Covariant Integral Quantizations and Their Applications to Quantum Cosmology

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Acta Polytechnica 56(3):173–179, 2016 © Czech Technical University in Prague, 2016 doi:10.14311/AP.2016.56.0173 available online at http://ojs.cvut.cz/ojs/index.php/ap

COVARIANT INTEGRAL QUANTIZATIONS AND THEIR APPLICATIONS TO QUANTUM COSMOLOGY

Jean Pierre Gazeau

Astroparticle and Cosmology, Univ Paris Diderot, Sorbonne Paris Cité, Paris 75205, France correspondence: [email protected]

Abstract. We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on group representation and probabilistic aspects of these constructions. Simple phase space examples illustrate the procedure: plane (Weyl-Heisenberg symmetry), half-plane (aﬃne symmetry). Interesting applications to quantum cosmology (“smooth bouncing”) for Friedmann-Robertson-Walker metric are presented and those for Bianchi I and IX models are mentioned. Keywords: integral quantization; covariance; POVM; aﬃne group; Weyl-Heisenberg group; coherent states; FRW model; smooth bouncing.

1. Introduction define POVM-based integral quantizations of functions defined on a measure space, their subsequent Group theory is one of the favorite domains of J. Pa- semi-classical counterparts (i.e., lower symbols) and tera and P. Winternitz. This contribution to the cele- discuss their possibilities when they are compared bration of the Jiri and Pavel 80th birthday emphasizes with other methods. Covariant integral quantizations the role of group representation theory in a certain are then considered when the measure space is homo- type of quantization procedures. These methods [1–9] geneous for some group, and we give the example of are named (covariant) integral quantizations and offer the half-plane viewed as the affine group of the real a large scope of possibilities for mapping a classical line. Section 4 is devoted to another example, the mathematical model of a physical system to a quan- integral quantization of functions on the plane viewed tum model. They are based on normalized positive as a coset of the Weyl-Heisenberg group. In Section 5 operator-valued measures (POVM) and present at we illustrate the method with recent applications to least two attractive advantages: quantum cosmology where the initial singularity is (1.) By employing all resources of integration and regularized by using the affine integral quantization. distributions, they are most appropriate when we Finally we sketch in Section 6 a more philosophical have to deal with some geometric singularities. point of view on the approaches described in this contribution. (2.) They afford a semi-classical phase space portrait A large part of the content has already been pub- of quantum states and quantum dynamics together lished, as exemplified by the above citations. On the with a probabilistic interpretation. other hand the original side of the paper lies in the Moreover, their recent applications to quantum cos- synthetic presentation of the method together with mology [10–15] has yielded interesting results as: some new ideas like the general (and promising) construction sketched in Section 3.2. • consistent quantum dynamics of isotropic, anisotropic non-oscillatory and anisotropic models of 2. Universe; What is quantization? 2.1. • singularity resolution; Canonical quantization The basic procedure of quantization of a classical me- • unitary dynamics without boundary conditions; chanical model is well known and is named “canonical" • consistent semi-classical description of involved (Heisenberg, Born, Jordan, Dirac, Weyl, etc). Starting quantum dynamics. from a phase space or symplectic manifold, e.g. R2, e.g. the phase space for the motion of particle on the The aim of the present article is to give an overview line, it consists in the maps of the methods, their main characteristics, and their 2 promising applications to early cosmology. In Sec- R 3 (q, p), {q, p} = 1 7→ self-adjoint (Q, P ), (1) tion 2 we give a rapid survey of canonical quantiza- with the canonical commutation rule (ccr) [Q, P ] = tion with the problems raised by this procedure, and i I, and we define a few basic requirements that any quan- ~ tization procedure should fullfill. In Section 3 we f(q, p) 7→ f(Q, P ) 7→ (Sym f)(Q, P ). (2)

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Beyond the ordering problem [16–18] raised by the sec- (3.) Reality — A real f is mapped to a self-adjoint ond step (symmetrisation), one should keep in mind operator Af in H or, at least, a symmetric operator, that [Q, P ] = i~I holds true with self-adjoint Q, P , (4.) Covariance — If X is endowed with symmetry, only if both have continuous spectrum (−∞, +∞), the latter should be preserved in a “certain” sense. and there is uniqueness of the solution, up to unitary equivalence (von Neumann). But then what about Of course, further requirements are necessarily singular f, e.g., the angle arctan(p/q)? What about added, depending on the mathematical structures barriers or other impassable boundaries? The mo- equipping X and C(X) (e.g., measure, topology, man- tion on a circle? In a bounded interval? On the ifold, closure under algebraic operations, time evolu- half-line? Despite their elementary aspects, these ex- tion or dynamics, etc). Moreover, a physical inter- amples leave open many questions both on mathemat- pretation should be advanced about measurement of ical and physical levels, irrespective of the manifold spectra of classical f ∈ C(X) or quantum Af ∈ A(H) quantization procedures, like Path Integral Quantiza- to which are given the status of observables. Finally, tion (Feynman, thesis, 1942), Geometric Quantization an unambiguous classical limit of the quantum physi- with Weyl (1927), Groenewold (1946), Kirillov (1961), cal quantities should exist, the limit operation being Souriau (1966), Kostant (1970), Deformation Quan- associated to a change of scale. tization with Moyal (1947), Bayen, Flato, Fronsdal, Lichnerowicz, Sternheimer (1978), Fedosov (1985), 3. Integral Quantization(s) Kontsevich (2003), Coherent state or anti-Wick or 3.1. Integral quantization: Toeplitz quantization with Klauder (1961), Berezin general setting and POVM (1974). Let (X, ν) be a measure space. Suppose that there Of course, the canonical procedure is universally exists an X-labeled family of bounded operators on a accepted in view of its numerous experimental valida- Hilbert space H resolving the identity I: tions, one of the most famous and simplest one going back to the early period of Quantum Mechanics with Z X 3 x 7→ M(x) ∈ L(H), M(x) dν(x) = I, (4) the quantitative prediction of the isotopic effect in X vibrational spectra of diatomic molecules [19]. These where the equality holds in a weak sense. The case data validated the canonical quantization, contrary we are specially interested in occurs when the M(x)’s to the Bohr-Sommerfeld ansatz (which predicts no are positive semi-definite and unit trace, i.e., when isotopic effect). Nevertheless this does not prove that another method of quantization fails to yield the same M(x) ≡ ρ(x) (density operator). (5) prediction [3]. Moreover, as already mentioned above, the canonical quantization is difficult, if not impos- Then, if X is a space with suitable topology, the map sible, in many circumstances. As a matter of fact, Z the canonical or the Weyl-Wigner integral quantiza- B(X) 3 ∆ 7→ ρ(x) dν(x) (6) ∆ tion maps f(q) to f(Q) (resp. f(p) to f(P )), and so are unable to cure any kind of classical singularity. may define a normalized positive operator-valued mea- Clearly, quantization procedures may be intractable sure (POVM) on the σ-algebra B(X) of Borel sets. when different phase space geometries and/or topolo- The quantization of complex-valued functions f(x) gies are considered. Self-adjointness of basic operators on X is the linear map: is not guaranteed anymore. Z f 7→ Af = M(x)f(x) dν(x), (7) 2.2. Quantization: minimal requirements X In our viewpoint, any quantization of a set X (e.g., understood as the sesquilinear form, a phase space or something else) and functions on it Z should meet four basic requirements: Bf (ψ1, ψ2) = hψ1|M(x)|ψ2if(x) dν(x), (8) X (1.) Linearity — it is a linear map defined on a dense subspace of H. If f is real and at least semi-bounded, and if the M(x)’s are posi- Q : C(X) 7→ A(H), Q(f) ≡ Af , (3) tive operators, then the Friedrich’s extension of Bf where: univocally defines a self-adjoint operator. If f is not • C(X) is a vector space of complex-valued func- semi-bounded, there is no natural choice of a self- tions f(x) on set X, i.e., a “classical” mathemat- adjoint operator associated with Bf . We then need ical model; more information on H to solve this subtlety. Covariance implies symmetry. Symmetry suggests • A(H) is a vector space of linear operators (ignor- a mathematical group G. A symmetry structure on ing domain limitations) in some complex Hilbert X presupposes that some group G acts on X. Our space H, i.e, a “quantum” mathematical model. hypothesis here is that X is an homogeneous space (2.) Unity — The map (3) is such that the function for G, which means that X ∼ G/H for H a subgroup f = 1 is mapped to the identity operator I on H. of G. We now examine two possible cases.

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3.1.1. X = G: quantizing the group integrable. We choose the following one. Let G be a Lie group with left invariant Haar measure eipx x dµ(g), and let g 7→ U(g) be a unitary irreducible L2( ∗ , dx) 3 ψ(x) 7→ U(q, p)ψ(x) = √ ψ R+ q q representation (UIR) of G in a Hilbert space H. Let (17) M be a bounded operator on H. Suppose that the As a density operator we choose the rank one pro- operator jector ρ = |ψihψ| where the fiducial vector, i.e., the Z ‘wavelet” ψ is in L2( , dx) ∩ L2( , dx/x) and its † R+ R+ R := M(g) dµ(g), M(g) := U(g)MU(g) , (9) affine transport produces the overcomplete family of G affine coherent states (ACS) is defined in a weak sense. From the left invariance |q, pi = U(q, p)|ψi. (18) of dµ(g) the operator R commutes with all operators U(g), g ∈ G, and so, from Schur’s Lemma, R = cMI The ACS integral quantization reads as with Z Z dq dp f(q, p) 7→ A = f(q, p)|q, pihq, p| (19) cM = tr ρ0M(g) dµ(g), (10) f c G R+×R ρ where the unit trace positive operator ρ0 is chosen R ∞ 2 dx with cρ = |ψ(x)| . More details and new devel- in order to make the integral convergent. If cM is 0 x finite and not zero, then we get the resolution of the opments are given in the recent [20]. Applications of identity: this method to the resolution of the singularity problem on a quantum level for a series of cosmological Z models (FRW, Bianchi I, Bianchi IX), for which the M(g) dν(g) = I, dν(g) := dµ(g)/cM. (11) G volume — expansion pair variables form the above half-plane, are given in the recent papers [10–15] and For this reason, the operator M to be U transported will be summarised in Section 5. will be designated (admissible) fiducial operator. The method becomes more apparent when the representa- 3.1.2. X = G/H: quantizing a non-trivial tion U is square-integrable in the following sense (for group coset example, it is the case for the affine group of the real In the absence of square-integrability over G (the line, see below). Suppose that there exists a density trivial illustration is the Weyl-Heisenberg group which operator ρ which is admissible in the sense that underlies in particular the canonical quantization, see Z Section 4), there exists a definition of square-integrable cρ = dµ(g) tr ρρ(g) < ∞, (12) representations with respect to a left coset manifold G X = G/H, with H a closed subgroup of G (it is the center in the Weyl-Heisenberg case), equipped with with ρ(g) = U(g)ρU †(g). Then the resolution of the a quasi-invariant measure ν [2]. For a global Borel identity (11) holds with M(g) and c = c . This M ρ section σ : X → G of the group, let ν be the unique allows a covariant integral quantization of complex- σ quasi-invariant measure defined by valued functions on the group dν (x) = λσ(x), x dν(x), (20) Z dµ(g) σ f 7→ Af = ρ(g)f(g) dν(g), dν(g) := , (13) where λ(g, x)dν(x) = dν(g−1x) for g ∈ G with G cρ † λ(g, x) obeying the cocycle condition λ(g1g2, x) = U(g)Af U (g) = AU(g)f (covariance), (14) −1 λ(g1, x)λ(g2, g1 x). Let U be a UIR which is square integrable mod(H) with an admissible ρ, i.e., where R 0 −1 0 c := tr(ρρ (x)) dν (x) < ∞, with ρ (x) = U(g)f(g ) := f(g g ) (15) ρ X σ σ σ U(σ(x))ρU(σ(x))†. Then we have the resolution of 2 is the regular representation if f ∈ L (G, dµ(g)). This the identity and the resulting quantization leads to a non-commutative version of the original 1 Z manifold structure for G. f 7→ Af = f(x)ρσ(x) dνσ(x) (21) cρ X Example: affine CS integral quantization In Covariance holds here too in a certain sense (see [2, this example, the group G is the affine group of the Chapter 11]). real line and is also viewed as the phase space for the Besides the Weyl-Heisenberg group, another exam- motion on the half-line, i.e., the half-plane {(q, p) ∈ ∗ ple concerns the motion on the circle for which G is R+ × R}. The group law is defined as the group of Euclidean displacements in the plane, i.e., 2 p the semi-direct product R oSO(2), and the subgroup (q, p)(q , p ) = qq , 0 + p , (16) 0 0 0 q H is isomorphic to R [22]. Other examples involve the relativity groups, Galileo, Poincaré [2], 1 + 1 Anti and the left invariant measure is the symplectic dq dp. de Sitter (unit disk and SU(1, 1))[21], 1 + 1 and 3 + 1 This group has two UIR’s which are both square de Sitter [23].

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3.2. An example of construction (iii)The method produces in essence a regularizing of original operator M or ρ effect, at the exception of certain choices, like the Let U be a UIR of G and $(x) be a function on the Weyl-Wigner (i.e., canonical) integral quantization. coset X = G/H. Suppose that it allows to define a (iv)The method, through POVM choices, offers the $ bounded operator Mσ on H through the operator- possibility to keep a fully probabilistic content. As valued integral a matter of fact, the Weyl-Wigner integral quanti- Z zation does not rest on a POVM. $ Mσ = $(x)U(σ(x)) dνσ(x). (22) X 4. Weyl-Heisenberg covariant Then, under appropriate conditions on the “weight” integral quantization(s) function $(σ(x)) such that U be a UIR which is The Weyl-Heisenberg group is defined as GWH = $ square integrable mod(H) and Mσ is admissible in { (s, z) | s ∈ R, z ∈ C } with multiplication law the above sense, the family of transported operators $ $ † (s, z)(s0, z0) = s + s0 + Im(zz¯0), z + z0 (26) Mσ (x) := U(σ(x))Mσ U (σ(x)) resolves the identity. Let H be a separable (complex) Hilbert space with 3.3. Semi-classical portraits orthonormal basis e0, e1, . . . , en ≡ |eni,... , and cor- Some quantization features, e.g., spectral properties responding lowering and raising operators defined in of Af , may be derived or at least well grasped from the usual way functional properties of the lower (Lieb) or covariant √ (Berezin) symbol (it generalizes Husimi function or a|eni = n|en−1i, a|e0i = 0, √ Wigner function) † † a |eni = n + 1|en+1i, [a, a ] = I, ˇ Af 7→ f(x) := tr M(x)Af , (23) It is well known that the Weyl-Heisenberg group has a unique non-trivial UIR U, up to unitary equivalence, When M = ρ (density operator) this new function each element in this equivalence class corresponding is the local average of the original f with respect to to a non-zero real number like the Planck constant. the probability distribution tr(ρ(x)ρ(x0)) Consider the center C = { (s, 0) | s ∈ R } of GWH. Z Then, the set X is the coset X = GWH/C ∼ C with ˇ 0 0 0 2 f(x) 7→ f(x) = f(x ) tr ρ(x)ρ(x ) dν(x ). (24) measure d z/π. Choosing the trivial section C 3 z 7→ X σ(z) = (0, z), to each z corresponds the (unitary) displacement (∼ Weyl) operator The Bargmann-Segal-like map f 7→ fˇ which, gen- † eralizes the Berezin or heat kernel transform when D(z) := U(σ(z)) = eza −za¯ , X = G is a Lie group, is in general a regularization −1 † of the original, possibly extremely singular, f, and D(−z) = (D(z)) = D(z) . (27) the classical limit itself means that, given one or more The noncommutativity of quantum mechanics is en- scale parameter(s) and a distance d(f, fˇ), we have (i) coded in the relation

d(f, fˇ) → 0 as → 0. (25) 0 1 (zz¯0−zz¯ 0) 0 (i) D(z)D(z ) = e 2 D(z + z )

¯0 0 3.4. Comments = e(zz −zz¯ )D(z0)D(z), (28) Beyond the freedom (think of analogy with Signal i.e., z 7→ D(z) is a projective representation of the Analysis where different techniques are complemen- abelian group C. The standard (i.e., Schrödinger- tary) allowed by integral quantization, the advantages Klauder-Glauber-Sudarshan) CS are defined as of the method with regard to other quantization procedures in use are of four types. |zi = D(z)|e0i. (29) (i)The minimal amount of constraints imposed on We now adopt the construction sketched in Section 3.2. the classical objects to be quantized. Let $(z) be a function on the complex plane obeying (ii)Once a choice of (positive) operator-valued mea- $(0) = 1. Suppose that it allows to define a fidu- sure has been made, which must be consistent with cial operator M$ on H through the operator-valued experiment, there is no ambiguity in the issue, con- integral Z d2z trarily to other method(s) in use (think in particular M$ = $(z)D(z) . (30) π of the ordering problem). To one classical model C corresponds one and only one quantum model. Of Then, the family of displaced M$(z) := course different choices of (P)OVM are requested to D(z)M$D(z)† under the unitary action D(z) resolves be physically equivalent, e.g., leading to the same the identity experimental predictions, if they are aimed to de- Z d2z M$(z) = I. (31) scribe one specific system. π C 176 vol. 56 no. 3/2016 Covariant Integral Quantizations and Quantum Cosmology

It is a direct consequence of D(z)D(z0)D(z)† = where a(t) is the scale factor and the ωi’s are right- 0 0 2 ezz¯ −zz¯ D(z0), of R ezξ¯−zξ¯ d ξ = πδ2(z), and of invariant dual vectors. The resolving of the Hamilto- C π $(0) = 1 with D(0) = I. The resulting quantiza- nian constraint leads to a model of singular universe tion map is given by analogous to a particle moving on the half-line (0, ∞). Indeed, canonical coordinates (q, p) can be introduced Z d2z in terms of the volume V ∼ a3 and the expansion rate f 7→ A$ = M$(z)f(z) f π V˙ /V ∼ a/a˙ through the relations V = q2/(1−w) and C −(1+w)/(1−w) Z d2z K = (3/8)(1 − w)pq . Clearly, q > 0 and = $(z)D(z)fs[f](z) , (32) p ∈ . Then the reduced Hamiltonian reads as π R C where are involved the symplectic Fourier transforms {q, p} = 1, h(q, p) = α(w)p2 + 6kq˜ β(w), q > 0. fs and its space reverse fs with k˜ = (R dω)2/3k, α(w) = 3(1−w)2/32 and β(w) = 2 Z ¯ d ξ 2(3w + 1)/(3(1 − w)). k = 0, −1 or 1 (in suitable unit f [f](z) = ezξ−zξ¯ f(ξ) , f [f](z) = f [f](−z) s π s s of inverse area) depending on whether the universe is C flat, open or closed. Assuming a closed universe with Both are unipotent fs[fs[f]] = f and fs[fs[f]] = f. radiation content w = 1/3 and k = +1, the classical We have the following covariance properties isotropy singularity is cured on the quantum level by • Translation covariance: using affine CS quantization [10]. Indeed the latter regularizes the kinetic term by adding a repulsive A$ = D(z )A$ D(z )†. (33) f(z−z0) 0 f(z) 0 centrifugal potential which prevents the system from • Parity covariance reaching the value q = 0. Precisely, with the choice of a fiducial vector ψ, $ $ Af(−z) = PAf(z)P ∀f ⇐⇒ $(z) = $(−z) ∀z, (34) 1 2 2 1 2 K(ψ) 1 h = p + 6q 7→ Ah = P + P∞ n 24 24 24 Q2 where P = n=0(−1) |enihen| is the parity operator. + 6M(ψ)Q2, (39) • Complex conjugation covariance d P ≡ −i , Qφ(x) ≡ xφ(x). † dx A$ = A$ ∀f ⇐⇒ $(−z) = $(z) ∀z, f(z) f(z) (35) The constants K(ψ) and M(ψ) are > 0 for all ψ and their appearance is due to the affine CS quantization. • For rotational covariance we define the unitary rep- Moreover, for a ψ such that K(ψ) ≥ 3 the quantum resentation θ 7→ U (θ) of 1 on the Hilbert space 4 T S Hamiltonian A is essentially self-adjoint, giving a H as the diagonal operator h unique unitary evolution: there is no need of boundary i(n+ν)θ conditions. UT(θ)|eni = e |eni, The semi-classical dynamics is ruled by the ACS where ν is arbitrary real. From the matrix elements mean value of the quantum Hamiltonian : of D(z) one proves easily the rotational covariance

property hq, p|Ah|q, pi † iθ Z U (θ)D(z)U (θ) = D(e z), (36) 0 0 0 0 2 0 0 T T = c(ψ) dq dp hq , p |q, pi h(q , p ), (40) × and its immediate consequence on the nature of M$ R+ R $ and the covariance of Af : the equality with a displacement of the equilibrium point of the potential at Q4 = 1 K . The constant c(ψ) is > 0 U (θ)A$U (−θ) = A$ , (37) eq 144 M T f T T (θ)f for all ψ. In Figure 1 contour plots of phase space with T (θ)f(z) := f e−iθz, holds true if and only trajectories for classical Hamiltonian h(q, p) (left) and if semi-classical Hamiltonian (right) defined by (40) are $(eiθz) = $(z) ∀z, θ, compared. As the result of affine quantization and restoring $ i.e., if and only if M is diagonal. arbitrariness on w and k, we obtain two corrections to the Friedmann equation which reads in its semi- 5. Applications of ACS classical version as quantization in quantum a˙ 2 1 kc2 8πG + c2a2 (1 − w)2A + B = ρ, (41) cosmology a P V 2 a2 3c2 Let us consider the Friedmann-Robertson-Walker model filled with barotropic fluid with equation of where A and B are positive factor dependent on ψ and state p = wρ. The line element is given by: can be adjusted at will in consistence with (so far very hypothetical!) observations. The first correction is 2 2 2 2 i j ds = −N(t) dt + a(t) δijω ω , (38) the repulsive potential, which depends on the volume,

177 Jean Pierre Gazeau Acta Polytechnica

enthusiasm in search for symmetry. He is also indebted to H. Bergeron, E. Czuchry, and P. Małkiewicz, for having developed with them a large part of the material exposed in the present contribution.

References [1] J.-P. Gazeau, Coherent States in Quantum Physics, Wiley-VCH, Berlin, 2009. [2] S. T. Ali, J.-P Antoine, and J.-P. Gazeau, Coherent States, Wavelets and their Generalizations 2d edition, Figure 1. Contour plots of the classical and semiclas- Theoretical and Mathematical Physics, Springer, New sical Hamiltonians for the closed Friedmann universe. York (2013), specially Chapter 11. The singularity q = 0 on the left is replaced with a bounce on the right. [3] H. Bergeron, J.-P. Gazeau, and A. Youssef, Are the Weyl and coherent state descriptions physically equivalent? Phys. Lett. A 377 598(2013); and this excludes non-compact universes from quan- arXiv:1102.3556 tum modelling. We notice that as the singularity is [4] H Bergeron and J.-P Gazeau, Integral quantizations approached a → 0, this potential grows faster (∼ a−6) with two basic examples, Annals of Physics (NY) 344 than the density of fluid (∼ a−3(1+w)) and therefore 43 (2014); arXiv:1308.2348 at some point the contraction must come to a halt. [5] H. Bergeron, E. M. F. Curado, J.-P. Gazeau, and Ligia Second, the curvature becomes dressed by the factor M. C. S. Rodrigues, Quantizations from (P)OVM’s, J. B. This effect could in principle be observed far away Phys.: Conf. Ser. (2014); arXiv:1310.3304 from the quantum phase. However, we do not observe [6] M. Baldiotti, R. Fresneda, and J.-P. Gazeau, Three the intrinsic curvature neither in the geometry nor in examples of covariant integral quantization, Proceedings the dynamics of space. Nevertheless, for a convenient of Science (2014). choice of ψ, this factor ≈ 1 Also note that the form of the repulsive potential [7] J.-P. Gazeau and B. Heller, Positive-Operator Valued Measure (POVM) quantization, Axioms (Special issue does not depend on the state of fluid filling the uni- on Quantum Statistical Inference) 4 1 (2015); verse: the origin of singularity avoidance is quantum http://www.mdpi.com/2075-1680/4/1/1 geometrical. The applications of the same quantization methods [8] M. Baldiotti, R. Fresneda, and J.-P. Gazeau, About to more involved cosmological models like Bianchi I Dirac&Dirac constraint quantizations, Invited comment, Phys. Scr. 90 074039 (2015). and Bianchi IX are described in [11–15]. For other approaches to quantum cosmology based [9] R. Fresneda and J.-P Gazeau, Integral quantizations on affine symmetry see [24–26]. with POVM and some applications, in Proceedings of the 30th International Colloquium on Group Theoretical Methods eds. J. Van der Jeugt et al, Journal of Physics: 6. Conclusion Conference Series 597 (2015) 012037. To conclude with an allusion to more “philosophical" [10] H. Bergeron, A. Dapor, J.-P. Gazeau, and P. perspectives, we think that integral quantization is Małkiewicz, Smooth big bounce from affine quantization, just a part of a world of mathematical models for one Phys. Rev. D 89 083522 (2014); arXiv:1305.0653 phenomenon. Indeed, the physical laws are expressed in terms of combinations of mathematical symbols, [11] H Bergeron, A Dapor, J.-P. Gazeau, and P Małkiewicz, Smooth Bounce in Affine Quantization of Bianchi I, and these combinations make sense for people who Phys. Rev D 91 124002 (2015); arXiv:1501.07718 have learnt these elements of mathematical language. This language is in constant development, enhance- [12] H. Bergeron, E. Czuchry, J.-P. Gazeau, P. ment, since the set of phenomenons which are acces- Małkiewicz, and W. Piechocki, Smooth quantum sible to our understanding is constantly broadening. dynamics of the Mixmaster universe, Phys. Rev. D 92 These combinations take place within a mathemat- 061302(R) (2015); arXiv:1501.02174 ical model. A model is usually scale dependent. It [13] H. Bergeron, E. Czuchry, J.-P. Gazeau, P. depends on a ratio of physical (i.e., measurable) quan- Małkiewicz, and W. Piechocki, Singularity avoidance in tities, like lengths, time(s), sizes, impulsions, actions, quantum Mixmaster universe, Phys. Rev D 92 124018 energies, etc. Changing scale for a model amounts to (2015); arXiv:1501.07871 “quantize” or “de-quantize” toward (semi-)classicality. [14] H. Bergeron, E. Czuchry, J.-P. Gazeau, and P. One changes perspective like one would change glasses Małkiewicz, Inflationary aspects of quantum Mixmaster to examine one’s environment. universe, submitted; arXiv:1511.05790 [15] H. Bergeron, E. Czuchry, J.-P. Gazeau, and P. Acknowledgements Małkiewicz, Vibronic framework for quantum The author is indebted to Jiri Patera and Pavel Winter- Mixmaster universe, Phys. Rev D 93 064080 (2016); nitz for inviting him at many occasions to share their arXiv:1512.00304

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[16] M. Born and P. Jordan, On quantum mechanics, Zs. [22] R. Fresneda, J.-P. Gazeau, and D. Noguera, f. Phys. 34 858 (1925). Covariant integral quantization of the motion on the [17] B. S. Agarwal and E. Wolf, Calculus for Functions of circle, in preparation. Noncommuting Operators and General Phase-Space [23] J.-P. Gazeau, A. Izadi, A. Rabei, and M. Takook, Methods in Quantum Mechanics, Phys. Rev. D 2 2161; Covariant Integral Quantization of the motion on 1 + 1 2187; 2206 (1970). de Sitter space-time, in preparation. [18] F. A. Berezin and M. A. Shubin, The Schrödinger [24] M. Fanuel and S. Zonetti, Affine Quantization and Equation, Kluwer Academic Publishers, Dordrecht, 1991. the Initial Cosmological Singularity, Eur. Phys. Lett. [19] G. Herzberg, Molecular Spectra and Molecular 101, 10001 (2013); Structure: Spectra of Diatomic Molecules, 2nd. ed., [25] J. R. Klauder, An Affinity for Affine Quantum Krieger Pub., Malabar, FL, 1989. Gravity, Proc. Steklov Institute of Mathematics 272, [20] J.-P. Gazeau and R. Murenzi, Covariant Affine 169-176 (2011); gr-qc/1003.261 Integral Quantization(s), accepted to J. Math. Phys. [26] J. R. Klauder, Enhanced Quantization, Particles, (2016), arXiv:1512.08274 Fields & Gravity, World Scientific, Singapore, 2015. [21] J.-P. Gazeau and M. del Olmo, Unit disk & SU(1,1) integral quantizations, in preparation.

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