Abstract
Abstract
We simplify and complete the construction of fully O(D)-equivariant fuzzy spheres
S
Λ
d
, for all dimensions
d
≡
D
−
1
, initiated in Fiore and Pisacane (2018 J. Geom. Phys.
132 423–51). This is based on imposing a suitable energy cutoff on a quantum particle in
R
D
subject to a confining potential well V(r) with a very sharp minimum on the sphere of radius r = 1; the cutoff and the depth of the well diverge with
Λ
∈
N
. As a result, the noncommutative Cartesian coordinates
x
‾
i
generate the whole algebra of observables
on the Hilbert space
; applying polynomials in the
x
‾
i
to any
we recover the whole
. The commutators of the
x
‾
i
are proportional to the angular momentum components, as in Snyder noncommutative spaces.
, as carrier space of a reducible representation of O(D), is isomorphic to the space of harmonic homogeneous polynomials of degree Λ in the Cartesian coordinates of (commutative)
R
D
+
1
, which carries an irreducible representation
π
Λ
of
O
(
D
+
1
)
⊃
O
(
D
)
. Moreover,
is isomorphic to
π
Λ
U
s
o
(
D
+
1
)
. We resp. interpret
,
as fuzzy deformations of the space
of (square integrable) functions on S
d
and of the associated algebra
of observables, because they resp. go to
as Λ diverges (with
ℏ
fixed). With suitable
ℏ
=
ℏ
(
Λ
)
⟶
Λ
→
∞
0
, in the same limit
goes to the (algebra of functions on the) Poisson manifold
T
∗
S
d
; more formally,
yields a fuzzy quantization of a coadjoint orbit of
O
(
D
+
1
)
that goes to the classical phase space
T
∗
S
d
.
Subject
General Physics and Astronomy,Mathematical Physics,Modeling and Simulation,Statistics and Probability,Statistical and Nonlinear Physics
Cited by
1 articles.
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