Novel Complete Non-compact Symmetries for the Wheeler-DeWitt Equation in a Wormhole Scalar Model and Axion-Dilaton String Cosmology
aa r X i v : . [ h e p - t h ] A ug Novel Complete Non-compact Symmetries for the Wheeler-DeWitt Equation in aWormhole Scalar Model and Axion-Dilaton String Cosmology
Rub´en Cordero ∗ Departamento de F´ısica, Escuela Superior de F´ısica y Matem´aticas del I. P. N.,Unidad Profesional Adolfo L´opez Mateos, Edificio 9, 07738 M´exico D.F., M´exico.
Victor D. Granados † Departamento de F´ısica, Escuela Superior de F´ısica y Matem´aticas del I. P. N.,Unidad Profesional Adolfo L´opez Mateos, Edificio 9, 07738 M´exico D.F., M´exico.
Roberto D. Mota ‡ Departamento de ICE de la Escuela Superior de Ingenier´ıa Mec´anica y El´ectrica del I. P. N.,Unidad Culhuacan. Av. Santa Ana No. 1000,San Francisco Culhuacan, Coyoacan M´exico D. F.,C. P. 04430, M´exico. (Dated: November 8, 2018)We find the full symmetries of the Wheeler-DeWitt equation for the Hawking and Page wormholemodel and an axion-dilaton string cosmology. We show that the Wheeler-DeWitt Hamiltonianadmits an U (1 ,
1) hidden symmetry for the Hawking and Page model and U (2 ,
1) for the axion-dilaton string cosmology. If we consider the existence of matter-energy renormalization, for each ofthese models we find that the Wheeler-DeWitt Hamiltonian accept an additional SL (2 , R ) dynamicalsymmetry. In this case, we show that the SL (2 , R ) dynamical symmetry generators transform thestates from one energy Hilbert eigensubspace to another. Some new wormhole type-solutions forboth models are found. PACS numbers: 98.80.Qc; 11.30.Pb; 11.30.-j;
I. INTRODUCTION
The study of the early universe has become one of themore intense research areas in physics. General relativitydescribes the universe at scales larger than the Planckscale and it is expected that quantum mechanics has tobe taken into account at least at these small scales.In the quantum cosmology framework the whole uni-verse is represented by means of a wave function. Thequantum cosmology formalism, including the definitionof the wavefunction of the universe, its configurationspace and its evolution according to the Wheeler-DeWittequation, was set up in the late 1960s [1–5].The development of quantum cosmology started atthe beginning of 1980 when was proposed that the uni-verse could be spontaneously nucleated out of nothing[6], where nothing means the absence of space and time.After nucleation the universe enters to a phase of in-flationary expansion and continues its evolution to thepresent. However there are several important questionsthat remain to be solved like the appropriate boundaryconditions for the Wheeler-DeWitt equation. In the caseof quantum mechanics there is an external setup andthe boundary conditions can be imposed safely, but in ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] χ and it comesfrom the third rank field strength corresponding to theKalb-Ramond field, the other one is called the dilaton φ . The physical consequences of the axion in a curvedspacetime has been investigated in the aim of findingpossible indirect evidences of low energy string theory[16–18]. The dilaton is very important in string theorysince it defines the string coupling constant g s as e φ/ ,it determines the Newton constant, the gauge couplingconstants and Yukawa couplings.It is a well known fact that symmetries are very impor-tant to understand several properties of diverse theories.In particular, it is very interesting to investigate the un-derlying symmetries of the Hawking and Page wormholemodel and axion-dilaton string cosmology. For the firstmodel, a U (1) symmetry generated by the “angular mo-mentum” is present. For the second model, Maharana[19, 20] showed that the “angular part” of the Wheeler-DeWitt equation is invariant under the SO (2 ,
1) groupof transformations. For both models the angular symme-tries were employed to reduce the Wheeler-DeWitt equa-tion to a one-dimensional radial equation. However, inthis work we show that these systems have larger sym-metry groups.In this paper we consider the axion-dilaton stringcosmology studied by Maharana and find the U (2 , U (1 ,
1) symmetry. Also, for each of these models weshow that the Wheeler-DeWitt Hamiltonian accept anadditional SL (2 , R ) dynamical symmetry when matter-energy renormalization is allowed. In this case, we provethat the SL (2 , R ) dynamical symmetry transform statesfrom an energy Hilbert eigensubspace to another energyeigensubspace. The paper is organized as follows. Insection 2 we find the U (1 ,
1) and SL (2 , R ) symmetriesfor the wormhole scalar model. In section 3, by choos-ing a factor ordering different from the used in [19, 20]we show that the groups U (2 ,
1) and SL (2 , R ) are sym-metries for the axion-dilaton string cosmology. In sec-tion 4, for both models, we find some new solutions forthe Wheeler-DeWitt equation, including wave packets.By imposing the Hartle-Hawking boundary conditions,wormhole type-solutions are found. Finally, in section 5,we give our concluding remarks. II. SYMMETRIES FOR THE HAWKING ANDPAGE WORMHOLE SCALAR MODEL
Hawking and Page considered the Wheeler-DeWittequation for the massless scalar field φ in a Friedmann-Robertson-Walker (FRW) spacetime H ψ ( a, φ ) = 12 (cid:18) a ∂∂a a ∂∂a − a ∂ ∂φ − a (cid:19) ψ ( a, φ ) = 0 , (1)whose independent solutions are [12] ψ ( a, φ ) = J ± i m ( ia / e imφ . (2)Notice that there exists a solution for each integer m , corresponding to the angular momentum eigenvalue: L ψ ( a, φ ) ≡ − i ∂ψ ( a,φ ) ∂φ = mψ ( a, φ ). Thus, the Wheeler-DeWitt equation has an infinite number of eigenstates.We define the creation and annihilation operators a = 1 √ (cid:18) − sinh φ (cid:18) a + ∂∂a (cid:19) + cosh φa ∂∂φ (cid:19) , (3)¯ a = 1 √ (cid:18) − sinh φ (cid:18) a − ∂∂a (cid:19) − cosh φa ∂∂φ (cid:19) , (4) a = 1 √ (cid:18) cosh φ (cid:18) a + ∂∂a (cid:19) − sinh φa ∂∂φ (cid:19) , (5)¯ a = 1 √ (cid:18) cosh φ (cid:18) a − ∂∂a (cid:19) + sinh φa ∂∂φ (cid:19) , (6)which satisfy the commutation relations [ a µ , ¯ a ν ] = G µν =diag( − , µ, ν = 0 ,
1. By means of the coordinatetransformation x = a sinh φ and y = a cosh φ , these op-erators become a = 1 √ (cid:18) − x + ∂∂x (cid:19) , ¯ a = − √ (cid:18) x + ∂∂x (cid:19) , (7) a = 1 √ (cid:18) y + ∂∂y (cid:19) , ¯ a = 1 √ (cid:18) y − ∂∂y (cid:19) . (8)The angular momentum operator is L = − i ∂∂φ = − i (cid:18) y ∂∂x + x ∂∂y (cid:19) (9)= − i ( a ¯ a − a ¯ a ) , (10)and the Hamiltonian (1) can be written as H = 12 (cid:20) ∂ ∂y − ∂ ∂x − ( y − x ) (cid:21) (11)= ¯ a a − ¯ a a − . (12)The creation and annihilation operators defined aboveallow to find the U (1 ,
1) hidden symmetry generators¯ a a , ¯ a a , ¯ a a and ¯ a a , which commute with theHamiltonian operator H . Since the group U (1 ,
1) is equalto SU (1 , × U (1) [26], we find that the U (1) generatoris J = − ¯ a a + ¯ a a , (13)= − a ¯ a + a ¯ a − , (14)= − ( H + 1) , (15)and the non-trivial SU (1 ,
1) traceless generators are J = ¯ a a + 12 J , J = ¯ a a , (16) J = ¯ a a , J = ¯ a a − J . (17)These symmetry operators are such that J = J ,[ J , H ] = 0, [ J , J µν ] = 0 and [ J µν , H ] = 0.If we consider the possibility of a matter-energy renor-malization by introducing an arbitrary constant [7], equa-tion (1) can be rewritten in the following form12 (cid:18) a ∂∂a a ∂∂a − a ∂ ∂φ − a − E (cid:19) ψ Em ( a, φ ) = 0 . (18)Notice this equation enforces us to introduce the energy E to label the wavefunction. Since the operators J µν commute with the Hamiltonian then they do not changethe energy E but the angular momentum quantum num-ber m . Thus, these generators transform the degeneratestates corresponding to a given energy between them-selves, in particular those for the zero-energy E = 0.Also we can define the set of operators K + ≡ (cid:0) − ¯ a + ¯ a (cid:1) , (19) K − ≡ (cid:0) − a + a (cid:1) , (20) K ≡
12 ( − ¯ a a + ¯ a a + 1) = − H , (21)which satisfy the commutation relations[ K + , K − ] = − K , [ K , K ± ] = ± K ± . (22)This means that the operators K , K + and K − closethe SL (2 , R ) dynamical Lie algebra. A direct calcula-tion shows that the Casimir operator ˆ K ≡ K ( K − − K + K − is related to the angular momentum L as K = −L − . From this result, the common eigenfunc-tions for the Hamiltonian K and the Casimir K oper-ators of the SL (2 , R ) algebra can be chosen as those ofthe Hamiltonian and the angular momentum operators.Thus, from equation (21) we get K | E m i = − E | E m i ,and from the second commutation relation we show that K K ± | E m i = − (cid:0) E ∓ (cid:1) K ± | E m i . These results implythat K ± | E m i ∝ | E ∓ m i . Hence, the operators (19)and (20) acting on the states | E m i change the energyand leave fixed the angular momentum quantum num-ber. If we restrict the solutions to those of the Wheeler-DeWitt equation without matter-energy renormalization,we must consider the states with E = 0, and the above SL (2 , R ) dynamical symmetry is not relevant. III. SYMMETRIES OF AXION-DILATONSTRING COSMOLOGY
We begin summarizing some important points of theMaharana papers [19, 20] which are relevant to our work.In these references it has been found the SO (2 ,
1) sym-metry of axion-dilaton string cosmology derived from theaction in the Einstein frame S = Z d x √− g (cid:18) R − ∂ µ φ∂ µ φ − e φ ∂ µ χ∂ µ χ (cid:19) , (23)where R is the scalar curvature, √− g is the determinantof the metric g µν , and φ and χ are the dilaton and axionfields, respectively. The homogeneous and isotropic FRWmetric for closed universes ( k = 1) ds = − dt + a ( t ) (cid:18) dr − r + r d Ω (cid:19) , (24) was assumed, where a ( t ) is the scalar factor and t is thecosmic time. The corresponding Wheeler-DeWitt equa-tion is H Ψ := 12 (cid:18) ∂ ∂a + pa ∂∂a − a + 1 a ˆ C (cid:19) Ψ = 0 , (25)where in order to solve the ordering ambiguity between a and ∂/∂a , it was adopted the prescription p = 1. Sincethe action S is invariant under the SO (2 ,
1) transforma-tions (S-duality), also H is invariant under these trans-formations. ˆ C is the SO (2 ,
1) Casimir operator, whichexpressed in the pseudospherical coordinate system x = a sinh α cos β, y = a sinh α sin β, z = a cosh α, (26)is just the Laplace-Beltrami operator given byˆ C = − α ∂∂α (cid:18) sinh α ∂∂α (cid:19) − α ∂ ∂β . (27)The axion and dilaton fields can be written in terms ofthe pseudospherical coordinates (26) as χ = sinh α cos β cosh α + sinh α sin β , e − φ = 1cosh α + sinh α sin β . (28)The explicit solutions for the Wheeler-DeWitt con-straint (25) on the pseudosphere were obtained from the SO (2 ,
1) group theory by identifying that the correct se-ries involved in quantum cosmology is the continuous one[20]. These areΨ( a, α, β ) = J ± i ν ( ia / Y m − + iλ (cosh α, β ) , (29)where Y m − + iλ (cosh α, β ) = e imβ P m − + iλ (cosh α ), and ν = ( λ + ), are the eigenfunctions for the non-compact operator ˆ C and the compact generator − i∂ β ,with P m − + iλ (cosh α ) the associated Legendre polynomi-als (also called toroidal functions). Notice that for thiscase, by varying λ and m there exists and infinite degen-eracy.One of the main results of this paper is to find the fullsymmetries for the Wheeler-DeWitt equation (25). Thisis based on recognizing that equation (25) in the coor-dinates (26) with factor ordering p = 2 can be writtenas H Ψ = 12 (cid:18) ∂ ∂z − ∂ ∂y − ∂ ∂x − ( z − y − x ) (cid:19) Ψ = 0 . (30)Some aspects about the factor ordering operator in thecontext of string cosmology are important to remark.When one considers the Wheeler-DeWitt equation in thestring frame the operator ordering is usually fixed byT-duality invariance of the Hamiltonian. This selectionof the factor ordering is important in the graceful exitproblem of quantum string cosmology of pre-big bangscenario. In the case of homogeneous and isotropic cos-mology without the axion, the T-duality is just the scalefactor duality a → a and this requirement constraint thechoice of factor ordering to p = 1 [21, 22]. However, thescale factor duality is not adequate in the graceful exit inpre-big bang string cosmology when quantum loop cor-rections are taken into account [23]. Besides, the pres-ence of an homogeneous axion field or spatial curvature iscompatible with S-duality but breaks T-duality (O(d,d)symmetry) [24, 25]. In our case, the use of Einstein framehelps to show S-duality of the theory but it is not usefulto fix the factor ordering because a → a under S-duality[19].If we want to preserve reparametrization invariance ofthe Hamiltonian, following the arguments presented in[21], we need p = 2 because in our case the minisuper-space is three-dimensional unlike to that found in [12],where the minisuperspace is bidimensional and thereforethe adequate factor ordering results to be p = 1.For this model we propose the set of creation and an-nihilation operators a = 1 √ (cid:18) − z + ∂∂z (cid:19) , ¯ a = − √ (cid:18) z + ∂∂z (cid:19) , (31) a = 1 √ (cid:18) y + ∂∂y (cid:19) , ¯ a = 1 √ (cid:18) y − ∂∂y (cid:19) , (32) a = 1 √ (cid:18) x + ∂∂x (cid:19) , ¯ a = 1 √ (cid:18) x − ∂∂x (cid:19) . (33)These operators satisfy the commutation relations[ a µ , ¯ a ν ] = G µν = diag( − , , µ, ν = 0 , , , and fac-torize the Hamiltonian as follows H = − ¯ a a + ¯ a a + ¯ a a + 32 . (34)The operators (31)-(33) allow us to define the angularoperators J z = − i (¯ a a − ¯ a a ) = − i (cid:18) y ∂∂x − x ∂∂y (cid:19) , (35) J y = − i (¯ a a − ¯ a a ) = i (cid:18) z ∂∂y + y ∂∂z (cid:19) , (36) J x = − i (¯ a a − ¯ a a ) = − i (cid:18) x ∂∂z + z ∂∂x (cid:19) . (37)These operators satisfy the SO (2 ,
1) commutation rela-tions[ J x , J y ] = − iJ z , [ J z , J x ] = iJ y , [ J y , J z ] = iJ x , (38)and reproduce the Casimir operator ˆ C = − J z + J y + J x .This result reflects a non-compact symmetry (S-duality)on the angular part of the Hamiltonians (25) or (30).The creation and annihilation operators (31)- (33) al- low to define the set of second-order operators J = ¯ a a + 13 J , J = ¯ a a − J (39) J = ¯ a a − J , J = ¯ a a , (40) J = ¯ a a J = ¯ a a , (41) J = ¯ a a J = ¯ a a , (42) J = ¯ a a (43) J = − ¯ a a + ¯ a a + ¯ a a , = − a ¯ a + a ¯ a + a ¯ a − J and the Wheeler-DeWittHamiltonian (30) are related by J = H − . We provethat operators J and J µν satisfy the commutation rela-tions [ J , J µν ] = 0 . (45)This means that the operators J µν are the symmetriesof the Hamiltonian H . In fact, the operators J µν are thenon trivial generators of SU (2 , J is the U (1)generator [26].In a similar way to the wormhole model we can in-troduce a matter-energy renormalization. Hence, theWheeler-DeWitt equation (25) takes the form12 (cid:18) ∂ ∂a + 2 a ∂∂a − a + 1 a ˆ C − E (cid:19) Ψ Eλm = 0 . (46)By using the spherical functions Y m − + iλ (cosh α, β ) = e imβ P m − + iλ (cosh α ) as the correct wavefunctions for theCasimir operator ˆ C , we propose Ψ Eλm to have the form h aαβ | Eλm i = e imβ P m − + iλ (cosh α ) W Eλ ( a ) . (47)This allows us to find the solution to the Wheeler-DeWitt equation with matter-energy renormalization forthe scale factor W Eλ ( a ). It is given by W Eλ ( a ) ≡ c a − M − E , Λ ( a )+ c a − W − E , Λ ( a ) , (48)where M and W are the Whittaker functions, and Λ ≡ √ λ . Notice that to set the correct coefficients, weneed to impose on the functions W Eλ ( a ) one of the wellknown proposals for the boundary conditions [7–11].The toroidal functions P m − + iλ ( x ) [27], can be ex-pressed in terms of the hypergeometric functions [20] P mj ( x ) = 1Γ(1 − m ) (cid:18) x − x + 1 (cid:19) m F (cid:18) − j, j + 1; 1 − m ; 1 − x (cid:19) , (49)with j = − + iλ . They satisfy the orthogonality rela-tions [28] Z ∞ P m − + iλ ( x ) P m − − iλ ′ ( x ) dx = δ ( λ − λ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( iλ )Γ (cid:0) + iλ − m (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (50)and the completeness relation Z ∞ P m − + iλ ( x ) P m − − iλ ( x ′ ) dλ = δ ( x − x ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( iλ )Γ (cid:0) + iλ − m (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (51)Since operators J µν commute with the Hamiltonian H , it is immediate to show that the functions J µν | Eλm i are also eigenfunctions of the Wheeler-DeWitt equation(46). Taking into account that the h aαβ | Eλm i are acomplete set of functions [28], we can use them to expand J µν | Eλm i . Thus, J µν | Eλm i = ∞ X m ′ = −∞ Z ∞ C m,m ′ ( λ, λ ′ ) | Eλ ′ m ′ i dλ ′ . (52)However, the analytical expression for the functions C m,m ′ ( λ, λ ′ ) is very difficult to obtain due to the inte-grand involves both toroidal and Whittaker functions.We define the new set of operators K + = 12 (cid:0) − ¯ a + ¯ a + ¯ a (cid:1) , (53) K − = 12 (cid:0) − a + a + a (cid:1) , (54) K = 12 (cid:18) − ¯ a a + ¯ a a + ¯ a a + 32 (cid:19) = H , (55)which satisfy the SL (2 , R ) commutation relations[ K + , K − ] = − K , (56)[ K , K ± ] = ±K ± . (57)The Casimir operator ˆ K for this SL (2 , R ) algebra andthe Casimir operator ˆ C for the SO (2 ,
1) angular momen-tum are related by ˆ K = − ˆ C − . These results al-low to find the action of the operators K − and K + onthe non-zero energy eigenstates of the Wheeler-DeWittequation, K ∓ | E λ m i ∝ | E ∓ λ m i . (58)Thus, if we restrict to the solutions of the Wheeler-DeWitt equation (30) with E = 0, the generators K + , K − and K of the SL (2 , R ) algebra do not play a rele-vant role as a symmetry group for the Wheeler-DeWittequation. However, the SU (2 ,
1) symmetry generatorsare relevant for any energy E , and describe the degener-acy in the quantum numbers λ and m . IV. WAVE PACKETS SOLUTIONS FOR THESCALAR AND AXION-DILATON STRINGCOSMOLOGY MODELS
In a similar way to that followed to calculate wavepackets for the Kantowski-Sachs spacetime [29], we can construct wave packets solutions by integrating over thequantum number mψ W DW = Z ∞−∞ e i ( mφ + γ ) J ± i m ( ia / dm. (59)This allows us to obtain ψ wpHP = A , e ± a cosh(2 φ + γ , ) , (60)where A and γ are constants. These functions aresolutions for the Wheeler-Dewitt equation (1) for theHawking-Page scalar model. The solution with the mi-nus sign is the only which is a gaussian wave packetand satisfy the Hartle-Hawking boundary condition (”no-boundary proposal”) [7], i. e. the wave function ofthe universe is regular at a → a → ∞ . Thus, the gaus-sian wave packet represents a wormhole solution for theWheeler-Dewitt equation [12].For the string cosmology model, we find the wave pack-ets Ψ wpSC = A ± e ± a cosh(2 α + γ ± ) , (61)which are solutions for the Wheeler-Dewitt equations( H ± )Ψ = 0. These are completely analogous to thosefor the Hawking-Page scalar model (60). Therefore, thesolution with the minus sign corresponds to a whorm-hole type-solutions for the axion-dilaton string cosmologymodel.Also, we can show that the functionsΨ SC = e ± ( z + y ) √ x (cid:16) C I ( x /
2) + C K ( x / (cid:17) , (62)are solutions for the Wheeler-DeWitt equation H Ψ = 0,being I ( x /
2) and K ( x /
2) the modified Bessel func-tions and C and C constants. In these solutions if weinterchange the x - and y -coordinates, the resulting func-tions are also solutions for the Wheeler-DeWitt equa-tion. However, only the solution with the minus sign and C = 0 is regular at the origin (this is because the onlymodified Bessel function which leads to a regular wavefunction as a → I ( x / = Ce − ( z + y ) √ xI ( x /
2) (63)and Ψ = Ce − ( z + x ) √ yI ( y /
2) (64)are wormhole type-solutions for the axion-dilaton stringcosmology model. To our knowledge the solutions abovein this section do not have been reported in the literature.From the point of view of the group theory, it is moreimportant the U (1) generator than the dynamical equa-tion (in this case, the Wheeler-Dewitt equation). Forthe Hawking-Page and the axion-dilaton string cosmol-ogy models, we find the following solutions ψ hpU (1) = e ± a , Ψ stU (1) = e ± a , (65)which satisfy the Wheeler-DeWitt equations( H ∓ ψ = 0 and ( H ∓ )Ψ = 0, respectively. Infact, these solutions are annihilated by either of the twoforms for the U (1) generators J and J , respectively.These are manifestly Lorentz invariant under rotationsaround the z -axis and under the SO (2 ,
1) group, respec-tively. However, from the quantum cosmology point ofview, these solutions do not represent any interestingscenario because they do not involve the scalar field ( φ )or the axion-dilaton ( χ − φ ) fields. V. CONCLUDING REMARKS
In this paper we have found the symmetries related tothe Wheeler-DeWitt Hamiltonian for the Hawking andPage wormhole and the axion-dilaton quantum cosmol-ogy. We have shown that the Wheeler-DeWitt Hamil-tonian for the wormhole model has the U (1 ,
1) non-compact symmetry which describe the degeneracy of thestates with or without energy-matter renormalization.Also we have found that the Wheeler-DeWitt Hamilto-nian accept an additional SL (2 , R ) dynamical symmetrywhen energy-matter renormalization is considered. Inthis case, we showed that the SL (2 , R ) dynamical sym-metry generators transform the states from an energyHilbert eigensubspace to another.The factor ordering is frequently chosen by conve-nience [30–33]. For the axion-dilaton string cosmologywe have set the factor ordering p = 2, which is necessaryto preserve reparametrization invariance of the Hamil-tonian. Indeed, this choice was crucial in order to findthe closed Lie algebras representing the hidden symme-tries of the axion-dilaton string cosmology. A similarsetting has been taken by Pioline, et. al. [34] in or-der to fix the conformal symmetry SO (2 ,
1) for the one-dimensional Wheeler-DeWitt equation. For the axion-dilaton string cosmology Hamiltonian, the non-compact hidden symmetries U (2 ,
1) and SL (2 , R ) are permissible.The U (2 ,
1) symmetry is valid whenever energy-matterrenormalization is or not present. Also, in this case, the SL (2 , R ) hidden symmetry transforms the states from anenergy Hilbert eigensubspace to another.The zero-energy eigenfunctions (2) and (29) are par-ticular cases of the huge degeneracy on the states ψ Em or Ψ Eλm when matter-energy renormalization is consid-ered. The huge degeneracy of the wave functions on thesecondary quantum numbers m or λm for the systemsstudied in this paper are fully described by the non-compact symmetries U (1 ,
1) or U (2 , SL (2 , R ) symmetries are suitable torelate the states with different principal quantum numberbut maintaining the secondary quantum numbers fixed.Other symmetries have been found for the Wheeler-DeWitt equation in different physical settings. For ex-ample, the SO (2 ,
1) conformal group has been foundfor the one-dimensional radial Wheeler-DeWitt equationwith cosmological constant [34]. We emphasize that thesymmetries for the Wheeler-DeWitt equation found inthis work are for the complete Hamiltonian.Wormhole type-solution for the Hawking-Page model,equation (60), and for the axion-dilaton string cosmology,equations (61)-(64), were found. To our knowledge thesesolutions do not have been reported in previous works.Finally, our procedure can be applied to find the sym-metries of the Wheeler-DeWitt equation for other sys-tems like multidimensional quantum wormholes [35, 36],or the Kantowski-Sachs quantum cosmological model[37, 38], which is work in progress.
VI. ACKNOWLEDGMENTS
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