Broadening quantum cosmology with a fractional whirl
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Modern Physics Letters A © World Scientific Publishing Company
Broadening quantum cosmology with a fractional whirl
S. M. M. Rasouli
Departamento de F´ısica,Centro de Matem´atica e Aplica¸c˜oes (CMA - UBI),Universidade da Beira Interior,Rua Marquˆes d’Avila e Bolama, 6200-001 Covilh˜a, Portugal.andDepartment of Physics,Qazvin Branch,Islamic Azad University,Qazvin, [email protected]
S. Jalalzadeh
Departmento de F´ısica,Universidade Federal de Pernambuco,Pernambuco, PE 52171-900, [email protected]
P. V. Moniz
Departamento de F´ısica,Centro de Matem´atica e Aplica¸c˜oes (CMA - UBI),Universidade da Beira Interior,Rua Marquˆes d’Avila e Bolama, 6200-001 Covilh˜a, [email protected]
Received (Day Month Year)Revised (Day Month Year)We start by presenting a brief summary of fractional quantum mechanics, as means toconvey a motivation towards fractional quantum cosmology . Subsequently, such appli-cation is made concrete with the assistance of a case study. Specifically, we investigateand then discuss a model of stiff matter in a spatially flat homogeneous and isotropicuniverse. A new quantum cosmological solution, where fractional calculus implicationsare explicit, is presented and then contrasted with the corresponding standard quantumcosmology setting.
Keywords : quantum cosmology; fractional quantum mechanics; fractional Schr¨odingerequation; fractional Wheeler–DeWitt equation; Caputo derivative; Riesz derivative.PACS Nos.: 05.40.Fb, 04.60.-m, 98.80.Qc 1 anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz
1. Introduction
A particular branch of mathematical analysis, albeit not yet widely explored, isfractional calculus, in which the power of the differentiation operator can be anyrational (or even a real or a complex) number. Indeed, investigations into frac-tional calculus started in the 17th century, which is almost the same temporalframe as the integer-order calculus started to develop. However, in the early stagesof the evolution of these two branches of mathematics, due to the lack of practi-cal evidence in physics (and mechanics, specifically), the former developed muchmore slowly than the latter. This was the setting until Mandelbrot realized thatan exciting connection between fractional Brownian motion and Riemann-Liouvillefractional calculus could be reasonably established. After that remark, applicationsof fractional calculus to study phenomena in engineering as well as other variousfields of science have increased, and therefore the equations of fractional calculushave played a significant role in describing, e.g., anomalous transport, super-slowrelaxation, diffusion-reaction processes. In addition, areas of contemporary en-gineering were also explored within fractional calculus. Furthermore, the physical world includes not only the conservative systems butalso the non-conservative ones (due to the presence of friction). Advanced proce-dures like classical Hamiltonian (or Lagrangian) mechanics (which is formulatedbased on the derivative of integer order) could be replaced by fractional Hamilto-nian (and Lagrangian) equations of motion, aiming in investigating analytically anypeculiar consequences of the non-conservative forces.
Similarly, in the context of quantum physics, there might be suggestive physi-cal reasons to reformulate the theory based on fractional calculus. For instance, ifrestricting to only consider the Brownian paths, it is impossible to analyze otherpertinent many phenomena. Such consideration in quantum physics leads to gen-eralized versions of the well-known Schr¨odinger equation (SE). More concretely, byincluding non-Brownian trajectories in the path integral derivation, either the space -fractional, time -factional and space-time -factional versions of the standard SE havebeen established.
Let us introduce them briefly and then, in the next section,we further specify (though shortly) about their formalism: • In deriving the space-fractional SE, the Feynman-Hibbs path integral pro-cedure has been extended by Laskin such that the Gaussian probabilitydistribution was replaced by L´evy’s. More precisely, the second-orderspatial derivative in the standard SE is then modified to the fractional-order α , whilst the time sector remains unchanged. This equation is characterizedby the L´evy index α , 0 < α ≤
2, such that for the particular case where α = 2, the standard SE is recovered. Moreover, the Laplacian operatoracquires a fractional Riesz derivative. By applying some examples of specific potential fields, this (space) frac-tional quantum system has been coherently exlored.
37, 38 • From employing the Caputo fractional derivative,
4, 5, 13 the first-order timeanuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main
Broadening quantum cosmology with a fractional whirl derivative associated with the standard SE has been generalized to beingfractional. Therefore, the time-fractional SE was obtained. The associ-ated Hamiltonian is non-Hermitian and not local in time. Indeed, in thisprocedure, analogous to the map between the standard versions of the dif-fusion equation and the SE, the time-fractional diffusion equation was alsomapped into the time-fractional SE. We should note that, in the latter,the spatial derivative, the same as the standard SE, is still the second or-der. Indeed, unlike the standard SE and the space-fractional SE (whichboth obey the Markovian evolution law), the time-fractional SE describesa non-Markovian evolution in quantum physics. As applications of thetime-fractional SE, a free particle and a particle in a particular potentialwell have been investigated. • The term “space-time fractional SE” has been used for the first time byWang and Xu, where they employed a combination of the proceduresused by Naber and Laskin to establish this equation. More precisely,this is the most generalized version of the standard SE, which is retrievedby including both the Caputo fractional derivative and the quantum Rieszfractional operator. The space-time fractional QM was applied to studya free particle and an infinite square potential well. Moreover, Dong andXu have represented the work of Wang and Xu (although with a minordifference) and then applied the space-time fractional SE to study a quan-tum particle in a δ -potential well. Furthermore, not only the space-timefractional SE has recently been represented by Laskin within his ownmethod but also the important features of the fractional QM approacheshave been outlined. In the therein established framework, the space-timefractional SE includes not only dimensional parameters but also a set of di-mensionless fractality parameters. These latter parameters are responsiblefor modeling the spatial and temporal fractality. Recently, fractional quantum mechanics (FQM) has been considered as a tool toexplore features within cosmology. Namely, fractional quantum cosmology (FQC). Ithas been emerging gradually, subsequently pointing to fascinating opportunitiesand ‘lateral’ connections with what were unforeseen mathematics and physics do-mains, with this review herewith following a particular (canonical) route as chartedin another publication. Before proceeding, let us point that this review paper is organized as follows.In Section 2, we briefly review fractional quantum mechanics (publications withmuch more detail can be found in the bibliography
25, 26, 29, 30 ). Subsections 2.1 and2.2 convey the summary line of FQM extensions within spatial -FQM and space-time -FQM, respectively. Section 3 reports the study of a cosmological model withinthe scope of time -FQM. Finally, in section 4 we present a brief discussion andoutlook with respect canonical FQC, as unveiled in this review paper and a recentcontribution as well. anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz
2. A brief review of fractional quantum mechanics
In this section, we would like to introduce and summarize the space-fractionalSE and the space-time fractional SE. In order to retrieve more details of the men-tioned frameworks and their applications, we suggest the readers consider additionalreferences, indicated in our bibliography.
Space fractional quantum mechanics
The Hamiltonian of a system in classical mechanics is given by H ( p , r ) := p m + V ( r ) , (1)where r and p , respectively, stand for the space coordinates and the correspondingmomentum, associated with a particle with mass m ; the potential energy, in general,is assumed to be a function of the space coordinates, i.e., V = V ( r ). In analogywith classical mechanics, the Hamiltonian of quantum mechanics can be extractedfrom (1), provided that the corresponding quantities are taken as operators (whichare denoted with an over-hat):ˆ H (ˆ p , ˆ r ) := ˆ p m + ˆ V (ˆ r ) . (2)Pondering the empirical physical (classical) fact realized in the square depen-dence of the momentum in equations (1) and (2), such feature has motivated re-searchers to explore diverse functions of the kinematic term, such that it wouldbe nevertheless consistent with the fundamental principles of classical mechanicsand quantum mechanics. Such an investigation was launched for the first time byLaskin, who chose the Feynman path integral approach (to quantum mechanics)as an appropriate procedure. Let us be more precise. In the Laskin approach, theBrownian-like quantum mechanical trajectories applied in the Feynman’s frame-work, are replaced by L´evy-like ones. a More concretely, applying a natural gener-alization
25, 28, 30 of equation (1) as H α ( p , r ) := D α | p | α + V ( r ) , < α ≤ , (3)(where D α is taken as a generalized coefficient carrying dimension [ D α ] =erg − α cm α sec − α ) and letting p → ˆ p and r → ˆ r , the (space) fractional quan-tum dynamics is constructed. Obviously, in the particular case where α = 2 and D α = 1 / (2 m ), equation (3) reduces to (1). Namely, the fractional quantum me-chanics (FQM) established through applying the L´evy path integral is a naturalgeneralization of the Feynman path integral. Consequently, the space-fractional SE(including the space derivatives of order α ), which is a generalized version of thestandard one, is obtained via applying the Feynman path integral over Brownian a Although, it should be noted that the path integral over the L´evy paths has formerly been broughtup by Kac. anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Broadening quantum cosmology with a fractional whirl trajectories. In summary, in the particular case where α = 2, the equations of thestandard (non-fractional) quantum mechanics are recovered from the correspondingones associated with the space-fractional quantum mechanics.Now, we briefly introduce the space-fractional SE. In analogy with the sameprocedure that yields the standard SE, let us admit E := D α | p | α + V ( r ) , < α ≤ , (4)where E and p can transform as E → i ~ ∂∂t , p → − i ~ ∇ , (5)where ∇ = ∂∂ r and ~ is the Planck’ s constant over 2 π . Therefore, the space-fractional SE can be obtained by employing the well-known transformations in (5)into equation (3) and using them
25, 28, 30 to eventually extract the wave function ψ ( r , t ) : i ~ ∂ψ ( r , t ) ∂t = D α ( − ~ ∆) α/ ψ ( r , t ) + V ( r , t ) ψ ( r , t ) , < α ≤ . (6)In equation (6), ψ ( r , t ) and △ := ∇ . ∇ are the wave function in space represen-tation and the Laplacian, respectively. Moreover, the fractional (quantum) Rieszderivative
30, 42 in 3 D , ( − ~ ∆) α/ , is given by( − ~ ∆) α/ ψ ( r , t ) = 1(2 π ~ ) Z d pe i p · r ~ | p | α ϕ ( p , t ) , (7)where ϕ ( p , t ) is the wave function in momentum representation, which is related to ψ ( r , t ) by 3 D Fourier transforms.In summary, replacing r and p by quantum mechanical operators ˆ r and ˆ p ,equation (3) leads to the fractional Hamilton operator ˆ H α (ˆ p , ˆ r ):ˆ H α ( ˆp , ˆr ) := D α | ˆp | α + V ( ˆr ) , < α ≤ . (8)Therefore, the space-fractional SE equation can also be rewritten in the operatorform as i ~ ∂ψ ( r , t ) ∂t = ˆ H α (ˆ p , ˆ r ) ψ ( r , t ) . (9)In what follows, it is worthy to present some features of FQM, which are consid-ered as the generalized version of those known in the standard quantum mechanics:(i) The fractional Hamiltonian operator is a self-adjoint or Hermitian operator inthe space with scalar product; (ii) The parity conservation law for the FQM isgiven by ˆP H α = H α ˆP , where ˆP denotes the inversion operator; (iii) It has beenshown that ∂ρ ( r , t ) ∂t + div j ( r , t ) = 0 , (10)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz where ρ ( r , t ) := ∗ ψ ( r , t ) ψ ( r , t ) and j ( r , t ) denote the quantum mechanical probabilitydensity and the fractional probability current density vector, respectively, where thelatter is defined as j := 1 α (cid:18) ψ ˆv ∗ ψ + ∗ ψ ˆv ψ (cid:19) , < α ≤ . (11)Notwithstanding the formal similarities with the standard formulation, in equation(11) the velocity operator ˆv = d ˆr /dt is specifically given instead by ˆv = αD α | ˆp | α/ − ˆp . (12)Let us close this subsection by focusing on the special case where the Hamiltoniandoes not depend explicitly on time. In this case, the space-fractional SE (6) issatisfied by the special solution and in one dimension ψ ( x, t ) = (cid:20) exp (cid:18) − iEt ~ (cid:19)(cid:21) φ ( x ) , (13)provided that H α φ ( x ) = − D α ( ~ ∇ ) α φ ( x ) + V ( x ) φ ( x ) = Eφ ( x ) , < α ≤ , (14)which is the time-independent fractional SE. Equation (14) implies that the prob-ability to find the particle at x is equal to | φ | , which is independent of time. Space-time fractional quantum mechanics
As mentioned, the space-time fractional SE has been originally established byWang and Xu and then further discussed by Dong and Xu. In this subsection,let us introduce this interesting framework from within the approach of Laskin, in which the wording “time fractional QM” has been used because the time deriva-tive of the SE associated with both the “standard QM” and the “space FQM”is substituted by a fractional time derivative (namely, the Caputo fractional timederivative).The space-time fractional SE can be considered as a generalized version of equa-tion (9): ~ β i β ∂ βt ψ ( r , t ) = ˆ H α,β (ˆ p β , ˆ r ) ψ ( r , t ) , < α ≤ , < β ≤ , (15)where ˆ H α,β (ˆ p β , ˆ r ) = D α,β | ˆp β | α + V ( ˆr , t ) , (16)in which, ˆ r and ˆ p = − i ~ β ∂∂ r denote the 3 D quantum operator of coordinate and3 D time fractional quantum momentum operator, respectively. Moreover, in (15), ~ β and D α,β are two scale coefficients with physical dimensions [ ~ β ] = erg . sec β and[ D α,β ] = erg − α . cm α . sec − αβ ; ∂ βt denotes the left Caputo fractional derivative oforder β : ∂ βt f ( t ) = 1Γ(1 − β ) Z t dτ f ′ ( τ )( t − τ ) β , < β ≤ , (17)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Broadening quantum cosmology with a fractional whirl where f ′ ( τ ) ≡ df ( τ ) dτ . It should be reminded that in the time fractional QM, theoperator ˆ H α,β (ˆ p β , ˆ r ) introduced by equation (16) is not the Hamilton operator ofthe considered quantum mechanical system, and it was therefore called the pseudo-Hamilton operator. However, it is straightforward to show that ˆ r and ˆ p β areHermitian operators. Moreover, assuming V ( ˆr , t ) = V ( − ˆr , t ), we find that theparity conservation law is satisfied, i.e., ˆP ˆ H α,β = ˆ H α,β ˆP .In this review, we abstain from detailing the applications of the three frameworksof FQM, since much is available in the literature and we point to some in thebibliography. However, in what follows, let us mention a few general features: (1) In special cases, the most generalized FQM yields the following well-known cases(let us focus on a one-dimensional case): • For α = 2 and β = 1, we have D α,β = 1 / (2 m ), p β = p and ~ β = ~ ,therefore the traditional SE is recovered. • Assuming 1 < α ≤ β = 1, we find that D α,β → D α , and thereforethe space fractional SE is recovered. • Assuming α = 2 and 0 < β ≤
1, we find that D α,β → D ,β and there-fore the time fractional SE is retrieved, which can be considered as analternative route to that established by Naber. (2) Notwithstanding the previous item, let us outline a few shortcomings of thetime fractional QM. It has been shown that the space fractional QM
25, 28, 30 supports all the QM fundamentals. Whilst, the following QM characteristicsare violated by the time fractional QM: (i) Quantum superposition law; (ii)Unitarity of evolution operator; (iii) Probability conservation law; (iv) Existenceof stationary energy levels of quantum system. Moreover, as mentioned, in spiteof the standard and the space fractional QM, whose dynamics are governed bya Hamilton operator, the operator ˆ H α,β associated with the time fractional QMis a pseudo-Hamilton (namely, its eigenvalues are not the energy levels of a timefractional quantum system).(3) Let us here remark a few advantages of the time fractional QM. Indeed, investi-gating it highlighted the importance and fundamental beauty of the standard aswell as space fractional QM (some of them mentioned above); the time fractionalQM is however a great challenge for studying and discovering new mathemati-cal tools, which have not yet been applied in the the standard as well as spacefractional QM. Concretely, in order to investigate dissipative quantum systemsinteracting with environment, the time fractional QM can be considered as anenticing approach.
24, 31–33
3. Fractional quantum cosmology case study
As a simple model of application of FQM towards canonical quantum cos-mology (i.e., (canonical) fractional quantum cosmology ), let us consider aanuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz
Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe with a line element ds = − N ( t ) dt + e x ( t ) (cid:20) dr − kr + r d Ω (cid:21) , (18)where N ( t ) is the lapse function, e x ( t ) represents the scale factor and k = ± , t . Note that the compactness of Σ t requires thatthe 3-volume V k is finite. The ADM action functional of the gravitational sectorplus matter fields (in this work, we will consider a perfect fluid with the barotropicequation of state p = ωρ , where ω is a constant and ρ , p stand for the energy densityand pressure, respectively) in the formalism developed by Schutz is S = πG Z t f t i dt Z Σ t d xN √ h h (3) R + K ij K ij − K i + Z t f t i dt Z Σ t d xN √ hp, (19)where (3) R , K ij and h ij denote the Ricci scalar, the extrinsic curvature, and theinduced metric of Σ t , respectively. In Schutz’s formalism, the fluid’s 4-velocity canbe expressed in terms of five potentials σ, ζ, ¯ β, θ and S u ν := 1 µ ( ǫ ,ν + ζ ¯ β ,ν + θ S ,ν ) , u ν u ν = − , (20)where µ denotes the specific enthalpy of the fluid, S is the specific entropy, and thepotentials ζ and ¯ β are connected with rotation which are absent of homogeneous andisotropic models. The action (19) corresponding to the line element (18), after somethermodynamical considerations and using the constraints for the fluid, reducesto S = Z t f t i " V k πG (cid:18) − e x ˙ x N + kN e x (cid:19) + ω V k e x e − S ω (1 + ω ) ω N ω ( ˙ ǫ + θ ˙ S ) ω dt, (21)where an overdot denotes differentiation with respect time coordinate t . It isstraightforward to show that the ADM Hamiltonian of our herein model is given by H ADM = ˙ xp + ˙ S p S + ˙ ǫp ǫ − L ADM = ˜ N h − πm V k e ω − x p + e S V kω p ωǫ − k V k m π e (3 ω − x i , (22)where m P = 1 / √ G is the Planck mass and the new lapse function ˜ N is defined by˜ N := Ne x . (23)Moreover, p := − V k e x πGN ˙ x, p ǫ := V k e x N ω (1 + ω ) ω ( ˙ ǫ + θ ˙ S ) ω e − S ω , (24)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Broadening quantum cosmology with a fractional whirl are the conjugate momenta of the scale factor x and ǫ , respectively. Note that themodel has constraints of the second kind, i.e., p θ = 0 and p S = θp ǫ .It is easy to show that employing the additional canonical transformations p T := − e S V ωk m P p ωǫ , T := m P V ωk p S e −S p − ( ω +1) ǫ , ¯ ǫ := ǫ − ( ω − p S p ǫ , ¯ p ǫ := p ǫ , (25)it simplifies the ADM Hamiltonian to H ADM = ˜ N (cid:20) − πm V k e ω − x p − m P p T − k V k m π e (3 ω − x (cid:21) . (26)In equation (26), it is seen that the momentum p T is the only remaining canonicalvariable (associated with the perfect fluid), which appears linearly. We should alsonote that the new variable T is dimensionless. The super-Hamiltonian constraint ofthe model is given by H := − πm V k e ω − x p − m P p T − k V k m π e (3 ω − x = 0 . (27)We can then obtain the Wheeler–DeWitt equation by imposing the standard quan-tization conditions on the canonical momenta:ˆ x = x, ˆ p = − i∂ x , ˆ T = T, ˆ p T = − i∂ T . (28)Moreover, we assume that the super-Hamiltonian operator annihilate the wave func-tion i∂ T Ψ( x, T ) = − M e ω − x (cid:2) ∂ x − ( ω − ∂ x (cid:3) Ψ( x, T ) + k V k m P π e (3 ω − x Ψ( x, T ) , (29)where M is the dimensionless ‘mass’ parameter and we have used the Laplace–Beltrami operator ordering. Requiring that the Hamiltonian operator H in theright hand side of (29) to be self-adjoint, the inner product of two arbitrary wavefunctions Φ and Ψ must take the form h Φ | Ψ i = Z ∞−∞ e − ( ω − x Φ ∗ ( x, T )Ψ( x, T ) dx. (30)Redefining the wavefunction as ψ ( x, T ) := e ( ω − Ψ( x, T ) , (31)the Wheeler–DeWitt equation (29) transforms to i∂ T ψ ( x, T ) = − M e ω − x ∂ x ψ ( x, T ) + h π ( ω − V k m e ω − x + k V k m P π e (3 ω − x i Ψ( x, T ) , (32)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz with inner product given by h φ | ψ i = Z ∞−∞ φ ∗ ( x, T ) ψ ( x, T ) dx. (33)Consequently, the Wheeler–DeWitt equation (29) becomes in the form of theSchr¨odinger equation i∂ t ψ = Hψ .For simplicity and in order to obtain analytical solutions, from now on, let usproceed our consideration with only the spatially flat universe ( k = 0) filled with thestiff matter described with ω = 1. Therefore, for such a simple case, the Wheeler–DeWitt equation (29) reduces to the Schr¨odinger equation of a free particle in 1 D mini-superspace: i∂ T Ψ( x, T ) = − M ∂ x Ψ( x, T ) , (34)where V denotes the 3-volume associated with k = 0. The solution of the above ‘freeparticle’ Schr¨odinger equation could be represented by a superposition of momen-tum eigenfunctions, with coefficients given by the Fourier transform of the initialwavefunction Ψ( x, T ) = 1 √ π Z ∞−∞ Φ ( p ) e i ( px − ωt ) dp, (35)where ω := p M and Φ ( p ) is the Fourier transform of the wavefunction Ψ( x, ( p ) = (cid:18) a π (cid:19) e − a ( p − p ) , (36)where a, p ∈ R . Inserting the initial wave function (36) into (35) and performingthe Gaussian integration, we obtainΨ( x, T ) = 1 √ q (cid:18) a π (cid:19) e ip ( x − p M T ) e − q ( x − p M T ) , (37)where q := a + i T M . Therefore, the expectation value of the scale factor is givenby h x i = p o M T = 4 πp m V T. (38)In what follows, we would like to extend our herein model by applying the mostgeneralized fractional QM framework presented in subsection 2.2. Concretely, the‘space-time’ fractional counterpart of the Schr¨odinger–Wheeler–DeWitt equation(32) is given by i β ∂ βT ψ ( x, T ) = π V k m e ω − x ( − ∂ x ) α ψ ( x, T )+ h π ( ω − V k m e ω − x + k V k m P π e (3 ω − x i ψ ( x, T ) , < α ≤ , < β ≤ . (39)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Broadening quantum cosmology with a fractional whirl We should note that the time coordinate T as well as x are dimensionless andconsequently we have assumed the scaling coefficients in ‘space-time’ fractional SE(15) are unit, i.e., ~ β = D α,β = 1. Again, assuming our simple case with k = 0 and ω = 1, equation (39) reduces to i β ∂ βT Ψ( x, T ) = 12 M ( − ∂ x ) α Ψ( x, T ) , < α ≤ , < β ≤ . (40)Applying the Fourier transform to the wavefunction asΨ( x, T ) = 1 √ π Z ∞−∞ dpe ipx Φ( p, T ) , (41)we obtain i β ∂ βT Φ( p, T ) = 12 M | p | α Φ( p, T ) . (42)The solution of the above equation isΦ( p, T ) = E β (cid:18) i β M | p | α t β (cid:19) Φ ( p ) , (43)where Φ ( p ) = Φ( p, T = 0) and E β ( z ) := ∞ X n =0 z n Γ( βn + 1) , (44)is the Mittag–Leffler function. Inserting (43) into (41) yields the wavefunction ofthe ‘space-time’ fractional stiff matter flat universe:Ψ( x, T ) = 1 √ π Z ∞−∞ dpe ipx E β (cid:18) i β M | p | α t β (cid:19) Φ ( p ) . (45)If we choose the following ‘weight’ function, Φ ( p ) as the fractional generalizationof Gaussian ‘weight’ (36)Φ ( p ) = aν ν +1 ν Γ( ν ) ! e − a ν | p − p | ν , p > , ν ≤ α, (46)then the wavefunction becomesΨ( x, T ) = 1 √ π aν ν +1 ν Γ( ν ) ! Z ∞−∞ dpe ipx E β (cid:18) i β M | p | α t β (cid:19) e − a ν | p − p | ν . (47)Hence, the density of the probability of finding the universe to ‘occupy’ a scalefactor x is | Ψ( x, T ) | = aν ν +3 ν π Γ( ν ) Z ∞−∞ dp dp e i ( p − p ) x E β (cid:16) i β M | p | α t β (cid:17) E β (cid:16) ( − i ) β M | p | α t β (cid:17) e − a ν ( | p − p | ν + | p − p | ν ) . (48)anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main Rasouli, Jalalzadeh, Moniz
Note that for β = 1 the total probability is not conserved and it increases with time.For large values of β the total probability increases much faster. For that ‘space’fractional case, β = 1, the expectation value of the scale factor x is h x i = αp α − M t, (49)which shows that the center of the L´evy wave packet (47) moves with the groupvelocity αp α − M .
4. Discussion and Outlook
The bibliography of FQM, within the three approaches herewith pointed(namely, space-, time-, space-time-fractional), is now relevant and significant. Frac-tional calculus has indeed become a computational powerhouse in physics and manyengineering applications. Endeavouring towards cosmology (including in a quan-tum mechanical context) has also recently emerged.
In the review paper herein presented, we follow the ‘ouverture’ made in anotherpublication, with an emphasis on the canonical description. After a summary ofFQM, we presented a new solution for a FRW model, within the scope of time -FQMfor a Schr¨odinger-Wheeler-DeWitt equation.Our results are depicted, also mentioning the corresponding ones within thestandard Wheeler-DeWitt-SE for quantum cosmology; it can be easily appraisedhow the fractional feature (in the time derivative, specifically) brings significantalterations into the quantum dynamics of the universe. This assertion follows resultsin other publications and herein, hence the suggestive ’omen’ in our proposed title,broadening (canonical) quantum cosmology with a fractional agitation, like a ’whirl’of change. In particular, it is instructive to regard equations (3), (38), (48), (49) andcontrast the standard to the fractional ‘whirl’ brought into concrete observables.As far as subsequent work, in the herein footsteps and elsewhere, the covari-ant d’Alembertian (i.e., for both time and space derivatives in FQM, physicallyconsistent within geometrodynamics covariance) remains an open issue. So far, wehave only attempted ‘restricted’ explorations for either space or time fractionalderivatives, in separate, never together (surely not covariantly, as stressed). Theselimitations notwithstanding, the results so far are interesting and point that frac-tional calculus (even for simple case studies) can indeed open up new routes towardsmore realistic and so far unchallenged situations in quantum cosmology. Theoutlook, for (canonical) FQC is therefore warmly exciting and promising. The au-thors sincerely hope and trust that FQC will be a decisive route for explorationwithin the next decades.anuary 11, 2021 1:27 WSPC/INSTRUCTION FILE main
Broadening quantum cosmology with a fractional whirl Acknowledgments
The authors are grateful to J. Fabris and G. Calcani forletting us know of their earlier work. Likewise, we are most thankful to M. Or-tigueira for sharing details about his work and pointing to a rich collection of ref-erences within fractional calculus. In addition, PVM and SMMR acknowledge theFCT grants UID-B-MAT/00212/2020 and UID-P-MAT/00212/2020 at CMA-UBI.This article is based upon work conducted within the Action CA18108–Quantumgravity phenomenology in the multi-messenger approach–supported by the COST(European Cooperation in Science and Technology).
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