Within the framework of symmetric teleparallel fleft Qright gravity for a connection defined in the noncoincidence gauge we derive the WheelerDeWitt equation of quantum cosmology. Because the gravitational field equation in fleft Qright gravity admits a minisuperspace description the WheelerDeWitt equation is a single inhomogeneous partial differential equations. We assume the powerlaw fleft Qright f0Qmu model and with the application of linear quantum observables we calculate the wavefunction of the universe. Finally we investigate the effects of the quantum correction terms in the semiclassical limit.
['General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
SemiClassical limit and quantum corrections in noncoincidence powerlaw fQCosmology
{'Semi-Classical limit and quantum corrections in noncoincidence power-law f ( Q ) -Cosmology': "Andronikos Paliathanasis 1, 2, ∗  \n1 Institute of Systems Science, Durban University of Technology, Durban 4000, South Africa 2 Departamento de Matem'aticas, Universidad Cat'olica del Norte,  \nAvda. Angamos 0610, Casilla 1280 Antofagasta, Chile  \nWithin the framework of symmetric teleparallel f ( Q )-gravity for a connection defined in the non-coincidence gauge we derive the Wheeler-DeWitt equation of quantum cosmology. Because the gravitational field equation in f ( Q )-gravity admits a minisuperspace description the Wheeler-DeWitt equation is a single inhomogeneous partial differential equations. We assume the power-law f ( Q ) = f 0 Q µ model and with the application of linear quantum observables we calculate the wavefunction of the universe. Finally, we investigate the effects of the quantum correction terms in the semi-classical limit.", 'PACS numbers:': 'Keywords: Symmetric teleparallel f ( Q )-gravity; nonmetricity gravity; Wheeler-DeWitt equation; exact solutions;', '1. INTRODUCTION': 'Symmetric teleparallel general relativity [1-6] (STEGR) is a gravitational theory that is equivalent to General Relativity (GR). In STEGR, the geometry of the physical space is described by a metric tensor as in GR, but the autoparallels are defined by a symmetric and flat connection that inherits the symmetries of the metric tensor. This leads to different autoparallels than those of GR, which are constructed by the Levi-Civita connection.  \nAssume that ˜ Γ κ µν is a general connection. This can be decomposed [7] as ˜ Γ κ µν =    κ µ ν    +  \n2Γ κ [ µν ] +∆ κ µν , where    κ µ ν    is the Levi-Civita connection, Γ κ [ µν ] is the torsion tensor, and ∆ κ µν is the symmetric and flat nonmetricity part [8]. Since in STEGR, the connection is flat, it leads to a zero-valued curvature tensor R κ λµν = 0 and it is symmetric, i.e., T κ µν = 0. Thus, only the nonmetricity tensor survives, which leads to the nonmetricity scalar Q . The latter scalar substitutes the Ricci scalar of the Einstein-Hilbert Action Integral, leading to STEGR. As the Ricci scalar R and the nonmetricity scalar Q differ by a boundary term [1], it follows that STEGR is equivalent to GR.  \nNowadays, GR is challenged by the analysis of recent cosmological data [9-14]. This has led the scientific community to introduce alternative and modified theories of gravity. Within the symmetry and teleparallel theory, the simplest modification is the f ( Q )-gravity [15, 16], where the Lagrangian function for the gravitational Action Integral is a nonlinear function of the nonmetricity scalar Q . In the linear limit of the function f , the STEGR theory, with or without the cosmological constant term, is recovered. f ( Q )-gravity is the analogue in the framework of STEGR of other modifiend f -theories defined by the Levi-Civita connection or in teleparalellism, see for instance [17-24] and references therein.  \nThe dynamical degrees of freedom introduced by the nonlinear f ( Q ) can be attributed to scalar fields [25]. In the scalar field description, f ( Q )-theory is equivalent to a specific case of the scalar nonmetricity theory, where the scalar field is nonminimally coupled to gravity. Therefore, f ( Q )-gravity has properties similar to a Machian theory, even if the theory is not purely Machian. For more details, we refer the reader to [29, 30].  \nf ( Q )-theory suffers from two major problems in the cosmological perturbations of a spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) background geometry. In particular, it suffers from strong coupling and the appearance of ghosts [31, 32]. Despite the fact that f ( Q )-theory fails to explain the global evolution of the universe, it provides a unique mechanism leading to the construction of different toy models with interesting applications in gravitational physics. See, for instance, [33-46] and references therein. For other modifications of the STEGR theory, we refer the reader to [47-50]. Moreover, f ( Q )-theory is the simplest mathematical theory which can be applied to understand the effects of the selection of the connection in modified symmetric teleparallel theories.  \nThis investigation deals with quantum cosmology within the framwork of f ( Q )-theory and the effects of the quantum correction in the semiclassical limit [52]. Specifically, for  \nthe FLRW geometry, we assume a connection defined in the noncoincidence gauge where the field equations admit a minisuperspace description. For this specific connection, the field equations are derived from a point-like Lagrangian with three dependent variables: the scale factor of the FLRW geometry and two scalar fields, which attribute the degrees of freedom provided by the definition of the connection and the nonlinear function f ( Q ). For this two-scalar field cosmological model, we write the Wheeler-DeWitt equation (WDW) of quantum cosmology [53, 54]. We employ the theory of symmetries of differential equations [55] to define quantum observables. These are applied to define similarity transformations and derive the wavefunction of the universe.  \nIn the hydrodynamic description of quantum cosmology [56, 57], we make use of the quantum observables to write down conservation laws for the classical field equations and to calculate the classical solutions, because for the power-law f ( Q )-model the classical field equations form an superintegrable Hamiltonian system. The effects of the quantum correction terms related to the nonzero Ricci scalar term of the minisuperspace [53, 54], and of the Bohmian potential term [58-62], are discussed. It follows that the quantum corrections do not affect the general evolution and dynamics of the classical gravitational model. The structure of the paper is as follows.  \nIn Section 2 we briefly discuss STEGR and its modification, the f ( Q )-gravity. This gravitational theory has a scalar field description, the theory is equivalent to a specific case of the scalar-nonmetricity gravitational model. Furthermore, the effects of conformal transformations in the theory are investigated, from where we can understand the degrees of freedom introduced by the theory.  \nThe case of an isotropic and homogeneous universe in the framework of f ( Q )-gravity is presented in Section 3. We review previous results on the different families of connections in the FLRW background, and explore how the selection of the connection affects the gravitational model and the evolution of the physical space. Moreover, by applying scalar fields, we introduce the minisuperspace description for the field equations.  \nFor the connection where the Lagrangian of the field equations is point-like, that is, it describes the motion of classical particles in a curved space under a conservative force term, in Section 4, we derive the WDW equation, which is a single inhomogeneous partial differential equation. For the power-law f ( Q ) model in Section 5, we present the quantum observables for the WDW equation. The latter are used to construct similarity transfor-  \nmations aimed at reducing the WDW equation into an ordinary differential equation and providing a closed-form expression for the wavefunction of the universe. In Section 6, we employ the Madelung representation of quantum mechanics to derive the classical and semiclassical limits where quantum correction terms are introduced in the gravitational field equations. For the two models, we explicitly solve the reduced gravitational field equations using the Hamilton-Jacobi theory. It follows that the quantum correction terms do not affect the main dynamics of the classical solution. The two asymptotic limits of the classical cosmological solution describe the self-similar solution related to the power-law model, as well as a cosmological solution with two perfect fluids that do not interact and have constant equation of state parameters. Finally, in Section 7, we draw our conclusions.', '2. SYMMETRIC TELEPARALLEL f ( Q ) -GRAVITY': 'In this Section, we discuss the main properties and definitions of STEGR and introduce the gravitational model under our consideration, which is that of symmetric teleparallel f ( Q )-gravity.', '2.1. STEGR': "In the framework of STEGR, the physical space is described by a four dimensional metric tensor g µν , and the connection Γ κ µν which is symmetric, i.e. Γ κ µν = Γ κ νµ , and flat. Hence, there exists a point transformation x µ → x µ ' , such that all the components of the connection in the new coordinate system are zero, i.e. Γ κ ' µ ' ν ' = 0 [8].  \nBecause of these two fundamental properties for the connection Γ κ µν , the curvature tensor defined as  \nR κ λµν (Γ) = ∂ Γ κ λν ∂x µ -∂ Γ κ λµ ∂x ν +Γ σ λν Γ κ µσ -Γ σ λµ Γ κ µσ , (1)  \nand the torsion tensor which reads  \nT κ µν (Γ) = 1 2 ( Γ κ µν -Γ κ νµ ) , (2)  \nare always zero.  \nThe fundamental scalar of STEGR is nonmetricity scalar [1]  \nQ = Q κµν P κµν , (3)  \nin which Q κµν is the nonmetricity scalar Q κµν = ∇ κ g µν , that is,  \nQ κµν = ∂g µν ∂x κ -Γ σ κµ g σν -Γ σ κν g µσ , (4)  \nand P κ µν is defined as follow  \nP κ µν = 1 4 Q κ µν + 1 2 Q κ ( µ ν ) + 1 4 ( Q κ -˜ Q κ ) g µν -1 4 δ κ ( µ Q ν ) (5)  \nwhere the vector fields Q µ and ¯ Q µ are given by the expressions  \nQ µ = Q ν µν , ˜ Q µ = Q ν µν . (6)  \nThe gravitational Action Integral in STEGR is defined [1]  \nS = ∫ d 4 x √ -gQ. (7)  \nThe nonmetricity scalar Q for the connection Γ κ µν and the Ricci scalar ˆ R ( ˆ Γ ) , for the LeviCivita connection ˆ Γ κ µν of the metric tensor  \nˆ Γ κ µν = 1 2 g κλ ( g µκ,ν + g λν,µ -g µν,λ ) (8)  \nare related as Q = ˆ R + B , in which B is a boundary term [6]  \nB = -1 2 ˚ ∇ λ P λ , (9)  \nand P λ = P λ µν ¯ g µν . Consequently, the variation of the Action Integral (7) leads to the field equations of Einstein's GR which follow by the Einstein-Hilbert Action. Finally, operator ˚ ∇ λ in (9) remarks the covariant derivative with respect to the Levi-Civita connection ˆ Γ κ µν (8)", '2.2. f ( Q ) -gravity': "The simplest modification of the STEGR Action Integral (7) is the introduction of the cosmological constant. Nevertheless, inspired by other modifications of GR, the introduction of nonlinear terms in the nonmetricity scalar Q in (7) leads to the introduction of dynamical degrees of freedom which can influence the dynamics and describe the cosmic evolution.  \nIn this general concept we assume the gravitational Action Integral to be [15, 16]  \nS f ( Q ) = ∫ d 4 x √ -gf ( Q ) , (10)  \nwhere f ( Q ) is a smooth differentiable function, and when f '' ( Q ) = 0, the limit of STEGR with or without the cosmological constant term is recovered. With a prime ' we note total derivative with respect to the scalar Q , i.e. f ' ( Q ) = df ( Q ) dQ and f '' ( Q ) = d 2 f ( Q ) dQ 2 .  \nThe modified gravitational field equations related to the Action Integral (10) follows from the variation with respect to the metric tensor g µν .  \nThe gravitational field equations are [15, 16]  \n2 √ -g ∇ λ ( √ -gf ' ( Q ) P λ µν ) -1 2 f ( Q ) g µν + f ' ( Q ) ( P µρσ Q ρσ ν -2 Q ρσµ P ρσ ν ) = 0 . (11)  \nequivalently  \nf ' ( Q ) G µν -1 2 g µν ( f ( Q ) -f ' ( Q ) Q ) + 2 f '' ( Q ) P λ µν ∇ λ Q = 0 , (12)  \nwhere G µν is the equivalent of the Einstein tensor in STEGR, that is,  \nG µν = ( P µρσ Q ρσ ν -2 Q ρσµ P ρσ ν ) + 2 √ -g ∇ λ ( √ -gP λ µν ) . (13)  \nWhen Q = Q 0 , expression (12) reads  \nG µν +Λ eff g µν = 0 , (14)  \nwhere Λ eff is an effective cosmological constant term defined as Λ eff = 1 2 f ( Q 0 ) -f ' ( Q 0 ) Q 0 f ' ( Q 0 ) .  \nTherefore, for the case where Q = Q 0 and f ( Q 0 ) -f ' ( Q 0 ) Q 0 = 0, the vacuum solutions  \nof STEGR are recovered. On the other hand, for the case where Q = Q 0 and f ( Q 0 ) -f ' ( Q 0 ) Q 0 = 0, the effects of a nonzero cosmological constant are present in STEGR.  \n/negationslash  \nFurthermore, varying the Action Integral (10) with respect to the symmetric and flat connection Γ κ µν yields the equation of motion  \n∇ µ ∇ ν ( √ -gf ' ( Q ) P µν σ ) = 0 . (15)  \nThe connection is characterized as being defined in the 'coincidence gauge' when equation (15) is identically satisfied. On the other hand, we refer to the connection as being defined in the 'noncoincidence gauge' [15, 16].  \nThe transformation rule for the metric tensor follows that of a tensor field. However, the connection is not a tensor and thus has a different transformation rule. Specifically, the connection is coordinate dependent, though the tensors defined by the connection, such as the curvature tensor, the torsion tensor, and the nonmetricity tensor, are coordinateindependent.  \nWhen we consider a specific line element for the metric tensor, this corresponds to the definition of a proper coordinate system. Thus, the components of the connection may not be identically zero in these coordinates, and equation (15) is not trivially satisfied. This means that dynamical degrees of freedom are introduced due to the connection defined in the non-coincidence gauge. These dynamical degrees of freedom associated with the noncoincidence gauge are of geometric origin. As we shall see in the following section, they can be attributed to scalar fields.", '2.3. Equivalency with scalar field theory': "We introduce the Lagrange multiplier λ m , such that the gravitational Action Integral (10) reads  \nS f ( Q ) = ∫ d 4 x √ -g ( f ( Q ) + λ m ( Q -ˆ Q )) , (16)  \nwhere ˆ Q = ˆ Q ( x κ ) and is the functional expression for the nonmetricity scalar Q .  \nVariation with respect to the scalar Q , leads to the equation of motion δS f ( Q ) δQ = 0 , that is, λ m = -f ' ( Q ).  \nBy replacing in expression (16) it follows  \nS f ( Q ) = ∫ d 4 x √ -g ( f ' ( Q ) ˆ Q +( f ( Q ) -Qf ' ( Q )) ) . (17)  \nWe introduce the scalar field φ = f ' ( Q ), and the potential function V ( φ ) = ( Qf ' ( Q ) -f ( Q )), and the latter Action Integral is expressed in the following simpler form  \nS f ( Q ) = ∫ d 4 x √ -g ( φ ˆ Q -V ( φ ) ) . (18)  \nThis is analogous to O'Hanlon gravity [63] in the framework of symmetric teleparallel theory. Hence, we can say that the nonmetricity f ( Q )-gravity has properties similar to those of a Machian theory [64], although the theory is not purely Machian [29, 30].  \nThe Action Integral (18) is a particular case of the more general theory with a scalar field minimally coupled to gravity, that is, of the scalar-nonmetricity theory defined as  \nS STφ = ∫ d 4 x √ -g ( F ( φ ) 2 Q -ω ( φ ) 2 g µν φ ,µ φ ,ν -V ( φ ) ) , (19)  \nwhere the gravitational field equations are  \nF ( φ ) G µν +2 F ,φ φ ,λ P λ µν + g µν V ( φ ) + ω ( φ ) 2 ( g µν g λκ φ ,λ φ ,κ -φ ,µ φ ,ν ) = 0 . (20)  \nThe equation of motion for the scalar field reads  \nω ( φ ) √ -g g µν ∂ µ ( √ -g∂ ν φ ) + ω ,φ 2 g λκ φ ,λ φ ,κ + 1 2 F ,φ Q -V ,φ = 0 , (21)  \nwhile the equation of motion for the connection is  \n∇ µ ∇ ν ( √ -gF ( φ ) P µν σ ) = 0 . (22)  \nWe remark that for a linear function F ( φ ) = 2 φ , and ω ( φ ) = 0, the f ( Q )-gravity is recovered, and the field equations (12) become  \nφG µν +2 φ ,λ P λ µν + g µν V ( φ ) = 0 . (23)  \nMoreover, equation (21) is simultaneously satisfied by the definition of the scalar field potential V ( φ ), while the equation of motion for the connection is simplified to  \n∇ µ ∇ ν ( √ -gφP µν σ ) = 0 . (24)", '2.4. Conformal transformation': 'In the previous lines, we learned that f ( Q ) is a partially Machian theory, where, in the scalar field description, the theory is defined in the so-called Jordan frame. In the following lines, we discuss the effects of conformal transformations in f ( Q )-gravity and introduce the equivalent theory in the Einstein frame.  \nLet the two conformally related four-dimensional metrics be ¯ g µν and g µν , defined as  \n¯ g µν = e 2Ω( x κ ) g µν , ¯ g µν = e -2Ω( x κ ) g µν , (25)  \nwhere Ω( x κ ) is a smooth differentiable function. Ω ( x κ ) is known as the conformal factor and defines the transformations.  \nThe geometric quantities, which define the f ( Q )-gravity, for the conformally related metrics are related as [30]  \n¯ Q λµν = e 2Ω Q λµν +2Ω ,λ ¯ g µν . (26)  \nand  \n¯ P λ = ¯ P λ µν ¯ g µν = e -2Ω P λ +3Ω ,λ . (27)  \nTherefore, the nonmetricity scalars Q and ¯ Q are  \n¯ Q = ¯ Q λµν ¯ P λµν = e -2Ω Q + ( 2Ω ,λ P λ +6Ω λ Ω ,λ ) . (28)  \nAssume now the Action Integral (18) for the f ( ¯ Q ) -theory in the framework of the space with metric ¯ g µν . Then, the equivalent theory for the metric tensor g µν is given by the Action Integral [28]  \n¯ S f ( ¯ Q ) = ∫ d n x √ -g ( Q -1 2 B ln φ + 3 2 φ g µν φ ,µ φ ,ν -V ( ϕ ) φ 2 ) , (29)  \nwhere we have assumed that Ω = -1 2 ln φ , and scalar B is the boundary term which relates the Ricci scalar ˆ R ( ˆ Γ ) for the Levi-Civita connection with the nonmetricity scalar, given by expression (9). Indeed, the boundary term is defined as [28] B = -1 2 ˚ ∇ λ P, equivalently, [28]  \nB = ( ˜ Q κ -Q κ ) . (30)  \nWe observe that the boundary term B plays an important role in the conformal equivalent description of the theory, and it is another scalar field. The definition of the boundary term is directly related to the definition of the connection Γ and the equation of motion (15).  \nAt this point, it is important to mention that the conformal equivalency of the gravitational theory in the two frames relates only to the trajectory solutions for the field equations. In general, it is not an equivalence of physical properties. Nevertheless, for the scalar-nonmetricity theory, it has been found that conformal transformations preserve the main eras of the cosmological history and evolution.', '3. ISOTROPIC AND HOMOGENEOUS COSMOLOGY': 'In cosmological scales the physical space is assumed that it is homogeneous and isotropic described by the FLRW line element. The latter in spherical coordinates is expressed as follows  \nds 2 = -N ( t ) 2 dt 2 + a ( t ) 2 [ dr 2 1 -kr 2 + r 2 ( dθ 2 +sin 2 θdϕ 2 ) ] , (31)  \nin which k is the spatial curvature of the three-dimensional space; N ( t ) is the lapse function, a ( t ) is the scale factor; and H = 1 N ˙ a a , ˙ a = da dt , is the Hubble function. We assume the comoving observer u µ = 1 N δ µ t , thus, the Hubble function is related to the expansion rate θ = u µ ; µ by the expression H = 1 3 θ .', '3.1. Symmetries': 'In the coordinate system ( t, r, θ, ϕ ), the six isometries of the line element (31) are as follows  \nK 1 = sin ϕ∂ θ + cos ϕ tan θ ∂ ϕ , K 2 = -cos ϕ∂ θ + sin ϕ tan θ ∂ ϕ , K 3 = ∂ ϕ ,  \nK 4 = √ 1 -kr 2 ( sin θ cos ϕ∂ r + 1 r ( cos θ cos ϕ∂ θ -sin ϕ sin θ ) ∂ ϕ ) ,  \nK 5 = √ 1 -kr 2 ( sin θ sin ϕ∂ r + 1 r ( cos θ sin ϕ∂ θ + cos ϕ sin θ ) ∂ ϕ ) , K 6 = √ 1 -kr 2 ( cos θ∂ r -sin θ r ∂ ϕ ) .  \nFor k = 0, the latter symmetry vectors form the E 3 ⊗ SO (2) Lie algebra, otherwise for k = 0, the six symmetry vectors are the elements of the SO (4) Lie algebra.  \n/negationslash  \nThe requirement the connection to inherit the symmetry of the background geometry provides the following set of constraints  \nL K I Γ κ µν = 0 , (32)  \nwhere L K I is the Lie derivative with respect the vector field K I and I = 1 , 2 , 3 , 4 , 5 , 6.  \nThe definition of the Lie derivative depends on the transformation rule. Because the connection has a different transformation rule from that of tensor fields, the Lie derivative differs accordingly. Specifically, in terms of coordinates, the Lie derivative of the connection reads  \nL K I Γ κ µν = K I κ , µν +Γ κ µν,r K I r -K I κ ,r Γ r µν + K I s ,µ Γ κ sν + K I s ,ν Γ κ µs . (33)  \nBecause the connection is symmetric, expression (33) is simplified as  \nL K I Γ κ µν = ∇ ν ∇ µ ( K I κ ) -R κ µνλ K I λ . (34)  \nHowever, in symmetric teleparallel theory, the connection is flat, that is,  \nR κ µνλ (Γ) = 0 . (35)  \nonsequently, the requirement for the connection to inherit the symmetries of the background space is equivalent to the set of differential equations  \n∇ ν ∇ µ ( K I κ ) = 0 . (36)  \n/negationslash', '3.2. Symmetric and flat connection': 'For the FLRW line element (31) and the requirements (35), (36) it follows that there are found different families of symmetric connections Γ κ µν . One family of connection is defined for k = 0 and three families of connections are defined for k = 0 [65, 66].  \n/negationslash  \nThe common nonzero components for the four families of connections are  \nΓ r tr = Γ r rt = Γ θ tθ = Γ θ θt = Γ ϕ tϕ = Γ ϕ ϕt = k  \nr -  \n-γ ( t ) , Γ r rr = kr 1 -kr 2 , Γ r θθ = -r ( 1 -kr 2 ) , Γ r ϕϕ = -r sin 2 θ ( 1 -κr 2 ) , Γ θ rθ = Γ θ θr = Γ ϕ rϕ = Γ ϕ ϕr = 1 , Γ θ ϕϕ = sin θ cos θ , Γ ϕ θϕ = Γ ϕ ϕθ = cot θ.  \nFor k = 0, the additional nonzero components for the family Γ k are  \nΓ t tt = -k + ˙ γ ( t ) γ ( t ) , Γ t rr = γ ( t ) 1 -kr 2 Γ t θθ = γ ( t ) r 2 , Γ t ϕϕ = γ ( t ) r 2 sin 2 ( θ ) ,  \nby comparing the latter connections with the notation presented in [65], it follows that γ ( t ) = C 2 ( t ).  \nMoreover, for k = 0 , connection Γ A has the additional nonzero components  \nΓ t tt = γ ( t ) ,  \nwhere Γ A is connection Γ ( III ) Q in [65] and γ ( t ) = C 1 ( t ) .  \nConnection Γ B is defined for k = 0, with the nonzero components  \nΓ t tt = ˙ γ ( t ) γ ( t ) + γ ( t ) , Γ r tr = Γ r rt = Γ θ tθ = Γ θ θt = Γ ϕ tϕ = Γ ϕ ϕt = γ ( t ) ,  \nand it is connection Γ ( I ) Q of [65] with γ ( t ) = C 3 ( t )  \nFinally, for k = 0, family Γ C has the nonzero components  \nΓ t tt = -˙ γ ( t ) γ ( t ) , Γ t rr = γ ( t ) , Γ t θθ = γ ( t ) r 2 , Γ t ϕϕ = γ ( t ) r 2 sin 2 θ,  \nwhich is compared with connection Γ ( II ) Q of [65] and γ ( t ) = C 2 ( t ).  \nThe connection Γ A is the unique connection defined in the coincidence gauge, and the function γ ( t ) plays no role in the gravitational field equations. Nevertheless, the remaining three families of connections, Γ k , Γ B , and Γ C , are defined in the noncoincidence gauge. In the framework of f ( Q )-gravity, the function γ ( t ) is constrained by the equation of motion (15).', '3.3. Field equations in f ( Q ) -gravity': "In the following lines, we present the field equations for f ( Q )-gravity in an FLRW background geometry. The selection of the connection is crucial in the theory; thus, for each family of connections, a different set of field equations results.  \nFor nonzero spatial curvature and connection Γ k we calculate the nonmetricity scalar  \nQ ( Γ k ) = -6˙ a 2 N 2 a 2 + 3 γ a 2 ( ˙ a a + ˙ N N ) + 3˙ γ a 2 + k ( 6 a 2 + 3 γN 2 ( ˙ N N + ˙ γ γ -3˙ a a )) . (37)  \nThe gravitational field equations in f ( Q )-gravity are  \n0 = 3 f ' ( Q ) H 2 + 1 2 ( f ( Q ) -Qf ' ( Q )) -3 γ ˙ Qf '' ( Q ) 2 a 2 +3 k ( f ' ( Q ) a 2 -˙ Qf '' ( Q ) 2 γN 2 ) , (38)  \n0 = -2 N ( f ' ( QH ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) + γ ˙ Qf '' ( Q ) 2 a 2 -k ( f ' ( Q ) a 2 + 3 ˙ Qf '' ( Q ) 2 γN 2 ) , (39)  \nand the equation of motion for the connection  \n0 = ( f '' ˙ Q ) · ( 1 + k a 2 N 2 γ 2 ) + f '' ˙ Q ( ( 1 + 3 k a 2 N 2 γ 2 ) NH + ( 1 -k a 2 N 2 γ 2 ) ˙ N N +2 ˙ γ γ ) . (40)  \nWe remark that in the previous field equations, if we set k = 0, we recover the gravitational field equations for the connection Γ C .  \nFurthermore, for connection Γ A defined in the coincidence gauge, the nonmetricity scalar  \nis  \nQ ( Γ A ) = -6 H 2 , (41)  \nand the gravitational field equations are  \n0 = 3 H 2 f ' ( Q ) + 1 2 ( f ( Q ) -Qf ' ( Q )) , (42a)  \n0 = -2 N ( f ' ( Q ) H ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) . (42b)  \nFinally, for the family of connection Γ B the nonmetricity scalar reads  \nQ = -6 H 2 + 3 γ N ( 3 H -˙ N N 2 ) + 3˙ γ N 2 , (43)  \nwhile the gravitational field equations are  \n0 = 3 H 2 f ' ( Q ) + 1 2 ( f ( Q ) -Qf ' ( Q )) + 3 γ ˙ Qf '' ( Q ) 2 N 2 (44a)  \n0 = -2 N ( f ' ( Q ) H ) · -3 H 2 f ' ( Q ) -1 2 ( f ( Q ) -Qf ' ( Q )) + 3 γ ˙ Qf '' ( Q ) 2 N 2 . (44b)  \nBecause Γ B is defined in the noncoincidence gauge, the equation of motion for the connection is as below  \n0 = ( f '' ˙ Q ) · + N ˙ Q ( 3 H -˙ N N 2 ) f '' ( Q ) . (45)", '3.4. Minisuperspace description': "A novel property of these cosmological models is that they admit a 'minisuperspace' description. In particular, for each connection, there exists a Lagrangian function whose variation leads to the corresponding field equations.  \nFor connections Γ k (and Γ C for k = 0) the corresponding field equations follows from the variation of the (non-canonical) Lagrangian function  \nL ( Γ k ) = -3 N aφ ˙ a 2 +3 kNaφ -N 2 V ( φ ) + 3 k 2 N a 3 ˙ Ψ ˙ φ -3 2 aN ˙ Ψ ˙ φ, (46)  \nin which φ = f ' ( Q ), V ( φ ) = ( Qf ' ( Q ) -f ( Q )) and ˙ Ψ = 1 γ . We observe that the latter  \nLagrangian has kinetic components in the denominator.  \nSimilarly, for the connection Γ A the Lagrangian function is  \nL ( Γ A ) = -6 N aφ ˙ a 2 -Na 3 V ( φ ) . (47)  \nFinally for connection Γ B the corresponding Lagrangian of the field equations is  \nL ( Γ B ) = -3 N φa ˙ a 2 -3 2 N a 3 ˙ φ ˙ ψ -N 2 a 3 V ( φ ) , (48)  \nwhere now the scalar field ψ is related to the connection as ˙ ψ = γ . Lagrangian function (48) is of the form of a point-like dynamical system. In particular, it describes a constraint dynamical systems where the scale factor and two scalar fields play the role of the particles with interaction U eff = a 3 V ( φ ).  \nThe existence of the lapse function in the Lagrangians is necessary in order to reconstruct the constraint equation in each case.  \nBetween the above Lagrangian functions, only these who correspond on connections Γ A and Γ B describe dynamical system where expressed in the form L = 1 N K E -NU eff , where K is the kinetic energy, K E = 1 2 G AB ˙ q A ˙ q B and U eff is the effective potential. Thus, for these two dynamical systems we can write the Hamiltonian function in the form  \nH ≡ N ( K E + U eff ) = 0 . (49)  \nWe make use of this property in order to continue with the quantization of the field equations. The quantization of the field equations which correspond to the connection Γ A has been studied in detailed in . Because of the nature of the Lagrangian, Dirac's method for the quantization of constraint dynamical systems was applied in [67]. Due to the existence of second class constraints the quantization process was different from the usual WDW formalism, where only first class constraints exist. See also the more recent study [68].  \nThe field equations for the connection Γ B described by the point-like Lagrangian function (48) possess only first class constraints, which means that the quantization process is similar to that of the WDW formalism and different from that of the corresponding system for the connection Γ A .", '4. THE WHEELER-DEWITT EQUATION': 'In GR the Wheeler-DeWitt equation follows from the quantization of the constraint equation in the ADM formalism (for discussion we refer the reader to [53, 54, 69]). The WDW equation is a hyperbolic functional differential equation on superspace and represents a family of differential equations at different points. However, when there exist a minisuperspace description, the WDW reduces to a single differential equation.  \nIn the minisuperspace description the Hamiltonian constraint is expressed as  \nH = N [ 1 2 G AB P A P B + U ( q ) ] = N H ≡ 0 , (50)  \nin which G AB is the minisuperspace.  \nThe classical field equations are invariant under conformal transformations. This property should also hold for the WDW equation of quantum cosmology. The quantization P A = i /planckover2pi1 ∂ A leads to a differential equation of Klein-Gordon type, which is generally not conformally invariant. Therefore, the Klein-Gordon equation is replaced by the Yamabe equation to ensure conformal invariance. This is achieved by modifying the potential with the term n -2 8( n -1) R , where R is the Ricci scalar of the minisuperspace G AB with dimension n .  \nThe WDW equation reads  \nH Ψ = [ /planckover2pi1 2 ( 1 2 ∆ -n -2 8( n -1) R ) -U ( q ) ] Φ( q ) = 0 , (51)  \nin which ∆ = 1 √ -G ∂ A ( √ -GG AB ∂ B ) is the Laplace operator with respect to the minisuperspace G AB .  \nAs we shall see in the following a nonzero Ricci scalar R introduces to the Hamiltonian function of the semiclassical system a potential term which correspond to quantum corrections.', '4.1. Connection Γ B': 'From Lagrangian function (48) we calculate the three-dimensional minisuperspace  \nG AB =      6 φa 0 0 0 0 3 2 a 3 0 3 2 a 3 0      , G AB =      1 6 φa 0 0 0 0 2 3 a 3 0 2 3 a 3 0      . (52)  \nThus, the corresponding Hamiltonian function is expressed as  \nH = 1 12 φa P 2 a + 2 3 a 3 P φ P ψ -a 3 V ( φ ) , (53)  \nwith  \nP a = 6 φa ˙ a , P φ = 3 2 a 3 ˙ ψ and P ψ = 3 2 a 3 ˙ φ . (54)  \nMoreover, the Ricci scalar for the minisuperspace is derived  \nR = -3 4 a 3 φ , (55)  \nTherefore, the WDW equation (51) for the Hamiltonian function (53), Φ = 0, is expressed  \nH as  \n1 6 aφ Φ ,aa + 4 3 a 3 Φ ,φψ + 5 12 a 2 φ Φ ,a + 1 3 φa 3 Φ ,ψ + ( 3 32 a 3 φ -2 /planckover2pi1 2 a 3 V ( φ ) ) Φ = 0 . (56)', '4.2. Power-law theory': 'We assume that the function f ( Q ) is a power-law, i.e., f ( Q ) = f 0 Q µ . The power-law function provides a cosmological history that depends on the selection of the connection [70]. However, for all connections, the power-law f ( Q ) function is associated with the existence of self-similar cosmological solutions [71]. For the cosmological model defined by the connection Γ B , the self-similar scaling solution is always unstable, while the unique attractor describes the accelerated de Sitter universe. This physical property remains the same in the presence of matter [74].  \nIn the scalar field description of the f ( Q ) theory, the power-law model f ( Q ) = f 0 Q µ leads to the power-law potential function V ( φ ) = V 0 φ κ , where constants V 0 and κ are related with  \nf 0 , µ as  \nV 0 = ( µ -1) f -1 µ -1 0 µ -µ µ -1 , κ = µ µ -1 . (57)  \nRecall that for values of µ close to 1, the gravitational model describes small deviations from General Relativity. We refer the reader to the interesting discussion on the power-law f ( R ) theory presented in [75].  \nTherefore, the WDW equation (56) for the power-law model is simplified as  \n1 6 aφ Φ ,aa + 4 3 a 3 Φ ,φψ + 5 12 a 2 φ Φ ,a + 1 3 a 3 φ Φ ,ψ + ( 3 32 a 3 φ -V 0 /planckover2pi1 2 a 3 φ κ ) Φ = 0 . (58)  \nThe solution of the WDW equation (58) provides the wavefunction Φ ( a, φ, ψ ) for this specific cosmological model.  \nThe WDW equation is a linear inhomogeneous partial differential equation. In the following lines we employ the method of similarity transformations, also known as Lie symmetry analysis, such that to determine expressions for the wavefunction Φ which satisfy the partial differential equation (58).', '5. SIMILARITY TRANSFORMATIONS': 'Lie symmetry analysis is a robust method for the study of nonlinear differential equations. It provides a systematic way to investigate the algebraic properties and compute solutions for nonlinear and inhomogeneous differential equations [72, 73]. The main characteristic of Lie symmetry analysis is that it allows for the construction of one-parameter point transformations which leave the given differential equation invariant. From these point transformations, it is possible to derive similarity transformations used to facilitate solutions. This procedure is known as the reduction process, and in the case of partial differential equations, the existence of a Lie symmetry is equivalent to the existence of a quantum operator.  \nFor a review on the wide applications of symmetry analysis in classical gravitational physics and cosmology, we refer the reader to [55]. In the framework of the WDW equation, the mathematical approach for constructing quantum operators from Lie symmetry analysis is analytically described in [76]. This method has been widely used for constructing the wavefunction of the universe in a broad range of gravitational models [77-84].  \nThe application of the Lie symmetry analysis for the WDW equation (58), i.e. the  \nYamabe equation, leads to the following quantum operators  \nΞ 1 : ( ∂ ψ -β 1 /planckover2pi1 i ) Φ = 0 , (59) Ξ 2 : ( -( κ +1) 6 a∂ a + φ∂ φ -β 2 /planckover2pi1 i ) Φ = 0 , (60)  \nΞ 3 : ( ( κ +1) 6 ln φa∂ a -ln φ φ∂ φ + ( ψ -2 ( κ +1) 3 ln a ) ∂ ψ -( ( K -1) 8 ln φ + β 3 /planckover2pi1 i )) Φ = 0 . (61)  \nWith the use of the latter operators we are able to construct the following solution for the wavefunction of the universe,  \nΦ( a, φ, ψ ) = a -3 4 + 2 3 i /planckover2pi1 β 1 ( κ +1) φ -κ +1 8 + i /planckover2pi1 λ e -i /planckover2pi1 β 1 ψ ( Φ 1 0 J ¯ λ ( 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 ) +Φ 2 0 Y ¯ λ ( 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 )) , (62)  \nin which J ¯ λ , Y ¯ λ are the Bessel functions of the first and second kind respectively; and constants λ, ¯ λ are defined as  \nλ = 1 9 ( β 1 +9 β 2 + β 1 κ (2 + κ )) , (63)  \n¯ λ = i 9 √ β 1 /planckover2pi1 ( 4 β 1 /planckover2pi1 κ ( κ +2) + 9 i ( κ -1) -72 β 2 /planckover2pi1 ) . (64)  \nWavefunctions of the form in expression (62) have been explicitly derived before in the literature [83, 84]. Thus, the same analysis for the construction of a Hilbert space can be applied. We conclude the discussion here and shift our focus to the classical limit.  \nRecall that for large values of the argument of the Bessel functions, the asymptotic solution is  \nΦ( a, φ, ψ ) /similarequal a -3 2 + 2 3 i /planckover2pi1 β 1 ( κ +1) φ -κ +1 4 + i /planckover2pi1 λ e -i /planckover2pi1 β 1 ψ ( ¯ Φ 1 0 cos ( K ) + Φ 2 0 sin ( K ) ) , (65)  \nwith K = 2 3 /planckover2pi1 i √ 3 V 0 a 3 φ κ +1 2 -¯ λπ 2 -π 4 ; while in the limit a 3 φ κ +1 2 → 0, the asymptotic behaviour of the wavefunction reads  \nΦ( a, φ, ψ ) /similarequal a -3 4 + 2 3 i /planckover2pi1 β 1 ( κ +1)+3 ¯ λ φ -κ +1 8 + i /planckover2pi1 λ -κ +1 2 ¯ λ e -i β 1 /planckover2pi1 ψ . (66)  \nIn the following lines, we focus on the analysis of the semiclassical limit of quantum cosmology.', '6. SEMICLASSICAL LIMIT': 'In the Madelung representation [56] of quantum mechanics, that is, in the hydrodynamic approach, we write the wavefunction in the for Φ ( q ) = Ω ( q ) e i /planckover2pi1 S ( q ) , where Ω is the amplitude of the wavefunction. By replacing in the WDW equation (51) and separating the real and imaginary parts we end with the  \n1 2 G AB ( ∂S ∂q A )( ∂S ∂q B ) + U ( q ) + /planckover2pi1 2 ( n -2 8( n -1) R1 2Ω ∆(Ω) ) = 0 . (67)  \nIn the limit where /planckover2pi1 2 → 0, that is, in the WKB approximation, the Hamilton-Jacobi equation for the classical gravitational field equations is recovered.  \nThe new term V Q = -1 2Ω ∆(Ω) is known as the quantum potential in the de Broglie-Bohm representation of quantum mechanics [58, 59] and it depends only on the amplitude of the wavefunction. Furthermore, from the above expression it is clear how the curvature of the minisuperspace contributes in the semiclassical limit.  \nThe observables, Q 1 , Q 2 and Q 3 , lead to the conservation laws for the classical system  \nI 1 = P ψ , (68)  \nI 2 = -( κ +1) 6 aP a + φP φ , (69)  \nI 3 = ln φ ( ( κ +1) 6 aP a -φP φ ) + ( ψ -2 ( κ +1) 3 ln a ) P ψ -( K -1) 8 ln φ. (70)  \nin which P a = ∂S ∂a , P φ = ∂S ∂φ and P ψ = ∂S ∂ψ .', '6.1. Classical solution': 'The Hamilton-Jacobi equation for the classical system reads  \n1 6 aφ ( ∂S ∂a ) 2 + 4 3 a 3 ( ∂S ∂φ )( ∂S ∂ψ ) +2 V 0 a 3 φ κ = 0.  \nA 1 ( I 1 , I 2 , κ ) = 2 3 ( κ +1) ( I 1 ( κ +1) 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 )) ) , A 2 ( I 1 , I 2 , κ ) = I 1 ( κ +1) 2 -9 I 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 )) .  \nBy using the conservation laws we determine the closed-form solution for the action S ( a, φ, ψ ), that is,  \nS ( a, φ, ψ ) = 1 9 ∫ 9 I 2 + √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) -( κ +1) 2 I 1 φ dφ + 2 3 ( κ +1) ∫ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) -( κ +1) 2 I 1 a da +9 U 0 ( κ +1) ∫ ∫ φ κ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) dφ da + I 1 ψ. (71)  \nThus  \nP a = -2 ( I 1 ( κ +1) 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) ) 3 a ( κ +1) , (72)  \nP φ = -I 1 ( κ +1) 2 -9 I 2 -√ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 ) 9 φ , (73)  \nP ψ = I 1 , (74)  \nBy replacing the momentum terms from (54) we end with a system of three nonlinear firstorder differential equations.  \nIn the limit where a 6 φ κ +1 → 0, the field equations are described by the following system (we have assumed the lapse function to be a constant)  \nφa 2 ˙ a /similarequal -A 1 ( I 1 , I 2 , κ ) 6 , φa 3 ˙ ψ /similarequal -2 A 2 ( I 1 , I 2 , κ ) 27 , a 3 ˙ φ /similarequal 2 3 I 1 .  \nwhere  \nWe write the equivalent system  \n∫ 1 φ dφ /similarequal 4 I 1 A 1 ∫ 1 a da, ∫ dψ /similarequal 4 A 2 9 A 1 ∫ 1 a da, (75)  \nwhere the analytic solution is terms of the scale factor reads  \nφ ( a ) /similarequal a 4 I 1 A 1 , ψ ( a ) /similarequal 4 A 2 9 A 1 ln a . (76)  \nOn the other hand, when the term a 6 φ κ +1 dominates we end with the following system  \n˙ a /similarequal -√ -3 2 V 0 6 ( κ +1) aφ κ -1 2 , (77)  \n˙ ψ /similarequal -√ -2 V 0 √ 27 φ κ -1 2 , (78)  \n˙ φ /similarequal 2 3 I 1 a -3 . (79)  \nThe solution is real only when V 0 < 0. Thus, in this case in a similar way as before we end with the reduced system  \n∫ φ κ -1 2 dφ /similarequal -4 √ 2 ( κ +1) 3 | V 0 | I 1 ∫ a -4 da , (80)  \n√ ψ ( a ) /similarequal ∫ 4 ( κ +1) 3 φ κ -1 2 1 a da . (81)  \nwhich means that the asymptotic solution is  \nφ ( a ) /similarequal ( -4 √ 2 ( κ +1) I 1 √ 3 | V 0 | ) 2 κ +1 a -6 κ +1 , ψ ( a ) /similarequal a 3 1 -κ 1+ κ . (82)  \nRecall that this solution is valid when a 3 φ κ +1 2 → 0, i.e. ( -4 √ 2( κ +1) I 1 √ 3 | V 0 | ) 2 κ +1 → 0.', '6.1.1. Physical properties of the asymptotic solutions': 'In the following lines we investigate the physical properties of the asymptotic solutions derived before.  \nFrom the field equations of connection Γ B , it follows that  \nH 2 = 1 6 φ ( V ( φ ) -3 ˙ φ ˙ ψ ) (83)  \nHence, for the asymptotic solution at the limit a 6 φ κ +1 → 0, we derive  \nH 2 /similarequal H 1 0 ( I 1 , I 2 , κ ) a -6 -8 I 1 I 2 + H 1 0 ( I 1 , I 2 , κ, V 0 ) a 4 I 1 I 2 (1+ κ ) -8 I 1 I 2 . (84)  \nThis corresponds to a cosmological solution where the effective fluid in the framework of STEGR corresponds to two perfect fluids with constant equation parameters.  \nOn the other hand, in the limit where the term a 6 φ κ +1 dominates the asymptotic behavior for the Hubble function reads  \nH 2 /similarequal ¯ H 1 0 ( I 1 , I 2 , κ, V 0 ) a -6+ 12 1+ κ . (85)  \nThe latter describes the perfect fluid solution with constant equation of state parameter, which leads to a self-similar spacetime [71]. The equation of state parameter is defined as w eff = 3 -8 1+ κ , from where it follows that acceleration is occurred when -1 < κ < 5 3 .', '6.2. Quantum potential': 'We determine the solution of the field equations in the semiclassical regime where the quantum effects take place and affects the cosmological dynamics.  \nFrom the wavefunction (62) we calculate the quantum potential  \nV Q ( a, φ, ψ ) = V 0 Q ( I 1 , I 2 , κ ) φa 3 , (86)  \nin which V 0 Q ( I 1 , I 2 , κ ) is a constant related to the parameters I 1 , I 2 and κ .  \nConsequently, the Hamilton-Jacobi equation (67) becomes  \n1 6 aφ ( ∂ ˆ S ∂a ) 2 + 4 3 a 3 ( ∂ ˆ S ∂φ )( ∂ ˆ S ∂ψ ) +2 V 0 a 3 φ κ + 2 ¯ V 0 Q ( I 1 , I 2 , κ ) a 3 φ /planckover2pi1 2 = 0 .  \nThe conservation laws presented before are also conservation laws and for the latter Hamil-  \ntonian system with the quantum correction.  \nThus, the action reads  \nψ.  \nˆ S ( a, φ, ψ ) = 1 9 ∫ 9 I 2 + √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) -( κ +1) 2 I 1 φ dφ + 2 3 ( κ +1) ∫ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) -( κ +1) 2 I 1 a da +9 U 0 ( κ +1) ∫ ∫ φ κ √ ( κ +1) 2 ( I 1 ( ( κ +1) 2 I 1 -18 I 2 ) -27 V 0 a 6 φ κ +1 -27 ¯ V 0 Q /planckover2pi1 2 ) dφ da + I 1 (87)  \nWe observe that the latter action is of the same function form of (71) resulting the same gravitational field equations as before, with a rescale on the constants I 1 , I 2 and κ .  \nConsequently, the behaviour of the semiclassical solution is similar with that derived before for the classical solution without the quantum correction term.', '7. CONCLUSIONS': 'In this study, we reviewed the basic mathematical properties of nonmetricity f ( Q )-gravity. This gravitational model represents the simplest generalization of STEGR by introducing nonlinear terms of the nonmetricity scalar Q into the gravitational Action Integral. f ( Q )-gravity is a particular case of the more general nonmetricity-scalar theory, where a scalar field is nonminimally coupled to gravity. f ( Q )-gravity exhibits Machian properties, even though it is not purely Machian. Due to this characteristic, we discussed the effects of conformal transformations on the gravitational Action Integral and introduced the analogy of the Jordan and Einstein frames in nonmetricity theory.  \nWithin the cosmological framework of FLRW geometry, the theory provides four different sets of gravitational field equations. This arises because the connection in STEGR theory is not uniquely defined. The choice of connection leads to the introduction of geometrodynamical degrees of freedom in the field equations, which affects the dynamics of the cosmological parameters. For the coordinate system where the line element for FLRW is expressed in the usual form of (31), there is a family of connections defined in the so-called coincidence gauge. In this case, the connection depends on a gauge function, which does not  \naffect the cosmological dynamics. However, for the remaining three families of connections, the geometrodynamical degrees of freedom of the resulting gravitational equations can be attributed to two scalar fields. These two scalar field have a geometric origin related to the dynamical degrees of freedom introduced by the nonlinear function f , and the degrees of freedom introduced by the connection.  \nFor the connection where the two scalar fields define a canonical kinetic term, such that the gravitational field equations admit a minisuperspace description, we employed the quantization process to derive the WDW equation of quantum cosmology. We considered the power-law f ( Q ) = f 0 Q µ gravity, thus quantum observables were calculated using the theory of Lie symmetries. In this consideration we focused in a multi-scalar field cosmological model, either if the origin of the scalar field is geometric.  \nThe quantum operators constructed by the Lie symmetry analsysis were used to reduce the WDW equation and write a closed-form solution for the wavefunction. We focused on the effects of quantum correction terms in the semiclassical limit. Therefore, we calculated the action by solving the Hamilton-Jacobi equation for the classical system, both with and without quantum correction terms. From the derivation of the Action we were able to write the equivalent reduced classical system.  \nWe found that the quantum corrections do not affect the general evolution of the classical solution. However, the quantum potential can rescale the integration constants of the classical solutions. This is an important observation because it states that the initial value problem can be overcome by using the quantum corrections in the semi-classical limit.  \nAlthough f ( Q )-cosmology is challenged by strong coupling and the presence of ghosts, this geometric model provides a mathematical framework for a better understanding of the effects of connection choice in gravitational physics. Nonmetricity f ( Q )-theory distinguishes the connection from the metric, in contrast to GR, leading to new physics. 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