New Hamiltonian analysis of Regge Teitelboim minisuperspace cosmology
aa r X i v : . [ g r- q c ] J a n New Hamiltonian analysis of Regge Teitelboim minisuperspacecosmology
Rabin Banerjee a , b , Pradip Mukherjee c , d , Biswajit Paul a , ea S. N. Bose National Centre for Basic Sciences, JD Block, Sector III, Salt Lake City, Kolkata -700098, India c Department of Physics, Barasat Government College,Barasat, West Bengal b [email protected] d [email protected] e bisu [email protected] Abstract
A new Hamiltonian formulation of the minisuperspace cosmology following from the geodeticbrane gravity model introduced by Regge and Teitelboim is presented. The model is consideredin the framework of higher derivative theories which facilitates Hamiltonian formulation. Theanalysis is done using the equivalent first order approach. The gauge generator containing the exactnumber of gauge parameters is constructed. Equivalence between the gauge and reparametrizationsymmetries has been demonstrated. Complete gauge fixed computations have been provided andformal quantization is done indicating the Wheeler de Witt equation. Compatibility with existingresults is shown.
Introduction
Higher derivative(HD) theories were once introduced as a possible mechanism of renormalization. Byhigher derivative theory we mean those theories with Lagrangian depending on higher order timederivative of the fields than the first. Recently, interest in this field is rekindled due to the adventof higher order theories of gravitation. An interesting occurrence of higher derivative terms in theaction appears in General Relativity. There, usually, such terms are isolated as surface terms anddropped. However in case of gravity the surface term is always not ignorable e.g. the requirementof the Gibbons - Hawking term in the action. This is more so in the brane world scenario wherethe universe is viewed as a hypersurface immersed in a bulk. A classic model is due to Regge andTeitelboim (RT) [1] where gravitation is described as the world volume swept out by the motion of athree - dimensional brane in a higher dimensional Minkowski spacetime. Hamiltonian analysis of themodel and its quantization was further explored in [2–4]. Unlike the Einstein gravity, in the RT modelthe independent fields are the embedding functions rather than the metric. In the RT model secondderivatives of the fields appear in the action and like general relativity these higher derivative termsmay be clubbed in a surface term. In the usual formulation this surface term is dropped [3] therebyreducing the original model to a first order theory. However this makes the Hamiltonian formulationof the model problematic [3]. These problems are bypassed by introducing an auxiliary field [3]. Onthe other hand recently it has been pointed out that no such auxiliary field is needed if one includesthe surface term in the RT model containing higher derivative terms [4]. Obviously, therefore, theHamiltonian formulation of this model is far from closed. The present paper addresses this and relatedissues.Higher derivative theories were studied and used in different contexts over a long period of time[4–11, 14–25]. Though the classical Hamiltonian formulation of higher derivative theories was workedout by Ostrogradsky long ago [26] and has been refined over the years, specifically in the context ofgauge theories certain aspects of the Hamiltonian formulation were not adequately emphasised. Onesuch issue is the mismatch between the number of primary first class constraints and the number ofindependent gauge degrees of freedom in a higher derivative relativistic particle model [9]. Recentlyit has been demonstrated [11] that under an equivalent first order formalism [10] which is a variant ofthe Ostrogradsky approach, the well known algorithmic method of construction of the gauge generatorfor first order systems [27, 28] can be invoked to settle the issue. The Hamiltonian method developedin [11] of abstracting the independent gauge degrees of freedom of higher derivative systems has beenapplied to a number of particle and field theoretic models [11–13] successfully. Note in this contextthat the anasysis of the RT model in the ambit of higher derivative theory [4] was done from theOstrogradsky approach and this work is based on the minisuperspace model following from the RTtheory. The minisuperspace model carries the reparametrization invariance of the original RT gravitywhich appears as gauge invariance in the Hamiltonian analysis. It will naturally be interesting toapply the equivalent first order formalism of [11] to the RT model with the surface term. This will bethe subject of the present paper. Like [4] the analysis will be based on the minisuperspace model.Before finishing the introductory comments it will be appropriate to say a few words about theequivalent first order formalism. This method of treating higher derivative systems can be distin-guished easily from the usual Ostrogradsky approach. In both the approaches successive time deriva-tives of the coordinates are considered as phase space variables right upto one order less than thehighest derivative appearing in the Lagrangian. Corresponding momenta are introduced to completethe phase space. The relations between the ‘coordinates’ of the enlarged phase space is reflected in1he Ostrogradsky method in the choice of momenta which have to be defined in a particular way toaccount for the higher derivative nature. In contrast, in the equivalent first order formalism suchrelations are accommodated as Lagrangian constraints so that momenta are defined in the usual wayas is done for the first order theories. This introduces new restrictions on the variations in phasespace which is not apparent in the Ostrogradsky method. This difference was instrumental in theconstruction of the Hamiltonian gauge generator [11] that could explain the apparent mismatch in thenumber of independent gauge degrees of freedom with the number of independent primary first classconstraints reported in [9]. Note that in [9] the Ostogradsky approach of Hamiltonian formulation wasadopted. The equivalent first order formalism also provides a straightforward Hamiltonian procedurea la Dirac [29] to treat the singular systems endowed with gauge symmetry. The analysis of the RTmodel with the higher derivative terms from the point of view of the equivalent first order formalismis thus interesting in its own right.The structure of the paper is as follows. In section 2 a review of the cosmological model basedon RT gravity is provided. This will also help us in fixing notations. In section 3 Hamiltonianformulation of the RT cosmology is discussed. This is a new Hamiltonian formulation of the modelwhich like [4] retains the higher derivative term but, contrary to [3], is based on the equivalent firstorder formalism of treating higher derivative system rather than the usual Ostrogradsky approach.Analysis of independent gauge symmetries is given which is demonstrated to be consistent with theLagrangian (reparametrization) invariance of the model. An exact mapping between the gauge andreparametrization parameter has been worked out. Gauge fixing has been done and an appropriatesymplectic algebra in the form of the Dirac brackets between the phase space variables has been given.Using the strongly implemented(second class) constraints the phase space is reduced and the numberof independent phase space variables is found to be two. Finally formal quantization is indicated inthe usual way [30]. The Wheeler DeWitt(WDW) equation is constructed in the fully reduced phasespace. Its compatibility with the results existing in the literature [2] is demonstrated. Our conclusionsare given in section 4.
The RT model considers a d-dimensional brane Σ which evolves in a N dimensional bulk spacetime withfixed Minkowski metric η µν . The world volume swept out by the brane is a d + 1 dimensional manifold m defined by the embedding x µ = X µ ( ξ a ) where x µ are the local coordinates of the backgroundspacetime and ξ a are local coordinates for m . The theory is given by the action functional S [ X ] = Z m d d +1 ξ √− g ( β R − Λ) , (1)where β has the dimension [ L ] − d and g is the determinant of the induced metric g ab . Λ denotescosmological constant and R is the Ricci scalar. As has been already stated above, we will be confinedto the minisuperspace cosmological model following from the RT model.The standard procedure in cosmology is to assume that on the large scale the universe is homo-geneous and isotropic. These special symmetries enable the 4 dimensional world volume representingthe evolving universe to be embedded in a 5-dimensional Minkowski space time ds = − dt + da + a d Ω , (2)2here d Ω is the metric for unit 3 sphere. To ensure the FRW case we take the following parametricrepresentation for the brane x µ = X µ ( ξ a ) = ( t ( τ ) , a ( τ ) , χ, θ, φ ) , (3) a ( τ ) is known as the scale factor.After ADM decomposition with space like unit normals ( N = p ˙ t − ˙ a is the lapse function) n µ = 1 N ( − ˙ a, ˙ t, , , , (4)the induced metric on the world volume is given by, ds = − N dτ + a d Ω . (5)Now, one can compute the Ricci scalar which is given by R = 6 ˙ ta N ( a ¨ a ˙ t − a ˙ a ¨ t + N ˙ t ) . (6)With these functions we can easily construct the Lagrangian density as L = √− g (cid:18) β R − Λ (cid:19) . (7)The Lagrangian in terms of arbitrary parameter τ can be written as [4] L ( a, ˙ a, ¨ a, ˙ t, ¨ t ) = a ˙ tN (cid:16) a ¨ a ˙ t − a ˙ a ¨ t + N ˙ t (cid:17) − N a H . (8)Varying the action with respect to the field a ( τ ) we get the corresponding Euler Lagrange equation as ddτ (cid:18) ˙ a ˙ t (cid:19) = − N a ˙ t ( ˙ t − N a H )(3 ˙ t − N a H ) . (9)Note that the Lagrangian (8) contains higher derivative terms of the field a . However we can write itas [4] L = − a ˙ a N + aN (cid:16) − a H (cid:17) + ddτ a ˙ aN ! . (10)If we neglect the boundary term the resulting Lagrangian becomes usual first order one. As is wellknown the equation of motion is still given by (9). However the Hamiltonian analysis is facilitatedif we retain the higher derivative term. Thus our Hamiltonian analysis will proceed from (8). Notethat the higher order model was also considered in [4] where the Hamiltonian analysis was performedfollowing the Ostrogradsky approach. We on the contrary follow the equivalent first order approachof [11]. here H = Λ3 β , a constant quantity Hamiltonian analysis
This section contains the main results of the present paper. As stated above our aim is to developa new Hamiltonian analysis following from the Lagrangian (8) which is a second order theory. AHamiltonian analysis of the same model has been discussed in [4] from the Ostrogradsky approach.We on the other hand adopt the equivalent first order formalism which has been demonstrated tobe useful, specifically in treating the gauge invariances from the Hamiltonian point of view [11–13].The point of departure is to convert (8) to a first order theory by defining the first derivative of a and t as additional fields and including the following constraints into the Lagrangian with the helpof undetermined multipliers. These multipliers are then treated as new fields and the phase space isconstructed by the entire set of fields along with their conjugate momenta defined in the usual wayas is done for first order theories. Automatically primary constraints arise. The constraint analysisis then presented in detail. In addition to first class constraints the model also has second classconstraints. The second class constraints are then strongly implemented by substituting the Poissonbrackets by the corresponding Dirac brackets. Effectively the theory becomes a first class system withthe symplectic algebra given by these Dirac brackets of which a complete list has been given.The results derived so far are then used in two ways. First an analysis of the gauge invariancesof the model has been done and its connection with the reparametrization invariance of the actionhas been discussed. Secondly, the gauge redundancy of the model has been eliminated by choosing anappropriate gauge. The final Dirac brackets have been used to reduce the phase space and indicate aformal quantization of the model.In the equivalent first order formalism, we define the new fields as,˙ a = A ˙ t = T, (11)which also introduce new constraints in the system given by A − ˙ a ≈ T − ˙ t ≈ , (12)Now the HD Lagrangian (8) is transformed to the first order Lagrangian where the constraints (12)are enforced through the Lagrange multipliers λ a , and λ t as L ′ = aT ( T − A ) (cid:16) aT ˙ A − aA ˙ T + (cid:16) T − A (cid:17) T (cid:17) − (cid:16) T − A (cid:17) a H + λ a ( A − ˙ a ) + λ t (cid:0) T − ˙ t (cid:1) . (13)The Euler Lagrange equation of motion, obtained from the first order Lagrangian (13), by varyingw.r.t. a, A, t, T, λ a and λ t , are respectively given by2 a ( ˙ AT − AT ˙ T )( T − A ) + T ( T − A ) − a H ( T − A ) + ˙ λ a = 0 (14)3 a A ( ˙ AT − AT ˙ T )( T − A ) − ddτ a T ( T − A ) ! − a T ˙ T ( T − A ) + aAT ( T − A ) + a AH ( T − A ) + λ a = 0 (15)4 λ t = 0 (16)3 a T ( ˙ AT − AT ˙ T )( T − A ) + 2 a ˙ AT ( T − A ) − ddτ a AT ( T − A ) ! − a A ˙ T ( T − A ) + 2 aT ( T − A ) − aT ( T − A ) + λ t = 0 (17) A − ˙ a = 0 (18) T − ˙ t = 0 . (19)Eliminating the multipliers λ a , and λ t from the above equations we get back equation (9)In the Hamiltonian formulation adopted in the present paper the Lagrange multipliers are con-sidered formally as independent fields and the momenta corresponding to them are introduced inthe usual way. Here we denote the phase space coordinates by q µ = a, t, A, T, λ a , λ t and their cor-responding momenta as Π q µ = Π a , Π t , Π A , Π T , Π λ a , Π λ t with µ = 0 , , , , ,
5. We adopt the usualdefinition Π q µ = ∂L ′ ∂ ˙ q µ , (20)since the Lagrangian (13) is in the first order form. This is the point of departure of our Hamiltonianformulation from the Ostrogradsky formulation of [4].From the definition of the phase space variables, we get the following primary constraintsΦ = Π t + λ t ≈ = Π a + λ a ≈ = Π T + a T A ( T − A ) ≈ = Π A − a T ( T − A ) ≈ = Π λ t ≈ = Π λ a ≈ . (21)The nonzero Poisson brackets between the primary constraints are computed as { Φ , Φ } = 1 { Φ , Φ } = − aT A ( T − A ) { Φ , Φ } = 2 aT ( T − A ) { Φ , Φ } = 1 . (22)5aking the constraint combination Φ ′ = T Φ + A Φ , ≈ , (23)we find that Φ ′ commutes with all the constraints. The nonzero poisson brackets between the newlydefined primary set of constraints Φ , Φ , Φ ′ , Φ , Φ , Φ , become { Φ , Φ } = 1 { Φ , Φ } = 2 aT ( T − A ) { Φ , Φ } = 1 . (24)We can write down the canonical Hamiltonian as H can = Π q µ ˙ q µ − L ′ = − aT ( T − A ) + (cid:16) T − A (cid:17) a H − λ a A − λ t T. (25)The total Hamiltonian is given by H T = H can + Λ Φ + Λ Φ + Λ Φ ′ + Λ Φ + Λ Φ + Λ Φ . (26)Here Λ , Λ , Λ , Λ , Λ , Λ are undetermined Lagrange multipliers. Preserving the primary constraintsΦ , Φ , Φ in time ( { Φ i , H T } ≈
0) the following Lagrange multipliers get fixedΛ = 0Λ = T Λ = A. Whereas, conservation of Φ gives the following condition between Λ and Λ T ( T − A ) − a H (cid:16) T − A (cid:17) + Λ + Λ aT ( T − A ) = 0 . (27)Time preservation of the constraint Φ ′ gives rise to the following secondary constraintΨ = aT ( T − A ) − a H (cid:16) T − A (cid:17) + λ t T + λ a A ≈ . (28)Likewise, Φ yields the following secondary constraintΨ = aAT ( T − A ) − a H A ( T − A ) − λ a ≈ . (29)6onzero brackets for Ψ and Ψ with the other constraints are given below, { Φ , Ψ } = − T − a H (cid:0) T − A (cid:1) ( T − A ) { Φ , Ψ } = − aAT ( T − A ) − a H A ( T − A ) − λ a { Φ , Ψ } = − aT (2 A − T )( T − A ) + a H T ( T − A ) − λ t { Φ , Ψ } = − T { Φ , Ψ } = − A { Φ , Ψ } = − AT ( T − A ) + 3 a H A ( T − A ) { Φ , Ψ } = − aT (cid:0) T + 2 A (cid:1) ( T − A ) + a H T ( T − A ) { Φ , Ψ } = 1 . (30)Time preservation of Ψ trivially gives 0 = 0. A similar analysis involving Ψ yields, on exploiting(27), Λ = − (cid:0) T − a H (cid:0) T − A (cid:1)(cid:1) (cid:0) T − A (cid:1) a (3 T − a H ( T − A ))Λ = − (cid:0) T − a H (cid:0) T − A (cid:1)(cid:1) ( T − a H (cid:0) T − A (cid:1) )( T − A ) (3 T − a H ( T − A )) . (31)The iterative procedure is thus closed and no more secondary constraints or other relations are gen-erated.The above analysis reveals that of all the Lagrange multipliers Λ i , only Λ remains undeterminedin (26) signifying one independent gauge degree of freedom. This fact will be reflected in the gaugegenerator that has been constructed in section 3.1. It is interesting to note that this consistency isnot always obvious in the Ostrogradsky formulation, as we have already mentioned in connection withthe massive relativistic particle model [9].We have now altogether eight primary and secondary constraints. Computation of the Poissonbracket between these constraints shows that only Φ ′ is the first class constraint, whereas other sevenconstraints are apparently second class. The odd number of apparently second class constraints signalsthe existence of additional first class constraints. Indeed, the new constraint combinationΨ ′ = Ψ − Λ Φ − Λ Φ − Λ Φ − Λ Φ − Λ Φ , (32)leads to a secondary first class constraint. So now we have two first class constraints Φ ′ , Ψ ′ and sixsecond class constraints Φ , Φ , Φ , Φ , Φ and Ψ . The total number of phase space variables is7welve. The number of independent phase space variables is therefore 12 − (2 × ′ is the sole primaryfirst class constraint. The number of primary first class constraint matches with the residual numberof undetermined multiplier in the total Hamiltonian. This fact will be important in the constructionof the gauge generator.To study gauge symmetry of the system we need to get rid of the second class constraints. This isdone by the introduction of the Dirac brackets which enable us to set these constraints strongly zero.For simplicity of the calculation we remove them pair by pair. The Dirac bracket between the basicfields after removing Φ , Φ , Φ , Φ remains same as their corresponding Poisson brackets. SolvingΦ , Φ , Φ , Φ the new constraint structure becomes F = Φ ′ = T Φ + A Φ ≈ F = Ψ ′ = Ψ − Λ Φ ≈ S = Φ ≈ S = Ψ = aAT ( T − A ) − a AH ( T − A ) + Π a ≈ . (33)For simplicity we use new notations { F , F } and { S , S } where, the first pair denotes the set of firstclass constraint and second pair denotes the remaining set of second class constraints. Some detailsof this reduction are given below.To calculate Dirac brackets of the theory we first find out the Poisson brackets between the secondclass constraints which are written as∆ ij = { S i , S j } = − aT (cid:0) T − a H (cid:0) T − A (cid:1)(cid:1) ( T − A ) ǫ ij , (34)with ǫ = 1 and i, j = 1 ,
2. Dirac brackets are defined by { f, g } D = { f, g } − { f, S i } ∆ − ij { S j , g } . (35)We calculate the Dirac brackets between the basic fields which are given below(only the nonzerobrackets are listed) { a, A } D = − (cid:0) T − A (cid:1) aT (3 T − a H ( T − A )) { a, Π a } D = T + 2 A − a H (cid:0) T − A (cid:1) (3 T − a H ( T − A )) { a, Π A } D = − aA T − a H ( T − A ) { a, Π T } D = a (cid:0) T + 2 A (cid:1) T (3 T − a H ( T − A ))8 t, Π t } D = 1 { A, Π a } D = − A (cid:0) T − A (cid:1) (cid:0) T − a H (cid:0) T − A (cid:1)(cid:1) aT (3 T − a H ( T − A )) { A, Π A } D = 2 (cid:0) T − A (cid:1) T − a H ( T − A ) { A, Π T } D = A (cid:0) T + 2 A − a H (cid:0) T − A (cid:1)(cid:1) T (3 T − a H ( T − A )) { T, Π T } D = 1 { Π a , Π A } D = − a (cid:0) T + A T + a H ( T − A )(9 A − T ) (cid:1) ( T − A ) (3 T − a H ( T − A )) { Π a , Π T } D = aA (cid:0) T + 2 T A + a H ( T − A )( T + 6 A ) (cid:1) T ( T − A ) (3 T − a H ( T − A )) { Π A , Π T } D = − a T (cid:0) T + 2 A − a H ( T − A ) (cid:1) ( T − A ) (3 T − a H ( T − A )) . (36)The introduction of the above Dirac brackets allows the second class pair { S , S } to be stronglyimplemented. Note that the secondary first class constraint then becomes equal to the canonicalHamiltonian: F = Ψ = − H c = − T Π t − T A Π a ≈ . (37)Vanishing of the canonical Hamiltonian is a consequence of the reparametrisation invariance of thetheory. The equivalent first order formalism offers a structured algorithm for the abstraction of the gaugegenerator of the higher derivative system [11] which is based on the method presented in [27, 28] forthe first order systems. According to the Dirac conjecture [29] the gauge generator is G = X a ǫ a Φ a . (38)Here { Φ a } is the whole set of constraints and ǫ a are the gauge parameters. However not all thegauge parameters ǫ a are independent. The number of independent gauge parameters is equal to thenumber of independent primary first class constraints [27,28] . Demanding the commutativity of gaugevariation and time translation we get the following master equations δ Λ a = dǫ a dt − ǫ a ( V aa + Λ b C b aa ) (39)0 = dǫ a dt − ǫ a ( V aa + Λ b C b aa ) . (40)Here the indices a , b ... refer to the primary first class constraints while the indices a , b ... correspondto the secondary first class constraints. Λ a are the Lagrange multipliers multiplying the primary first9lass constraints in the expression of the total Hamiltonian and δ denotes gauge variation. Thecoefficients V aa and C b aa etc. are the structure functions of the involutive algebra, defined as { H can , Φ a } D = V ab Φ b { Φ a , Φ b } D = C abc Φ c . (41)Equations (39) give no new conditions as they can be shown to follow from (40) [27]. The latterequations actually impose restrictions on the gauge parameters. Using these the independent gaugeparameters can be identified. A new feature appears in case of the HD theories where in the equivalentfirst order formalism we define the time derivatives of the coordinates right upto one order less thanthe highest order appearing in the Lagrangian as independent fields. Thus the gauge variations heremust be consistent with this definition and we require conditions of the form δq n,α − ddt δq n,α − = 0 , ( α > , (42)where q n,α denotes the α -th order time derivative of q . The conditions (42) sometimes impose someextra condition on the gauge parameters and sometimes not [11–13]. Expressing the gauge parametersin terms of the independent elements of the set in (38) the most general form of the gauge generatoris constructed. Now we can write gauge variations of the basic fields as δ ǫ a q n,α = { q n,α , G } D . (43)where on the right hand side only the independent gauge parameters appear.After the short review of the basic methodology we come back to the present model. The gaugegenerator is defined as the linear combination of all the first class constraints which is written as, G = ǫ F + ǫ F . (44)Here ǫ and ǫ are the gauge parameters. From equations (41) we find that C = − − C and V = 1 are the only nonzero structure functions. Now using equation (40)the following relationbetween the gauge parameters is obtained ǫ = − Λ ǫ − ˙ ǫ . (45)So here ǫ may be chosen as the independent gauge parameter.At this stage we observe that there is one independent parameter in the gauge generator (44). Theconditions (42) following from the higher derivative nature is yet to be implemented. As has beenmentioned earlier this may or may not impose additional restriction on the gauge parameters. Thegauge transformations of the fields are given by δa = { a, G } D = − ǫ A (46) δt = − ǫ T (47) δA = ǫ A − ǫ (cid:0) T − a H (cid:0) T − A (cid:1)(cid:1) (cid:0) T − A (cid:1) a (3 T − a H ( T − A )) (48) δT = ǫ T (49) from now on we have to use only Dirac brackets since we removed all second class constraints. Poissson brackets aredenoted by { , } , whereas, { , } D refers to Dirac brackets ddτ δa = δA (50) ddτ δt = δT. (51)So the constraints (42) hold identically for the present model and impose no new condition on thegauge parameters. We find therefore that there is only one independent gauge transformation whichessentially is in conformation with the fact that there is only one independent primary first classconstraint.The gauge variations obtained from the Hamiltonian analysis can be exactly mapped to thereparametrization invariance of the model. Consider arbitrary infinitesimal change in the parame-ter τ → τ ′ = τ + σ . The action is invariant under this reparametrization. Now the fields transformas δa = − σaδt = − σt. (52)These are identical with the gauge variations (46) and (47) of a and t if σ is identified with ǫ . Theequivalence of gauge invariances with the reparametrization invariance of the model is thus established. After the reduction of phase space by the Dirac bracket procedure we are left with only the two firstclass constraints F and F . These first class constraints reflect the redundancy of the theory whichare connected by gauge transformations. In the above analysis our focus was on the abstraction ofthe gauge degrees of freedom. We now elucidate a formal quantisation prescription. A gauge fixing isdone and the appropriate WDW equation is written.The choice of gauge is arbitrary subject to the conditions that they must reduce the first classconstraints to second class. Also the constraint algebra should be nonsingular. As there are two firstclass constraints we need two gauge conditions. We take one of these to be the cosmic gauge ϕ = p T − A − ≈ . (53)The name derives from the fact that the resultant metric becomes the usual FLRW metric. As thesecond gauge condition we take ϕ = T − αa ≈ . (54)where the constant α is chosen so that α = H . The following calculations will show that these areappropriate gauge conditions.As usual the gauge conditions are treated as additional constraints which make the first classconstraints of the theory second class. For convenience, renaming the two first class constraints wewrite the complete set of constraints as Ω = F (55)11 = F (56)Ω = ϕ (57)Ω = ϕ . (58)Modifying the algebra by the Dirac brackets corresponding to this second class system we will beable to put all the second class constraints (Ω i , i = 1 , , ,
4) to be strongly equal to zero. These willcorrespond to operator relations in the corresponding quantum theory.Using the algebra (36) we can straightforwardly compute the algebra of the constraints Ω i . Theresults are given in the following tableTable 1: Constraint bracketsΩ Ω Ω Ω Ω − T Ω A ( α − H ) a (3 α − H ) − αA Ω − A ( α − H ) a (3 α − H ) Aαa (3 α − H ) Ω T αA − Aαa (3 α − H ) ij = { Ω i , Ω j } (59)Using the definition (35) we can calculate the final Dirac brackets. Nonzero Dirac brackets betweenthe phase space variables are { t, a } ∗ = 14 αa ( α − H ) { t, A } ∗ = α a A ( α − H ) { t, T } ∗ = 14 a ( α − H ) { t, Π a } ∗ = − a α + 34 αaA { t, Π A } ∗ = α ( α − H ) { t, Π t } ∗ = 1 { t, Π T } ∗ = − a α + 34 aA ( α − H ) . (60)With the introduction of the final Dirac brackets all the constraints (including the gauge conditions)become second class and strongly zero. We thus have the following conditions on the phase spacevariables Π T + a T A ( T − A ) = 012 A − a T ( T − A ) = 0 − Π t − TA Π a = 0 aAT ( T − A ) − a AH ( T − A ) + Π a = 0 p T − A = 1 T − αa = 0 (61)where use has been made of equations (33, 53, 54). From the final Dirac brackets (60) it is clearthat only the pair ( t, Π t ) is canonical. We thus identify this pair as the two independent phase spacedegrees of freedom found earlier by a standard count using the constraints of the system (see below32). To develop a quantum theory it is necessary to write down the whole theory with respect to thecanonical variables in the reduced phase. All the variables can be expressed in favour of ( t, Π t ) byappropriately solving the constraints which are now strongly implemented. The result is, T = αaA = p α a − A = α a Π T = − αa p α a − a = − a ( α − H ) p α a − . (62)where a is expressed as a = (cid:18) Π t α ( α − H ) (cid:19) . (63)Thus we find that all the phase space variables except t are given as function of Π t .The passage from the classical to quantum theory proceeds in the usual way. The phase spacevariables are lifted to operators in some Hilbert space and the conditions (62, 63) are now treated asoperator relations. The Dirac brackets are promoted to commutators according to the prescription. { B, C } ∗ → i ¯ h [ B, C ] . (64)The fundamental canonical algebra is thus (with ¯ h = 1 )[ t, t ] = [Π t , Π t ] = 0 , [ t, Π t ] = i. (65)We next proceed to formulate the WDW equation for the universe governed by the Lagrangian(10). Before that we write down the first class constraint F which is the canonical Hamiltonian as F = − H can = − − A Π t + T Π a A Π a = 0 (66)Considering the sate vector | Ψ i in the appropriate Hilbert space, the WDW equation may be writtenas, H can | Ψ i = 0 . (67)13sing the Schrodinger representation compatible with (65), we obtain,Π t = − i ∂∂t (68)Exploiting (66-68) and the expression for A given in (62) we obtain, after some algebra, the followingWDW equation, − ∂ ∂t | Ψ i = α a ( α − H ) | Ψ i . (69)Making a change of variables ξ = α H , the WDW equation may be reexpressed as, − ∂ ∂t | Ψ i = ξ ( ξ − H a | Ψ i . (70)The above equation exactly reproduces one piece of the bifurcated WDW equation found in the firstitem of [2] .Furthermore, introducing the conserved ‘energy’ ω by, ξ ( ξ − H a = ω (71)we may reexpress (70) by the standard equation, − ∂ ∂t | Ψ i = ω | Ψ i . (72)The expression for the conserved energy ω in (71) matches with the form given in [2]. It is now possibleto proceed with the quantisation as elaborated in [2].Before concluding this section it is worthwhile to mention the efficacy of the gauge choice (54).While the first gauge condition (53) is the standard cosmic gauge, the second one (54) has not beenconsidered earlier. We have shown that this simple choice (54) is a valid choice that yields the fullyreduced space of the model. Also, at the quantum level, the WDW equation subjected to this gaugefixing reproduces the expression obtained earlier in [2]. The minisuperspace cosmology following from the geodetic brane gravity model introduced by Reggeand Teitelboim [1] has been considered from the point of view of higher derivative theory followingCordero, Molgado and Rojas [4]. We have presented a new Hamiltonian formulation of the modelbased on the equivalent first order formalism [11–13]. This is different from the analysis of [4] wherethe usual Ostrogradsky approach is adopted. Not only that our equivalent first order formalism differsfrom the first order Hamiltonian formalism for the model obtained by dropping a boundary term fromthe action [3]. The latter is plagued with problems that can be eradicated only by the introductionof an auxiliary field. Our Hamiltonian formalism is free from such difficulties. Apart from this the Note that the other part of the bifurcated WDW involving the ‘ a ′ variable is nonexistent in the present analysis. Thisis because here we have only one (configuration space) independent degree of freedom (i.e. t ) instead of two variables ( t and a ) as occurs in [2]. This mismatch happens because, contrary to [2], the present analysis is done in a fully reducedspace where all constraints are eliminated all thefirst class constraints of the theory. The structure functions are worked out from the algebra of thefirst class constraints with respect to the Dirac brackets referred above. These structure functions areplugged in the master equation connecting the gauge parameters provided by the chosen algorithm.One relation is found between the two gauge parameters appearing in the gauge generator. Theadditional constraints following from the higher derivative nature were shown to hold identically.Thus only one gauge parameter was found to be independent. There was only one primary first classconstraint. So in this case the number of independent gauge parameters was found to be equal to thenumber of primary first class constraints. Exact mapping of the Hamiltonian gauge invariances withthe Lagrangian (reparametrization) invariances of the model has also been demonstrated.The canonical quantization of the model is discussed next. For this the redundancy of the phasespace was eliminated by choosing appropriate gauge fixing conditions. The familiar cosmic gaugewas chosen as one of the gauge conditions. But the second gauge was a new one different from thenonstandard gauge chosen in [4]. As subsequent analysis revealed this new gauge condition is a goodchoice. A detailed account of the complete gauge fixed calculations for the model has been presented.Formal quantization is obtained by promoting the phase space variables to operators in an assumedHilbert space. The phase space is reduced so that only two phase space variables remain independent;the number being equal to the number of degrees of freedom in phase space. The fundamentalcommutator is then obtained from the Dirac bracket between the varibles according to well knownprocedure [30]. The WDW equation which defines the quantum states of the universe correspondingto the Lagrangian is constructed. This equation and the energy expression are shown to match with15he existing literature [2]. Finally, we would like to mention a recent paper [32] where somewhatconclusions were obtained in the model considered here (1) augmented by an extrinsic curvature term. Acknowledgement
One of the authors (BP) gratefully acknowledges Claus Kiefer for discussions. He also acknowledgesthe Council of Scientific and Industrial Research (CSIR), Government of India, for financial assistance.
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