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 Research articleQuantum cosmology of a classically constrained nonsingular UniverseAbhik Kumar Sanyal Dept of Physics, Jangipur College, Murshidabad, West Bengal, 742213, India
PMC Physics A 2009, 3:5doi:10.1186/1754-0410-3-5 The electronic version of this article is the complete one and can be found online at: http://www.physmathcentral.com/1754-0410/3/5
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2009 Sanyal AbstractThe quantum cosmological version of a nonsingular Universe presented by Mukhanov and Brandenberger in the early nineties has been developed and the Hamilton Jacobi equation has been found under semiclassical (WKB) approximation. It has been pointed out that, parameterization of classical trajectories with semiclassical time parameter, for such a classically constrained system, is a nontrivial task and requires Lagrangian formulation rather than the Hamiltonian formalism. PACS Codes: 98.80.Qc, 04.50.Kd. 1 IntroductionIt transpires from Hawking-Penrose energy condition that the singularities of the general theory of relativity are unavoidable
classically. In the absence of a complete and satisfactory theory of quantum gravity, it is not clear what would be the nature
or the fate of singularities in the quantum domain. However, it might be possible in principle to get rid of the singularities
classically by imposing some quantum mechanical bounds on certain quantities. At the singularity, some of the curvatures,
eg., energy momentum and even the Riemann tensors diverge. Further, quantum field theory predicts that, the scattering cross
sections become infinity when all radiative corrections are taken into account. Renormalization makes individual terms manageable,
but the entire series diverges. So in order to get rid of the singularities, curvature invariant terms must be constrained
to take some upper limits. If Planck's length lpl is assumed to be the fundamental length, below which no length is measurable, then from dimensional argument, curvature invariants
must be bounded as, Since Weyl tensor vanishes for an isotropic Universe, so in its simplest form, LCH can be realized by constraining only a
couple of lower order curvature invariants, viz., I1 = 4R and
Here, the scalar field ϕ2 has been introduced to obtain nonsingular solutions, while ϕ1 bounds the curvature invariants for these nonsingular solutions. The potentials V1(ϕ1) and V2(ϕ2) should be chosen in a manner such that (i) for ϕi ≪ 1, the leading terms in the action gives back Einstein's theory and (ii) their asymptotic behaviour should be such that LCH may be realized, which requires V1(ϕ1) → constant as ϕ1→ ∞, and V2(ϕ2) → 0 as ϕ2 → ∞. For a at Robertson-Walker metric, they ([2] &[3] ) had obtained a first order differential equation containing fields and the potentials and had drawn a phase diagram in ϕ1-ϕ2 plane, for some particular choice of the potentials. The phase diagram indicates nonsingular solutions in all regions and de-Sitter phase in the asymptotic region. In a nut-shell, the results obtained [4] are the following. Firstly, all the homogeneous and isotropic cosmological solutions are nonsingular. Next, Two dimensional cosmological and Black-Hole solutions are nonsingular. Such Black-Hole solutions in particular, exhibit Schwarzchild solution at large radial co-ordinate and never evaporate completely [5] . Finally, there are evidence that nonsingular solutions may exist for four-dimensional Black-Holes and homogeneous-anisotropic cosmological models. Further, LCH has also been applied to dilaton-cosmology [6,7] and as a result, a class of spacially flat bouncing cosmological solutions have emerged. This construction is definitely appealing in itself, however, it also incorporates some additional features in quantum domain. In the path integral formulation of quantum gravity, one has to analytically continue the Lorentzian space of indefinite metric to the Euclidean space of positive definite metric in order to ensure convergence of the path integral. Nevertheless, since Euclidean action of the gravitational field is not bounded from below, the path integral never really converges. The action presented by Mukhanov and Brandenberger [2] and Brandenberger, Mukhanov and Sornborger [3] is bounded from below, and therefore, the corresponding path integral converges. In addition, since the initial and the final stages of cosmological evolution are de-Sitter, so one gets Lorentzian wormholes instead of Euclidean. Finally, as the authors had claimed that the theory can incorporate matter field and also can be extended to include anisotropic models, so it appears that the model is of greater interest with a wide spectrum of its applicability. In the theory under consideration, two scalar fields, one of which is non-dynamical, have been nonminimally coupled to gravity,
containing higher order curvature invariant term. Thus, along with the lapse function N(t), an additional Lagrange multiplier is required in connection with the non-dynamical Scalar field, leading to classically
constrained gravity theory. Study of Quantum cosmological aspects of such a constrained theory may reveal more important cosmological
implications. eg., in a recent work, such an attempt was made by Gabadadze and Shang [8]
for a different classically constrained theory of gravity, which has been found to admit solutions, absent from general theory
of relativity. However, the new solutions have some wonderful features like,-the spatially flat de-Sitter Universe can be
created from nothing, has boundaries, with vanishing total energy, etc. [9]
. So, it appears worth to study the quantum cosmological aspects of the theory under consideration. Therefore, the constrained
Hamiltonian dynamics of the theory under consideration has been studied along the line of Dirac's algorithm to find a constraint
free primary Hamiltonian. Despite the fact that the theory under consideration contains constraints in addition to the Hamiltonian
constraint of the standard theory of gravity, no attempt has been made to find the true degrees of freedom and to find the
expression for the Hamiltonian in the reduced phase space. Rather, it has been quantized in the same tune of Wheeler-deWitt
equation, The main motivation of this work is to study the connection of the quantum-cosmological equations for the theory under investigation with the Hamilton-Jacobi equations of the classical theory and in the process to point out the difficulty in finding the semiclassical time required to parameterize classical trajectories. It has been observed that it is nontrivial task to extract semiclassical time parameter for such a constrained system. It has been shown that the semiclassical time parameter extracted in the usual manner does not lead to classical constraint equation. Further, the semiclassical time parameter may be found through speculation, but it is complex and requires Wick rotation. It has also been shown that in order to parameterize classical trajectories by an unique real time parameter, it is required to analyze Lagrangian constrained dynamics rather than Hamiltonian. In the following section we have written down the classical field equations corresponding to the action under consideration
for isotropic and homogeneous Robertson-Walker space time. The Hamiltonian and the corresponding Wheeler-deWitt equation have
been found in section 3, after analyzing the constraint of the theory. Semiclassical approximation has been made by expressing
the wave function as, 2 FormalismThe action presented by Mukhanov and Brandenberger [2] , corresponding to a homogeneous and isotropic non-singular Universe, for which Weyl tensor vanishes, is given by,
where, V1(ϕ1) and V2(ϕ2) are the potentials corresponding to the pair of apparently non-dynamical scalar fields ϕ1 and ϕ2 respectively. LCH may be realized from the above action (1), only for some appropriate choice of the potentials. Action (1)
leads to Einstein's theory at small curvature, provided, Vi(ϕi) ~ Now, taking Robertson-Walker line element,
where N(t) is the lapse function, the Ricci scalar is given by,
and the next higher order curvature invariant term is,
So the action takes the following form,
where, the surface term is
while the ϕ2 variation equation is,
The Hamiltonian constraint equation, obtained by varying the action (3) with respect to N is,
Finally, varying the action with respect to the scale factor a, one gets,
In the above field equations
The phase diagram for a particular choice of the potentials was plotted by Mukhanov and Brandenberger [2]
, showing four different classes of trajectories. In the first, the trajectory starts from the de-Sitter phase as ϕ2 → - ∞ and evolves through to de-Sitter as ϕ2 → ∞. In the second, for small initial values of ϕ1 trajectory starts at ϕ2 = -∞ reaches a turning point and returns to ϕ2 = -∞. In the third, the trajectory shows periodic solutions about Minkowski space-time ϕ1 = ϕ2 = 0. Finally, in the fourth class, trajectories starting with small ϕ1 and 3 Analyzing the constraint and the Wheeler-deWitt equationIt is important to note that in the process of developing an action that might produce nonsingular cosmological solutions,
the determinant of the Hessian, 1. Firstly we remember that the existence of an infinite invariance group leads to first class constraints, while singular Lagrangians which do not possess a local gauge invariance leads to second class constraints. 2. The rank of Wij for a system having 2 N phase space variables (that does not contain time explicitly) is N. For a singular Lagrangian it is R <N. under this circumstances there exists a non-degenerate R × R matrix Wαa, such that 3. Now if a function F(q, p) be defined on the primary phase space (this will be defined shortly) Γp < Γ, then the restriction on F to Γp is achieved by replacing pr by gr(q, pα). If F vanishes identically after this replacement, then it is called weakly vanishing and is denoted by F ≈ 0. If in addition the gradient of F also vanishes, then it is called strongly vanishing and is denoted by F ≃ 0. 4. Primary constraints are expressed as ϕr(q, p) ≈ 0; and the constrained Hamiltonian, defined on the constrained phase space Γc, as,
One can write down the primary Hamiltonian defined on the primary phase space Γp, as,
where, λr are the Lagrange multipliers. 5. Next one has to check if the constraints are preserved in time, i.e.,
If the Poisson bracket of a constraint vanishes with all other at least weakly, then it turns out to be first class constraint. In that case, {ϕj, Hc} ≈ 0, and all these first class constraints are exhausted. The constraints ϕi which do not vanish modulo the constraint, leads to secondary constraints χk. 6. One needs to check the consistency condition of the secondary constraints again, i.e., if the constraints are preserved in time. Again for the secondary constraints, if the poisson brackets of some (say l), vanish with all other, then they are first class and for them {χl, Hc} ≈ 0. Otherwise they are second class in nature and the Lagrange multipliers are determined in the process. There are as many undetermined Lagrange multipliers as there are first class primary constraints. If all the constraints are second class then the primary Hamiltonian together with the primary and secondary constraints constitute the field equations, which may be solved in principle, leaving no arbitrariness in the solutions. 7. Finally, one can try to find the true degrees of freedom and express the Hamiltonian in the reduced phase space ΓR. However, it is a very difficult tusk, if not impossible in most of the situations. Even without finding the reduced Hamiltonian it is possible to quantize the theory. There is no unique prescription in this regard. Any quantum theory may be considered to be the correct one if it can go over to the classical counterpart through a suitable correspondence principle. With this background knowledge, let us proceed to construct the Hamiltonian of the theory. We understand that in order to
construct the primary Hamiltonian, one has to introduce all the constraints of the theory in the Hamiltonian through Lagrange
multiplier. Choosing
while
Since,
where, α is the Lagrange multiplier. Since, {C1, Hc} does not vanish even weakly, so C1 is a second class primary constraint. As there are no first class primary constraint, so there will be no undetermined Lagrange multiplier of the theory. Now the constraint must be preserved in time, i.e.,
This is a new second class constraint as long as
modulo the constraint (10). This is not a new constraint, rather it fixes the Lagrange multiplier
One can now easily check that we have obtained the correct Hamiltonian in the usual manner. The only exception appears to
be with
and write down the corresponding Hamiltonian,
It is interesting to note that the classical field equations (4) through (7) reduce to the vacuum Einstein's equations at any stage of cosmic evolution as ϕ1 = ϕ2 = 0. However, the primary Hamiltonian (12) once constructed in view of the constrained system under investigation, is different from (13) and does not ever reduce to the vacuum Einstein's equation. This fact has been manifested in the linear appearance of pa in Hp1. The fact that here pa appears linearly in the Hamiltonian, clearly differentiates all actions (corresponding to minimal and standard non-minimal coupling) with the present one. Our next attempt will be to canonically quantize the theory under consideration. There is no standard prescription to quantize
a classically constrained system and the only justification of a particular method of quantization lies in its success. The
meaning of the last sentence is that one should be able to find a correspondence principle to go over to the classical theory
under semiclassical limit. We shall write down the counterpart of the Wheeler-deWitt equation
Clearly, first order derivative of the wavefunction Ψ with respect to a, (ie.,
which is independent of the operator ordering parameter q(a). Not all quantum states |Ψ > of the Wheeler -deWitt equation (14) are allowed, since they are constrained by the quantum analogue of the classical constraints (10) and (11). However, since our aim is not to find the solution of (14), so we neither, are in search of true degrees of freedom nor incorporate the quantum analogue of the classical constraints. Rather, for the sake of comparison, we write down the Wheeler-deWitt equation corresponding to Einstein's gravity with a minimally coupled scalar field (see Appendix),
where, q takes care of some of the operator ordering ambiguities. We observe that, neither 4 Semiclasical approximationReparametrization invariance of the theory of gravity leads to the Hamiltonian constraint yielding Wheeler-deWitt equation.
Despite the fact that the theory under consideration has additional constraints, we have not made any attempt to find the
true degrees of freedom to construct the Hamiltonian in the reduced phase space ΓR. Rather, the corresponding quantum equation (14) has been constructed in the same tune of the standard Wheeler-deWitt equation
At energy below Planck scale, the wave function can be expressed as,
Let us now collect expression having same powers in M. For Einstein-Hilbert action with minimally coupled fields one gets to the power of M2 an expression that states that the Hamilton-Jacobi function S0 depends on three space only (see Appendix). Here the situation is quite different in the sense that we do not get expressions corresponding to the order M2. To the order M1, we have,
This is the Hamilton-Jacobi equation. It can be identified with the Hamiltonian constraint equation (6) only under an appropriate choice of semiclassical time parameter. Following two sections are devoted to find the semiclassical time parameter. 5 Semiclassical time parameter - standard techniqueSince in view of the Hamiltonian (12) all the velocities are now invertible and so following standard technique as in Einstein's gravity with minimally coupled scalar field (see Appendix, 9.2), the time parameter can be found as,
The problem is that, this choice of time parameter does not lead to classical constraint equation (6). This is because, One can try to find the same by expressing the time parameter as
where, b, c, d, j, l, m, u, v and n are arbitrary functions of a, ϕ1 and ϕ2. It is now possible to find
Substituting all these expressions in equation (6) and equating the coefficients of ▽S0 etc., between the equation thus formed and the Hamilton-Jacobi equation (16), one can finally arrive at the following time parameter,
With this technique of parametrization, classical constraint equation (6) is automatically reproduced, but the problem associated with this time parameter is that, there still exists an arbitrariness in the form of the arbitrary parameter n = n(a, ϕ1, ϕ2). Thus, the time parameter is not unique, and something else should be tried. It is interesting to note that simply by inspection one can choose a time parameter free from such arbitrariness as,
This time parameter reproduces equation (6), in view of Hamilton-Jacobi equation (16). However, this is purely intuitive on one hand and is imaginary on the other. So we must find an involved technique for this purpose. 6 Semiclassical time parameter - an involved techniqueAs already mentioned, equation (4) is an additional gravitational constraint equation of the theory under consideration, since it does not contain second order derivatives. The standard technique should be to differentiate equation (4) and then to compare it with equation (6). In the process, if the emerging equation is again a constraint equation, then it should be entered into the Lagrangian via Lagrange multiplier. The emerging constraint equation in this case is,
Hence, introducing this constraint equation (17) in action (3) via a Lagrange multiplier λ, we obtain,
The canonical momenta are found from the action (18) as,
We don't write down the field equations since we are not going for classical solutions. However, it is important to note that since variation with respect to λ gives back the constraint equation (17) and in view of (19) the classical constraint equation (6) remains unchanged, so, the primary Hamiltonian (12), the Wheeler-deWitt equation (14) and the Hamilton-Jacobi equation (16) are systematically reproduced. Further, identifying canonical momenta with corresponding derivatives of the Hamilton-Jacobi function, equation (12) can be found again from Hamilton-Jacobi equation (16). Now in order to obtain equation (6) from (16), let us use equations (17) and (19), which gives,
Equations (20), (21) and (22) are now combined to yield the correct and unique semiclassical time parameter,
One can now easily obtain 7 Including a source termIn this section we consider an additional source term in the action (1), in the form of a minimally coupled scalar field with Lagrangian density,
where, σ is the scalar field and U(σ) is an arbitrary potential. The action (1) now reads (taking
Corresponding field equations are (under variation of the above action with respect to ϕ1, ϕ2, σ, N and a and setting N = 1),
where,
Next we incorporate this equation (30) into the Lagrangian through the same Lagrange multiplier λ, as before. In view of the Lagrangian so formed, we obtain the same set of canonical momenta (19) along with an additional one, corresponding to the field σ, viz.,
Variation of the Lagrangian with respect to the Lagrange multiplier λ returns equation (30). So, in view of equations (19), (30) and (31), one can express all the velocities in terms of momenta, and thus the primary Hamiltonian free from arbitrariness of Lagrange multiplier is found as,
Corresponding Wheeler-deWitt equation
As before, we proceed to make semiclassical approximation by expressing Ψ[a, ϕ1, ϕ2, σ] as
M1 order term leads to Hamilton-Jacobi equation for the source free nonsingular gravitational field as,
One can easily identify this equation with the Hamilton-Jacobi equation (16). It leads to the classical source free gravitational constraint equation (6) under the same choice of the time parameter (23). Hence, source free Hamiltonian (12) can be found from it by identifying the canonical momenta with corresponding derivatives of S0, as before. The next, ie., M0 order term gives the following equation,
Now in order to identify this equation (36) with the functional Schrödinger equation for the matter field σ, let us define a function,
In view of (37), equation (36) reduces to
provided, the function D(a, ϕi) satisfies the following equation,
Since the right hand side of equation (38) is the quantum Hamiltonian operator for the scalar field σ operating on function f, in the background of curved space time, so it is clear that equation (38) is the functional Schrödinger equation for the field σ, propagating in the background of curved space time, under the same choice of the time parameter (23). Further, under the same choice of time parameter (23), equation (39) reduces to,
Hence, upto this order of approximation, we have,
where, S0 is obtainable from equation (35), f from (38) and D from (39) or (40). So at the end we observe that under the same choice of time parameter (23), one can parameterize the classical trajectories by reproducing classical source free gravitational equation (6) from the source free Hamilton-Jacobi equation (35) at one hand, while the M0 order term (36) can be identified with the functional Schrödinger equation (38) for the field σ propagating in the background of curved space time, on the other. This time parametrization is unique since equation (38) reduces to the functional Schrödinger equation only under the choice of the real time parameter (23) leaving no arbitrariness. This choice is correct since it fulfills both the requirements of obtaining functional Schrödinger equation and parametrization of the classical trajectories. 8 Concluding remarksIn a series of works, Brandenberger, Mukhanov, Sornborger and others [2-4] and [6] constructed a nonsingular gravitational action by invoking limiting curvature hypothesis (LCH) and studied its classical aspects. The action contains a couple of scalar fields, one of which (ϕ2) is required to guarantee nonsingular solutions, while the other (ϕ1) binds the curvature invariants. The corresponding potentials are chosen in such a manner that for |ϕi| ≪ 1, Einstein's theory is recovered at one hand and their asymptotic behaviour should be such that LCH may be realized on the other. In the present work, we have studied the quantum cosmological aspect of the theory and made a connection of the quantum cosmological equation with the Hamilton-Jacobi equation of the classical theory. The answer to the question that, why it is at all necessary to quantize a nonsingular Universe model, lies in fact that some new results have been unveiled in the process and a non-trivial technique of parameterizing the classical trajectories has been found. Summarily, the results are, 1.) Introduction of the scalar field ϕ1 in the action (1), only introduces yet another constraint equation (other than the Hamiltonian constraint). Thus it is essentially a classically constrained theory of gravity and so one has to deal with such constraints critically, which we have done using Dirac's algorithm. 2.) In the absence of the scalar fields, the classical field equations reduce to those corresponding to standard theory of
gravity. However, the Hamiltonian obtained after analyzing the constraint, does not give Einstein's theory back. Likewise,
the Wheeler-deWitt equation is free from first derivative terms 3.) The wave function is real for k = ± 1, unlike the situation encountered in standard gravity theory (see appendix), where, k = +1 leads to a complex wave function. 4.) The semiclassical time parameter required to parameterize classical trajectories can be found in a unique way only by using the technique of Lagrange undetermined multiplier in the Lagrangian constrained dynamics, rather than using Hamiltonian formalism. 9 AppendixParamerizing Einstein's gravity with a minimally coupled scalar field with semiclassical timeIn the appendix we shall recapitulate the standard technique [12-14]
, to parameterize classical trajectories of the Hamilton-Jacobi equation, with some appropriate choice of semiclassical time
parameter, corresponding to Einstein's gravity with a minimally coupled scalar field, and to find the functional Schrödinger
equation. The action for Einstein's gravity with a source term - a minimally coupled scalar field ϕ, (with
apart from a surface term. The
which corresponds to the Hamilton constraint equation,
Thus, the corresponding Wheeler-deWitt equation is
where, p takes care of some of the operator ordering ambiguities. 9.1 Expansion with power series of Planck's constant ħIn the standard WKB approximation, the wave functional Ψ is expressed as
This is Einstein-Hamilton-Jacobi (EHJ) equation, which is essentially equation (43), if one identifies pa with
and
The classical trajectories are parameterized by a time, which is a linear combination of (46) and (47). Thus,
Under this choice of time parameter (48), equation (45) produces Einstein's equation (42). This has been shown by Kiefer [12]
, taking, a = exp α and using the unit M = 1. It is important to note that the choice of the semiclassical time parameter does not involve operator ordering ambiguity.
Equation (45) can in principle be solved for S0 and to this order of approximation one obtains 9.2 Expansion with power series of Planck's mass MIn the present work, we have expanded S(a, ϕ) in the power series of Planck's mass, instead. The advantage is that, one can obtain a functional Schrödinger equation directly
from it. Substituting, S = MS0 + S1 + M-1S2 + ⋯ etc., in the Wheeler-deWitt equation (44) and collecting terms in different orders in M, we obtain for M2 order,
Now, in the absence of the matter field, equations (42) and (43) reduce to
respectively. Hence the EHJ equation (49) is essentially the vacuum Einstein equation (50), under the identification
Further in view of equation (49), we have,
Under the choice of the time parameter (52), equation (49) reduces to the vacuum Einstein's equation (50). Hence, expansion of S in the power series of M decouples gravity from the source and leaves it to behave classically. Equation (49) can be solved for S0, and up to this order of approximation,
Thus the wave-functional is well behaved at the classical singularity a → 0. The next order of approximation yields the following equation,
where, p takes care of some of the factor ordering ambiguities. Now choosing a function,
provided, D(a) satisfies the following equation,
Equation (55) can be identified with Tomonaga-Schwinger equation, which is essentially the functional Schrödinger equation
for the mater field propagating in the background of curved space-time, if the semi-classical time operator is identified
with (51). The right hand side of equation (55) represents quantum Hamilton operator
Equation (56) can be solved for D(a) as,
Substituting
m being the constant of integration. Finally, to this order of approximation, the wave-functional takes the form,
Equation (55) can in principle be solved for f(a, ϕ) and thus explicit form of the wave-functional can be obtained. Now to make semiclassical approximation to the functional
Schrödinger equation (55), let us express
Since we have already identified
Further, if one identifies
and the suppressed part of the WKB time parameter appearing in equation (48) reappears. This when combined with equation (51), one finally obtains the semiclassical time parameter as,
AcknowledgementsI would like to thank prof. R. Brandenberger and the reviewers for going through the manuscript and rendering some valuable comments and suggestions. References
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, etc. However, curvature invariants are infinite in number and there is no guarantee that imposing constraint on some lower
order curvature invariant terms, all the higher order terms will be bounded. Hence, 'Limiting Curvature Hypothesis' (LCH)
. Thus in its simplest form, ie., for homogeneous and isotropic space-time, Mukhanov and Brandenberger 
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, and then by expanding S in the powers of M instead of ħ. In the process, Hamilton-Jacobi equation has been found in section 4. In section 5, the usual procedure to extract the semiclassical
time parameter has been attempted. However, it fails to fulfill the basic requirement of reproducing classical (Hamiltonian)
constraint equation. Other naive techniques to find the time parameter have also been explored. Section 6 is devoted to follow
an involved technique to find the unique semiclassical time parameter, which is the main motivation of the present paper.
In section 7, a source term has been incorporated in the action and the same procedure has been followed. In the process,
the functional Schrödinger equation has been found and the uniqueness of the semiclassical time parameter has been proved
without ambiguity. The outcome of the present work has been summarized in section (8). Finally, we have added up an appendix
in section 9, to remind the readers about the semiclassical approximation in quantum cosmology.
, at |ϕi| ≪ 1, where, i = 1, 2. Further, to realize LCH, first requirement is to bound the curvature and the next is to obtain nonsingular solutions.
Curvature is bounded provided, V1(ϕ1) ~ ϕ1 at |ϕ1| ≫ 1 and de Sitter solution is obtained in the asymptotic region, provided, V2(ϕ2) ~ constant, at |ϕ2| ≫ 1.
, being the trace of the extrinsic curvature. The surface term is clearly different from the one that appears in Einstein-Hilbert
action. However, the two are the same, if ϕ2 vanishes, which has got a kinetic term in the action, and acts as a dynamical variable. It is important to notice that the
surface term is not affected by the presence of the scalar field ϕ1, since it acts as Lagrange multiplier only. In the above action N and ϕ1 act as Lagrange multipliers, resulting in a pair of constraint equations. Variation with respect to N gives the Hamiltonian constraint equation, but one can fix the gauge, N = 1, without loss of generality. However, we are not going to fix the other Lagrange multiplier ϕ1, since it has been invoked to bound the curvature invariants for the nonsingular universe guaranteed by the field variable
ϕ2. So, altogether we get four equations (not all independent), two of which are constraint equations and those obtained under
the variation with respect to ϕ2 and the scale factor a are the two field equations, where we set, N = 1, at the end. The constraint equation appearing under the variation of ϕ1 is,
and 
denote the derivatives of the potentials with respect to ϕ1 and ϕ2 respectively. For spatially flat, k = 0 case, the above field equations can be combined to yield a first order differential equation,
, along with ϕ2 ≥ 0, evolve towards de-Sitter at ϕ2 = ∞. In a nutshell, all the phase trajectories are either periodic about Minkowski space-time or else they asymptotically
approach to de Sitter space. Hence all solutions are nonsingular. It is important to note that if 
vanishes, ϕ2 turns out to be a constant and as a result 
vanishes and the resulting solutions again become singular. However, the condition 
, corresponding to the Lagrangian of the above action (3) vanishes, and so the action becomes singular. Vanishing of the determinant
of Hessian, signals the presence of constraint in the theory which should be analyzed step by step carefully. Dirac algorithm
can be solved for 
as 
, where, a → 1, ...., R, 
= M, M being the square of the Planck mass, one finds in view of the action (3) under the gauge choice N = 1,
is not invertible due to the constraint,
, so the above constraint vanishes weakly. Hence the primary Hamiltonian may be expressed as,
} = {C1, D1} = 
. As already pointed out that for the existence of nonsingular solutions 
provided 
, which looks like a new equation. However, this is the one, we were in search of, since ϕ1 can now be inverted through this equation. In any case, it is not an independent equation, since taking time derivative of
the first (constraint) equation (4) and using the second one (5), one can arrive at it. Thus the Hamiltonian, (12) being free
from constraints and being able to produce all the field equations, is the correct one. To make a comparison, let us take
Einstein-Hilbert action minimally coupled to a scalar field,
where, q(a) is an arbitrary function of a. Hence the first term of the Wheeler-deWitt equation corresponding to the Hamiltonian (12), after replacing 
by -iħ▽, turns out to be
) does not appear, while the same with respect to the scalar field (ie., 
) appears. Further, operator ordering in 
and 
appearing due to the presence of the second and the third terms in (12), again introduces first derivative terms (ie., 
and 
can be neglected at sufficiently small values of a, ie, at sufficiently early epoch. Hence, the Wheeler-deWitt equation may be kept free from the first derivative terms even
after removing some of the operator ordering ambiguities (unlike the situation encountered in standard and other nonstandard
theories of gravity), and is expressed as,
nor 
term appears in the Wheeler-deWitt equation (14), which may have some deep significance, not presently known.
. Expanding, S in the powers of M as, S = MS0 + S1 + M-1S2 + ⋯, and inserting it in the Wheeler-deWitt equation (14), one obtains
appearing in the Hamilton-Jacobi equation (16) is not obtainable from it and as a result, remains arbitrary. This is the
source of trouble that we encounter in the present situation, to parameterize classical trajectories with semiclassical time
parameter following usual procedure. Thus, the standard procedure does not work in the classically constrained system under
consideration.
and 
, using this expression as,
, 
from above time parameter (23) and substituting these in equation (16), one can reproduce equation (6). To check whether
we have found the correct and unique semiclassical time parameter let us continue by including a source term (in the form
of a dynamical scalar field) in the model.
, as before),
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expanding S in power series of M as, S = MS0 +S1 +M-1S2 + ⋯ etc., and then substituting all these in equation (33). Finally, equating terms having different orders of M to zero, we find for M2 order
, and thus is quite different from those obtained in view of standard and other nonstandard (nonminimally coupled) theories
of gravity.
, which is different from 
component of Einstein's equation is
and the functional S(a, ϕ) is expanded in the power series of the Planck's constant ħ as, S = S0 + ħS1 + ħ2S2 + ⋯ etc. Substituting all these in the Wheeler-deWitt equation (44), and collecting terms independent of ħ, one obtains,
. In order to parameterize classical trajectories, WKB time parameter is chosen in the following manner,
. Higher order terms of ħ produce fluctuations around the classical trajectory S0 and hence introduce corrections to the wave-functional 
. To parameterize classical trajectories, we identify,
and substituting 
and 
from it in the above equation (54), one obtains,
for the matter field operating on the function f(a, ϕ). Thus equation (55) reduces to
, in equation (56), one obtains,
, expand A in the power series of ħ as, A(a, ϕ) = A0 + ħA1 + ħ2A2+ ⋯, substitute it in equation (55) and collect terms independent of ħ, to get,
with the semiclassical time parameter 
, in equation (52), so the above equation takes the form,
with pϕ, it transpires that equation (62) is the Hamiltonian for the matter field in the background of curved space time, corresponding
to the classical action, 
in the isotropic and homogeneous space-time under consideration. Now, since 
, so
