Spin Random Fields: the pullback approach
( meaning, equivalences and results )
Maurizia ROSSI
(joint work with Paolo BALDI)
University of Rome “Tor Vergata”
Department of Mathematics
13 January 2014
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Spin random fields
January 13, 2014
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Outline
1
Introduction
What is a (isotropic) spin random field ?
Why do we study spin random fields ?
Spin theory: our point of view (sketch)
2
Random fields on SO(3)
3
Spin random fields:
Geometrical setting
Definitions and difficulties
A new tool: the pullback random field
Representation formula for Gaussian isotropic spin random fields
Open question: spin Brownian fields
The connection with the classical spin theory
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Spin random fields
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Introduction
What is a (isotropic) spin random field ?
Spin random fields: examples
For each s ∈ Z, we have the notion of spin s random field on the sphere S2 .
Example (Spin 0 random field)
A spin 0 random field T is just a random field on the sphere S2 . Roughly speaking,
for each point x of S2 we randomly choose a (possibly complex) number Tx .
We say that T is isotropic if its law is invariant w.r.t. any rotation (an element
∈ SO(3)) of the sphere S2 .
Example (Spin 1 random field)
A spin 1 random field T is a random section of the tangent bundle on the sphere
S2 . Roughly speaking, for each point x of S2 we randomly choose a vector Tx
belonging to the tangent plane to the sphere S2 at the point x.
We say that T is isotropic if its law is invariant w.r.t. the action of SO(3) on the
tangent bundle, which we shall explain later.
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Spin random fields
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Introduction
What is a (isotropic) spin random field ?
For any s, it is more difficult to give a rigorous picture for the realization of a spin
s random field.
Just to fix ideas:
We can see a spin s random field T as a random field on the sphere S2 that, for
each x ∈ S2 , takes as a value a curve living in the tangent plane to S2 at x.
Example (Spin 2 random field)
A spin s = 2 random field can be viewed as a random field on the sphere S2 that,
for each x ∈ S2 , takes as a value an ellipse living in the tangent plane to S2 at x.
Note that the ellipse is invariant w.r.t. planar rotations by an angle π(= 2π/s).
We say that T is isotropic if its law is invariant w.r.t. the action of SO(3) on the
spin 2 line bundle on S2 (more details later...).
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Spin random fields
January 13, 2014
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Introduction
Why do we study spin random fields ?
Spin random fields and CMB
We study spin random fields as they
• are interesting from a theoretical point of view;
• find applications in Cosmology e.g.
Actually, the Cosmic Microwave Background data can be viewed as a single
realization of an isotropic random section of a vector bundle on the sphere S2 .
More precisely, in modern cosmological setting,
the temperature of the CMB is modeling as an isotropic random field on the
sphere S2 (a spin 0 random field);
the (complex) polarization of the CMB is modeling as an isotropic spin 2 random
field on the sphere S2 .
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Introduction
Spin theory: our point of view (sketch)
Spin theory: our approach
We were quite inspired by Malyarenko (2011).
1) Motivations
Spin random fields are quite difficult to study (especially concerning theoretical
issues as definitions of isotropy, mean square continuity and so on..). Indeed they
live in complex structures as (homogeneous) line bundles.
2) Main idea
• To each spin s random field T on the sphere S2 we associate a random field X
on the group SO(3) (the pullback random field) , satisfying certain pathwise
invariance properties.
• We prove that T and X are equivalent, so we can study pullback random fields
which are easier to investigate (they are just particular scalar-valued random
fields).
3) ... and about the existing constructions ?
We prove the equivalence between the pullback approach and Geller-Marinucci’s
construction and Malyarenko’s construction respectively.
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Introduction
Spin theory: our point of view (sketch)
How to point out the pullback approach
2
Spin random
 fields on S

y
Random sections of homogeneous vector bundles
on homogeneous spaces X of a compact group G
xx

Pullback yy approach
Random fields on homogeneous spaces X of a compact group G
x


Random fields on SO(3)
To be more concrete, in this talk we explain the pullback approach in the
particular spin case, even if we investigated the general case of homogeneous
vector bundles (see http://arxiv.org/pdf/1310.0321.pdf).
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Spin random fields
January 13, 2014
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Random fields on SO(3)
Random fields on SO(3):
definitions ...
Let (Ω, F , P) be a probability space.
Definition (Random field on SO(3))
A random field T on SO(3) is a collection of (complex-valued) r.v.’s (Tg )g ∈SO(3)
on (Ω, F , P) such that the map
T : Ω × SO(3)−→C,
(ω, g ) 7→ Tg (ω)
(2.1)
is F ⊗ B(SO(3))-measurable, B(SO(3)) being the Borel σ-field on SO(3).
• T is said to be a.s. continuous if the functions g → Tg are a.s. continuous;
• T is said to be second order if E[|Tg |2 ] < +∞ for each g ∈ SO(3);
• T is mean square continuous if limh→g E[|Th − Tg |2 ] = 0.
• T is said to be a.s. square integrable if its sample paths g 7→ Tg belong to the
space of square integrable functions L2 (SO(3)) a.s.
We work with random fields that are at least a.s. square integrable.
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Random fields on SO(3)
... isotropy ...
SO(3) acts on L2 (SO(3)) with the left regular representation L, i.e. for g ∈ SO(3)
Lg : L2 (SO(3))−→L2 (SO(3)),
Lg f(·) := f(g−1 ·)
(2.2)
Let T be an a.s. square integrable
random field on SO(3). If f ∈ L2 (SO(3)),
R
consider the r.v. T (f ) := SO(3) Tg f (g ) dg = hT , f iL2 (SO(3)) .
Definition (Isotropic random field on SO(3))
The a.s. square integrable r.f. T on SO(3) is isotropic if the joint laws of the
random vectors
(T (f1 ), . . . , T (fm )) and (T (Lg f1 ), . . . , T (Lg fm ))
(2.3)
coincide for every g ∈ SO(3) and f1 , f2 , . . . , fm ∈ L2 (SO(3)).
This definition is somehow different from the usual one concerning the invariance
of the finite dimensional distributions, but if T is a.s. continuous, the two
definitions are equivalent (see Marinucci and Peccati (2013)).
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Spin random fields
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Random fields on SO(3)
... Fourier expansions
If T is a.s. square integrable, we can apply the Peter Weyl theorem (see Faraut
(2008)) to the functions g → Tg (ω). T have a Fourier expansion of the form
Tg =
X√
`≥0
2` + 1
`
X
a`m,n D`m,n (g)
(2.4)
m,n=−`
• the D ` ’s are the√Wigner RD matrices;
√
`
`
` (g ) dg =
• the r.v. am,n
2` + 1hT , Dm,n
i;
= 2` + 1 SO(3) Tg Dm,n
2
• the series converges in L (SO(3)) a.s.
If T is a.s. square integrable, isotropic and second order, then the convergence is
both in L2 (P) for every fixed g and in L2 (Ω × SO(3)) (see the book Marinucci
and Peccati (2011)).
→ Every a.s. square integrable, isotropic and second order random field on
SO(3) (... on a homogeneous space of a compact group) is mean square
continuous (see Marinucci and Peccati (2013)).
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Spin random fields
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Spin random fields:
Geometrical setting
Spin line bundles on S2 : construction
The sphere S2 is the homogeneous space of SO(3) and
S2 ∼
= SO(3)/K
(3.1)
where K ∼
= SO(2) is the isotropy group of the north pole o.
For each s ∈ Z, let χs be the s-th character of K . More precisely, if k ∈ K is a
rotation by an angle γk , then
(3.2)
χs (k) = eisγk
K acts on SO(3) × C by
k(g, z) := (gk, χs (k−1 )z),
k ∈ K, (g, z) ∈ SO(3) × C
(3.3)
The orbit θ(g , z) of (g , z) is given by θ(g , z) = {(gk, χs (k −1 )z) : k ∈ K } and the
space of the orbits is
SO(3) ×s C := {θ(g, z) : (g, z) ∈ SO(3) × C}
(3.4)
The spin −s line bundle on S2 is the triple ξs = (SO(3) ×s C, πs , S2 ), where
πs : SO(3) ×s C−→S2 ,
M. Rossi (Rome Tor Vergata)
θ(g, z)−→gK ∈ S2 .
Spin random fields
(3.5)
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Spin random fields:
Geometrical setting
• SO(3) ×s C is the total space;
• S2 is the base space;
• πs is the bundle projection;
• πs−1 (x) is the fiber on x ∈ S2 and is ∼
= C → ξs is a complex line bundle.
• The isomorphism πs−1 (x) ∼
= C induces a well-defined inner product on each fiber
as
(3.6)
hθ(g, z1 ), θ(g, z2 )iπs−1 (gK) := hz1 , z2 iC = z1 z2
• There exists a natural action of SO(3) on the spin −s line bundle ξs :
→ on the total space SO(3) ×s C
h ∈ SO(3)
hθ(g, z) := θ(hg, z),
(3.7)
→ and on the base space S2
h(gK) = hgK,
h ∈ SO(3)
(3.8)
The action commutes with the projection, i.e.
πs (hθ(g, z)) = hπs (θ(g, z))
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Spin random fields:
Geometrical setting
Example (s = 0)
ξ0 = (S2 × C, π0 , S2 ) is the trivial bundle; the action of SO(3) on ξ0 is simply the
action of SO(3) on the sphere S2 .
Example (s = −1)
ξ−1 = (SO(3) ×−1 C, π−1 , S2 ) is the tangent bundle; the action of SO(3) on ξ−1
could be viewed in this way:
• the action of SO(3) on the base space S2 is the standard one;
• the action of SO(3) on SO(3) ×−1 C can be described as follows.
SO(3) ×−1 C is the sphere S2 with all the tangent planes attached to it.
Focus on the action on the tangent plane Tx0 S2 at the north pole x0 : the rotation
h ∈ SO(3) acts on Tx0 S2 as rotating it and then translating it to the tangent
plane in x := hx0 .
Example (s = 1)
ξ1 = (SO(3) ×1 C, π1 , S2 ) is the cotangent bundle...
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Spin random fields:
Geometrical setting
Sections of the spin −s line bundle ξs
A section u of ξs is a map
u : S2 −→SO(3) ×s C
(3.9)
such that πs ◦ u = idS2 . In other words, u(x) ∈ πs−1 (x) for each x ∈ S2 . Moreover
S2 and SO(3) ×s C being topological spaces, we can speak about continuous
sections.
Recall that on each fiber πs−1 (x) ∼
= C we have a scalar product. We can define
the inner product between two sections u1 , u2 as follows
Z
hu1 , u2 i :=
hu1 (x), u2 (x)iπs−1 (x) dx
(3.10)
S2
and the L2 -space L2 (ξs ) accordingly. The action U of SO(3) on L2 (ξs ) is
Uh u(x) := hu(h−1 x), h ∈ SO(3) .
(3.11)
U is the representation of SO(3) induced by χs .
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Spin random fields:
Geometrical setting
Pullback functions (of type s) on SO(3)
A function f : SO(3) → C is said to be of type s if
f(gk) = χs (k−1 )f(g), g ∈ SO(3), k ∈ K
(3.12)
• A function f of type s uniquely defines a section u of ξs as follows:
u(x) := θ(g, f(g)), x = gK
(3.13)
Proposition (u ←→ f )
For each section u of ξs , there exists a unique function f of type s (3.12) such
that u has the form (3.13). We call f the pullback of u.
Furthermore, the section u is continuous if and only if its pullback f is continuous.
Given two sections u1 , u2 andRdenoting f1 , f2 their respective pullbacks
hu1 , u2 i = hf1 , f2 iL2s (SO(3)) := SO(3) f1 (g)f2 (g) dg. Therefore
L2 (ξs ) 3 u ←→isometry f ∈ L2s (SO(3))
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Spin random fields
(3.14)
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Spin random fields:
Geometrical setting
We recall the representation U of SO(3) on L2 (ξs ):
Uh u(x) := hu(h−1 x), h ∈ SO(3) .
(3.15)
U can be equivalently realized on L2s (SO(3)) as
Lh f(g) := f(h−1 g),
h ∈ SO(3) .
(3.16)
Moreover we have
Uh u(gK ) = hu(h−1 gK ) =
hθ(h−1 g , f (h−1 g )) = θ(g , f (h−1 g )) = θ(g , Lh f (g ))
so that, thanks to the uniqueness of the pullback function:
Proposition (Pullback respects the action)
If f is the pullback function of the section u then Lh f is the pullback function of
the section Uh u.
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Spin random fields:
Spin spherical harmonics:
Geometrical setting
orthonormal basis of L2 (ξs )
• f : SO(3) → C of type s, ∈ L2s (SO(3)) → its Fourier expansion is in terms of
the (−s)-columns of the Wigner’s D matrices, i.e.
f(g) =
X√
2` + 1
`
X
`
fm,−s
D`m,−s (g)
(3.17)
m=−`
`≥|s|
√
`
`
where fm,−s
= 2` + 1hf , Dm,−s
i and the series converges in L2 .
`
• Each function g → Dm,−s (g) is of type s and defines (up to a factor) the
section of the spin (−s) line bundle ξs given by
r 2` + 1
D`m,s (g) ,
x = gK ∈ S2 .
(3.18)
−s Y`,m (x) := θ g,
4π
−s Y`,m , ` ≥ |s|, m = −`, . . . , ` are the spin −s spherical harmonics. The
isometry L2s (SO(3)) ↔ L2 (ξs ) −→ −s Y`,m form an orthonormal basis for L2 (ξs ).
The Fourier expansion of u ∈ L2 (ξs ) ↔ f ∈ L2s (SO(3)) is given by
u(x) =
`
X X
u`,m
−s Y`,m (x),
√
`
u`,m = (−1)s−m 4πfm,−s
.
(3.19)
`≥|s| m=−`
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Spin random fields
January 13, 2014
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Spin random fields:
Spin random fields:
Definitions and difficulties
definitions
A spin random field is a random section of a spin line bundle on S2 . More
precisely, let (Ω, F , P) be a probability space.
Definition (Spin random field)
A spin (−s) random field T is a F ⊗ B(S2 )-measurable map
T : Ω × S2 −→SO(3) ×s C ,
(ω, x) 7→ Tx (ω)
(3.20)
such that for each ω, x 7→ Tx (ω) is a section of the spin (−s) line bundle ξs .
In other words for each x ∈ S2 , the random variable Tx ∈ πs−1 (x) = the fiber on x.
• T is said to be a.s. continuous if the sections x → Tx are a.s. continuous;
• T is said to be second order if E[kTx k2π−1 (x) ] < +∞ for each x ∈ S2 ;
s
• T is said to be a.s. square integrable if the sections x 7→ Tx ∈ L2 (ξs ) a.s. We
work with spin random fields that are at least a.s. square integrable.
• Other definitions, like isotropy and mean square continuity, require more
attention, the spin random fields living in line bundles.
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January 13, 2014
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Spin random fields:
Spin random fields:
Definitions and difficulties
isotropy and mean square continuity
Given an a.s. square integrable spin (−s) random field T and a section
u ∈ L2 (ξs ), define the random variable T (u) := hT , uiL2 (ξs ) .
Definition (Isotropic spin random field)
The a.s. square integrable spin (−s) random field T is said to be isotropic if for
every choice of square integrable sections u1 , u2 , . . . , um of ξs , the random vectors
T (u1 ), . . . , T (um ) and T (Ug u1 ), . . . , T (Ug um )
(3.21)
have the same law for every g ∈ SO(3).
• ...and about the mean square continuity for T ? The naive approach
lim E[kTx − Ty k2 ] = 0
y →x
is not completely meaningful as Tx , Ty ∈ different spaces, i.e. the fibers
πs−1 (x), πs−1 (y ), even if isomorphic.
• How to overcome this difficulty and make easier the investigation of spin
random fields?
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Spin random fields:
A new tool: the pullback random field
Our answer: the pullback random field
Let T be a random section of the spin-(−s)-line bundle ξs and for each ω, let
g 7→ Xg (ω) the pullback function of type s of the section x 7→ Tx (ω). The
random field X = (Xg )g ∈SO(3) is of type s, i.e.
Xgk = χs (k−1 )Xg , g ∈ SO(3), k ∈ K
(3.22)
Tx = θ(g, Xg ), x = gK ∈ S2
(3.23)
and satisfies
We call X the pullback random field of T .
The notion of pullback r.f. is natural and the following proposition is immediate.
Proposition (T and X are “equivalent”)
a) T is a.s. square integrable if and only if X is a.s. square integrable.
b) T is second order if and only if X is second order.
c) T is a.s. continuous if and only if its pullback random field X is a.s. continuous.
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Spin random fields
January 13, 2014
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Spin random fields:
A new tool: the pullback random field
Proposition (T isotropic ↔ X isotropic)
Let T be an a.s. square integrable spin (−s) radom field and X be its pullback
random field (of type s) on SO(3). Then X is isotropic ↔ T is isotropic.
The notion of pullback random field is natural but also necessary:
Definition (Mean square continuous spin random field)
The spin (−s) random field T is said to be mean square continuous if its pullback
random field X is mean square continuous, i.e.,
lim E[|Xh − Xg |2 ] = 0 .
h→g
(3.24)
As a consequence of the last proposition and definition, we have
Corollary
Every a.s. square integrable, second order and isotropic spin random field T is
mean square continuous.
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Spin random fields
January 13, 2014
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Spin random fields:
A new tool: the pullback random field
Let T be an a.s. square integrable spin −s random field and X its pullback
random field. The Fourier expansion of X is, a.s. in L2 (SO(3)),
Xg =
X√
`≥|s|
2` + 1
`
X
X`m D`m,−s (g) .
(3.25)
m=−`
√
`
where Xm` = 2` + 1hX , Dm,−s
i. If T is isotropic, then X is isotropic and the
following structural theorem is a consequence of well known general properties of
the random coefficients of isotropic random fields (see Baldi and Marinucci
(2006), Baldi and Trapani (2013), or Malyarenko (2011)).
Proposition
Let s ∈ Z, T be a spin (−s) random field and X its pullback random field. If T is
a.s. square integrable, isotropic and second order, then the random Fourier
coefficients Xm` of X in its stochastic expansion (3.25) are pairwise orthogonal and
the variance, c` , of Xm` does not depend on m. Moreover E[Xm` ] = 0 unless
` = 0, s = 0.
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January 13, 2014
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Spin random fields:
A new tool: the pullback random field
Construction of isotropic spin s random fields
• s = 0 → Let S be an appropriate Gaussian white noise on S2 preserving the real
character. f ∈ L2 (S2 ) left-K -invariant → random field T f on S2 defined as
Tfx := S(Lg f),
g ∈ SO(3) : gx0 = x .
(3.26)
T f turns out to be Gaussian, isotropic and mean square continuous.
Its covariance kernel is completely characterized by the function φf on SO(3)
given by
φf (g) = Cov (Tfgx0 , Tfx0 ) = hLg f, fi .
(3.27)
• If f = 1Hx0 the indicator function of the half-sphere, then Wxf := Tfx − Tfx0 is the
P. Lévy’s spherical Brownian motion.
• s 6= 0 → Let S be an appropriate isometry between L2 (P) and the L2 -space
generated by function of type s and −s. f : SO(3) → C square integrable and
bi-s-associated, i.e. f (k1 gk2 ) = χs (k1 )f (g )χs (k2 ), g ∈ SO(3), ki ∈ K → a
complex Gaussian and isotropic random field X of type s given by
Xfg := S(Lg f), g ∈ SO(3)
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Spin random fields
φf (g) = hLg f, fi
(3.28)
January 13, 2014
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Spin random fields:
A new tool: the pullback random field
We proved that previous constructions are indeed representation formulas, i.e.
Every real Gaussian isotropic random field on S2 and every complex
Gaussian isotropic spin random field can be obtained with these
constructions.
• Open question: what should be a spin s Brownian random field?
1) Look for a P.Lévy’s Brownian field W on SO(3), i.e. a centered Gaussian
random field on SO(3), vanishing at some point and whose covariance kernel
would be
Cov (Wg , Wh ) = 12 d(g , e) + d(h, e) − d(g , h)
(3.29)
where e is the identity in SO(3) and d is the distance function on SO(3).
... such a Brownian field on SO(3) does not exist (see Baldi and R. (2013)).
2) Understand what a distance on a spin line bundle would mean.
... open problem.
3) Look for a bi-s-associated function f generalizing the indicator function on the
half-sphere (which allows to obtain the spherical Brownian motion).
... every function we proved seems not to be the right one.
4) ...
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Spin random fields
January 13, 2014
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Spin random fields:
The connection with the classical spin theory
Geller and Marinucci’spin theory
The spin s concept was introduced by Newman and Penrose (1966):
a quantity u defined on S2 has spin weight s if, whenever a tangent vector ρ at
any point x on the sphere transforms under coordinate change by ρ0 = e iψ ρ, then
the quantity at this point x transforms by u 0 = e isψ u.
In Geller and Marinucci (2011) such a u is modeled as a section of a complex
line bundle on S2 , described by giving charts and fixing transition functions to
express the transformation laws under changes of coordinates.
More precisely, they define an atlas of S2 as follows. They consider the open
covering (UR )R∈SO(3) of S2 given by
Ue := S2 \ {x0 , x1 }
and
UR := RUe ,
(3.30)
where x0 =the north pole (as usual), x1 =the south pole. On Ue they consider the
usual spherical coordinates (ϑ, ϕ), ϑ =colatitude, ϕ =longitude and on any UR
the “rotated” coordinates (ϑR , ϕR ) in such a way that x in Ue and Rx in UR have
the same coordinates.
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Spin random fields:
The connection with the classical spin theory
The transition functions are defined as follows. For each x ∈ UR , let ρR (x) denote
the unit tangent vector at x, tangent to the circle ϑR = const and pointing to the
direction of increasing ϕR . If x ∈ UR1 ∩ UR2 , let ψR2 ,R1 (x) denote the (oriented)
angle from ρR1 (x) to ρR2 (x). The transition function from the chart UR1 to the
chart UR2 at x is defined by
eisψR2 ,R1 (x)
(3.31)
This means that if the coordinates at x on the chart UR1 are
( (ϑR1 (x), ϕR1 (x)), z ), then the coordinates at x on the chart UR2 are
( (ϑR1 (x), ϕR1 (x)), z ) → ( (ϑR2 (x), ϕR2 (x)), eisψR2 ,R1 (x) z ) .
(3.32)
This complex line bundle is our spin s line bundle ξ−s = (SO(3) ×−s C, π−s , S2 ).
Plan: 1) trivialize the line bundle ξ−s by choosing the same atlas as before on the
base space S2 and a smart atlas on the total space SO(3) ×−s C;
2) check that the transition functions for ξ−s given by the trivialization in 1) turn
out to be exactly the same as in (3.31).
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Spin random fields
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Spin random fields:
The connection with the classical spin theory
1) Trivialization of ξ−s = SO(3) ×−s C →
π−s
2
S
i) Choose as atlas on S2 the same as before { UR , (ϑR , ϕR ) }R∈SO(3) .
• Each rotation R ∈ SO(3) is described by its three Euler angles (αR , βR , γR ): R
maps the north pole x0 of S2 to the new location x = Rx0 whose spherical
coordinates are (βR , αR ), after rotating the tangent plane at x0 by an angle γR .
• For every R ∈ SO(3) and x ∈ UR , choose a representative element gxR of the
coset x = gxR K in this way: if R = e, then gxe := gx whose third Euler angle
γgx = 0. Otherwise
gxR := RgR−1 x .
(3.33)
ii) Choose an atlas on SO(3) ×−s C as follows.
−1
Let η ∈ π−s
(UR ) and x := π−s (η). Then η = θ(gxR , z) for a unique z ∈ C. Set
−1
the coordinates of η on π−s
(UR ) as
( (ϑR (x), ϕR (x)), z) .
M. Rossi (Rome Tor Vergata)
Spin random fields
(3.34)
January 13, 2014
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Spin random fields:
The connection with the classical spin theory
2) Transition functions
−1
iii) If η ∈ π−s
(UR1 ∩ UR−2 ) and x = π−s (UR1 ∩ UR−2 ), then there exists k ∈ K
such that
θ(gxR2 , z2 ) = η = θ(gxR1 , z1 ) = θ(gxR2 k, z1 ) = θ(gxR2 , χ−s (k)z1 )
which implies
z2 = χ−s (k)z1 ,
k = (gR−1 x )−1 R−1
2 R1 gR−1 x .
2
(3.35)
1
Therefore the transition function from the chart UR1 to UR−2 at x is
χ−s (k) .
(3.36)
Lemma
The rotational angle of k as in (3.35) is = −ψR2 ,R1 (x). Therefore our transition
functions and Geller and Marinucci’s ones coincide, i.e.
χ−s (k) = eisψR2 ,R1 (x) .
M. Rossi (Rome Tor Vergata)
Spin random fields
(3.37)
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Spin random fields:
The connection with the classical spin theory
Spin spherical harmonics: local data
Recall that through the pullback approach, the spin s spherical harmonic s Y`,m is
the section of ξ−s given by
r 2` + 1
`
Y
(x)
:=
θ
g,
D
(g)
,
x = gK ∈ S2 .
(3.38)
s `,m
m,−s
4π
By out trivialization of ξ−s , the local data on the chart Ue of s Y`,m are given by
the function defined, for x = (ϑ, ϕ) ∈ Ue , as
r
2` + 1 `
Dm,−s (ϕ, ϑ, 0) .
(3.39)
s Y`,m;e (x) :=
4π
and similarly we have the local data s Y`,m;R of s Y`,m on UR .
• We find the usual characterization of spin spherical harmonics
{s Y`,m;R }R∈SO(3) .
M. Rossi (Rome Tor Vergata)
Spin random fields
(3.40)
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Spin random fields:
The connection with the classical spin theory
Thank you for your attention!
M. Rossi (Rome Tor Vergata)
Spin random fields
January 13, 2014
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Spin random fields:
The connection with the classical spin theory
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M. Rossi (Rome Tor Vergata)
Spin random fields
January 13, 2014
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Spin random fields:
The connection with the classical spin theory
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Electron. J, Probab. 18 (2013), 1–10.
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M. Rossi (Rome Tor Vergata)
Spin random fields
January 13, 2014
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