Alternative quantization of the Hamiltonian in loop quantum cosmology II: Including the Lorentz term

Since there are quantization ambiguities in constructing the Hamiltonian constraint operator in isotropic loop quantum cosmology, it is crucial to check whether the key features of loop quantum cosmology are robust against the ambiguities. In this paper, we quantize the Lorentz term of the gravitational Hamiltonian constraint in the spatially flat FRW model by two approaches different from that of the Euclidean term. One of the approaches is very similar to the treatment of the Lorentz part of Hamiltonian in loop quantum gravity and hence inherits more features from the full theory. Two symmetric Hamiltonian constraint operators are constructed respectively in the improved scheme. Both of them are shown to have the correct classical limit by the semiclassical analysis. In the loop quantum cosmological model with a massless scalar field, the effective Hamiltonians and Friedmann equations are derived. It turns out that the classical big bang is again replaced by a quantum bounce in both cases. Moreover, there are still great possibilities for the expanding universe to recollapse due to the quantum gravity effect.


I. INTRODUCTION
An important motivation of the theoretical search for a quantum theory of gravity is the expectation that the singularities predicted by classical general relativity would be resolved by the quantum gravity theory. This expectation has been confirmed by the recent study of certain isotropic models in loop quantum cosmology (LQC) [1,2,3], which is a simplified symmetry-reduced model of a full background-independent quantum theory of gravity [4], known as loop quantum gravity (LQG) [5,6,7,8]. In loop quantum cosmological scenario for a universe filled with a massless scalar field, the classical singularity gets replaced by a quantum bounce [3,9,10]. Moreover, It is also revealed in the effective scenarios that there are great possibilities for a spatially flat FRW expanding universe to recollapse due to the quantum gravity effect [11]. However, as in the ordinary quantization procedure, there are quantization ambiguities in constructing the Hamiltonian constraint operator. Thus it is crucial to check the robustness of the key results against the quantization ambiguities. Moreover, since LQC serves as a simple arena to test ideas and constructions induced in the full LQG, it is important to implement those treatments from the full theory to LQC as more as possible.
In the previous paper [12] of this series of work, it has been shown that the above key features of LQC are robust against a quantum ambiguity arising from the quantization of the field strength of the gravitational connection. In this paper we will propose alternative versions of Hamiltonian operator for isotropic LQC, which inherit more features from full LQG comparing to the conven- * Electronic address: mayg@bnu.edu.cn tional one so far considered in the literatures. It is well known that the Hamiltonian constraint in the full theory is composed of two terms, the so-called Euclidean and Lorentz terms. In spatially flat and homogeneous models, the two terms are proportional to each other. Thus people usually rewrite the Lorentz term in the form of the Euclidean one classically and then quantize their combination [3]. However, this treatment is impossible in the full theory, where the Lorentz term has to be quantized in a form quite different from the Euclidean one [7,13]. The issue that we are considering is what would happen in the improved dynamics setting of LQC if one kept the distinction of the two terms as in full theory rather than mixed them. Could one construct an operator corresponding to the Lorentz term in a way similar to that in full LQG? If so, could the classical big bang singularity still be replaced by a quantum bounce in the new quantum dynamics? To answer these questions, two alternative Hamiltonian constraint operators including the Lorentz terms are constructed respectively in the improved scheme in this paper, which are both shown to have the correct classical limit by the semiclassical analysis. In the spatially flat FRW model with a massless scalar field, the effective Hamiltonians and Friedmann equations are derived in both case. It turns out that the classical big bang is again replaced by a quantum bounce. Moreover, there are still great possibilities for the expanding universe to recollapse due to the quantum gravity effect.
In the spatially flat and isotropic model, one has to first introduce an elementary cell V and restrict all integrations to this cell. Fix a fiducial flat metric o q ab and denote by V o the volume of the elementary cell V in this geometry. The gravitational phase space variables -the connections A i a and the density-weighted triads E a i -can be expressed as [14] A where ( o ω i a , o e a i ) are a set of orthonormal co-triads and triads compatible with o q ab and adapted to the edges of the elementary cell V. In terms of p, the physical triad and cotriad are given by The basic (nonvanishing) Poisson bracket is given by where κ = 8πG (G is the Newton's constant) and γ is the Barbero-Immirzi parameter.
To pass to the quantum theory, one constructs a kinematical Hilbert space H grav kin = L 2 (R Bohr , dµ Bohr ), where R Bohr is the Bohr compactification of the real line and dµ Bohr is the Haar measure on it [14]. The abstract *algebra represented on the Hilbert space is based on the holonomies of the connection A i a . In the Hamiltonian constraint of LQG, the gravitational connection A i a appears through its curvature F i ab . Since there exists no operator corresponding to c, only holonomy operators are well defined. Hence one is led to express the curvature in terms of holonomies. Similarly, in the improved dynamics setting of LQC [3], to express the curvature one employed the holonomies along an edge parallel to the triad o e a i of lengthμ |p| ≡ D, where D is a constant, with respect to the physical metric q ab , where I is the identity 2 × 2 matrix and τ i = −iσ i /2 (σ i are the Pauli matrices). Thus, the elementary variables could be taken as the functions exp(iμc/2) and the physical volume V = |p| 3/2 of the cell, which have unambiguous operator analogs. It is convenient to work with the v-representation. In this representation, states |v , constituting an orthonormal basis in H grav kin , is more directly adapted to the volume operatorV aŝ where ℓ 2 p = G and The action of exp(iμc/2) is given by exp(iμc/2)|v = |v + 1 .
Now let us consider the gravitational field coupled with a massless scalar field φ. The Hamiltonian of the matter field is given by H φ = |p| −3/2 p 2 φ /2, where p φ denotes the momentum of φ. Hence the total Hamiltonian constraint is given by H = H grav + H φ . The basic Poisson bracket for the matter field is given by The Hamiltonian evolution equations for the matter field readṗ which show that φ is monotonic function of the time parameter. So the scalar field can be regarded as internal time. To quantize the matter field, we can choose the standard Schrödinger representation for scalar field. The kinematical Hilbert space, H φ = L 2 (R, dφ), is the space of the square integrable functions on R endowed with the Lebesgue measure. Hence the kinematical Hilbert space of the gravitational field coupled with the scalar field is H kin = H grav kin ⊗ H φ . The elementary operators of the scalar field are defined by: In the following sections, we will construct two different Hamiltonian operators including the Lorentz term in the above quantum kinematic framework. Their classical limits are confirmed by calculating the expectation values of these new Hamiltonian operators with respect to suitable semiclassical states. By this approach we also obtain the effective descriptions of quantum dynamics in both cases. In the end we will summarize the results and discuss some of their ramifications.

II. ALTERNATIVE REGULARIZED HAMILTONIAN CONSTRAINTS
Because of homogeneity, we can assume that the lapse N is constant and from now onwards set it to be one. The gravitational Hamiltonian constraint is given by The symmetry-reduced expressions read For the spatially flat isotropic model, one has classically the identity Hence one obtains the classically equivalent expression of Eq. (11) as where Both Eqs. (11) and (14) are alternative expressions of the Hamiltonian constraint classically equivalent to the one currently employed in the literatures (see e.g. [3]).
Since the expressions (11) and (14) inherit more features of the Hamiltonian constraint in the full theory, we will take them separately as the starting-points of our quantization. To this aim, the first step is to give their regularized expressions which would be suitable for quantization. Note that the improved quantum operator representing the Euclidean Hamiltonian constraint H E (1) was first introduced in [3], and its regularized formulation reads Now our task is to give the regularized formulations of the Lorentz terms T S (1) and T (1) in Eqs. (14) and (11) respectively. Let us first deal with the symmetry-reduced Lorentzian term T S (1). By Thiemann's trick [7,13], whereǫ abc is the Levi-Civita density, Eq. (15) can be written as Moreover it is easy to show that Hence Eq. (18) can be written as where Putting Eqs. (16) and (22) together, we obtain the regularized Hamiltonian constraint corresponding to (14) as Let us now turn to the original Lorentz term in Eq. (11): Though this term was considered in some early literature [15], here we will treat it in the new improved quantization framework [3]. As in the full theory [7,13], the extrinsic curvature can be written as where is the integrated trace of K i a . Hence Eq. (24) can be reexpressed as Moreover, we have the following identity Using the identities (20) and (28), Eq. (27) can be written as where Putting Eqs. (16) and (30) together, we obtain the regularized Hamiltonian constraint corresponding to (11) as

III. ALTERNATIVE HAMILTONIAN CONSTRAINT OPERATORS
Since both regularized Hamiltonian constraints (23) and (31) are now expressed in terms of elementary variables and their poisson brackets, which have unambiguous quantum analogs, it is straightforward to write down the quantum operatorsĤ S,μ grav andĤ F,μ grav . However, the limitμ → 0 of these operators do not exist, not only for the Euclidean termĤ E,μ (1), but also for the Lorentzian termTμ S (1) orTμ F (1). In fact, even in the full theory, there are no local operators representing connections and curvatures. To get unambiguous operators, one should have recourse to the area gap as in the improved scheme [3], whereμ is given byμ here ∆ = 4 √ 3πγℓ 2 p is a minimum nonzero eigenvalue of the area operator [10]. The Euclidean Hamiltonian constraint operatorĤ E (1) corresponding to (16) is given in the improved scheme by [3] where, for clarity, we have suppressed hats over the operators sin(μc/2), cos(μc/2) and sgn(v)/μ 3 . In the vrepresentation where v := sgn(p)|p| 3/2 2πγℓ 2 H E (1) acts on the basis |v of H grav kin aŝ where Now we turn to the Lorentz part. The regularized expression (22) enables us to define a self-adjoint operator on H grav kin in the improved scheme corresponding to (15) asT Its action on the basis |v readŝ where Hence we obtain the action of the Hamiltonian constraint operator corresponding to (14) on |v aŝ where On the other hand, the regularized expression (30) enables us to define the other operator on H grav kin corresponding to (24) aŝ whereB It is easy to see from Eq. (42) thatT F (1) is a symmetric operator. Its action on |v readŝ where here Hence the action of the Hamiltonian constraint operator corresponding to (11) on |v is given bŷ where Thus, both of the new proposed Hamiltonian constraint operators in Eqs. (41) and (47) are also difference operators with constant steps in eigenvalues of the volume operatorV . But they contain more terms with steps of different size comparing to the operator (33).
The Hamiltonian constraint of the scalar field has been quantized in the literatures as [3,14] where the action of |p| −3/2 on |v reads |p| −3/2 |v = 3 2 Thus we obtain alternative total Hamiltonian operators respectively asĤ Note that in both quantum dynamics the scalar field φ can be used as emergent time. But the different expressions of gravitational Hamiltonian operators may lead to different physics, which may be examined in some aspects. We will consider their classical limit and effective dynamics in next section.

IV. CLASSICAL LIMIT AND EFFECTIVE DYNAMICS
It has been shown in [11,16] that the improved Hamiltonian constraint operator constructed in [3] has the correct classical limit. In this section, we will show that the two Hamiltonian constraint operators constructed in this paper also have the correct classical limit. Moreover, the effective Hamiltonian incorporating higher order quantum corrections can also be obtained. In order to do the semiclassical analysis, it is convenient to introduce a new variable: is the energy density of the scalar field. Let us first consider the gravitational part. Since there are uncountable basis vectors, the natural Gaussian semiclassical states live in the algebraic dual space of some dense set in H grav kin . A semiclassical state (Ψ (bo,vo) | peaked at a point (b o , v o ) of the gravitational classical phase space reads: where d is the characteristic "width" of the coherent state. For practical calculations, we use the shadow of the semiclassical state (Ψ (bo,vo) | on the regular lattice with spacing 1 [17], which is given by where ǫ = 1/d and we choose v o = N ∈ Z. Since we consider large volumes and late times, the relative quantum fluctuations in the volume of the universe must be very small. Therefore we have the restrictions: 1/N ≪ ǫ ≪ 1 and b o ≪ 1. One can check that the state (55) is sharply peaked at (b o , v o ) and the fluctuations are within specified tolerance [11,16]. The semiclassical state of matter part is given by the standard coherent state where σ is the width of the Gaussian. Thus the whole semiclassical state reads (Ψ (bo, vo) | (Ψ (φo,p φ ) |.
The task is to use this semiclassical state to calculate the expectation values of the two Hamiltonian operators in Eqs. (50) and (51) to a certain accuracy. In the calculation, one gets the expression with the absolute values, which is not analytical. To overcome the difficulty we separate the expression into a sum of two terms: one is analytical and hence can be calculated straightforwardly, while the other becomes exponentially decayed out. We first see the expectation value of the Hamiltonian oper-atorĤ F , which inherits more features of the full theory.
The expectation values of the two terms ofĤ F grav in Eq. (47) are respectively calculated as In the calculation of Ĥ φ , one has to calculate the expectation value of the operator |p| −3/2 , which is given by [11]: For clarity, we will suppress the label o in the following. Collecting these results we can obtain an effective Hamiltonian with the relevant quantum corrections of order ǫ 2 , 1/v 2 ǫ 2 , 2 /σ 2 p 2 φ as: Hence the classical constraint (53) is reproduced up to small quantum corrections and therefore the Hamiltonian operatorĤ F has correct classical limit. We can further obtain the Hamiltonian evolution equation of v by taking its Poisson bracket with where ρ c = 3/(κγ 2 ∆). The vanishing of the effective Hamiltonian constraint (59) gives For the classical region, b ≪ 1 and ρ ≪ ρ c , we have from Eq. (61) where The modified Friedmann equation can then be derived as It is easy to see that if one neglects the small quantum corrections in the classical region, χ F ≪ 1 for ρ ≪ ρ c , one gets which reduces to the standard Friedmann equation. However, quantum geometry effects lead to a modification of the Friedmann equation especially at the scales when ρ becomes comparable to ρ c . Remarkable changes to the classical theory happen when the Hubble parameter in Eq. (64) vanishes by If we consider only the leading order contribution in Eq.
(63), this can happen when Thus, when energy density of the scalar field reaches to the leading order critical energy density ρ F c = ρ c /4(1 + γ 2 ), the universe bounces from the contracting branch to the expanding branch. The quantum bounce implied by (64) is shown in Fig. 1 .
In a similar way, we can calculate the expectation value of the other Hamiltonian operator (50) as well. The effective Hamiltonian corresponding toĤ S with the relevant quantum corrections is obtained as Hence the classical constraint (53) is again reproduced up to small quantum corrections. The corresponding modified Friedmann equation can then be derived as where The Hubble parameter in Eq. (69) can also vanish when Thus the quantum dynamics given byĤ S has qualitatively similar feature of that given byĤ F . However, there are quantitative differences between them. For the leading order effective theory ofĤ S , when energy density of the scalar field reaches to the critical energy density ρ S c = 4(1 + γ 2 )ρ c , the universe bounces from the contracting branch to the expanding branch.

V. DISCUSSION
We have successfully constructed two versions of Hamiltonian operator for isotropic LQC in the improved scheme, where the Lorentz term is quantized in two approaches different from the Euclidean one. The treatments of the Lorentz term of Hamiltonian in the full LQG can be properly implemented in one of our constructions, which inherits more features of the full theory. One of the Hamiltonian operators is self-adjoint and the other is symmetric. Both of them are shown to have the correct classical limit by the semiclassical analysis. Hence the alternative Hamiltonian operators that we proposed can provide good arenas to test the ideas and constructions of the quantum dynamics in full LQG. In the spatially flat FRW model with a massless scalar field, the effective Hamiltonians and Friedmann equations are derived in both case. Although there are quantitative differences between the two versions of quantum dynamics, qualitatively they have the same dynamical features. Especially, the classical big bang is again replaced by a quantum bounce in both cases. For instance, in the leading order effective theory ofĤ F , the universe would bounce from the contracting branch to the expanding branch when the energy density of scalar field reaches to the critical ρ F c = ρ c /4(1 + γ 2 ). Therefore, the key feature of LQC for the resolution of the big bang singularity is still maintained for the new quantum dynamics inheriting more features of the full theory. Recall that the quantum bounce happens at ρ c for the quantum dynamics in [3]. Thus the new quantum dynamics here lead to some quantitatively different critical energy density for the bounce.
On the other hand, the discussion in [11] can be carried out similarly. It is easy to see from Eqs. (64) and (69) that the Hubble parameter in both cases may also vanish by the vanishing of χ F and χ S respectively, whence the asymptotic behavior of the quantum geometric fluctuations plays a key role for the fate of the universe. By the ansatz ǫ = λ(r)v −r(φ) with 0 ≤ r(φ) ≤ 1, Eqs. (63) and (70) imply that there are great possibilities for the expanding universe to undergo a recollape in the future. The recollape can happen provided 0 ≤ r < 1 in the large scale limit. Suppose that the semiclassicality of our coherent state is maintained asymtotically so that the quantum fluctuation 1/ǫ of v cannot increase as v unboundedly as v approaches infinity. Thus the recollape is in all probability as viewed from the parameter space of r(φ). Taking the effective dynamics ofĤ F as an example, in the scenario when r = 0 asymtotically, besides the quantum bounce when the matter density ρ increases to the Planck scale, the universe would also undergo a recollapse when ρ decreases to ρ F coll ≈ 8(1 + γ 2 )ǫ 2 ρ F c . Therefore, the quantum fluctuations also lead to a cyclic universe in this case. The cyclic universe in this effective scenario is illustrated in Fig. 2. This amazing possibility that quantum gravity manifests herself in the large scale cosmology was first revealed in [11]. Nevertheless, the condition that the semiclassicality is maintained in the large scale limit has not been confirmed. Hence further numerical and analytic investigations to the properties of dynamical semiclassical states in the alternative quantum dynamics are still desirable. It should be noted that in some simplified completely solvable models of LQC (see [9] and [18]), the dynamical coherent states could be ob- tained, where r(φ) approaches 1 in the large scale limit.
While those treatments lead to the quantum dynamics different from ours, they raise caveats to the conjectured recollapse.
To summarize, the quantum dynamics of LQC in the improved scheme is extended in order to inherit more features from the full LQG. The key features of LQC in this model, that the big bang singularity is replaced by a quantum bounce and there are great possibilities for an expanding universe to recollapse, are robust against the quantization ambiguities with the extensions. This result further supports the expectation that the above features of LQC originate not only in the symmetric model but also from the fundamental LQG.