# Particle Production via Dirac Dipole Moments in the Magnetized and Nonmagnetized Exponentially Expanding Universe

Authors: Semra Gurtas Dogan; Ganim Gecim; Yusuf Sucu
Journal: Advances in High Energy Physics
 (2019)

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## Abstract

In the present paper, we solve the Dirac equation in the 2+1 dimensional exponentially expanding magnetized by uniform magnetic field and nonmagnetized universes, separately. Asymptotic behaviors of the solutions are determined. Using these results we discuss the current of a Dirac particle to discuss the polarization densities and the magnetization density in the context of Gordon decomposition method. In this work we also calculate the total polarization and magnetization, to investigate how the magnetic field affects the particle production. Furthermore, the electric and the magnetic dipole moments are calculated, and based on these, we have discussed the effects of the dipole moments on the charge distribution of the universe and its conductivity for both the early and the future time epoch in the presence/absence of a constant magnetic field and exponentially expanding spacetime.


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## Body

## 1. Introduction

One of the most interesting and important results of the formulation of the relativistic quantum mechanics in curved spacetime is the particle creation event in the expanding universe which was firstly discussed by Parker for the scalar particles and Dirac particles [1–3]. So, he computed the number density of the created particles by means of the Bogolibov transformation by using the out vacuum states constructed from the solutions of the relativistic particles wave equations. After these important works of Parker, the solutions of the relativistic particle wave equations have extensively been studied in various 3+1 dimensional spacetime backgrounds [4–19]. Using the WKB approach, the number density and renormalized energy-momentum tensor of the created spin-1/2 particle in the spatially flat (3+1)-dimensional Friedmann-Robertson-Walker (FRW) spacetime have been calculated [20]. The effect of the scalar particle creation on the collapse of a spherically symmetric massive star was investigated and it was demonstrated that the collapsing process was not independent from the particle creation rate [21]. Moreover, in [22] thermodynamics laws and equilibrium conditions are discussed in the presence of particle creation in the context of the (3+1)-dimensional Chern-Simons gravity theory. Recently, creation of the massless fermion in the Bianchi type-I spacetime investigated and showed that the massless particles can be created during the early anisotropic expansion epoch [23].The Gordon decomposition of the Dirac currents is another useful tool for discussing the particle creation phenomena [4, 9, 10, 24]. In the decomposition method, the Dirac currents constructed from the solutions of the Dirac equations are separated into three parts, the convective, polarization with three components, and magnetization with three components, in the 3+1 dimensional spacetime. This method includes some complexities stemming from the 3+1 dimensional spacetime. Using this method in a 2+1 dimensional curved spacetime, the densities of the particle currents are separated into three parts, as in the 3+1 dimensional spacetime, but polarization density has two components and the magnetization density has only one component [24], and, moreover, as the Dirac spinor can be defined by only two components, the computations in the 2+1 dimensional spacetime become more simple than that of 3+1 dimensional spacetime. Because of the simplicity stemming from the dimensions, the dipole moments that are computed from the polarization and magnetization densities of the Dirac electron under influence in a constant magnetic field are easily computed and their result are, furthermore, compatible with the current experimental results [25]. With these motivations, in this study, we solve the Dirac equation in the 2+1 dimensional exponentially expanding magnetized by uniform magnetic field and nonmagnetized universes, separately, and discuss the particle creation event by means of the Dirac currents written in terms of these solutions. As a result, we observe that the polarization and magnetization parts of the currents are affected differently whether the exponentially expanding universe is magnetized or not and find expressions for the electric and magnetic dipole moments by integrating the polarization and magnetization densities on hypersurface.The outline of the work is as follows; in Section2, we, at first, discuss the Dirac equation solutions in the 2+1 dimensional exponentially expanding universe. In Section 3, the Dirac equation is solved in the 2+1 dimensional exponentially expanding universe with a constant magnetic field. In Section 4, we derive the components of Dirac currents for the solutions obtained in Sections 2 and 3 and also compute the polarization density, the magnetization density, total polarization (electric dipole moment), and the total magnetization (magnetic dipole moment). Finally, the last section, conclusion, includes a discussion about the results of this work.


## 2. Dirac Particle in the 2+1 Dimensional Exponentially Expanding Universe

The behavior of the electron in 2+1 dimensional curved space is represented by the covariant form of the Dirac equation [24], which is important application in curved spacetime [26–33](1)iσ¯νx∂ν-Γν+ieAνΨx=mΨx,where Ψ(x) =Ψ1Ψ2 is the Dirac spinorial wave function with two components that are positive and negative energy eigenstates, m is the mass of Dirac particle, e is the charge of the Dirac particle, and Aν are 3-vectors of electromagnetic potential. Using triads, e(i)ν(x), Dirac matrices that dependent on spacetime, σ¯ν(x), are written in terms of the constant Dirac matrices, σ¯i;(2)σ¯νx=eiνxσ¯i.So, we choose the constant Dirac matrices, σ¯i, in the flat spacetime as follows:(3)σ¯i=σ¯0,σ¯1,σ¯2with(4)σ¯0=σ3,σ¯1=iσ1,σ¯2=iσ2,where σ1, σ2, and σ3 are Pauli matrices. The spin connection, Γν(x), for the diagonal metrics is defined as(5)Γνx=-14gταΓνβασ¯τx,σ¯βx,where Γνβα is the Christoffel symbol given as follows [34]:(6)Γνβα=12gλα∂gβλ,ν+∂gνλ,β-∂gβν,λ.Also, the metric tensor gβν(x) is written in terms of triads as follows:(7)gβνx=eβixeνjxηij,where β and ν are curved spacetime indices run from 0 to 2, i and j are flat spacetime indices run 0 to 2, and η(i)(j) is the signature with (1,-1,-1).The (2+1) dimensional de Sitter spacetime metric can be written as [35](8)ds2=dt2-e2Htdr2+r2dϕ2where H is Hubble parameter. From (3)-(8), the spin connections for the metric read(9)Γ0=0,Γ1=-H2eHtσ¯0σ¯1,Γ2=-12rHeHtσ¯0σ¯2+σ¯1σ¯2.Using (2), (8), and (9), then the Dirac equation in the 2+1 dimensional exponentially expanding universe becomes(10)σ¯0∂t+H+im+e-Htσ¯1∂r+12r+σ¯2r∂ϕΨx=0.Letting (4) and (10) and the Dirac spinor, Ψ(x) =Ψ1Ψ2, we write the Dirac equation in explicit form as follows:(11)∂t+im+Hψ1+ie-Ht∂r+12r-ir∂ϕψ2=0∂t-im+Hψ2-ie-Ht∂r+12r+ir∂ϕψ1=0.To find the solutions of (11), thanks to the separation of variables method, the wave function components can be defined as(12)Ψ1Ψ2=eikϕT1tR1rT2tR2r.By these definitions, the Dirac equation is separated into the following two differential equation systems:(13)ddr+12r+krR2r=-λR1rddr+12r-krR1r=λR2rand(14)ddt+im+HT1t=iλe-HtT2tddt-im+HT2t=iλe-HtT1t,where λ is a separation constant, and we find the solutions of (13) in terms of the Bessel and confluent hypergeometric functions as follows:(15)R1r=AJk-1/2λr=Ae-iλrλr/2k-1/2Γk+1/2F11k,2k;i2rλ,R2r=BJk+1/2λr=Be-iλrλr/2k+1/2Γk+3/2F11k+1,2k+2;i2rλ.On the other hand, to solve (14), we must define a new variable such as z=λ/He-Ht. With the definition n=-im/H, the solutions of (14) are obtained in terms of Bessel functions or confluent hypergeometric functions:(16)T1z=Cz3/2Jn+1/2z=Cz3/2e-izz/2n+1/2Γn+3/2F11n+1,2n+2;i2zT2z=Dz3/2Jn-1/2z=Dz3/2e-izz/2n-1/2Γn+1/2F11n,2n;i2zThen, the wave function, Ψ(x), can be written as(17)Ψ=Nz3/2eikϕJn+1/2zJk-1/2λrJn-1/2zJk+1/2λrwhere N is normalization constant. To find the normalization constant, we use the Dirac-delta normalization condition [4, 36, 37]:(18)∫gΨ¯σ¯0Ψdσ=δλ-λ′where g is determinant of the metric tensor and Ψ¯=Ψ†σ¯0 and dσ=drdϕ for the surface t=constant [25]. Thus, the normalization constant is computed as(19)N2=H24πzλJn-1/2zJn+1/2z,where we use the following relation [4, 36, 37]:(20)∫0∞rJk+1/2λrJk+1/2λ′rdr=1λδλ′-λ.


## 3. Dirac Particle in the 2+1 Dimensional Exponentially Expanding Magnetized Universe

It is interesting in discussing if the universe is under influence in an external constant magnetic field in the beginning time. Therefore, an electromagnetic potential can be chosen asA0=0 and A→(r,ϕ)=1/2B0rϕ∧ for a constant magnetic field in 2+1 dimensional spacetime. Then, the Dirac equation in the 2+1 dimensional exponential expanding universe with a constant magnetic field becomes (21)σ¯0∂t+H+im+e-Htσ¯1∂r+12r+σ¯212r∂ϕ+ieB02Ψx=0and, thus, the explicit form of the equation is written as(22)∂t+im+Hψ1+ie-Ht∂r+12r-ir∂ϕ+eB02ψ2=0∂t-im+Hψ2-ie-Ht∂r+12r+ir∂ϕ-eB02ψ1=0To solve (22), we use the same procedure as the section before. The solutions of the equations are(23)Ψ=Nz3/2eikϕρJn+1/2zWκ,ηρJn-1/2zρ-2η-1e2B02-4λ2-eB02λρWκ,ηρ-e2B02-4λ2λρWκ+1,ηρwhere κ=-ekB0/e2B02-4λ2, η=k-1/2, ρ=e2B02-4λ2r, and N is normalization constant. As all the contributions for particle creation and dipole moments are taking place from the boundaries, we can write the wave function in the following asymptotic form [36]:(24)Ψ~Nz3/2eikϕρκ-1/2e-ρ/21Γn+3/2z2n+1/21Γn+1/2z2n-1/2ρ-2η-1e2B02-4λ2-eB0-2ρe2B02-4λ2-eB02λρ.Then, the normalization constant can be obtained from (18) as follows: (25)N2~H2η+1/221-4n2cosnπ1z2η+1/22-κ2Γ2κ+1-4n2Γ2κ-2,where we use Γ-n+3/2Γn+3/2=π(1-4n2)/4cos⁡(nπ) and Γ-n+1/2Γn+1/2=π/cos⁡(nπ).


## 4. Dirac Currents

The 2+1 dimensional Dirac current is written as(26)Jν=Ψ¯σ¯νxΨwhere Ψ¯ is Hermitian conjugate of the Dirac spinor Ψ and equal to Ψ¯=Ψ†σ¯0=Ψ†σ3 [24]. As shown in [24], (26) is expressed in explicit form as follows:(27)Jτ=12mΨ¯σ¯τυt,rΨ,υ-12mΨ¯i2gτυ∂υ⃡-eAτΨ-i4mΨ¯σ¯υt,r,σ¯,υτt,rΨ-i2mΨ¯σ¯υt,rΓυ,σ¯τt,rΨ-i4mΨ¯σ¯,υυt,r,σ¯τt,rΨThe components of the Dirac current in the 2+1 dimensional exponential expanding universe, J0 and Jk, are(28)J0=12m∂kΨ¯σ¯0kt,rΨ-12mΨ¯i2∂0⃡-qA0Ψ-i2mrexp⁡-HtΨ¯σ¯1σ¯0Ψand(29)Jk=12m∂0Ψ¯σ¯0kt,rΨ+12m∂lΨ¯σ¯lkt,rΨ-12mΨ¯i2∂k⃡-qAkΨ+i3H2mrexp⁡-HtΨ¯σ¯0σ¯kΨ+δk2i3H2mrexp⁡-HtΨ¯σ¯1σ¯2Ψwhere k,l=1,2, σ¯0k=i/2σ¯0,σ¯kt,r, σ¯kl=i/2σ¯kt,r,σ¯lt,r, and δk2 is Dirac-delta function. Also these components can be rewritten in terms of the convective, the polarization, and magnetization parts as follows:(30)J0=∂kPk+ρconvective-i2mrexp⁡-HtΨ¯σ¯1σ¯0Ψand(31)Jk=∂0Pk+∂lMlk+Jkconvective+i3H2mrexp⁡-HtΨ¯σ¯0σ¯kΨ+δk2i3H2mrexp⁡-HtΨ¯σ¯1σ¯2Ψwhere P0k are polarization densities and Mlk is magnetization density, and their explicit forms are given by(32)P0k=12mΨ¯σ¯k0t,rΨand(33)Mlk=12mΨ¯σ¯lkt,rΨ,respectively. From these relations, the total polarizations, pl0, and magnetization, μ, are defined as(34)pl0=∫P0kdΣkland(35)μ=∫MkldΣkl,where dΣkl is an hypersurface for t=constant and dΣkl=gd2x=e2Htrdrdϕ [25].Now, we are going to discuss the Dirac currents and the dipole moments expressions for the exponentially expending universe. So, inserting (17) and its conjugate into (26), we compute the components of the Dirac currents in asymptotic region as follows:(36)J0≈H22λ2π2z2r,J1≈H3z22n+12-z24π2λ32n+1rsin⁡λr-kπ2,J2≈iH3z2z2-2n+124π2λ32n+1rsin⁡λr-kπ2.Similarly, substituting (17) and its conjugate in (32) and (33), the components of the polarization densities and the magnetization density are written as follows:(37)P1=AzJk+1/2λrJk-1/2λr,P2=Bz1rJk+1/2λrJk-1/2λr,M12=λ4r8πm3Jk+1/2λrJk+1/2λr-Jk-1/2λrJk-1/2λr,respectively, where we use the following abbreviations:(38)Az=iH3z38πλ2mJn+1/2zJn-1/2z-Jn-1/2zJn+1/2z,Bz=-iAz.Giving to the polarization densities and magnetization density depending spacetime coordinates, we can say that particle production event takes place. To calculate the total polarizations and magnetization, we insert (37), the polarization densities, and magnetization density, into (34) and (35), respectively, and later integrate them on the hypersurface. Then, we obtain the total polarization densities (electric dipole moments) and magnetization density (magnetic dipole moment) as follows: (39)p1=0,p2=πλH2Bzz2,μ=0,where we use the integral representation of Bessel function [36] and p1 and p2 are total polarizations, i.e., electric dipole moment components, and also μ is total magnetization, i.e., magnetic dipole moment. From these results, we see that the particle creation events are affected from only p2 total polarization density, electric dipole moment. On the other hand, in the limit t→-∞z→∞, p2 vanishes by the following way: (40)p2→zH4λm2cos2⁡z-nπ/2-1sin⁡2z-nπ→0,that is, there is not particle production in this limit or the universe has a symmetric charge distribution in the beginning time and, of course, the universe has not any dipole moments, but, in the limit t→+∞z→0, p2 becomes(41)p2→eħH4λmc2z2-1-i2mc2/ħH21-i2mc2/ħH→-e2λδexp⁡-iδ,where ħ is Planck constant, c is the speed of light, and δ=2mc2/ħH, and, thus, the universe has a permanent complex dipole moment, which it oscillates with Zitterbewegung frequency, 2mc2/ħ.To calculate the Dirac current components for the Dirac particle in the 2+1 dimensional exponentially expanding magnetized universe, we insert (24) and its conjugate in (26). Thus, we find that the current components are(42)J0≈H2k2z2k2-κ2Γ2κ+e2B021-4n2Γ2κ-2z3ρ2κ-1e-ρz+1-4n2zk2-κ2ρ22k-ρk+κ+2kρ2J1≈i4nH3eB0k2κπk2-κ2z4ρ2κ-2e-ρ2k-ρk+κ+2kρJ2≈4H3k3z4πk2-κ2ρ-2kk+κ+2kρz2k2-κ2Γ2κ+1-4n2Γ2κ-2ρ2κ-3e-ρ.Using (24) in (32) and in (33), the components of polarization and magnetization can be found as follows:(43)P1≈2H3eB0k2z4κk2-κ2πm2k-ρk+κ+2kρ2z2λ2κ2Γ2κ+e2B021-4n2Γ2κ-2ρ2κ-2e-ρP2≈i4nH3k3z4πmk2-κ22k-ρk+κ+2kρz2k2-κ2Γ2κ+1-4n2Γ2κ-2ρ2κ-3e-ρM12≈2H4eB0k3z54n2-1κπmk2-κ24z2λ2κ2Γ2κ+eB01-4n2Γ2κ-2ρ2κ-2e-ρ×z1-4n2+1zk2-κ2ρ22k-ρk+κ+2kρ2To calculate the total polarizations and total magnetization, we insert (43) into (34) and (35). Then, using the integral representation of Bessel function [36], we find the following dipole moments expressions: (44)p1≈He3B03k4z2κ3m2k-κ2Γ2κz2k2-κ2Γ2κ+1-4n2Γ2κ-2,p2≈i2nHe4B04k5z2κ4m2k2-2κ2+4kκ+κ-kΓ2κ-1z2k2-κ2Γ2κ+1-4n2Γ2κ-2,μ≈H2e3B03k5k2-κ2κ3mz3.From these expressions, we see that, in finite time intervals, the particle creation is influenced by both polarization and magnetization components. On the other hand, in the limit t→∞z→0 the magnetic dipole moment, μ, goes to zero faster than the electric dipole moment components, p1 and p2. Therefore, the electric dipole moments in the particle creation events become more dominant than the magnetic dipole moment in finite time intervals if there exists in an external constant magnetic field.


## 5. Summary and Conclusion

We exactly solve the Dirac equation in existence of the exponentially expanding magnetized and nonmagnetized universe and, from these solutions, derive some expressions for the Dirac current components and dipole moments. The particle creation in the exponentially expanding universe is only affected by thep2 polarization in the finite time interval. However, this component goes to zero in the limit t→-∞, i.e., in the beginning of the universe, but, in the limit t→+∞, the universe has a permanent complex dipole moment oscillating with Zitterbewegung frequency, 2mc2/ħ: p2≃-e/2λδexp⁡-iδ. The complexity of the dipole moment points out the conductivity of the exponentially expanding universe. Also, the universe has the electric and magnetic dipole moments which are dependent on time in existence of an external constant magnetic field with the expansion such that, in the limit t→-∞, the dipole moment expressions become infinite, but, in the limit t→∞, they go to zero. The dependence on time of the polarization and magnetization shows that the particle creation happens. Furthermore, in the limit t→∞  z→0, the particle creation events are affected only via the polarization because the magnetization, m, goes to zero faster than p1 and p2. From the point of view, we point out that the exponential expansion of the universe causes a particle creation, a permanent complex electric dipole moment and asymmetric charge distribution, but, in existence of an external constant magnetic field with exponential expansion in time, the universe charge distribution is get and getting symmetric and thus all the dipole moments become zero as t→∞.



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*Source: 5208712-2019-07-22.xml*