Acta Polytechnica


doi:10.14311/AP.2016.56.0173
Acta Polytechnica 56(3):173–179, 2016 © Czech Technical University in Prague, 2016

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: gazeau@apc.univ-paris7.fr

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
Group theory is one of the favorite domains of J. Pa-
tera and P. Winternitz. This contribution to the cele-
bration of the Jiri and Pavel 80th birthday emphasizes
the role of group representation theory in a certain
type of quantization procedures. These methods [1–9]
are named (covariant) integral quantizations and offer
a large scope of possibilities for mapping a classical
mathematical model of a physical system to a quan-
tum model. They are based on normalized positive
operator-valued measures (POVM) and present at
least two attractive advantages:

(1.) By employing all resources of integration and
distributions, they are most appropriate when we
have to deal with some geometric singularities.

(2.) They afford a semi-classical phase space portrait
of quantum states and quantum dynamics together
with a probabilistic interpretation.

Moreover, their recent applications to quantum cos-
mology [10–15] has yielded interesting results as:

• consistent quantum dynamics of isotropic, aniso-
tropic non-oscillatory and anisotropic models of
Universe;

• singularity resolution;
• unitary dynamics without boundary conditions;
• consistent semi-classical description of involved
quantum dynamics.

The aim of the present article is to give an overview
of the methods, their main characteristics, and their
promising applications to early cosmology. In Sec-
tion 2 we give a rapid survey of canonical quantiza-
tion with the problems raised by this procedure, and
we define a few basic requirements that any quan-
tization procedure should fullfill. In Section 3 we

define POVM-based integral quantizations of func-
tions defined on a measure space, their subsequent
semi-classical counterparts (i.e., lower symbols) and
discuss their possibilities when they are compared
with other methods. Covariant integral quantizations
are then considered when the measure space is homo-
geneous for some group, and we give the example of
the half-plane viewed as the affine group of the real
line. Section 4 is devoted to another example, the
integral quantization of functions on the plane viewed
as a coset of the Weyl-Heisenberg group. In Section 5
we illustrate the method with recent applications to
quantum cosmology where the initial singularity is
regularized by using the affine integral quantization.
Finally we sketch in Section 6 a more philosophical
point of view on the approaches described in this
contribution.
A large part of the content has already been pub-

lished, as exemplified by the above citations. On the
other hand the original side of the paper lies in the
synthetic presentation of the method together with
some new ideas like the general (and promising) con-
struction sketched in Section 3.2.

2. What is quantization?
2.1. Canonical quantization
The basic procedure of quantization of a classical me-
chanical model is well known and is named “canonical"
(Heisenberg, Born, Jordan, Dirac, Weyl, etc). Starting
from a phase space or symplectic manifold, e.g. R2,
e.g. the phase space for the motion of particle on the
line, it consists in the maps

R2 3 (q,p), {q,p} = 1 7→ self-adjoint (Q,P), (1)

with the canonical commutation rule (ccr) [Q,P] =
i~I, and

f(q,p) 7→ f(Q,P) 7→ (Sym f)(Q,P). (2)

173

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Jean Pierre Gazeau Acta Polytechnica

Beyond the ordering problem [16–18] raised by the sec-
ond step (symmetrisation), one should keep in mind
that [Q,P] = i~I holds true with self-adjoint Q, P,
only if both have continuous spectrum (−∞, +∞),
and there is uniqueness of the solution, up to unitary
equivalence (von Neumann). But then what about
singular f, e.g., the angle arctan(p/q)? What about
barriers or other impassable boundaries? The mo-
tion on a circle? In a bounded interval? On the
half-line? Despite their elementary aspects, these ex-
amples leave open many questions both on mathemat-
ical and physical levels, irrespective of the manifold
quantization procedures, like Path Integral Quantiza-
tion (Feynman, thesis, 1942), Geometric Quantization
with Weyl (1927), Groenewold (1946), Kirillov (1961),
Souriau (1966), Kostant (1970), Deformation Quan-
tization with Moyal (1947), Bayen, Flato, Fronsdal,
Lichnerowicz, Sternheimer (1978), Fedosov (1985),
Kontsevich (2003), Coherent state or anti-Wick or
Toeplitz quantization with Klauder (1961), Berezin
(1974).

Of course, the canonical procedure is universally
accepted in view of its numerous experimental valida-
tions, one of the most famous and simplest one going
back to the early period of Quantum Mechanics with
the quantitative prediction of the isotopic effect in
vibrational spectra of diatomic molecules [19]. These
data validated the canonical quantization, contrary
to the Bohr-Sommerfeld ansatz (which predicts no
isotopic effect). Nevertheless this does not prove that
another method of quantization fails to yield the same
prediction [3]. Moreover, as already mentioned above,
the canonical quantization is difficult, if not impos-
sible, in many circumstances. As a matter of fact,
the canonical or the Weyl-Wigner integral quantiza-
tion maps f(q) to f(Q) (resp. f(p) to f(P)), and so
are unable to cure any kind of classical singularity.
Clearly, quantization procedures may be intractable
when different phase space geometries and/or topolo-
gies are considered. Self-adjointness of basic operators
is not guaranteed anymore.

2.2. Quantization: minimal requirements
In our viewpoint, any quantization of a set X (e.g.,
a phase space or something else) and functions on it
should meet four basic requirements:
(1.) Linearity — it is a linear map

Q : C(X) 7→A(H), Q(f) ≡ Af, (3)

where:
• C(X) is a vector space of complex-valued func-

tions f(x) on set X, i.e., a “classical” mathemat-
ical model;

• A(H) is a vector space of linear operators (ignor-
ing domain limitations) in some complex Hilbert
space H, i.e, a “quantum” mathematical model.

(2.) Unity — The map (3) is such that the function
f = 1 is mapped to the identity operator I on H.

(3.) Reality — A real f is mapped to a self-adjoint
operator Af in H or, at least, a symmetric operator,

(4.) Covariance — If X is endowed with symmetry,
the latter should be preserved in a “certain” sense.

Of course, further requirements are necessarily
added, depending on the mathematical structures
equipping X and C(X) (e.g., measure, topology, man-
ifold, closure under algebraic operations, time evolu-
tion or dynamics, etc). Moreover, a physical inter-
pretation should be advanced about measurement of
spectra of classical f ∈C(X) or quantum Af ∈A(H)
to which are given the status of observables. Finally,
an unambiguous classical limit of the quantum physi-
cal quantities should exist, the limit operation being
associated to a change of scale.

3. Integral Quantization(s)
3.1. Integral quantization:

general setting and POVM
Let (X,ν) be a measure space. Suppose that there
exists an X-labeled family of bounded operators on a
Hilbert space H resolving the identity I:

X 3 x 7→ M(x) ∈L(H),
∫
X

M(x) dν(x) = I, (4)

where the equality holds in a weak sense. The case
we are specially interested in occurs when the M(x)’s
are positive semi-definite and unit trace, i.e., when

M(x) ≡ ρ(x) (density operator). (5)

Then, if X is a space with suitable topology, the map

B(X) 3 ∆ 7→
∫

∆
ρ(x) dν(x) (6)

may define a normalized positive operator-valued mea-
sure (POVM) on the σ-algebra B(X) of Borel sets.

The quantization of complex-valued functions f(x)
on X is the linear map:

f 7→ Af =
∫
X

M(x)f(x) dν(x), (7)

understood as the sesquilinear form,

Bf (ψ1,ψ2) =
∫
X

〈ψ1|M(x)|ψ2〉f(x) dν(x), (8)

defined on a dense subspace of H. If f is real and
at least semi-bounded, and if the M(x)’s are posi-
tive operators, then the Friedrich’s extension of Bf
univocally defines a self-adjoint operator. If f is not
semi-bounded, there is no natural choice of a self-
adjoint operator associated with Bf. We then need
more information on H to solve this subtlety.
Covariance implies symmetry. Symmetry suggests

a mathematical group G. A symmetry structure on
X presupposes that some group G acts on X. Our
hypothesis here is that X is an homogeneous space
for G, which means that X ∼ G/H for H a subgroup
of G. We now examine two possible cases.

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vol. 56 no. 3/2016 Covariant Integral Quantizations and Quantum Cosmology

3.1.1. X = G: quantizing the group
Let G be a Lie group with left invariant Haar measure
dµ(g), and let g 7→ U(g) be a unitary irreducible
representation (UIR) of G in a Hilbert space H. Let
M be a bounded operator on H. Suppose that the
operator

R :=
∫
G

M(g) dµ(g), M(g) := U(g)MU(g)†, (9)

is defined in a weak sense. From the left invariance
of dµ(g) the operator R commutes with all operators
U(g), g ∈ G, and so, from Schur’s Lemma, R = cMI
with

cM =
∫
G

tr
(
ρ0M(g)

)
dµ(g), (10)

where the unit trace positive operator ρ0 is chosen
in order to make the integral convergent. If cM is
finite and not zero, then we get the resolution of the
identity:∫

G

M(g) dν(g) = I, dν(g) := dµ(g)/cM. (11)

For this reason, the operator M to be U transported
will be designated (admissible) fiducial operator. The
method becomes more apparent when the representa-
tion U is square-integrable in the following sense (for
example, it is the case for the affine group of the real
line, see below). Suppose that there exists a density
operator ρ which is admissible in the sense that

cρ =
∫
G

dµ(g) tr
(
ρρ(g)

)
< ∞, (12)

with ρ(g) = U(g)ρU†(g). Then the resolution of the
identity (11) holds with M(g) and cM = cρ. This
allows a covariant integral quantization of complex-
valued functions on the group

f 7→ Af =
∫
G

ρ(g)f(g) dν(g), dν(g) :=
dµ(g)
cρ

, (13)

U(g)AfU†(g) = AU(g)f (covariance), (14)

where (
U(g)f

)
(g′) := f(g−1g′) (15)

is the regular representation if f ∈ L2(G, dµ(g)). This
leads to a non-commutative version of the original
manifold structure for G.

Example: affine CS integral quantization In
this example, the group G is the affine group of the
real line and is also viewed as the phase space for the
motion on the half-line, i.e., the half-plane {(q,p) ∈
R∗+ ×R}. The group law is defined as

(q,p)(q0,p0) =
(
qq0,

p0
q

+ p
)
, (16)

and the left invariant measure is the symplectic dq dp.
This group has two UIR’s which are both square

integrable. We choose the following one.

L2(R∗+, dx) 3 ψ(x) 7→
(
U(q,p)ψ

)
(x) =

eipx
√
q
ψ
(x
q

)
(17)

As a density operator we choose the rank one pro-
jector ρ = |ψ〉〈ψ| where the fiducial vector, i.e., the
‘wavelet” ψ is in L2(R+, dx) ∩L2(R+, dx/x) and its
affine transport produces the overcomplete family of
affine coherent states (ACS)

|q,p〉 = U(q,p)|ψ〉. (18)

The ACS integral quantization reads as

f(q,p) 7→ Af =
∫
R+×R

dq dp
cρ

f(q,p)|q,p〉〈q,p| (19)

with cρ =
∫∞

0 |ψ(x)|
2 dx
x
. More details and new devel-

opments are given in the recent [20]. Applications of
this method to the resolution of the singularity prob-
lem on a quantum level for a series of cosmological
models (FRW, Bianchi I, Bianchi IX), for which the
volume — expansion pair variables form the above
half-plane, are given in the recent papers [10–15] and
will be summarised in Section 5.

3.1.2. X = G/H: quantizing a non-trivial
group coset

In the absence of square-integrability over G (the
trivial illustration is the Weyl-Heisenberg group which
underlies in particular the canonical quantization, see
Section 4), there exists a definition of square-integrable
representations with respect to a left coset manifold
X = G/H, with H a closed subgroup of G (it is the
center in the Weyl-Heisenberg case), equipped with
a quasi-invariant measure ν [2]. For a global Borel
section σ : X → G of the group, let νσ be the unique
quasi-invariant measure defined by

dνσ(x) = λ
(
σ(x),x

)
dν(x), (20)

where λ(g,x)dν(x) = dν(g−1x) for g ∈ G with
λ(g,x) obeying the cocycle condition λ(g1g2,x) =
λ(g1,x)λ(g2,g−11 x). Let U be a UIR which is
square integrable mod(H) with an admissible ρ, i.e.,
cρ :=

∫
X

tr(ρρσ(x)) dνσ(x) < ∞, with ρσ(x) =
U(σ(x))ρU(σ(x))†. Then we have the resolution of
the identity and the resulting quantization

f 7→ Af =
1
cρ

∫
X

f(x)ρσ(x) dνσ(x) (21)

Covariance holds here too in a certain sense (see [2,
Chapter 11]).

Besides the Weyl-Heisenberg group, another exam-
ple concerns the motion on the circle for which G is
the group of Euclidean displacements in the plane, i.e.,
the semi-direct product R2 o SO(2), and the subgroup
H is isomorphic to R [22]. Other examples involve
the relativity groups, Galileo, Poincaré [2], 1 + 1 Anti
de Sitter (unit disk and SU(1, 1)) [21], 1 + 1 and 3 + 1
de Sitter [23].

175



Jean Pierre Gazeau Acta Polytechnica

3.2. An example of construction
of original operator M or ρ

Let U be a UIR of G and $(x) be a function on the
coset X = G/H. Suppose that it allows to define a
bounded operator M$σ on H through the operator-
valued integral

M$σ =
∫
X

$(x)U(σ(x)) dνσ(x). (22)

Then, under appropriate conditions on the “weight”
function $(σ(x)) such that U be a UIR which is
square integrable mod(H) and M$σ is admissible in
the above sense, the family of transported operators
M$σ (x) := U(σ(x))M$σ U†(σ(x)) resolves the identity.

3.3. Semi-classical portraits
Some quantization features, e.g., spectral properties
of Af, may be derived or at least well grasped from
functional properties of the lower (Lieb) or covariant
(Berezin) symbol (it generalizes Husimi function or
Wigner function)

Af 7→ f̌(x) := tr
(
M(x)Af

)
, (23)

When M = ρ (density operator) this new function
is the local average of the original f with respect to
the probability distribution tr(ρ(x)ρ(x′))

f(x) 7→ f̌(x) =
∫
X

f(x′) tr
(
ρ(x)ρ(x′)

)
dν(x′). (24)

The Bargmann-Segal-like map f 7→ f̌ which, gen-
eralizes the Berezin or heat kernel transform when
X = G is a Lie group, is in general a regularization
of the original, possibly extremely singular, f, and
the classical limit itself means that, given one or more
scale parameter(s) �(i) and a distance d(f, f̌), we have

d(f, f̌) → 0 as �(i) → 0. (25)

3.4. Comments
Beyond the freedom (think of analogy with Signal
Analysis where different techniques are complemen-
tary) allowed by integral quantization, the advantages
of the method with regard to other quantization pro-
cedures in use are of four types.

(i)The minimal amount of constraints imposed on
the classical objects to be quantized.
(ii)Once a choice of (positive) operator-valued mea-
sure has been made, which must be consistent with
experiment, there is no ambiguity in the issue, con-
trarily to other method(s) in use (think in particular
of the ordering problem). To one classical model
corresponds one and only one quantum model. Of
course different choices of (P)OVM are requested to
be physically equivalent, e.g., leading to the same
experimental predictions, if they are aimed to de-
scribe one specific system.

(iii)The method produces in essence a regularizing
effect, at the exception of certain choices, like the
Weyl-Wigner (i.e., canonical) integral quantization.

(iv)The method, through POVM choices, offers the
possibility to keep a fully probabilistic content. As
a matter of fact, the Weyl-Wigner integral quanti-
zation does not rest on a POVM.

4. Weyl-Heisenberg covariant
integral quantization(s)

The Weyl-Heisenberg group is defined as GWH =
{(s,z) | s ∈ R, z ∈ C} with multiplication law

(s,z)(s′,z′) =
(
s + s′ + Im(zz̄′),z + z′

)
(26)

Let H be a separable (complex) Hilbert space with
orthonormal basis e0,e1, . . . ,en ≡ |en〉, . . . , and cor-
responding lowering and raising operators defined in
the usual way

a|en〉 =
√
n|en−1〉, a|e0〉 = 0,

a†|en〉 =
√
n + 1|en+1〉, [a,a†] = I,

It is well known that the Weyl-Heisenberg group has a
unique non-trivial UIR U, up to unitary equivalence,
each element in this equivalence class corresponding
to a non-zero real number like the Planck constant.
Consider the center C = {(s, 0) | s ∈ R} of GWH.
Then, the set X is the coset X = GWH/C ∼ C with
measure d2z/π. Choosing the trivial section C 3 z 7→
σ(z) = (0,z), to each z corresponds the (unitary)
displacement (∼ Weyl) operator

D(z) := U(σ(z)) = eza
†−z̄a,

D(−z) = (D(z))−1 = D(z)†. (27)

The noncommutativity of quantum mechanics is en-
coded in the relation

D(z)D(z′) = e
1
2 (zz̄

′−z̄z′)D(z + z′)

= e(zz̄′−z̄z
′)D(z′)D(z), (28)

i.e., z 7→ D(z) is a projective representation of the
abelian group C. The standard (i.e., Schrödinger-
Klauder-Glauber-Sudarshan) CS are defined as

|z〉 = D(z)|e0〉. (29)

We now adopt the construction sketched in Section 3.2.
Let $(z) be a function on the complex plane obeying
$(0) = 1. Suppose that it allows to define a fidu-
cial operator M$ on H through the operator-valued
integral

M$ =
∫
C
$(z)D(z)

d2z
π
. (30)

Then, the family of displaced M$(z) :=
D(z)M$D(z)† under the unitary action D(z) resolves
the identity ∫

C
M$(z)

d2z
π

= I. (31)

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vol. 56 no. 3/2016 Covariant Integral Quantizations and Quantum Cosmology

It is a direct consequence of D(z)D(z′)D(z)† =
ezz̄

′−z̄z′D(z′), of
∫
C e

zξ̄−z̄ξ d2ξ
π

= πδ2(z), and of
$(0) = 1 with D(0) = I. The resulting quantiza-
tion map is given by

f 7→ A$f =
∫
C

M$(z)f(z)
d2z
π

=
∫
C
$(z)D(z)fs[f](z)

d2z
π
, (32)

where are involved the symplectic Fourier transforms
fs and its space reverse fs

fs[f](z) =
∫
C
ezξ̄−z̄ξf(ξ)

d2ξ
π
, fs[f](z) = fs[f](−z)

Both are unipotent fs[fs[f]] = f and fs[fs[f]] = f.
We have the following covariance properties

• Translation covariance:

A$f(z−z0) = D(z0)A
$
f(z)D(z0)

†. (33)

• Parity covariance

A$f(−z) = PA
$
f(z)P ∀f ⇐⇒ $(z) = $(−z) ∀z,

(34)
where P =

∑∞
n=0(−1)

n|en〉〈en| is the parity opera-
tor.

• Complex conjugation covariance

A$
f(z)

=
(
A$f(z)

)† ∀f ⇐⇒ $(−z) = $(z) ∀z,
(35)

• For rotational covariance we define the unitary rep-
resentation θ 7→ UT(θ) of S1 on the Hilbert space
H as the diagonal operator

UT(θ)|en〉 = ei(n+ν)θ|en〉,

where ν is arbitrary real. From the matrix elements
of D(z) one proves easily the rotational covariance
property

UT(θ)D(z)UT(θ)† = D(eiθz), (36)

and its immediate consequence on the nature of M$
and the covariance of A$f : the equality

UT(θ)A$f UT(−θ) = A
$
T(θ)f, (37)

with T(θ)f(z) := f
(
e−iθz

)
, holds true if and only

if
$(eiθz) = $(z) ∀z,θ,

i.e., if and only if M$ is diagonal.

5. Applications of ACS
quantization in quantum
cosmology

Let us consider the Friedmann-Robertson-Walker
model filled with barotropic fluid with equation of
state p = wρ. The line element is given by:

ds2 = −N(t)2dt2 + a(t)2δijωiωj, (38)

where a(t) is the scale factor and the ωi’s are right-
invariant dual vectors. The resolving of the Hamilto-
nian constraint leads to a model of singular universe
analogous to a particle moving on the half-line (0,∞).
Indeed, canonical coordinates (q,p) can be introduced
in terms of the volume V ∼ a3 and the expansion rate
V̇ /V ∼ ȧ/a through the relations V = q2/(1−w) and
K = (3/8)(1 −w)pq−(1+w)/(1−w). Clearly, q > 0 and
p ∈ R. Then the reduced Hamiltonian reads as

{q,p} = 1, h(q,p) = α(w)p2 + 6k̃qβ(w), q > 0.

with k̃ = (
∫

dω)2/3k, α(w) = 3(1−w)2/32 and β(w) =
2(3w + 1)/(3(1 −w)). k = 0,−1 or 1 (in suitable unit
of inverse area) depending on whether the universe is
flat, open or closed. Assuming a closed universe with
radiation content w = 1/3 and k = +1, the classical
isotropy singularity is cured on the quantum level by
using affine CS quantization [10]. Indeed the latter
regularizes the kinetic term by adding a repulsive
centrifugal potential which prevents the system from
reaching the value q = 0. Precisely, with the choice of
a fiducial vector ψ,

h =
1
24
p2 + 6q2 7→ Ah =

1
24
P 2 +

K(ψ)
24

1
Q2

+ 6M(ψ)Q2, (39)

P ≡−i
d

dx
, Qφ(x) ≡ xφ(x).

The constants K(ψ) and M(ψ) are > 0 for all ψ and
their appearance is due to the affine CS quantization.
Moreover, for a ψ such that K(ψ) ≥ 34 the quantum
Hamiltonian Ah is essentially self-adjoint, giving a
unique unitary evolution: there is no need of boundary
conditions.
The semi-classical dynamics is ruled by the ACS

mean value of the quantum Hamiltonian :

〈q,p|Ah|q,p〉

= c(ψ)
∫
R+×R

dq′dp′
∣∣〈q′,p′|q,p〉∣∣2h(q′,p′), (40)

with a displacement of the equilibrium point of the
potential at Q4eq =

1
144

K
M
. The constant c(ψ) is > 0

for all ψ. In Figure 1 contour plots of phase space
trajectories for classical Hamiltonian h(q,p) (left) and
semi-classical Hamiltonian (right) defined by (40) are
compared.
As the result of affine quantization and restoring

arbitrariness on w and k, we obtain two corrections
to the Friedmann equation which reads in its semi-
classical version as(ȧ

a

)2
+ c2a2P (1 −w)

2A
1
V 2

+ B
kc2

a2
=

8πG
3c2

ρ, (41)

where A and B are positive factor dependent on ψ and
can be adjusted at will in consistence with (so far very
hypothetical!) observations. The first correction is
the repulsive potential, which depends on the volume,

177



Jean Pierre Gazeau Acta Polytechnica

Figure 1. Contour plots of the classical and semiclas-
sical Hamiltonians for the closed Friedmann universe.
The singularity q = 0 on the left is replaced with a
bounce on the right.

and this excludes non-compact universes from quan-
tum modelling. We notice that as the singularity is
approached a → 0, this potential grows faster (∼ a−6)
than the density of fluid (∼ a−3(1+w)) and therefore
at some point the contraction must come to a halt.
Second, the curvature becomes dressed by the factor
B. This effect could in principle be observed far away
from the quantum phase. However, we do not observe
the intrinsic curvature neither in the geometry nor in
the dynamics of space. Nevertheless, for a convenient
choice of ψ, this factor ≈ 1
Also note that the form of the repulsive potential

does not depend on the state of fluid filling the uni-
verse: the origin of singularity avoidance is quantum
geometrical.

The applications of the same quantization methods
to more involved cosmological models like Bianchi I
and Bianchi IX are described in [11–15].

For other approaches to quantum cosmology based
on affine symmetry see [24–26].

6. Conclusion
To conclude with an allusion to more “philosophical"
perspectives, we think that integral quantization is
just a part of a world of mathematical models for one
phenomenon. Indeed, the physical laws are expressed
in terms of combinations of mathematical symbols,
and these combinations make sense for people who
have learnt these elements of mathematical language.
This language is in constant development, enhance-
ment, since the set of phenomenons which are acces-
sible to our understanding is constantly broadening.
These combinations take place within a mathemat-
ical model. A model is usually scale dependent. It
depends on a ratio of physical (i.e., measurable) quan-
tities, like lengths, time(s), sizes, impulsions, actions,
energies, etc. Changing scale for a model amounts to
“quantize” or “de-quantize” toward (semi-)classicality.
One changes perspective like one would change glasses
to examine one’s environment.

Acknowledgements
The author is indebted to Jiri Patera and Pavel Winter-
nitz for inviting him at many occasions to share their

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.

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	Acta Polytechnica 56(3):173–179, 2016
	1 Introduction
	2 What is quantization?
	2.1 Canonical quantization
	2.2 Quantization: minimal requirements

	3 Integral Quantization(s)
	3.1 Integral quantization: general setting and POVM
	3.1.1 X=G: quantizing the group
	3.1.2 X=G/H: quantizing a non-trivial group coset

	3.2 An example of construction of original operator M or rho
	3.3 Semi-classical portraits
	3.4 Comments

	4 Weyl-Heisenberg covariant integral quantization(s)
	5 Applications of ACS quantization in quantum cosmology
	6 Conclusion
	Acknowledgements
	References