Semiclassical and quantum behavior of the Mixmaster model in the polymer approach for the isotropic Misner variable
Abstract
We analyze the semiclassical and quantum behavior of the Bianchi IX Universe in the Polymer Quantum Mechanics framework, applied to the isotropic Misner variable, linked to the space volume of the model. The study is performed both in the Hamiltonian and field equations approaches, leading to the remarkable result of a still singular and chaotic cosmology, whose Poincaré return map asymptotically overlaps the standard Belinskii–Khalatnikov–Lifshitz one. In the quantum sector, we reproduce the original analysis due to Misner, within the revised Polymer approach and we arrive to demonstrate that the quantum numbers of the pointUniverse still remain constants of motion. This issue confirms the possibility to have quasiclassical states up to the initial singularity. The present study clearly demonstrates that the asymptotic behavior of the Bianchi IX Universe towards the singularity is not significantly affected by the Polymer reformulation of the spatial volume dynamics both on a pure quantum and a semiclassical level.
1 Introduction
The Bianchi IX Universe [1, 2] is the most interesting among the Bianchi models. In fact, like the Bianchi type VIII, it is the most general allowed by the homogeneity constraint, but unlike the former, it admits an isotropic limit, naturally reached during its evolution [3, 4] (see also [5, 6]), coinciding with the positively curved Robertson–Walker geometry. Furthermore, the evolution of the Bianchi IX model towards the initial singularity is characterized by a chaotic structure, first outlined in [7] in terms of the Einstein equations morphology and then reanalyzed in Hamiltonian formalism by [8, 9]. Actually, the Bianchi IX chaotic evolution towards the singularity constitutes, along with the same behavior recovered in Bianchi type VIII, the prototype for a more general feature, characterizing a generic inhomogeneous cosmological model [10] (see also [6, 11, 12, 13, 14, 15]).
The original interest toward the Bianchi IX chaotic cosmology was due to the perspective, de facto failed, to solve the horizon paradox via such a statistical evolution of the Universe scale factors, from which derives the name, due to C. W. Misner, of the socalled Mixmaster model. However, it was clear since from the very beginning, that the chaotic properties of the Mixmaster model had to be replaced, near enough to the initial singularity (essentially during that evolutionary stage dubbed Planck era of the Universe), by a quantum evolution of the primordial Universe. This issue was first addressed in [8], where the main features of the WheelerDe Witt equation [16] are implemented in the scenario of the Bianchi IX cosmology. This analysis, besides its pioneering character (see also [6, 17]), outlined the interesting feature that a quasiclassical state of the Universe, in the sense of large occupation numbers of the wave function, is allowed up to the initial singularity.
More recent approaches to Canonical Quantum Gravity, like the socalled Loop Quantum Gravity [18], suggested that the geometrical operators, areas and volumes, are actually characterized by a discrete spectrum [19]. The cosmological implementation of this approach led to the definition of the concept of “BigBounce” [20, 21] (i.e. to a geometrical cutoff of the initial singularity, mainly due to the existence of a minimal value for the Universe Volume) and could transform the Mixmaster Model in a cyclic Universe [22, 23]. The complete implementation of the Loop Quantum Gravity to the Mixmaster model is not yet available [24, 25, 26], but general considerations on the volume cutoff led to characterize the semiclassical dynamics as chaosfree [27]. A quantization procedure, able to mimic the cutoff physics contained in the Loop Quantum Cosmology, is the socalled Polymer quantum Mechanics, de facto a quantization on a discrete lattice of the generalized coordinates [28] (for cosmological implementations see [29, 30, 31, 32, 33, 34, 35, 36]).
Here, we apply the Polymer Quantum Mechanics to the isotropic variable \(\alpha \) of the Mixmaster Universe, described both in the Hamiltonian and field equations representation. We first analyze the Hamiltonian dynamics in terms of the so called semiclassical polymer equations, obtained in the limit of a finite lattice scale, but when the classical limit of the dynamics for \(\hbar \rightarrow 0\) is taken. Such a semiclassical approach clearly offers a characterization for the behavior of the mean values of the quantum Universe, in the spirit of the Ehrenfest theorem [37]. This study demonstrates that the singularity is not removed and the chaoticity of the Mixmaster model is essentially preserved, and even enforced in some sense, in this cutoff approach.
Then, in order to better characterize the chaotic features of the obtained dynamics, we translate the Hamiltonian polymerlike dynamics, in terms of the modified Einstein equations, in order to calculate the morphology of the new Poincaré return map, i.e. the modified BKL map.
We stress that, both the reflection rule of the pointUniverse against the potential walls and the modified BKL map acquire a readable form up to first order in the small lattice step parameter. Both these two analyses clarify that the chaotic properties of the Bianchi IX model survive in the Polymer formulation and are expected to preserve the same structure of the standard General Relativity case. In particular, when investigating the field equations, we numerically iterate the new map, showing how it asymptotically overlaps to the standard BKL one.
The main merit of the present study is to offer a detailed and accurate characterization of the Mixmaster dynamics when the Universe volume (i.e. the isotropic Misner variable) is treated in a semiclassical Polymer approach, demonstrating how this scenario does not alter, in the chosen representation, the existence of the initial singularity and the chaoticity of the model in the asymptotic neighborhoods of such a singular point.
Finally, we repeat the quantum Misner analysis in the Polymer revised quantization framework and we show, coherently with the semiclassical treatment, that the Misner conclusion about the states with high occupation numbers, still survives: such occupation numbers still behave as constants of motion in the asymptotic dynamics.
 (i)
In [30], where the polymer approach is applied to the anisotropy variables \(\beta _{\pm }\), the Mixmaster chaos is removed like in [27] even if no discretization is imposed on the isotropic (volume) variable. Furthermore, approaching the singularity, the anisotropies classically diverge and their discretization does not imply any intuitive regularization, as de facto it takes place there. The influence of the discretization induced by the Polymer procedure, can affect the dynamics in a subtle manner, nonnecessarily predicted by the simple intuition of a cutoff physics, but depending on the details of the induced symplectic structure.
 (ii)
In Loop Quantum Gravity, the spectrum of the geometrical spatial volume is discrete, but it must be emphasized that such a spectrum still contains the zero eigenvalue. This observation, when the classical limit is constructed including the suitably weighted zero value, allows to reproduce the continuum properties of the classical quantities. Such a point of view is wellillustrated in [38], where the preservation of a classical Lorentz transformation is discussed in the framework of Loop Quantum Gravity. This same consideration must hold in Loop Quantum Cosmology, for instance in [20], where the position of the bounce is determined by the scalar field parameter, i.e., on a classical level, it depends also on the initial conditions and not only on the quantum modification to geometrical properties of spacetime.
We note that picking, as the phasespace variable to quantize, the scale factor \(a = e^{\alpha }\), or any other given power of this, the corresponding Polymer discretization is somehow forced to become singularityfree, i.e. we observe the onset of a Big Bounce. However, in the present approach, based on a logarithmic scale factor, the polymer dynamical scheme offers an intriguing arena to test the Polymer Quantum Mechanics in the Minisuperspace, even in comparison with the predictions of Loop Quantum Cosmology [18].
Finally, we are interested in determining the fate of the Mixmaster model chaoticity, which is suitably characterized by the Misner variables representation and whose precise description (i.e. ergodicity of the dynamics, form of the invariant measure) is reached in terms of the MisnerChitrélike variables [13, 39, 40], which are double logarithmic scale factors. Although these variables mix the isotropic and anisotropic Misner variables to some extent, the MisnerChitré timelike variable in the configurational scale, once discretized would not guarantee a minimal value of the universe volume. Thus the motivations for a Polymer treatment of the Mixmaster model in which only the \(\alpha \) dynamics is deformed relies on different and converging requests, coming from the cosmological, the statistical and the quantum feature of the Bianchi IX universe.
The paper is organized as follows. In Sect. 2 we introduce the main kinematic and dynamic features of the Polymer Quantum Mechanics. Then we outline how this singular representation can be connected with the Schrödinger one through an appropriate Continuum Limit.
In Sect. 3 we describe the dynamics of the homogeneous Mixmaster model as it can be derived through the Einstein field equations. A particular attention is devoted to the Bianchi I and II models, whose analysis is useful for the understanding of the general Bianchi IX (Mixmaster) dynamics.
The Hamiltonian formalism is introduced in Sect. 4, where we review the semiclassical and quantum properties of the model as it was studied by Misner in [8].
Section 5 is devoted to the analysis of the polymer modified Mixmaster model in the Hamiltonian formalism, both from a semiclassical and a quantum point of view. Our results are then compared with the ones derived by some previous models.
In Sect. 6 the semiclassical behavior of the polymer modified Bianchi I and Bianchi II models is developed through the Einstein equations formalism, while the modified BKL map is derived and its properties discussed from an analytical and numerical point of view in Sect. 7.
In Sect. 8 we discuss two important physical issues, concerning the link of the polymer representation with Loop Quantum Cosmology and the implications of a polymer quantization of the whole Minisuperspace variables on the Mixmaster chaotic features.
Finally, in Sect. 9 brief concluding remarks follow.
2 Polymer quantum mechanics
The Polymer Quantum Mechanics (PQM) is a representation of the usual canonical commutation relations (CCR), unitarily nonequivalent to the Schrödinger one. It is a very useful tool to investigate the consequences of the assumption that one or more variables of the phase space are discretized. Physically, it accounts for the introduction of a cutoff. In certain cases where the Schrödinger representation is welldefined, the cutoff can be removed through a certain limiting procedure and the usual Schrödinger representation is recovered, as shown in [29] and summed up in Sect. 2.4.
Many people in the Quantum Gravity community think that there is a maximum theoretical precision achievable when measuring spacetime distances, this belief being backed up by valuable although heuristic arguments [41, 42, 43], so that the cutoff introduced in PQM is assumed to be a fundamental quantity. Some results of Loop Quantum Gravity [44] (the discrete spectrum of the area operator) and String Theory [45] (the minimum length of a string) point clearly in the direction of a minimal length scale scenario, too.
PQM was first developed by Ashtekar et al. [28, 46, 47] who also credit a previous work of Varadarajan [48] for some ideas. It was then further refined also by Corichi et al. [29, 49]. They developed the PQM in the expectation to shed some light on the connection between the Planckianenergy Physics of Loop Quantum Gravity and the lowerenergy Physics of the Standard Model.
2.1 The Schrödinger representation
2.2 The polymer representation: kinematics
The \(\hat{V}(\lambda )\) operators form a \(\mathscr {C}^{*}\)Algebra and the mathematical theory of \(\mathscr {C}^{*}\)Algebras [51] provide us with the tools to characterize the Hilbert space which the wave functions \(\psi (p)\) belong to. It is given by the Bohr compactification \(\mathbb {R}_b\) of the real line [52]: \(\mathscr {H}_{\text {poly}} = L^2(\mathbb {R}_b,d\mu _H)\), where the Haar measure \(d\mu _H\) is the natural choice.
2.3 Dynamics
2.4 Continuum limit
3 The homogeneous Mixmaster model: classical dynamics
With the aim of investigating in Sect. 7 the modifications to the Bianchi IX dynamics produced by the introduction of the Polymer cutoff, in this section we briefly review some relevant results obtained by Belinskii, Khalatnikov and Lifshitz (BKL) regarding the classical Bianchi IX model. In particular they found that an approximate solution for the Bianchi IX dynamics can be given in the form of a Poincaré recursive map called BKL map. This map is obtained as soon as the exact solutions of Bianchi I and Bianchi II are known. Hence, we will first derive the dynamical solution to the classical Bianchi I and II models.
3.1 Bianchi I
3.2 Bianchi II
A period in the evolution of the Universe when the r.h.s of (19) can be neglected, and the dynamics is Bianchi Ilike, is called Kasner regime or Kasner epoch. In this section we show how Bianchi II links together two different Kasner epochs at \(\tau \rightarrow \infty \) and \(\tau \rightarrow \infty \). A series of successive Kasner epochs, where one cosmic scale factor increases monotonically, is called a Kasner era.
The old (the unprimed ones) and the new indices (the primed ones) are related by the BKL map, that can be given in the form of the new primed Kasner indices \(p_l',p_m',p_n'\) and \(\varLambda '\) as functions of the old ones. To calculate it we need at least four relations between the new and old Kasner indices \(f_i(p_l,p_m,p_n,\varLambda ,p'_l,p'_m,p'_n,\varLambda ') = 0\) with \(i = 1,\dots ,4\). Then we can invert these relations to find the BKL map.
3.3 Bianchi IX
Here we briefly explain how the BKL map can be used to characterize the Bianchi IX dynamics. This is possible because it can be shown [13] that the Bianchi IX dynamical evolution is made up piecewise by Bianchi I and IIlike “blocks”.
Again, we assume an initial Kasner regime, where the negative Kasner index \(p_1\) is associated with the ldirection. Thus, the “exploding” cosmic scale factor in the r.h.s. of (38) is identified again with a. By retaining in the r.h.s. of (38) only the terms that grow towards the singularity, we obtain a system of ordinary differential equations in time completely similar to the one just encountered for Bianchi II (29), with the only caveat that now there is no initial condition (i.e. no choice of the initial Kasner indices) such that the initial or final Kasner regimes are stable towards the singularity.
This because all the cosmic scale factors \(\{a,b,c\}\) are treated on an equal footing in the r.h.s. of (38). After every Kasner epoch or era, no matter on which axis the negative index is swapped, there will always be a perturbation in the Einstein equations (38) that will make the Kasner regime unstable. We thus have an infinite series of Kasner eras alternating and accumulating towards the singularity.
4 Hamiltonian formulation of the Mixmaster model
In this section we summarize some important results obtained applying the Arnowitt, Deser and Misner (ADM) Hamiltonian methods to the Bianchi models. The main benefit of this approach is that the resulting Hamiltonian system resembles the simple one of a pinpoint particle moving in a potential well. Moreover it provides a method to quantize the system through the canonical formalism.
4.1 Quantization of the Mixmaster model
Near the Cosmological Singularity some quantum effects are expected to influence the classical dynamics of the model. The use of the Hamiltonian formalism enables us to quantize the cosmological theory in the canonical way with the aim of identify such effects.
5 Mixmaster model in the polymer approach
Here the Hamiltonian formalism introduced in Sect. 4 is used for the analysis of the dynamics of the modified Mixmaster model. We choose to apply the polymer representation to the isotropic variable \(\alpha \) of the system due to its deep connection with the volume of the Universe. As a consequence we will need to find an approximated operator for the conjugate momentum \(p_\alpha \), while the anisotropy variables \(\beta _\pm \) will remain unchanged from the standard case.
5.1 Semiclassical analysis
5.2 Comparison with previous models
As we stressed in Sect. 5.1 our model turns out to be very different from Loop Quantum Cosmology [61, 62], which predicts a Big Bouncelike dynamics. Moreover, if the polymer approach is applied to the anisotropies \(\beta _\pm \), leaving the volume variable \(\alpha \) classical [30], a singular but nonchaotic dynamics is recovered.
5.3 Quantum analysis
5.4 Adiabatic invariant
6 Semiclassical solution for the Bianchi I and II models
As we have already outlined in Sect. 3, to calculate the BKL map it is necessary first to solve the Einstein’s Eqs for the Bianchi I and Bianchi II models.
Therefore, in Sect. 6.1 we solve the Hamilton’s Eqs in Misner variables for the Polymer Bianchi I, in the semiclassical approximation introduced in Sect. 5.1.
Then in Sect. 6.2 we derive a parametrization for the Polymer Kasner indices. It is not possible anymore to parametrize the three Kasner indices using only one parameter u as in (27), but two parameters are needed. This is due to a modification to the Kasner constraint (26).
In Sect. 6.3 we calculate how the Bianchi II Einstein’s Eqs are modified when the PQM is taken into account. Then we derive a solution for these Eqs, using the lowest order perturbation theory in \(\mu \). We are therefore assuming that \(\mu \) is little with respect to the Universe volume cubic root.
6.1 Polymer Bianchi I
6.2 Parametrization of the Kasner indices
 \(0< u < 1\)
 \(\tfrac{1}{2}< u < 0\)
 \(1< u < \frac{1}{2}\)
 \(2< u < 1\)
 \(u < 2\)
In Fig. 6 the values of the ordered Kasner indices are displayed for the (u, Q)parametrization (96), where \(Q\in \left[0,1 \right]\). Because the range \(u > 1\) is not easily plottable, the equivalent parametrization in \(u\in \left[0,1 \right]\) was used. We notice that the roles of \(p_m\) and \(p_n\) are exchanged for this range choice.
6.3 Polymer Bianchi II
Here we apply the method described Sect. in 6.1 to find an approximate solution to the Einstein’s equations of the Polymer Bianchi II model. We start by selecting the \(V_B\) potential appropriate for Bianchi II (60) and we substitute it in the Hamiltonian (56) (in the following we will always assume the time gauge \(N=1\)).
Now we find an approximate solution to the system (97), with the only assumption that \(\mu \) is little compared to the cubic root of the Universe volume. We are entitled then to exploit the standard perturbation theory and expand the solution until the first nonzero order in \(\mu \). Because \(\mu \) appears only squared \(\mu ^2\) in the Einstein’s Eq. (97), the perturbative expansion will only contain even powers of \(\mu \).
Equation (100) are two almost uncoupled nonhomogeneous ODEs. Equation (100a) is a linear second order nonhomogeneous ODE with nonconstant coefficients and, being completely uncoupled, it can be solved straightly. Equation (100b) is solved by mere substitution of the solution of (100a) in it and subsequent double integration.
To solve (100a) we exploited a standard method that can be found, for example, in [67, Lesson 23]. This method is called reduction of order method and, in the case of a second order ODE, can be used to find a particular solution once any non trivial solution of the related homogeneous equation is known.
Now that we know \(q_l^0\), \(q_l^1\), \(q_m^0\) and \(q_m^1\), the complete solution for \(q_l\), \(q_m\) and \(q_n\) is found through (98) and (99) by mere substitution.
7 Polymer BKL map
In this last section, we calculate the Polymer modified BKL map and study some of its properties.
In Sect. 7.1 the Polymer BKL map on the Kasner indices is derived while in Sect. 7.2 some noteworthy properties of the map are discussed.
Finally in Sect. 7.3 the results of a simple numerical simulation of the Polymer BKL map over many iterations are presented and discussed.
7.1 Polymer BKL map on the Kasner indices
Here we use the method outlined in Sects. 6.1 to 6.3 to directly calculate the Polymer BKL map on the Kasner indices.
First, we look at the asymptotic limit at \(\pm \infty \) for the solution of Polymer Bianchi II (98). As in the standard case, the Polymer Bianchi II model “links together” two Kasner epochs at plus and minus infinity. In this sense, the dynamics of Polymer Bianchi II is not qualitatively different from the standard one. We will tell the quantities at plus and minus infinity apart by adding a prime \(\Box '\) to the quantities at minus infinity and leaving the quantities at plus infinity unprimed. The two Kasner solutions at plus and minus infinity can be still parametrized according to (23).
where the functions \(f_{l,m,n,\varLambda }\) and \(f'_{l,m,n,\varLambda }\) correspond to the r.h.s. of (106) and (107) respectively and \(\varvec{c}\) is a shorthand for the set \(\{c_1,\dots ,c_{12}\}\).
Now, in complete analogy with (35), we look for an asymptotic condition. We don’t need to solve the whole system (108), but only to find a relation between the old and new Kasner indices and \(\varLambda \) at the first order in \(\mu ^2\). In practice, not all of the (108) relations are actually needed to find an asymptotic condition.
Now, we need other three conditions to derive the Polymer BKL map. One is provided by the sum of the primed Kasner indices at minus infinity. This is the very same both in the standard case (24) and in the Polymer case.
7.2 BKL map properties
Now, we study the asymptotic behavior of the polymer BKL map (116a) for u going to infinity. This limit is relevant to us for two reasons. Firstly, In Sect. 5.1 we already pointed out that infinitely many bounces of the \(\beta \)point against the potential walls happen until the singularity is reached. This means that the Polymer BKL map is to be iterated infinitely many times.
Secondly, we suppose the Polymer BKL map [or at least its u portion (116a)] to be ergotic, so that every open set of the parameter space is visited with non null probability. This assumption is backed up by the observation that the Polymer BKL map (116a) tends asymptotically to the standard BKL map (39) (that is ergodic), as shown in the following Sect. 7.3 through a numerical simulation.
The plateau (117) is higher and steeper the more Q is close to zero. This physically means that there is a sort of “centripetal potential” that is driving the \(\beta \)point off the corners and towards the center of the triangle. We infer that this “centripetal potential” is somehow linked to the velocitylike quantity \(v(\tau )\) that appears in the polymer Bianchi II Einstein’s Eq. (97). Because of this plateau, the mechanism that drives the Universe away from the corner, implicit in the standard BKL map (39), seems to be much more efficient in the quantum case. For \(Q\rightarrow 0\), the plateau tends to disappear: \(\lim _{\begin{array}{c} u\rightarrow \infty \\ Q\rightarrow 0 \end{array}} u'(u,Q) = \infty \).
As a side note, we remember that it has not been proved analytically that the standard BKL map is still valid deep inside the corners. As a matter of fact, the BKL map can be derived analytically only for the very center of the edges of the triangle of Fig. 2, because those are the only points where the Bianchi IX potential is exactly equal to the Bianchi II potential. The farther we depart from the center of the edges, the more the map looses precision. At any rate, there are some numerical studies for the Bianchi IX model [68, 69] that show how the standard BKL map is valid with good approximation even inside the corners.
The analysis of the behavior of the Polymer BKL map for the Q parameter (116b) is more convoluted. Little can be said analytically about the overall behavior across multiple iterations because of its evident complexity. For this reason, in Sect. 7.3 we discuss the results of a simple numerical simulation that probes the behavior of the map (116) over many iterations.
The most important point to check is if the dominion of definition (95) for Q is preserved by the map. We remember that for \(Q \approx 1\) the perturbation theory, by the means of which we have derived the map, is not valid anymore. So every result in that range is to be taken with a grain of salt.
That said, as soon as we consider the behavior of u, and particularly the plateau of Fig. 7, we notice that the Universe cannot “indulge” much time in “bigu regions”: even if it happens to assume a huge value for u, this is immediately dampened to a much smaller value at the next iteration. The net result is that Q is almost always decreasing as it is also strongly suggested by the numerical simulation of Sect. 7.3.
Summarizing, the Polymer BKL map on Q (116b), apart from a small set of initial conditions in the region where the perturbation theory is failing \(Q \approx 1\), preserves the dominion of definition of Q (95) and is decreasing at almost any iteration.
7.3 Numerical simulation

An initial couple of values for (u, Q) is chosen inside the dominion \(u\in [1,\infty )\) and \(Q\in [0,0.96]\).

We remember from Sects. 3.3 and 6.2 that, at the end of each Kasner era, the u parameter becomes smaller than 1 and needs to be remapped to values greater than 1. This marks the beginning of a new Kasner era. This remapping is performed through the relations listed on page 15. In the standard case, because the standard BKL map is just \(u \rightarrow u  1\), the u parameter cannot, for any reason, become smaller than 0. For the Polymer Bianchi IX, however, it is possible for u to become less than zero. This is why we have derived many remapping relations, to cover the whole real line.

Many values (\(\approx 2^{18}\)) for the initial conditions, randomly chosen in the interval of definition, were tested. We didn’t observe any “anomalous behavior” in any element of the sample. This meaning that all points converged asymptotically to the standard BKL map as is discussed in the following.

The polymer BKL map (116) is “well behaved” for any tested initial condition \(u\in [1,\infty )\) and \(Q\in [0,0.96]\): the dominion of definition for u and Q are preserved.

The line \(Q = 0\) is an attractor for the Polymer BKL map: for any tested initial condition, the Polymer BKL map eventually evolved until becoming arbitrarily close to the standard BKL map.

The Polymer BKL map on Q is almost everywhere decreasing. It can happen that, especially for initial values \(Q \approx 1\), for very few iterations the map on Q is increasing, but the overall behavior is almost always decreasing. The probability to have an increasing behavior of Q gets smaller at every iteration. In the limit of infinitely many iterations this probability goes to zero.

Since the Polymer BKL map (116) tends asymptotically to the standard BKL map (39), we expect that the notion of chaos for the standard BKL map, given for example in [56], can be applied, with little or no modifications, to the Polymer case, too (although this has not been proven rigorously).
8 Physical considerations
In this section we address two basic questions concerning the physical link between Polymer and Loop quantization methods and what happens when all the Minisuperspace variables are discrete, respectively.
First of all, we observe that Polymer Quantum Mechanics is an independent approach from Loop Quantum Gravity. Using the Polymer procedure is equivalent to implement a sort of discretization of the considered configurational variables. Each variable is treated separately, by introducing a suitable graph (de facto a onedimensional lattice structure): the group of spatial translations on this graph is a U(1) type, therefore the natural group of symmetry underlying such quantization method is U(1) too, differently from Loop Quantum Gravity, where the basic group of symmetry is SU(2).
It is important to stress that Polymer Quantum Mechanics is not unitary connected to the standard Quantum Mechanics, since the Stone  Von Neumann theorem is broken in the discretized representation. Even more subtle is that Polymer quantization procedures applied to different configurational representations of the same system are, in general, not unitary related. This is clear in the zero order WKB limit of the Polymer quantum dynamics, where the formulations in two sets of variables, which are canonically related in the standard Hamiltonian representation, are no longer canonically connected in the Polymer scenario, mainly due to the nontrivial implications of the prescription for the momentum operator.
When the Loop Quantum Gravity [18, 70] is applied to the Primordial Universe, due to the homogeneity constraint underlying the Minisuperspace structure, it loses the morphology of a SU(2) gauge theory (this point is widely discussed in [71, 72]) and the construction of a kinematical Hilbert space, as well as of the geometrical operators, is performed by an effective, although rigorous, procedure. A discrete scale is introduced in the Holonomy definition, taken on a square of given size and then the curvature term, associated to the AshtekarBarberoImmirzi connection, (the socalled Euclidean term of the scalar constraint) is evaluated on such a square path. It seems that just in this step the Loop Quantum Cosmology acquires the features of a polymer graph, associated to an underlying U(1) symmetry. The real correspondence between the two approaches emerges in the semiclassical dynamics of the Loop procedure [46], which is isomorphic to the zero order WKB limit of the Polymer quantum approach. In this sense, the Loop Quantum Cosmology studies legitimate the implementation of the Polymer formalism to the cosmological Minisuperspace.
However, if on one hand, the Polymer quantum cosmology predictions are, to some extent, contained in the Loop Cosmology, on the other hand, the former is more general because it is applicable to a generic configurational representation, while the latter refers specifically to the AshtekarBerberoImmirzi connection variable.
Thus, the subtle question arises about which is the proper set of variables in order that the implementation of the Polymer procedure mimics the Loop treatment, as well as which is the physical meaning of different Polymer dynamical behaviours in different sets of variables. In [73] it is argued, for the isotropic Universe quantization, that the searched correspondence holds only if the cubed scale factor is adopted as Polymer variable: in fact this choice leads to a critical density of the Universe which is independent of the scale factor and a direct link between the Polymer discretization step and the Immirzi parameter is found. This result assumes that the Polymer parameter is maintained independent of the scale factor, otherwise the correspondence above seems always possible. In this respect, different choices of the configuration variables when Polymer quantizing a cosmological system could be mapped into each other by suitable redefinition of the discretization step as a function of the variables themselves. Here we apply the Polymer procedure to the Misner isotropic variable and not to the cubed scale factor, so that different issues with respect to Loop Quantum Gravity can naturally emerge. The merit of the present choice is that we discretize the volume of the Universe, without preventing its vanishing behavior. This can be regarded as an effective procedure to include the zerovolume eigenvalue in the system dynamics, like it can happen in Loop Quantum Gravity, but it is no longer evident in its cosmological implementation.
Thus, no real contradiction exists between the present study and the BigBounce prediction of the Loop formulation, since they are expectedly two independent physical representations of the same quantum system. As discussed on the semiclassical level in [74], when using the cubed scale factor as isotropic dynamics, the Mixmaster model becomes nonsingular and chaos free, just as predicted in the Loop Quantum Cosmology analysis presented in [27]. However, in such a representation, the vanishing behavior of the Universe volume is somewhat prevented a priori, by means of the Polymer discretization procedure.
Finally, we observe that, while in the present study, the Polymer dynamics of the isotropic variable \(\alpha \) preserves (if not even enforces) the Mixmaster chaos, in [30] the Polymer analysis for the anisotropic variables \(\beta _{\pm }\) is associated to the chaos disappearance. This feature is not surprising since the two approaches have a different physical meaning: the discretization of \(\alpha \) has to do with geometrical properties of the spacetime (it can be thought as an embedding variable), while the implementation of the Polymer method to \(\beta _{\pm }\) really affects the gravitational degrees of freedom of the considered cosmological model.
Nonetheless, it becomes now interesting to understand what happens to the Mixmaster chaotic features when the two sets of variables (\(\alpha \) and \(\beta _\pm \)) are simultaneously Polymer quantized. In the following subsection, we provide an answer to such an intriguing question, at least on the base of the semiclassical dynamics.
8.1 The polymer approach applied to the whole Minisuperspace
9 Conclusions
In the present study, we analyzed the Mixmaster model in the framework of the semiclassical Polymer Quantum Mechanics, implemented on the isotropic Misner variable, according to the idea that the cutoff physics mainly concerns the Universe volume.
We developed a semiclassical and quantum dynamics, in order to properly characterize the structure of the singularity, still present in the model. The presence of the singularity is essentially due to the character of the isotropic variable conjugate momentum as a constant of motion.
On the semiclassical level we studied the system evolution both in the Hamiltonian and field equations representation, generalizing the two original analyses in [7, 8], respectively. The two approaches are converging and complementary, describing the initial singularity of the Mixmaster model as reached by a chaotic dynamics, that is, in principle, more complex than the General Relativity one, but actually coincides with it in the asymptotic limit.
This issue is a notable feature, since in Loop Quantum Cosmology is expected that the Bianchi IX model is chaosfree [26, 27] and it is well known [28] that the Polymer semiclassical dynamics closely resembles the average feature of a Loop treatment in the Minisuperspace. However, we stress that, while the existence of the singularity in Polymer Quantum Mechanics appears to be a feature depending on the nature of the adopted configuration variables, nonetheless the properties of the Poincaré map of the model is expected to be a solid physical issue, independent on the particular representation adopted for the system.
The canonical quantization of the Mixmaster Universe that we performed in the full Polymer quantum approach, i.e. writing down a Wheeler–DeWitt equation in the momentum representation, accordingly to the socalled Continuum Limit discussed in [29], is completely consistent with the semiclassical results. In fact, the Misner demonstration, for the standard canonical approach, that states with high occupation numbers can survive to the initial singularity, remains still valid in the Polymer formulation, here presented. This issue confirms that the cutoff we introduced in the configuration space on the isotropic Misner variable does not affect the cosmological character of the Mixmaster model.
This result appears rather different from the analysis in [30], where the polymer approach has been addressed for the anisotropic Misner variables (the real degrees of freedom of the cosmological gravitational field) with the emergence of a nonchaotic cosmology. Such a discrepancy suggests that the Polymer regularization of the asymptotic evolution to the singularity produces more profound modifications when it touches physical properties than geometrical features. Actually, the isotropic Misner variable can be suitably interpreted as a good time variable for the system (an embedding variable in the language of [75, 76]), while the Universe anisotropies provide a precise physical information on the considered cosmological model.
Despite this consideration about the gaugelike nature of the Misner isotropic variable, which shed light on the physics of our dynamical results, nonetheless we regard as important to perform further investigations on the nature of the singularity when other variables are considered to characterize the Universe volume, since we expect that, for some specific choice the regularization of the BigBang to the BigBounce must take place (see for instance [74]). However, even on the basis of the present analysis, we suggest that the features of the Poincaré map of the Bianchi IX model and then of the generic cosmological singularity (locally mimicked by the same Bianchi IXlike time evolution) is a very general and robust property of the primordial Universe, not necessarily connected with the existence of a cutoff physics on the singularity.
Footnotes
 1.
In this case the word “semiclassical” means that our superHamiltonian constraint was obtained as the lowest order term of a WKB expansion for \(\hbar \rightarrow 0\).
Notes
Acknowledgements
G.P. would like to thank Evgeny Grines(Lobachevsky State University of Nizhny Novgorod, Department of Control Theory and Dynamics of Systems) for the valuable assistance in determining some of the mathematical properties of the Polymer BKL map.
We would also like to thank Eleonora Giovannetti for her contribution to the analysis of Sect. 8
References
 1.L.D. Landau, E.M. Lifshitz, The Classical Theory of Fields, Fourth Edition: Volume 2 (Course of Theoretical Physics Series), 4th edn. (ButterworthHeinemann, 1980). http://www.worldcat.org/isbn/0750627689
 2.C. Misner, K. Thorne, J. Wheeler, Gravitation (W. H. Freeman, London, 1973)Google Scholar
 3.L. Grishchuk, A. Doroshkevich, V. Iudin, Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki 69, 1857 (1975)ADSGoogle Scholar
 4.A. Doroshkevich, V. Lukash, I. Novikov, Sov. Phys. JETP 37, 739 (1973)ADSGoogle Scholar
 5.A.A. Kirillov, G. Montani, Phys. Rev. D 66, 064010 (2002). https://doi.org/10.1103/PhysRevD.66.064010 CrossRefADSGoogle Scholar
 6.G. Montani, M.V. Battisti, R. Benini, G. Imponente, Int. J. Mod. Phys. A 23(16n17), 2353 (2008). https://doi.org/10.1142/S0217751X08040275 CrossRefADSGoogle Scholar
 7.V. Belinskii, I. Khalatnikov, E. Lifshitz, Adv. Phys. 19(80), 525 (1970). https://doi.org/10.1080/00018737000101171 CrossRefADSGoogle Scholar
 8.C.W. Misner, Phys. Rev. 186, 1319 (1969). https://doi.org/10.1103/PhysRev.186.1319 CrossRefADSGoogle Scholar
 9.C.W. Misner, Phys. Rev. Lett. 22, 1071 (1969). https://doi.org/10.1103/PhysRevLett.22.1071 CrossRefADSGoogle Scholar
 10.V. Belinskii, I. Khalatnikov, E. Lifshitz, Adv. Phys. 31(6), 639 (1982). https://doi.org/10.1080/00018738200101428 CrossRefADSGoogle Scholar
 11.A. Kirillov, JETP 76(3), 355 (1993)ADSGoogle Scholar
 12.G. Montani, Class. Quant. Grav. 12(10), 2505 (1995). http://stacks.iop.org/02649381/12/i=10/a=010
 13.G. Montani, M. Battisti, R. Benini, Primordial Cosmology (World Scientific, 2011). http://books.google.it/books?id=XRLv7PVocC
 14.J.M. Heinzle, C. Uggla, Class. Quant. Grav. 26, 075016 (2009). https://doi.org/10.1088/02649381/26/7/075016 CrossRefADSGoogle Scholar
 15.J.D. Barrow, F.J. Tipler, Physics Reports 56(7), 371 (1979). https://doi.org/10.1016/03701573(79)900978. http://www.sciencedirect.com/science/article/pii/0370157379900978
 16.B.S. DeWitt, Phys. Rev. 160, 1113 (1967). https://doi.org/10.1103/PhysRev.160.1113 CrossRefADSGoogle Scholar
 17.R. Graham, H. Luckock, Phys. Rev. D 49, 2786 (1994). https://doi.org/10.1103/PhysRevD.49.2786 MathSciNetCrossRefADSGoogle Scholar
 18.F. Cianfrani, O. Lecian, M. Lulli, G. Montani, Canonical Quantum Gravity: Fundamentals and Recent Developments (World Scientific, 2014). https://books.google.co.jp/books?id=OuF1ngEACAAJ
 19.C. Rovelli, L. Smolin, Nuclear Physics B 442(3), 593 (1995). https://doi.org/10.1016/05503213(95)00150Q. http://www.sciencedirect.com/science/article/pii/055032139500150Q
 20.A. Ashtekar, T. Pawlowski, P. Singh, Phys. Rev. Lett. 96(14), 141301 (2006)MathSciNetCrossRefADSGoogle Scholar
 21.A. Ashtekar, T. Pawlowski, P. Singh, Phys. Rev. D 73(12), 124038 (2006)MathSciNetCrossRefADSGoogle Scholar
 22.G. Montani, A. Marchi, R. Moriconi, Phys. Lett. B 777, 191 (2018). https://doi.org/10.1016/j.physletb.2017.12.016 CrossRefADSGoogle Scholar
 23.J.D. Barrow, C. Ganguly, Phys. Rev. D 95(8), 083515 (2017). https://doi.org/10.1103/PhysRevD.95.083515 MathSciNetCrossRefADSGoogle Scholar
 24.A. Ashtekar, A. Henderson, D. Sloan, Class. Quant. Grav. 26, 052001 (2009). https://doi.org/10.1088/02649381/26/5/052001 CrossRefADSGoogle Scholar
 25.A. Ashtekar, A. Henderson, D. Sloan, Phys. Rev. D 83, 084024 (2011). https://doi.org/10.1103/PhysRevD.83.084024 CrossRefADSGoogle Scholar
 26.M. Bojowald, G. Date, G.M. Hossain, Class. Quant. Grav. 21, 3541 (2004). https://doi.org/10.1088/02649381/21/14/015 CrossRefADSGoogle Scholar
 27.M. Bojowald, G. Date, Phys. Rev. Lett. 92, 071302 (2004). https://doi.org/10.1103/PhysRevLett.92.071302 CrossRefADSGoogle Scholar
 28.A. Ashtekar, J. Lewandowski, Class. Quant. Grav. 18, L117 (2001). https://doi.org/10.1088/02649381/18/18/102 CrossRefADSGoogle Scholar
 29.A. Corichi, T. Vukasinac, J.A. Zapata, Phys. Rev. D 76, 044016 (2007). https://doi.org/10.1103/PhysRevD.76.044016 CrossRefADSGoogle Scholar
 30.O.M. Lecian, G. Montani, R. Moriconi, Phys. Rev. D 88(10), 103511 (2013). https://doi.org/10.1103/PhysRevD.88.103511 CrossRefADSGoogle Scholar
 31.R. Moriconi, G. Montani, Phys. Rev. D 95(12), 123533 (2017). https://doi.org/10.1103/PhysRevD.95.123533 MathSciNetCrossRefADSGoogle Scholar
 32.M.V. Battisti, O.M. Lecian, G. Montani, Phys. Rev. D 78, 103514 (2008). https://doi.org/10.1103/PhysRevD.78.103514 MathSciNetCrossRefADSGoogle Scholar
 33.G.M. Hossain, V. Husain, S.S. Seahra, Phys. Rev. D 81(2), 024005 (2010)CrossRefADSGoogle Scholar
 34.G. De Risi, R. Maartens, P. Singh, Phys. Rev. D 76(10), 103531 (2007)CrossRefADSGoogle Scholar
 35.S.M. Hassan, V. Husain, Class. Quant. Grav. 34(8), 084003 (2017)CrossRefADSGoogle Scholar
 36.S.S. Seahra, I.A. Brown, G.M. Hossain, V. Husain, J. Cosmol. Astropart. Phys. 2012(10), 041 (2012)CrossRefGoogle Scholar
 37.J. Sakurai, J. Napolitano, Modern Quantum Mechanics (AddisonWesley, 2011). https://books.google.it/books?id=N4IAQAACAAJ
 38.C. Rovelli, S. Speziale, Phys. Rev. D 67, 064019 (2003). https://doi.org/10.1103/PhysRevD.67.064019 MathSciNetCrossRefADSGoogle Scholar
 39.A.A. Kirillov, G. Montani, Phys. Rev. D 56, 6225 (1997). https://doi.org/10.1103/PhysRevD.56.6225 CrossRefADSGoogle Scholar
 40.G. Imponente, G. Montani, Phys. Rev. D 63, 103501 (2001). https://doi.org/10.1103/PhysRevD.63.103501 MathSciNetCrossRefADSGoogle Scholar
 41.H. Salecker, E.P. Wigner, Phys. Rev. 109, 571 (1958). https://doi.org/10.1103/PhysRev.109.571 MathSciNetCrossRefADSGoogle Scholar
 42.G. AmelinoCamelia, Phys. Lett. B 477, 436 (2000). https://doi.org/10.1016/S03702693(00)002318 CrossRefADSGoogle Scholar
 43.S. Hossenfelder, Living Rev. Rel 16, 2 (2013). https://doi.org/10.12942/lrr20132 CrossRefGoogle Scholar
 44.A. Ashtekar, P. Singh, Class. Quant. Grav. 28, 213001 (2011). https://doi.org/10.1088/02649381/28/21/213001 CrossRefADSGoogle Scholar
 45.J.E. Lidsey, D. Wands, E.J. Copeland, Phys. Rept. 337, 343 (2000). https://doi.org/10.1016/S03701573(00)000648 CrossRefADSGoogle Scholar
 46.A. Ashtekar, S. Fairhurst, J.L. Willis, Class. Quant. Grav. 20, 1031 (2003). https://doi.org/10.1088/02649381/20/6/302 CrossRefADSGoogle Scholar
 47.A. Ashtekar, J. Lewandowski, H. Sahlmann, Class. Quant. Grav. 20(1), L11 (2003). URL http://stacks.iop.org/02649381/20/i=1/a=103
 48.M. Varadarajan, Phys. Rev. D 61, 104001 (2000). https://doi.org/10.1103/PhysRevD.61.104001 MathSciNetCrossRefADSGoogle Scholar
 49.A. Corichi, T. Vukasinac, J.A. Zapata, Class. Quant. Grav. 24, 1495 (2007). https://doi.org/10.1088/02649381/24/6/008 CrossRefADSGoogle Scholar
 50.E. Binz, S. Pods, The Geometry of Heisenberg Groups: With Applications in Signal Theory, Optics, Quantization, and Field Quantization. Mathematical surveys and monographs (American Mathematical Society, 2008). https://books.google.co.jp/books?id=yIP0BwAAQBAJ
 51.W. Arveson, An Invitation to C*Algebras. Graduate Texts in Mathematics (Springer New York, 1998). https://books.google.co.jp/books?id=zRep59fhoGkC
 52.W. Rudin, Fourier Analysis on Groups (Wiley, 2011). https://books.google.it/books?id=k3RNtFSBH8gC
 53.J. Wainwright, A. Krasiński, Gen Relativ. Gravit. 40(4), 865 (2008). https://doi.org/10.1007/s1071400705744 CrossRefADSGoogle Scholar
 54.W. contributors. BKL Singularity—Wikipedia, The Free Encyclopedia (2017). https://en.wikipedia.org/w/index.php?title=BKL_singularity&oldid=799333893. [Online; accessed 26December2017]
 55.J.D. Barrow, Phys. Rev. Lett. 46, 963 (1981). https://doi.org/10.1103/PhysRevLett.46.963 MathSciNetCrossRefADSGoogle Scholar
 56.J.D. Barrow, Phys. Rep. 85(1), 1 (1982). https://doi.org/10.1016/03701573(82)901715. http://www.sciencedirect.com/science/article/pii/0370157382901715
 57.I.M. Khalatnikov, E.M. Lifshitz, K.M. Khanin, L.N. Shchur, YaG Sinai, Fundam. Theor. Phys. 9, 343 (1984). https://doi.org/10.1007/9789400964693_18 CrossRefGoogle Scholar
 58.E.M. Lifshitz, I.M. Lifshitz, I.M. Khalatnikov, Soviet J. Exp. Theor. Phys. 32, 173 (1971)ADSGoogle Scholar
 59.M. Szydlowski, A. Krawiec, Phys. Rev. D 47(12), 5323 (1993)MathSciNetCrossRefADSGoogle Scholar
 60.R. Arnowitt, S. Deser, C.W. Misner, Phys. Rev. 117, 1595 (1960). https://doi.org/10.1103/PhysRev.117.1595 MathSciNetCrossRefADSGoogle Scholar
 61.A. Ashtekar, T. Pawlowski, P. Singh, K. Vandersloot, Phys. Rev. D 75, 024035 (2007). https://doi.org/10.1103/PhysRevD.75.024035 MathSciNetCrossRefADSGoogle Scholar
 62.A. Ashtekar, E. WilsonEwing, Phys. Rev. D 79, 083535 (2009). https://doi.org/10.1103/PhysRevD.79.083535 MathSciNetCrossRefADSGoogle Scholar
 63.M.V. Battisti, Submitted to: Phys. Rev. D (2008)Google Scholar
 64.M. Battisti, G. Montani, Phys. Lett. B 681, 179 (2009). https://doi.org/10.1016/j.physletb.2009.10.003 CrossRefADSGoogle Scholar
 65.G. Montani, Int. J. Mod. Phys. D 13, 1029 (2004). https://doi.org/10.1142/S0218271804004967 CrossRefADSGoogle Scholar
 66.Y. Elskens, M. Henneaux, Nuclear Phys. B 290(Supplement C), 111 (1987). https://doi.org/10.1016/05503213(87)901805. http://www.sciencedirect.com/science/article/pii/0550321387901805
 67.M. Tenenbaum, H. Pollard, Ordinary Differential Equations. Dover Books on Mathematics Series (Dover Publications, Incorporated, 2012). https://books.google.it/books?id=29utVed7QMIC
 68.A.R. Moser, R.A. Matzner, M.P. Ryan, Ann. Phys. 79(2), 558 (1973). https://doi.org/10.1016/00034916(73)900973. http://www.sciencedirect.com/science/article/pii/0003491673900973
 69.B.K. Berger, Living Rev. Relativ. 5(1), 1 (2002). https://doi.org/10.12942/lrr20021 MathSciNetCrossRefADSGoogle Scholar
 70.T. Thiemann, in Approaches to Fundamental Physics (Springer, Berlin, 2007), pp. 185–263Google Scholar
 71.F. Cianfrani, G. Montani, Phys. Rev. D 82, 021501 (2010). https://doi.org/10.1103/PhysRevD.82.021501 CrossRefADSGoogle Scholar
 72.F. Cianfrani, G. Montani, Phys. Rev. D 85, 024027 (2012). https://doi.org/10.1103/PhysRevD.85.024027 CrossRefADSGoogle Scholar
 73.C. Mantero, G. Montani, F. Bombacigno (2018). arXiv:1806.10364
 74.S. Antonini, G. Montani (2018). arXiv:1808.01304
 75.C.J. Isham, K.V. Kuchar, Ann. Phys. 164(2), 288 (1985)CrossRefADSGoogle Scholar
 76.C.J. Isham, K.V. Kuchar, Ann. Phys. 164(2), 316 (1985)CrossRefADSGoogle Scholar
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