for electromagnetic waves propagating in a medium with specificvalues for its relative permittivity and relative permeability. The form of those media parame-ters are calculated, implying that a Majorana quantum cosmology can be studied in an analogueelectromagnetic system.
PACS numbers:Keywords:
I. INTRODUCTION
The aim of this work is to show that there exists acorrespondence between the seemingly different physi-cal models of a cosmological model using dark energyand Maxwell equations. The link between these two for-malisms appears when one considers the quantum versionof the cosmological model [1] using the Breit prescription[2] for spin particles.The representation of dark energy using a quintessencefield with a potential, allows us to describe the cosmologi-cal dynamics in fashion which is analogous to the descrip-tion of the dynamics of a relativistic particle. In the caseof quintessence described by a free massless scalar field ina Universe with cosmological constant, the resultant the-ory may be quantized by using a Klein-Gordon scheme,giving rise to the Wheeler-DeWitt equation [3–13]. How-ever, using the Breit prescription [2], the same model canbe quantized as a spinorial theory. This procedure yieldsa Majorana version for the Quantum cosmology whichhappens to be supersymmetric [1].In addition of the cosmological implications of suchtheory, the aim of this article is to show its direct cor-respondence with the description of propagating electro-magnetic fields in a medium using Maxwell equations.We can identify the relative permittivity and perme-ability of the medium with parameters of the quantumcosmological model. As we show, this implies that theSupersymmetric Majorana quantum cosmology can bestudied in an analogue electromagnetic system using ei-ther normal materials or negative-index metamaterials(NIMs) [14, 15]. ∗ Electronic address: [email protected] † Electronic address: [email protected]
II. QUANTUM COSMOLOGY WITH DARKENERGY
Consider an isotropic and homogeneous Friedmann-Robertson-Walker (FRW) spacetime with a line element[16] ds = dt − a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) , (1)where a ( t ) is the scale factor, and the curvature constant k = ± ,
0. The evolution of a FRW cosmology withcosmological constant Λ, interacting with a quintessence(massless scalar) field φ ( x β ) characterized by a poten-tial V ( φ ), can be found by using Einstein equations [1](8 πG/c = 1, where G is the gravitational constant and c is the speed of light)2 ¨ aa + (cid:18) ˙ aa (cid:19) + ka + 12 ˙ φ − V ( φ ) = 0 , (2)3 (cid:18) ˙ aa (cid:19) + 3 ka − (cid:18)
12 ˙ φ + V ( φ ) (cid:19) = 0 , (3)where we introduced V ( φ ) = V ( φ ) − Λ. Also, the Klein-Gordon equation for the quintessence field is¨ φ + 3 ˙ aa ˙ φ + dV ( φ ) dφ = 0 . (4)The above system describes the evolution of aFriedmann–Robertson–Walker–Quintessence (FRWQ)Universe.In Ref. [1] was shown that the Lagrangian L F = r ¯ V ( ξ, θ ) e ξ (cid:16) ˙ θ − ˙ ξ (cid:17) , (5)gives rise to all three equations (2), (3) and (4), withvariables ξ = ln(2 √ a / / θ = 3 φ/ (2 √
6) [here ˙ ξ = dξ/dλ and ˙ θ = dθ/dλ , with λ is an arbitrary parameter],and the general potential¯ V ( ξ, θ ) = 3 (cid:18) (cid:19) / k e ξ/ − (cid:18) (cid:19) e ξ V ( θ ) . (6)This Lagrangian (5) shows that the FRWQ cosmologyevolves as a relativistic particle moving in two dimen-sional spacetime under the influence of potential ¯ V .It can be proved that the Lagrangian (5) gives riseto the FRWQ equations by recalling that the Jacobi–Maupertuis and Fermat principles [17] yield identicalequations of motion in classical mechanics and geomet-rical (ray) optics except for the fact that Fermat princi-ple also produces a constraint equation. Notice that La-grangian (5) can be written as the one for a relativisticparticle in a two-dimensional conformally flat spacetime L F = s g µν (cid:18) dx µ dλ (cid:19) (cid:18) dx ν dλ (cid:19) , (7)with the metric g µν = Ω η µν (where η µν is the flat space-time metric, and x = θ and x = ξ ), and the conformalfactorΩ ≡ p ¯ V e ξ = " (cid:18) (cid:19) k e ξ/ − V ( θ ) e ξ / , (8)Thus, the FRWQ system is equivalent to a relativisticparticle moving in a two-dimensional conformally flatspacetime, where the quintessence field plays the role ofan effective time.The above results allow to find the quantum version ofthe FRWQ cosmology, through quantization of the La-grangian (5). We restrict ourselves to cases where ¯ V > ξ , θ and ¯ V . All of this implies that ∂ θ g µν = 0, or ∂ θ ¯ V = 0, describing a constant potential V ( θ ) (becom-ing essentially equal to a cosmological constant). Thus,the associated Noether conservation law implies that ˙ θ does not change sign, and θ can be used as the evolutiontime variable. Using this θ –time, it can be shown at clas-sical level [19] that the Hamiltonian for Lagrangian (5)is H = √ g p − g π = √ g p π / Ω , where π isthe canonical momentum, and √ g = Ω. This classicalHamiltonian is used to construct the quantum Hamilto-nian operator H as [1] H = Ω / p p Ω / , ˆ p = p − g ˆ π = − i Ω ∂∂ξ , (9)where ˆ p is the momentum operator and ˆ π = − i∂ ξ . There-fore, the quantum equation that describes the quantiza-tion of the FRWQ system is i ~ ∂ Ψ ∂θ = H Ψ , (10) where Ψ is the wavefunction for the quantum FRWQcosmology. The quantum equation (10) emerges becauseof the direct correspondence between the FRW geom-etry and the quintessence scalar field at a Fermat-likeLagrangian level. To find a solution, we need a quanti-zation procedure to solve the square-root of the Hamil-tonian operator (9). It is costumary to solve the square-root using a spinless particle approach to get a Klein-Gordon equation [20], which gives origin to the Wheeler–DeWitt Super-Hamiltonian formalism. However, noticethat there is no restriction for the quantization schemepossible to be used. In principle, we can solve the square-root using matrices, obtaining the quantization of a rela-tivistic particle which leads to the Dirac equation. Thisprocedure was introduced by Breit [2], that shows thatthere is a correspondence between the Dirac and the rel-ativistic pointlike particle Hamiltonians. Breit’s inter-pretation [2] identifies the Dirac matrices as π/H → α ,and p − ( π/H ) → β . These identifications are consis-tent with the postulates of Dirac electron’s theory [2, 21].By the Breit’s prescription, the Hamiltonian operator (9)becomes [1] H = Ω / ( α · ˆ p + β ) Ω / , (11)where α and β are the two-dimensional flat spacetimeDirac matrices, as the effective curvature is already takeninto account in Ω (with ~ = 1). Moreover, now the wave-function Ψ [in Eq. (10)] is a two-dimensional spinor.With this Hamiltonian, and defining the wavefunctionΦ = √ ΩΨ, we finally find from Eq. (10) the spinor quan-tum equation [1, 22] iγ ∂ Φ ∂θ + iγ ∂ Φ ∂ξ = ΩΦ . (12)where γ = β and γ = γ α . The above equation corre-sponds to a Quantum Cosmology theory for the FRWQsystem, modelling now the Universe as a spin particle ina two-dimensional conformally flat spacetime [1].In order to obtain real wavefunctions Φ, the matri-ces in Eq. (12) should correspond to the two-dimensionalMajorana representation γ = (cid:18) − ii (cid:19) , γ = (cid:18) i − i (cid:19) . (13)In this form, Eq. (12) becomes a set of supersymmetricequations of quantum mechanics [1, 23–27], which canonly be obtained in the Majorana picture. The implica-tions of this system were thoroughly studied in Ref. [1],showing that Eq. (12) with matrices (13) represents a Su-persymmetric Majorana quantum cosmology. This canbe seen by definingΦ( θ, ξ ) = (cid:18) ϕ + ( ξ ) e Eθ ϕ − ( ξ ) e − Eθ (cid:19) , (14)to find from (12) the set of supersymmetric equations ofquantum mechanics [1, 22–27] Q ± ϕ ± = Eϕ ∓ . (15)with the operators Q ± = ± d ξ + Ω. These equations aresupersymmetric with the two spinor components beingsuper–partners of each other. Each wavefunction satisfies H ± ϕ ± = E ϕ ± , with the Hamiltonians operators H ± = − d ξ + W ± , and potentials W ± = ∓ d ξ Ω+Ω [1]. Also, Q ± correspond to supercharge operators. This theory canproduce new versions of quantum cosmological models.For example, it can be shown that in flat curvature case,the Universe behaves as a diatomic molecule subject tothe Morse potential [1]. III. CORRESPONDENCE TO MAXWELLEQUATIONS
In this section we focus the correspondence betweenthe theory (12) and electromagnetism. In order to makethis correspondence manifest, let us consider Maxwellequations in a medium with permittivity ǫ and perme-ability µ in the absence of charges [28] ∂ D ∂t = ∇ × H ,∂ B ∂t = −∇ × E , (16)with the electric field E , the magnetic field B , the dis-placement field D = ǫ E , the magnetization field H = B /µ (chosing the speed of light c = 1). Furthermore, ∇ · D = 0 and ∇ · B = 0.In order to show the correspondence with the Quan-tum Cosmology model, we assume that the permittiv-ity and the permeability are not constant. Also, let usconsider a two-dimensional spacetime system (one tem-poral and one spatial dimension), with spatial variationsin, let us say, the ˆ e z -direction. We choose transversefields B ( t, z ) = B ( t, z )ˆ e x , and D ( t, z ) = D ( t, z )ˆ e y , suchthat B · ˆ e z = 0 = D · ˆ e z , and B · D = 0. Besides, thetime-independent permittivity and permeability have thesame spatial dependence, ǫ = ǫ ( z ) and µ = µ ( z ). Then,Maxwell equations (16) acquire the form ∂ D ∂t = − B ˆ µ ∂ ˆ µ∂z + 1ˆ µ ∂B∂z ,∂B∂t = − D ˆ ǫ ∂ ˆ ǫ∂z + 1ˆ ǫ ∂ D ∂z . (17)where D = µ D , the relative permittivity is ˆ ǫ = ǫ/ǫ ,the relative permeability is ˆ µ = µ/µ , and ǫ and µ arethe free-space permittivity and permeability respectively(with ǫ µ = 1).By introducing the spinorΦ = (cid:18) B D (cid:19) , (18)we can rewrite system (17) in a simple way as iγ ∂ Φ ∂t + i Γ ∂ Φ ∂z = Γ Φ , (19) with γ given in (13), and the matricesΓ = (cid:18) i/ ˆ µ − i/ ˆ ǫ (cid:19) , Γ = (cid:18) − ˆ µ ′ / ˆ µ
00 ˆ ǫ ′ / ˆ ǫ (cid:19) , (20)with ˆ µ ′ = ∂ ˆ µ/∂z and ˆ ǫ ′ = ∂ ˆ ǫ/∂z . The spinor form (19)of Maxwell equations is general for a two-dimensionalspacetime. For any general permittivity and permeabil-ity, Eq. (19) does not coincide with the Quantum cos-mology equation (12). For example, for vacuum, Γ = 0.However, we can show that there exists a regime inwhich Maxwell equations and the Supersymmetric Ma-jorana Quantum Cosmology coincide. Let us consider thefollowing form for the relative permittivity and relativepermeabilityˆ ǫ ( z ) ≈ λ ( z ) , ˆ µ ( z ) ≈ − λ ( z ) , (21)in terms of a function λ to be determined. We focusour attention in the regime when | λ | ≪
1. The relativepermeability and permittivity of this material are almostequal, p ˆ µ/ ˆ ǫ ≈ − λ . With the relative permittivity andpermeability given by (21), the matrices (20) becomesΓ ≈ (cid:0) λ (cid:1) γ + iλ , Γ ≈ dλdz − i dλ dz γ , (22)with the matrix γ given by (13), and the two-dimensional unit matrix . In general, for any ˆ ǫ andˆ µ , Eq. (19) does not have a conserved current ¯Φ γ µ Φ =Φ † Φ + Φ † γ γ Φ. Nonetheless, for materials satisfying(21) and (22), Eq. (19) can have a conserved currentif | λ | ≪ B and D are much larger than the variation range of λ , i.e., λ ( ∂ Φ /∂z ) ≪ ( dλ/dz )Φ. Using these approximations,Maxwell equation (19) can be written in an approximatedform as iγ ∂ Φ ∂t + iγ ∂ Φ ∂z = dλdz Φ , (23)which has the usual conservartion law for a Dirac equa-tion. With all of the above, the correspondence betweenMaxwell equations (23) and the Supersymmetric Quan-tum Majorana Cosmologies equation (12) is now evidentby redefining t = θ ,ξ = z ,λ = λ ( ξ ) = Z Ω dξ = − V e − ξ Ω . (24)This last equation implies that this correspondence isonly valid in the regime Ω e − ξ ≪ | V | , which is the limitneeded in order to keep current conservation.Results (24) establish the complete correspondence be-tween a Supersymmetric Majorana Quantum cosmolog-ical model and Maxwel equations. Materials satisfying(21) must have almost equal relative permeability andpermittivity, and they can be both positive or both neg-ative. In the former case, we are describing a normalmaterial with such a property. In case that both perme-ability and permittivity are negative, we are describing aNIM [14, 15, 29]. In general, the refraction index is n ( ξ ) = p ˆ ǫ ( ξ )ˆ µ ( ξ ) ≈ ± (cid:18) − V e − ξ Ω (cid:19) , (25)where the positive sign describes a normal material, andthe negative refractive index represents a NIM in whicha wave propagates backwards.There are materials with relative permittivity and rela-tive permeability that can have the form (21). Compos-ite ferrites [30–33] can, under appropriated conditions,achieve almost matching permeability and permittivityvalues by shining radiation of different frequencies onthe material. This implies that the above results canbe tested in an analogue fashion using those materials.For example, for a spatially flat cosmology, the Maxwellequations and the Supersymmetric Majorana quantumcosmology coincide for λ ( ξ ) = p − V /
32 exp(2 ξ ), withconstant V <
0, and refraction index n ( ξ ) ≈ ± [1 +3 V exp(4 ξ ) / ξ ≪ ln(11 / | V | ).Furthermore, the boundary conditions for Φ given inEq. (18) (which depend on Ω) should be suitable for sim-ulating a quantum cosmology. For k = 0, the boundaryconditions for ξ → ∞ ( a → ∞ ) can be established to ob-tain a vanishing magnetic field at infinity in one spatialdimension (see Ref. [1]).Our proposal is in the same spirit than similar onesfor analogue optical systems for quantum cosmologies[34, 35], and for gravity in general (see for exampleRefs. [36–39]). However, our result establishes an “op-tical” analogue for a new kind of spinor quantized cos-mological model. The proposed analogue electromag-netic media that correspond to the quantum cosmology istime-independent but space-dependent, which is an ap-proach opposite to previous attempts [34]. Relations (21)are satisfied by certain tunable metamaterials [40, 41]and composite ferrites [30–33] used to operate at a widerrange of frequencies. All of the above makes of this Ma-jorana Supersymetric quantum cosmological model a sys-tem worth to be studied by studying wave propagationin Maxwell equations in the appropriated media. [1] S. A. Hojman and F. A. Asenjo, Phys. Rev. D , 083518(2015).[2] G. Breit, Proc. Nat. Acad. Sci. , 555 (1928).[3] B. S. DeWitt, Phys. Rev. , 1113 (1967).[4] J. J. Halliwell and S. W. Hawking, Phys. Rev. D , 1777(1985).[5] F. G. Alvarenga and N. A. Lemos, Gen. Rel. Grav. ,681 (1998).[6] G. Oliveira-Neto, Phys. Rev. D , 107501 (1998).[7] N. A. Lemos, J. Math. Phys. , 1449 (1996).[8] G. A. Monerat, E. V. Corrˆea Silva, G. Oliveira-Neto, L.G. Ferreira Filho and N. A. Lemos, Phys. Rev. D ,044022 (2006).[9] G. Oliveira-Neto, G. A. Monerat, E. V. Corrˆea Silva, C.Neves and L.G. Ferreira Filho, Int. J. Theor. Phys. ,2991 (2013).[10] B. Vakili, Ann. Phys. (Berlin) , 359 (2010).[11] R.-N. Huang, arXiv:1304.5309v2 (2013).[12] J. B. Barbour and N.’O Murchadha,arXiv:gr-qc/9911071v1 (1999).[13] S. W. Hawking and Z. C. Wu, Phys. Lett. , 15(1985).[14] N. Engheta and R. W. Ziolkowski, Metamaterials:Physics and Engineering Explorations (John Wiley &Sons, 2006).[15] G. V. Eleftheriades and K. G. Balmain,
Negative-Refraction Metamaterials: Fundamental Principles andApplications (John Wiley & Sons, 2005).[16] B. Ryden,
Introduction to Cosmology (Addison Wesley,2003).[17] S. A. Hojman, S. Chayet, D. N´u˜nez, and M. A. Roque,
J. Math. Phys. , 1491 (1991)[18] A. Saa, Class. Quantum Grav. , 553 (1996). [19] A. Hanson, T. Regge and C. Teitelboim, ConstrainedHamiltonian Systems (Accademia Nazionale dei Lincei,1976).[20] S. P. Gavrilov and D. M. Gitman, Class. Quantum Grav. , L133 (2000).[21] S. Savasta and O. Di Stefano, arXiv:0803.4013v1(2008); S. Savasta, O. Di Stefano and O. M. Marag`o,arXiv:0905.4741v1 (2009).[22] C. A. Rubio, F. A. Asenjo and S. A. Hojman, Symmetry , 860 (2019).[23] P. D. D’Eath, Supersymmetric Quantum Cosmology ,(Cambridge Monographs on Mathematical Physics,Cambridge University Press, 1996).[24] P. Vargas Moniz
Quantum Cosmology - The Supersym-metric Perspective - Vol. 1: Fundamentals , Lect. NotesPhys. 803 (Springer, Berlin Heidelberg 2010).[25] F. Cooper, A. Khare, R. Musto and A. Wipf, Ann. Phys. , 1 (1988).[26] M. de Crombrugghe and V. Rittenberg, Ann. Phys. ,99 (1983).[27] F. Cooper, A. Khare and U. Sukhatme,
Supersymme-try in Quantum Mechanics , (World Scientific,Singapore,2001).[28] J. D. Jackson,
Classical Electrodynamics (John Wiley &Sons, Inc. New York, 1975).[29] G. A. Kraftmakher and V. S. Butylkin, Tech. Phys. Lett. , 230 (2003).[30] Z. Zheng , H. Zhang , J. Q. Xiao and F. Bai, IEEE Trans-actions on Magnetics , 4214 (2013).[31] H. Su et al. , J. Appl. Phys. , 17B301 (2013).[32] L. B. Kong, Z. W. Li, G. Q. Lin and Y. B. Gan, IEEETransactions on Magnetics , 6 (2007).[33] A. Thakur, P. Thakur and J.-H. Hsu, Scripta Materialia , 205 (2011).[34] N. Westerberg et al. , New J. Phys. , 075003 (2014).[35] A. B. Batista et al. , Phys. Rev. D , 063519 (2002).[36] M. Visser C. Barcel´o and S. Liberati, Gen. Rel. Grav. ,1719 (2002).[37] C. Barcel´o, S. Liberati and M. Visser, Living Rev Relativ. , 3 (2011).[38] M. Ornigotti, S. Bar-Ad, A. Szameit and V. Fleurov,Phys. Rev. A , 013823 (2018). [39] D. Faccio, F. Belgiorno, S. Cacciatori, V. Gorini,S. Liberati and U. Moschella, Analogue Gravity Phe-nomenology: Analogue Spacetimes and Horizons, fromTheory to Experiments (Springer, 2013).[40] K. Bi et al. , PLoS ONE , e012733 (2015).[41] P. S. Grant et al. , Phil. Trans. R. Soc. A373