Download Symplectic Geometry and Geometric Quantization

Symplectic Geometry and Geometric Quantization
Sophie de Buylr , Stéphane Detournay♥ and Yannick Voglaire♣
r Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures–sur–Yvette, France
♥ Universita’ degli Studi di Milano and INFN, sezione di Milano, 16 Via G. Celoria, 20133 Milano, Italy
♣ Université Catholique de Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium
[email protected] — [email protected] — [email protected]
Abstract: We review in a pedagogical manner the geometrical formulation of classical mechanics in the framework of symplectic geometry and the geometric quantization that associate to a classical system a quantum one. These notes are based on
Lectures given at the 3rd Modave Summer School in Mathematical Physics by Sophie
de Buyl and Stéphane Detournay.
Contents
2
Chapter 1
Foreword
The aim of these lectures will be to concisely present a mathematical approach to the question of quantization
of a physical system, called geometric quantization. To begin with, we will introduce how mathematicians
formalize the notion of a classical mechanical system. Then we will show how they define and perform its
quantization in this framework. The question of quantization consists in assigning a quantum system to a
classical one. This problem is still very timely, since there is in general no unique way of doing so. The
different approaches then try to be as general and natural as possible. The general idea is that the quantization
procedure should preserve the initial structure of the classical system as much as possible. Namely, if a
classical system possesses a symmetry -represented by a hamiltonian action (to be defined later) of this group
on the symplectic manifold modelling the classical phase space-, one would like the associated quantum
system to form a unitary representation of this group. If the action is transitive, this representation should
be irreducible. The latter condition reflects the constraint that the quantization of an elementary classical
system, when possible, should yield an elementary quantum system, these systems being defined as those
which cannot be decomposed in smaller parts without breaking the symmetry. As we will see, to reach these
objectives we will be led to introduce and use a lot of mathematical tools, such as symplectic manifolds,
hamiltonian actions, moment maps, line bundles, Cech cohomology, Chern classes, Kahler manifolds and
polarizations. The prerequisites are a basic knowledge of differential geometry (manifolds, exterior calculus
on manifolds, (co-)tangent bundles, Lie derivative, pull-back, push-forward essentially), that can be found
e.g. in [?] and of the philosophy of quantum mechanics. These lectures are mainly based on [?] and [?].
3
Chapter 2
Geometric formulation of Classical
Mechanics
2.1
Basic notions and examples
The fundamental object used to represent a classical phase space is a symplectic manifold. It consists in a
pair (M, ω), where M is a differentiable manifold and ω a closed 2-form on M (dω = 0) such that ω x is non
degenerate ∀x ∈ M, i.e. if x ∈ M and ω x (Y, Z) = 0 ∀Z ∈ T x M, then Y = 0.
The 2-form ω is called a symplectic structure on M. Being non degenerate, it establishes a linear isomorphism ∀x ∈ M between T x M and T x? M:
T x M → T x? M : X → iX ω := ω(X, . ) .
Examples
1. Euclidean space: M = R2n with coordinates (q1 , ..., qn , p1 , ..., pn ), ω = d pi ∧ dqi .
2. Cotangent bundle : M = T ? N, where N is a manifold. M is the phase space of a system whose configuration space is N. M can be endowed with coordinates (qa , pb ), where at each point x = (q1 , ..., qn ) in
N the components of a form α ∈ T x? N are (p1 , ..., pn ) (i.e. α = pi dqi ).
The symplectic form is given locally by ω = dqi ∧ d pi . It is closed but also exact (i.e. there exists a
one-form θ such that ω = −dθ). Indeed, let π : T ? N → N be the canonical projection, with π(ξ x ) = x
if ξ x ∈ T x? N. The Liouville or canonical one-form θ on M is defined as
θξx (X) := hξ x , (π∗ )αx Xi
(2.1)
where (i) ξ x ∈ T ? N (we here make a common abuse of notation, using the symbol ξ x either for an
element of T ? N – where it is to be understood as (x, ξ x ) – or for an element of T x? N), (ii) X ∈ T ξx T ? N =
T ξx M, (iii) π? : T (T ? N) → T N is the differential of π and (iv) h., .i is the duality between T x N and
T x? N. The 2-form ω = −dθ defines the canonical symplectic structure of the cotangent bundle. Let’s
4
work this out in local coordinates. Let U ⊂ N be an open subset with local coordinates (q1 , · · · , qn ).
We get local coordinates (q1 (x), · · · , qn (x), p1 , · · · , pn ) on π−1 (U) ⊂ T ? N for a point ξ x in T ? N, with
ξ x = pi dqix .
(2.2)
X = ai ∂qi + b j ∂ p j .
(2.3)
In local coordinates, X ∈ T ξx can be written
Hence, using
(π? )αx
∂
∂
= i
i
∂q
∂q
,
(π? )αx
∂
= 0,
∂pi
(2.4)
one has with (??), (??) and (??)
θξx (X) = hpi dqix , a j
∂
i = pi ai = pi dqi (X).
∂q j
(2.5)
In local coordinates, for α x ∈ π−1 (U), the Liouville one-form thus reads
θ = pi dqi
(2.6)
ω|π−1 (U) = dqi ∧ d pi .
(2.7)
and hence, defining ω = −dθ one has
The 2-form ω is in this case globally defined, its expression in local coordinates being given by (??).
3. Coadjoint orbits. The latter play an important role namely in Kirillov’s orbit method, to which we will
allude in Sect.??. Let G be a connected Lie group with Lie algebra G. Let G? be its dual, i.e. the space
of real linear forms on G. The group acts on G? by the so-called coadjoint action:
Ad? = G × G? → G?
(2.8)
with Adg? f = f ◦ Adg−1 , g ∈ G, f ∈ G? , i.e.
hAdg? f, Xi = h f, Adg−1 Xi,
(2.9)
with Adg−1 X = g−1 Xg, X ∈ G. If f ∈ G? , let θ f be the coadjoint orbit of f in G? , defined as
θ f = AdG? f := {Adg? f | g ∈ G}.
(2.10)
If x ∈ θ f ⊂ G? , the tangent space T x θ f is spanned by the vectors X̃ x for any X ∈ G, where
X̃ x := x ◦ adX =: x ◦ [X, . ].
(2.11)
Indeed, we have
d
hAde?−tX x, i =
dt |0
=
5
d
hx, Ade−tX i
dt |0
hx, [X, Y]i.
(2.12)
One may define for all X x∗ , Y x∗ ∈ T x θ f , and ∀x ∈ θ f ⊂ G?
ω x (X x∗ , Y x∗ ) = hx, [X, Y]i , ∀X, Y ∈ G.
(2.13)
One may show (but it’s somewhat lengthy, see e.g. [?, ?, ?]) that ω indeed defines a symplectic structure
on θ f , i.e. that it is well-defined, nondegenerate and closed.
To close with this section, let us mention the important Darboux theorem: if (M, ω) is a symplectic manifold,
then ∀x ∈ M there exists an open neighborhood U of x in M and local coordinates (qi , p j ), called canonical
coordinates on U so that
ω|U = dqi ∧ d pi .
(2.14)
It thus states that any symplectic manifold locally looks like a cotangent bundle. However, in general, the
one-form θ need not be globally defined. In particular, on a compact manifold a global symplectic potential
θ such that ω = −dθ does not exist by virtue of Stokes’ theorem.
2.2
Observables and Poisson algebra
The state of a system in classical mechanics is specified by a point in phase space. An observable is then
simply a real-valued function on the manifold. If (M, ω) is a symplectic manifold, one can associate to each
such function f ∈ C ∞ (M) a vector field X f such that1
i(X f )ω = d f,
(2.15)
where i is the contraction operator, which means that ω x (X f , Y) = d f (Y) = Y( f ) for all Y ∈ T x M. This
relation defines X f , since ω is nondegenerate ∀x ∈ M. It is called the hamiltonian vector field generated by
f . Conversely, a vector field X on M is said hamiltonian if there exists a function fX such that i(X)ω = d fX .
The set of hamiltonian vector fields is denoted by Ham(M, ω).
If f , g ∈ C ∞ (M), one defines the Poisson bracket { f, g} by
{ f, g} = ω(X f , Xg ).
(2.16)
{ f, g} = iX f ω(Xg ) = d f (Xg ) = Xg ( f ) = −X f (g).
(2.17)
Using (??), we may rewrite this as
In canonical coordinates, in which the symplectic form is given by (??), the hamiltonian vector fields and
Poisson bracket read as2
Xg =
∂g ∂
∂g ∂
−
∂pi ∂qi ∂qi ∂pi
{ f, g} = −
∂ f ∂g
∂ f ∂g
+
∂pi ∂qi ∂qi ∂pi
The Poisson bracket enjoys the following properties, ∀ f, g, h ∈ C ∞ (M):
references use the convention i(X f )ω = −d f , which changes some signs from place to place, see below.
expressing ω(Xg , X) = dg(X) in local canonical coordinates
1 Some
2 by
6
(2.18)
(2.19)
(i) skewsymmetry: { f, g} = −{g, f }
(ii) Jacobi identity: { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0}
(iii) derivation: { f g, h} = f {g, h} + { f, h}g
Properties (i) and (ii) give to the set of smooth functions on M the structure of an infinite-dimensional Lie
algebra, while the additional property (iii) endow it with a Poisson algebra structure. The additional property
(iv) X{ f,g} = −[X f , Xg ],
where [., .] is the Lie bracket of vector fields, makes the application f → X f an anti-homomorphism of Lie
structure3 . Property (i) follows from the fact that ω is a 2-form, i.e. an antisymmetric tensor, (ii) is a consequence of dω = 0, using the definition of the exterior derivative4 (iii) can be shown by direct computation
and (iv) follows from the identities i([X, Y])ω = LX (i(Y)ω) − i(Y)LX ω and LX = i(X) ◦ d + d ◦ i(X), where
L denotes the Lie derivative. For detailed proofs, see [?, ?, ?].
2.3
A flight over Hamiltonian and Lagrangian mechanics
In the Hamiltonian formulation of mechanics, a system with configuration space N is characterized by its
phase space, given by a symplectic manifold (M = T ? N, ω) defining the kinematics (i.e. the state of a system
at a given time), and a Hamiltonian function H ∈ C ∞ (M) governing its dynamics (i.e. how it evolves). A
triple (M, ω, H) is called a Hamiltonian system. Hamilton’s equation take the following concise form in this
context:
i(XH )ω = dH.
(2.20)
This equation determines a hamiltonian vector field XH whose integral curves (i.e. the curves whose tangent
vector at each point x is (XH ) x ) are the classical trajectories c(t) for given initial data c(0). By definition, one
has
XH|c(s) =
d
c(t) =: ċ(s).
dt |t=s
(2.21)
In canonical coordinates, a curve or trajectory is written c(t) = (qi (t), p j (t)), with the obvious abuse of notation
qi (t) = qi (c(t)), and ċ(t) = q̇i ∂q∂ i + ṗi ∂p∂ i . On the other hand, (??) gives
XH =
∂H ∂
∂H ∂
− i
,
i
∂pi ∂q
∂q ∂pi
(2.22)
so that (??) gives the familiar Hamilton’s equations
q̇i =
∂H
∂pi
,
ṗi = −
∂H
.
∂qi
(2.23)
Of course, (??) supplemented with the fact that classical trajectories are integral curves of XH are not Godgiven, and derive from variational principles. The physical/classical path of a general mechanical system
3 Remark
that if we set the signs such that i(X f )ω = −d f , see (??), then one gets X{ f,g} = [X f , Xg ], i.e. a homomorphism of Lie
structures.
Pr+1
P
4 dω(X , . . . , X
r+1 X ω(X , · · · , X̂ , · · · , X
i+ j
1
r+1 ) = i=1 (−1)
i
1
i
r+1 ) + i< j (−1) ω([Xi , X j ], X1 , · · · , X̂i , · · · , X̂ j , · · · , Xr+1 )
7
is the one that minimizes an quantity called action. Without entering into too much details, let us say a
word about the Lagrangian formulation of mechanics in a geometrical context. In that case, the dynamics is
governed by a smooth function L : T N → R called Lagrangian. Recall that in the Hamiltonian formalism,
the dynamics was encoded in H, which is a smooth function H : T ? N → R, in some sense the “dual” of L. If
γ(t) is a curve in N, its action is
S =
Zt f
L(γ(t), γ̇(t))dt,
(2.24)
ti
and the classical trajectory γc (t) in N is the one that minimizes S . Working in local coordinates in U ⊂ N,
(qi ), inducing local coordinates (q1 , · · · , qn , v1 , · · · , vn ) on T U yields the familiar Euler-Lagrange equations
(see e.g. [?] p64)
∂L
d ∂L
(γc (t), γ˙c (t)) − k (γc (t), γ˙c (t)) = 0
k
dt ∂v
∂q
(2.25)
How can we connect both formulations? As expected, the operation that does the job is the Legendre transform, which relates tha variational (Euler-Lagrange) and symplectic (Hamilton) formulations of the equations
of motion. Usually, one establish the relations with relations which are of the form H = pq̇ − L, but in general
we have to be careful about the fact that H and L are not functions defined on the same space.
The Legendre transform associated to L ∈ C ∞ (T N) is a map
F L : T N→T ? N
(2.26)
defined ∀x ∈ N by
(F L ) x = T x N→T x? N : v →(F L ) x (v) =: p,
v ∈ T x N, p ∈ T x? N
(2.27)
such that in local coordinates (qi ) on U ⊂ N
v = va
∂L
∂
→(F L ) x (v) = b dqb .
a
∂q
∂v
(2.28)
∂L
) = (qa , pb ),
∂vb
(2.29)
We also write
F L (qa , vb ) = (qa ,
which is also called fiber derivative, since it acts trivially on the base of the fiber bundle while acting as a
derivation on the fiber. If ω is the canonical symplectic structure of T ? N, one may define a two-form on T N
ωL = (F L )∗ ω
(2.30)
HL : T N→R
(2.31)
(HL ) x (v) = h(F L ) x (v), vi − L x (v) ∀v ∈ T x N.
(2.32)
and a function on T N
such that ∀x ∈ N, one has
In that way, one may show (see [?], p69) that every tangent vector X to a path γ(t) satisfying Euler-Lagrange’s
equations also satisfies
i(X)ωL = dHL ,
8
(2.33)
i.e. Hamilton’s equations. This sketches the relation between the two formulations. Of course the situation is
more complicated when ωL can be degenerate at some points, which happens when
∂2 L
∂va ∂vb
= 0 at some points
(see [?, ?, ?]).
Before closing this section, let us emphasize on an important property of the Hamiltonian flow, which consists
of integral curves of XH . Thes statement is the following: any function f ∈ C ∞ (M) such that { f, H} = 0 is
constant along integral curves of XH , and vice-versa. These functions are called integrals of motion. Let us
prove this. Let Ht (x) be the flow of XH :
XH|x =
d
Ht (x) ,
dt |0
H0 (x) = x.
(2.34)
∀t ∈ R.
(2.35)
One would like to show that
d
f (Ht (x)) = 0
dt
We immediately have that
d
f (Ht (x)) = (XH f ) x = {H, f }(x) = 0
dt |0
Let g(x) =
d
dt |0 f (Ht (x)).
at t = 0.
(2.36)
One then has
0 = H s∗ g(x) := (g ◦ H s )(x)
=
=
=
d
H ∗ f (Ht (x))
dt |0 s
d
f (Ht (H s (x)))
dt |0
d
f (Hu (x)) ∀s
dt |u=s
(2.37)
which proves the statement. We used the fact that a one-parameter group of diffeomorphisms satisfies H s ◦
Ht = H s+t . In particular, the Hamiltonian function is an integral of motion.
2.4
Canonical transformations
In Riemannian geometry, isometries of a Riemannian manifold (M, g) are the diffeomorphisms preserving
the metric, in the sense
φ∗ g = g,
(2.38)
where here ∗ denotes the pull–back of the metric g, or in infinitesimal form
LX g = 0,
(2.39)
X being a Killing vector field, generating isometries. Their counterparts in symplectic geometry are called
canonical transformations or symplectomorphisms, defined as the diffeomorphisms of (M, ω) preserving the
symplectic form:
φ∗ ω = ω
(2.40)
LX ω = 0
(2.41)
or
9
As usual, (??) is a shorthand notation for [φ∗ ω|φ(x) ]|x = ω|x , φ∗ denoting the pull-back of forms.
One can show that (??) is the same as preserving the Poisson bracket of functions (see [?] p26 or [?] Sect.
C.3):
{φ∗ f, φ∗ g} = φ∗ { f, g}
(2.42)
We have the following proposition: a vector field X on (M, ω) generates a 1-parameter group of local symplectomorphisms of (M, ω) if and only if X ∈ aut(M, ω), where by definition,
aut(M, ω) = {vector fields X | d(i(X)ω) = 0}.
(2.43)
This follows directly from the identity LX = i(X) ◦ d + d ◦ i(X) (see also [?] for a more precise proof).
Remember that we defined the set of Hamiltonian vector fields, denoted by Ham(M, ω), as
Ham(M, ω) = {vector fields X | ∃ f s.t. i(X)ω = d f }.
(2.44)
Obviously, Ham(M, ω) ⊂ aut(M, ω), since d2 = 0. Therefore, to each function f ∈ C ∞ (M) is associated a
vector field X f generating a one-parameter group of symplectomorphisms.
2.5
Lie groups of symplectomorphisms
The presence of symmetries in a classical system translates in the context of symplectic geometry in the
presence of an action of a Lie group G on the phase space M. Let M be a smooth manifold and G a Lie group.
The group G is called a Lie transformation group of M if to each g ∈ G is associated a diffeomorphism σg
of M such that
1. σg1 σg2 (x) = σg1 g2 (x)
∀x ∈ M, ∀g1 , g2 ∈ M
2. the map σ : G × M → M : (g, x) → σg (x) is C ∞ .
The group homomorphism G → Diff(M) : g→σg is called the action of G on M. It is called effective if e ∈ G
is the only element of G which leaves each x ∈ M fixed. It is said transitive if ∀x ∈ M, σG (x) = {σg (x)|g ∈ G}
is equal to M. If G = Lie(G), each X ∈ G defines a one-parameter group of diffeomorphisms ψtX :
ψtX (x) = σetX (x).
(2.45)
The fundamental vector field5 X̃ associated to X ∈ G is defined as
X̃ x =
d
σe−tX (x).
dt |0
(2.46)
One can prove (see [?], pp. 4,34 and [?, ?]) that the map G→ vector fields: X → X̃ is a Lie algebra homomorphism:
]
[X̃, Ỹ] = [X,
Y].
5 Usually,
(2.47)
the fundamental vector field associated to X is denote X ∗ , to avoid confusion with the dual or pull–back, we prefer here the
notation X̃.
10
An action σ : G→Diff(M) is called a symplectic action if σg is a symplectomorphism for all g ∈ G, i.e.
σ∗g ω = ω
∀g ∈ G.
(2.48)
From now on, we’ll restrict ourselves to symplectic actions. Following what we have seen in the previous
section, the vector field X̃ ∈ aut(M, ω), ∀X ∈ G, since it generates a symplectomorphism. Thus d(i(X̃)ω) = 0.
If moreover, ∀X ∈ G a function λX such that
i(X̃)ω = dλX ,
(2.49)
exists, i.e. the fundamental vector fields X̃ are hamiltonian ∀X ∈ G, then the action is said almost hamiltonian.
In that case one has
X̃ = −{λX , . } .
(2.50)
as follows from (??) and (??). Remark that with the notation (??), one has the equality X̃ = XλX (where the
‘big’ X of XλX does not refer the element X ∈ G). If moreover the correspondence X → λX is a Lie algebra
homomorphism (i.e. λkY+Z = kλY + λZ , ∀k ∈ R, λ[Y,Z] = {λY , λZ }), then the action is said hamiltonian. Once
one is given an almost hamiltonian action of G on M, one can always render it hamiltonian by extending G
to G̃ = G × R and making an appropriate choice for λX (see [?], p39 and [?, ?]).
Example: coadjoint orbits
Let us now take M = θ f , see Sect.??. Being the orbit in g∗ of f under the group G, M obviously
admits an action of G:
σ : G × M→M : (g, x)→σ(g).x
,
(2.51)
whose fundamental vector fields are given by (??). One can show that G actually acts by symplectomorphisms, that is
σ(g)∗ ω = ω.
(2.52)
Next, consider the action of a one-parameter subgroup of G, σ(exp(tZ)), with Z ∈ g. This gives a
one-parameter group of symplectomorphisms of the orbit M. It can be checked that its associated
fundamental vector field Z̃ is Hamiltonian and
i(Z̃)ω = −dλZ
,
where λZ (x) = hx, Zi.
(2.53)
Furthermore, the correspondence g → C ∞ (M) can be shown to be a Lie algebra homomorphism.
Therefore, the action of G on M given by (??) is Hamiltonian.
The importance of coadjoint orbits rests on the fact that the only symplectic spaces admitting a
Hamiltonian action of G are the covering spaces of the coadjoint orbits of G [?, ?, ?] .
11
2.6
Moment maps, hamiltonian systems with symmetry and integrals
of motion
Assume that a Lie group G has a hamiltonian action σ on (M, ω), and set, ∀Y ∈ G, Ỹ = XλY , for a certain
λY ∈ C ∞ (M). The moment map is the map
λ : M → G? : x → λ(x)
with
hλ(x), Yi := λY (x).
(2.54)
The importance of the moment map appears namely when considering a hamiltonian system with G–symmetry,
consisting in a hamiltonian system (M, ω, H) and a hamiltonian action of G on M leaving the Hamiltonian
invariant:
σ∗g H = H
∀g ∈ G.
(2.55)
Indeed, in this case, the invariance condition of H gives infinitesimally
LỸ H
=
0
=
Ỹ(H)
=
−{λY , H}.
(2.56)
Thus the functions λY , called the moment functions are integrals of motion forming through their Poisson
bracket a Lie algebra isomorphic to that of G.
Example: Let us consider a simple example to illustrate this [?]. Take M = R2n = {x = (p, q) = (pi , q j )}.
Consider the hamiltonian action of the one-parameter group G = {gt } on M, with
,
gt : p j → p j
q j →q j + t.
(2.57)
The corresponding Hamiltonian field is
n
X ∂f
d
d
f (gt (x)) =
f (p1 , · · · , pn , q1 + t, · · · , qn + t) =
:= X f = −{λX , f }.
dt |0
dt |0
∂q j |x
j=1
(2.58)
With (??), one finds that
λX =
n
X
p j.
(2.59)
j=1
The invariance condition of H under G is H(pi , q j ) = H(pi , q j + t), or infinitesimally
n
P
j=1
∂
H(pk , qi )
∂q j
= 0. This
simply states that H can only depend upon the q j ’s through the differences qi − q j . When this the case, one
can check that
{H, λX } = 0
(2.60)
which expresses the well-known fact that the total momentum is conserved in a system invariant under translations.
12
Chapter 3
Geometric Quantization
The issue of assigning a quantum system to a classical one, i.e. quantization, is still very timely since there
is no unique way of proceeding. The various approaches then try to be as general and natural as possible. It
is also asked that the quantization procedure preserve as much as possible the initial structure of the classical
system. For instance if a classical system possesses a symmetry group – through an hamiltonian action of this
group on the symplectic manifold modelling the classical phase space – one would like that the associated
quantum system belong to an unitary representation of this group. Moreover, if the action of the group is
transitive, this representation should be irreducible.
3.1
General Philosophy
We have seen that the mathematical geometrical framework to study a classical system is symplectic geometry. The state of a system is specified by a point on a symplectic manifold (M, ω) which is called the phase
space, the observables are smooth functions on this manifold and their algebra is endowed with an additional
structure given by the Poisson bracket { , }.
Our purpose is to associate to such a classical system a quantum system. This means that we want to establish
a correspondence between the classical quantities and the quantum ones. The classical states — p ∈ M —
should become rays of an Hilbert space H (i.e. equivalence classes of elements of H, with v ∼ w if v = λw,
where λ is a scalar) and the classical observables — f on M — must correspond to self–adjoint operators
O f on this Hilbert space. The quantum observables are also endowed with a Poisson structure, namely the
commutator [ , ], see Table ??. These remarkable similarities led Dirac to formulate its quantification rules,
asking for the linear correspondence f → O f between classical and quantum observables in such a way that:
O1 = Id ,
[O f , Og ] = i~ O{ f,g} .
(3.1)
For instance, if we apply this to the canonical coordinates on R2n , we could get
qa → Oqa = qa ,
pa → O pa = −i~
13
∂
,
∂qa
(3.2)
Table 3.1: Classical System vs. Quantum System
Phase Space
States
Algebra of Observables
Structure on the Algebra of Observables
Classical System
Quantum System
symplectic manifold (M, ω)
Hilbert space H
point of M
ray of H
∞
f ∈ C (M)
Self–Adjoint Linear Operator on H
Poisson bracket
Commutator
where the operators act on the functions in L2 (Rn ) (depending only on the qa ). Then to extend this correspondence to operators, for example quadratic on the pa , it is necessary to make ordering choices. This method
therefore clearly depends on the coordinate choices which is a bad feature.
As mentioned above, an important issue in the quantization procedure is also to preserve the structure of the
classical phase space as much as possible. In particular, if a group acts on the classical system (M, ω) through
an Hamiltonian action, a ‘good’ quantization will translate this into the fact that the quantum states belong
to irreducible representations of this group. In the example of quantizing the canonical coordinates of R2n
mentioned above, the subspace of functions that depend only on the qa ’s (and not on the pa ’s) is clearly an
invariant subset under the action of the Oqa ’s and the O pa ’s given by equations (??). A solution to obtain
an irreducible representation of R2n (i.e. the group of invariance of the classical system1 ) is to restrict the
space of functions to quantize to be the subspace of C ∞ (R2n ) that depend only on the qa ’s, i.e. C ∞ (Rn ). More
generally, if the quantization procedure furnish a reducible representation of group, a solution is to restrict
the quantization of observables to a Poisson subalgebra A of C ∞ (M). It is generally not an easy task to select
the appropriate subalgebra A.
The program of geometric quantization is a procedure of quantization based on the ideas of Dirac that is
applicable to any symplectic manifold, i.e. any phase space, which is independent of coordinate choices and
which ‘keeps’ tracks of the symmetry of the classical system with an Hamiltonian action G [in the sense that
the quantum states form irreducible representation of G]. A first step in the geometric quantization procedure,
namely the prequantization, consists in forgetting the irreducibility condition. It is an elegant construction
involving line bundles and connections on these line bundles and but unfortunately, when one ask for the irreducibility condition to hold, this complicate the story. The will be done by the introduction of a polarization
of the space of functions on M (the aim is to select a subspace of functions which are quantize) as explained
in the sequel.
Geometric Quantization procedure : Let (M, ω) be a classical system and A a sub-algebra of observables.
A quantum system (H, O) is said to be associated with this classical system if
1 not
really hamiltonian, this will be explained at the end of section ??
14
Q1. H is a complex separable Hilbert space. Its elements ψ are the quantum wave functions and the rays
{λψ |λ ∈ C} are the quantum states.
Q2. O is an application that maps a classical observable f ∈ A into an self-adjoint operator O f on H such
that
O1. Oλ f +g = λO f + Og ,
O2. O1 = IdH ,
O3. [O f , Og ] = i~O{ f,g} ,
O4. if there is a transitive and strongly hamiltonian action of a group G on M, then H forms an irreducible
representation space of this group.
3.2
Mathematical Preliminaries
This section is devoted to introduce notions needed for the geometric quantization procedure. As explained
after, a line bundle is the correct object to define the states of a system, the observables are then connections
on this line bundle. Finally, as the quantum states should belong to an Hilbert space, the notion of hermiticity
on the fiber is introduced.
3.2.1
Line Bundle
A (complex) line bundle is a vector bundle whose fibers are one–dimensional (complex) vector spaces. Explicitly, it consists in a triple (L, π, M) such that
F1. L is a differentiable manifold and π : L → M is a smooth surjective application,
F2. ∀x ∈ M, the fiber E x = p−1 (x) possesses a structure of one dimensional complex vector space,
F3. There exist an open covering {Ua |a ∈ A} of M and smooth functions sa : Ua → L such that
(a) ∀a ∈ A, π ◦ sa = IdUa ,
(b) ∀a ∈ A, the application
ψa : Ua × C → π−1 (Ua ) : (m, z) → z sa (m) ,
(3.3)
is a diffeomorphism.
The functions sa satisfying the condition (a) are called local sections of L and a collection {(Ua , sa )| a ∈ A} a
local system for L. The space of global sections, s : M → L : m → s(m), is denoted Γ(L). A line bundle is
depicted in Figure ??.
15
L
π−1 (Ua )
ψa
Ua × C
π
sa
M
Ua
Figure 3.1: Line Bundle
Remark 1: the condition F3 (b) implies that the sections are nowhere vanishing, otherwise ψa could not be a
diffeomorphism.
Remark 2: The notation sa (m) denotes an element of L but it is also sometimes used to designate its corresponding element in the fiber at the point m, i.e. we have sa (m) = (m, sa (m)) and the meaning is clear according to the context. For instance in equation (??), z sa (m) (where sa (m) ∈ L) means z.sa (m) = z.(m, sa (m)) =
(m, z.sa (m)) ∈ L. This abuse of notation will often be used in the sequel.
Transition functions
On the intersection Uab := Ua ∩ Ub of two open sets Ua and Ub , a transition function can be defined by
Ψab = ψ−1
a ◦ ψb : U ab × C → U ab × C : (m, z) → (m,
sb (m)
z) .
sa (m)
We can therefore define the functions gba : Uab → C? = C/{0} : m → gba (m) =
sb (m)
sa (m)
such that Ψab (m, z) =
(m, zcba ). These functions satisfy
gaa
=
1 on Ua ,
gab gba
=
1 on Uab if not empty,
gab gbc gca
=
1 on Uabc := Ua ∪ Ub ∪ Uc if not empty .
(3.4)
The three conditions (??) are called cocycle conditions. For later purpose, these conditions can further be
expressed by setting
gab := e2πihab ,
hab : Uab → C smooth functions,
16
(3.5)
as
haa
=
0 on Ua ,
hab
=
−hba on Uab if not empty,
hab + hbc + hca
=: Cabc ∈ Z on Uabc = Ua ∪ Ub ∪ Uc if not empty .
(3.6)
Cabc is constant on Uabc (since it is smooth on a connected set and can only take discrete values) and is totally
antisymmetric in its arguments.
Remark: An equivalent definition of a line bundle can be given in terms of the transition functions. The
condition F3 is replaced by asking that there exists an open covering {Ua |a ∈ A} of M such that on each
intersection Uab , there exists a transition function Ψab : Uab × C → Uab × C : (m, z) → (m, gba (m)z) such that
the conditions (??) are fulfilled.
Trivialisation
The collection {Ua , ψa } defined by a local system {Ua , sa } is called a local trivialisation of L since over each
open set Ua , the diffeomorphism ψa gives to the fiber E x (for all x ∈ Ua ) a “trivial” structure, i.e. it establishes
a isomorphism between π−1 (Ua ) and the direct product Ua × C. A line bundle is said to be trivial if L is
globally diffeomorphic to M × C.
Cech Cohomology
The Cech cohomology that we will now introduce, provides a useful tool to classify line bundles . Let G be
an abelian group and {Ua |a ∈ A} a contractible2 open covering of a smooth manifold M. A p–cohain on M
with values in G is a rule c which assigns to each collection Ua0 , ..., Ua p of (p + 1) open sets in the covering
with non empty intersection (Ua0 ∪ ... ∪ Ua p , φ) an element ca0 ...a p in G so that it is totally antisymmetric in
its arguments.3 The co–boundary δc of a p–cochain c is the (p + 1)–cochain defined by
δcα0 ..α p+1 :=
p+1
X
(−1)i cα0 ...α̂i ...α p+1 ,
i=0
where α̂i means that the indice αi is omitted. One may check that δ2 = 0.
Example: Let us illustrate the notion of cochain and co–boundary on a very simple example. We
consider a manifold which is the union of three open sets U0 , U1 , U2 and the abelian group of
natural number N, +. A 1–cochain is a rule that assigns to {U0 , U1 }, {U1 , U2 }, {U0 , U2 }, {U1 , U0 },
{U2 , U1 } and {U2 , U0 } some natural numbers c01 , c12 , c02 , −c01 , −c12 and −c02 . Let us take a
concrete example c01 = 3, c12 = −1, c02 = 4. The co–boundary of c is a 2–cochain whose values
2 Contractible
3 Note
means that any close curve in Ua can be smoothly deformed to a point.
that the order in which the open sets are given matters for the rule c.
17
(3.7)
on the sets {U0 , U1 , U2 }, {U1 , U0 , U2 }, {U2 , U1 , U0 }, {U0 , U2 , U1 }, {U2 , U0 , U1 } and {U1 , U2 , U0 },
is given by the equation (??). For instance, one has δc012 = c12 − c02 + c01 = −2.
p
The pth –Cech cohomology space HCech
(M, G) is defined by
p
HCech
(M, G) :=
{p − cochains c | δc = 0}
{p − cochains c | c = δb}
(3.8)
p
Theorem : The pth –cohomology space HCech
(M, G) is independent of the choice of the contractible covering
p
p
of M. Moreover, if G = R or C, HCech
(M, G) ∼ HDe
Rham (M, G) (canonical isomorphism).
Let us now return to the transition functions of a line bundle and their cocycle conditions. Equation (??)
expresses that C is a 2–cochain. By explicit computation one may check that
Cbcd − Cacd + Cabd − Cabc = 0 if Uabcd , 0 .
(3.9)
Thus C is a 2–cocycle for the Cech cohomology of M with values in Z (δC = 0). For line bundle (L, π, M),
2
let c(L) denote the class of C in HCech
(M, Z):
c(L) := {2 cocycles c0 | c0 = C + δb, b is a 1 − cochain} ,
(3.10)
it is called the Chern class of the line bundle.
Theorem : The line bundle (L, π, M) is trivial iff c(L) = 0. Moreover, two line bundles are equivalent iff they
have same Chern class and there is a bijection between the set of equivalence classes of line bundles on M
and H 2 (M, Z) (the latter classify the former).
3.2.2
Connection on a Line Bundle
A connection is an application ∇ that associates to each vector field X ∈ V(M)C an endomorphism ∇X :
Γ(L) → Γ(L) such that
C1. ∇X+Y s = ∇X s + ∇Y s,
C2. ∇ f X s = f ∇X s,
C3. ∇X f s = X( f )s + f ∇X s,
for all s ∈ Γ(L), X, Y ∈ V(M)C and f ∈ C ∞ (M)C . The linear operator ∇X is the covariant derivative along X
for the connection ∇. Note that the condition C2 implies that ∇X s(m) depends on X only at the point m while
condition C3 indicates that ∇X s(m) depends on locally on the section s. We can focus on a open set Ua . Since
the connection is linear in X, we have for each sa ,
∇X sa = hθa , Xisa
for some 1–form θa ∈ Ω1C (Ua ). Therefore, for an arbitrary section (which can be written locally as s = fa sa
on Ua where fa ∈ C ∞ (Ua )), we have4
∇X s = (X( fa ) − 2πihαa , Xi fa )sa .
4 multiplication
of a section by a function means multiplication in the fiber, i.e. f s(m) = (m, f s(m)) locally
18
(3.11)
The set of 1–forms αa on the Ua is called 1–form connection. This 1–form connection (Ua , θa ) is not globally
defined on M.
On a non–empty intersection Uab , we have sa = gba sb and therefore
∇X sa
=
(X(cba ) − 2πihαb , Xigba )sb
=
−2πihαa , Xigba sb
which implies that
αb − αa =
1 dgba
.
2πi gba
(3.12)
The equation (??) implies that dαa|Uab = dαb|Uab . The 2–form Ω defined by
Ω|Ua = −dαa
∀a ,
(3.13)
is therefore globally well defined.
3.2.3
Curvature on a Line Bundle
The curvature of a connection is generally defined as the operator Curv∇ (X, Y) : Γ(L) → Γ(L) such that
Curv∇ (X, Y) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y] .
This operator is skewsymmetric and C ∞ (M)C –linear in X and Y and therefore should be given by a 2–form
on M called 2–form curvature. Explicitly, we have
Curv∇ (X, Y)sa
=
∇X ∇Y sa − ∇Y ∇X sa − ∇[X,Y] sa
=
∇X (−2πihαa , Yisa ) − ∇Y (−2πihαa , Xisa ) + 2πihαa , [X, Y]isa
=
−2πiX(hαa , Yi)sa + 2πihαa , Yi2πihαa , Xisa
+2πiY(hαa , Xi)sa + 2πihαa , Xi2πihαa , Yisa
+2πihαa , [X, Y]isa
=
−2πi dαa (X, Y)sa
=
2πi Ω(X, Y)sa
19
3.2.4
Hermitian Structure on a Line Bundle
An hermitian structure on a line bundle (L, π, M) is a correspondence x → H x = ( , ) x where H x is a positive
definite sesquilinear inner product5 on E x C, such that
x → H x (s(x), s(x)) ∈ C ∞ (M) ∀s ∈ Γ(L) .
(3.14)
[s(x) stands for (x, s(x)) as usual].
A connexion ∇ on a line bundle is called hermitian iff
X(H(s, s0 )) = H(∇ x s, s0 ) + H(s, ∇ x s0 ), ∀X ∈ V(M), s, s0 ∈ Γ(L) .
(3.15)
Remark 1: The curvature of a hermitian connexion is always real.6 Indeed, let {(Ua , sa )} be a local system,
{(Ua , αa )} a connexion ∇ and X be a real vector field. Condition (??) implies that [by writing H x (sa (x), sa (x)) =
(sa , sa )],
X((sa , sa ))
→ ᾱa − αa
=
(∇X sa , sa ) + (sa , ∇ x sa )
=
(−2πihαa , Xisa , sa ) + (sa , −2πihαa , Xisa )
=
2πihαa , Xi(sa , sa ) − 2πihαa , Xi(sa , sa )
=
α(sa , sa )(X)
1 α(sa , sa )
,
2πi (sa , sa )
=
(3.16)
where ᾱa is such that hᾱa , Xi := hαa , Xi. By taking the exterior derivative of both sides, we find that
dᾱa − dαa = 0, this implies that dαa is real ∀a and therefore Ω is real.
Remark 2: Let’s start from an arbitrary local system {(Ua , s0a )}. One may construct another system {(Ua , sa )}
p
with sa = s0a / (s0a , s0a ) such that (sa , sa ) = 1 everywhere ∀a. If the αa ’s are connexions 1–forms, this implies
that the αa ’s are real. But this in turn implies that the transition functions gab satisfy
1
=
(sa , sa ) = (sb gba , sb gba ) = gba gba (sb , sb )
=
gba gba .
(3.17)
Thus, every line bundle endowed with a hermitian connexion admits a local system whose transition functions
are U(1)–valued.
5 It
is an application (., .) : Γ(L) × Γ(L) → C such that ∀p, q, r ∈ E x and all z ∈ C,
H1. (p, p) ≥ 0,
H2. (p, q) = (q, p),
H3. (p, zq + r) = z(p, q) + (p, r),
H4. (p, p) = 0 implies p = 0.
6A
vector field X ∈ VC (M) = V(M) ⊗ C V(M) ⊕ iV(M) is said real if X( f ) ∈ C ∞ (M) ∀ f ∈ C ∞ (M). Writing X = A + iB,
A, B ∈ V(M), this implies as anticipated that B = 0. A 1–form α ∈ Ω1 (M)C is said real if hα, Xi ∈ C ∞ (M) ∀ real X.
20
3.3
Prequantization
As mentioned above, the first step in the quantization program, prequantization, is a simplification of the full
program described in section ??: it consists in forgetting the condition of irreduciblility Q2.O4. Our starting
point is a classical system, i.e. a symplectic manifold which models the classical phase space, (M, ω) of dimension 2n. We would first like to construct a Hilbert space from it in order to associate quantum states to
classical states. Next, we will consider the quantum observables and it will turn out that they can be defined
as connections on line bundles (as we will see, the quantum states should then be understand as line bundles).
We will also see that not any classical system (M, ω) can be prequantize and a condition of prequantization
will be given.
States
Let us consider a symplectic manifold (M, ω) of dimension 2n. The most natural vector space to consider
is the one of smooth functions on M. The symplectic form ω provides a volume form which allows us to
integrate over M, we have therefore a scalar product,
hφ1 , φ2 i =
Z
ψ̄1 ψ2 µ
M
where ψ1 and ψ2 are functions with compact support and µ is
µ =
=
1 n
ω
n!
dq1 ∧ dq2 ∧ ... ∧ dqn ∧ d p1 ∧ d pn .
(−1)n(n−1)/2
(3.18)
This construction provides a pre–hilbert space and its completion is an Hilbert space. This first step is very
easy and natural. However, the construction of quantum observables will lead to review the
Observables
Next, we want to define operators on this Hilbert space associated with the classical observables (i.e. functions
f on M). We know that to each function f on M, a vector field X f ∈ V LH (M) can be assigned. The most
natural correspondence is therefore7 f → O(1)f = ~i X f . However, a direct observation — a constant is sent to
zero — forbids this correspondence. Let us try a simple change that corrects this fact,
f → O(2)f =
~
Xf + f .
i
Unfortunately, the condition Q2 O3 is now no longer respected. Indeed
[O(2)f , O(2)
g ](ψ) =
7 the
[−i~X f + f, −i~Xg + g](ψ)
=
−~2 [X f , Xg ](ψ) − i~(X f (gψ) + f Xg (ψ) − Xg ( f ψ) − gX f (ψ))
=
−~2 [X f , Xg ](ψ) + 2i~{ f, g}ψ ,
indice (1) on O f just means that it is our first attempt for a correspondence
21
and
= ~2 X{ f,g} (ψ) + i~{ f, g}ψ
i~O(2)
(ψ)
{ f,g}
= −~2 [X f , Xg ](ψ) + i~{ f, g}ψ,
which implies that the condition Q2 O3 is clearly not fulfilled. To cure this fact, let us try the correspondence
O(3)f =
~
X f − hα, X f i + f
i
(3)
8
for some 1–form α. The condition [O(3)f , O(3)
g ] ψ = i~ O{ f,g} yields
0
~
~
= −~2 ([X f , Xg ] + X{ f,g} ) − (X f α(Xg ) − Xg α(X f ) − α([X f , Xg ])) + (X f (g) − Xg ( f ) + { f, g})
i
i
= −dα(X f , Xg ) + { f, g}
= i~ (dα + ω)(X f , Xg ).
This fixes the 1–form α to satisfy,
dα = −ω .
When ω is exact, any choice of α such that ω = −dα gives a prequantization. [This applies in particular to the
case of a cotangent bundle T ? N where one can choose for α the Liouville 1–form on T ? N.] In the general
case, ω is closed but not exact. If {Ui }i∈I is a covering of the manifold by contractible open sets, the Poincaré
lemma tells us that there exist 1–forms α j on U j such that
ω|U j = −dα j .
(3.19)
On U j one can take, for a function f on M, the operator O j| f as follows,
O j| f = −i~X f − α j (X f ) + f .
We have succeeded in assigning a quantum operator O j| f to each function f (i.e. classical operator) on each
open set U j . Now comes the question of how to piece those formulas together? On the intersection U j ∩ Uk ,
we have
dα j − dαk = 0 → α j − αk = d f jk ,
(3.20)
where f jk is a smooth function on U j ∩ Uk . Then on U j ∩ Uk (exercise), we have
i
i
Ok| f (e ~ fk j h) = e ~ fk j O j| f (h) , ∀h ∈ C ∞ (U j ∩ Uk ) .
Thus viewing
i
lk j = e ~ fk j
(3.21)
Ok| f = lk j ◦ O j| f ◦ lk−1j on U j ∩ Uk , ∀ f .
(3.22)
as a multiplication operator on C ∞ (Uk ∩ U j ),
8 remember
that dα(X, Y) = Xα(Y) − Yα(X) − α([X, Y]).
22
We want of course the action of the operators Oi| f to be independent on the open subset on which we are (in
particular, on U jk , both O j| f and Ok| f must give the same operator acting on wave functions). Therefore, we
may no longer consider as wave functions simply functions on the manifold, as seen from equation (??).
The way out is the following. Assume one may construct a line bundle (L, π, M) whose transitions functions
gk j are precisely the lk j . Let {(Ui , si )} be a local system for this bundle. A general section is specified by
s = fi si on each Ua . On the intersection U j ∩ Uk , we have
fk = f j gk j .
(3.23)
If s = { f j | j ∈ I} is an element of Γ(L), we define ∀ f :
O f s := {O j| f , j ∈ I} .
(3.24)
The element O f s is itself a section because (using , and g jk = l jk ),
Ok| f fk = Ok| f gk j f j = gk j O j| f g−1
k j gk j f j
→ (Ok| f fk ) = gk j (O j| f f j ) .
(3.25)
We have thus defined, for each f ∈ C ∞ (M), an operator O f on Γ(L) (remember that the reason for having to
do so basically originates in the fact that the one–form α in (??) may not be globally defined), such that
[O f , Og ] =
{[O j| f , O j|g ] f j , j ∈ I}
=
{−i~O j|{ f,g} f j , j ∈ I}
=
−i~O{ f,g} s
(3.26)
The question is now: when can we construct such a line bundle, i.e. a line bundle whose transition functions
are given by the l jk which depend on the f jk – see equation (??), thus on the α j – see equation (??), and
therefore on the ω|U j – see equation (??). More precisely, we have seen that the transition functions gab of a
line bundle satisfy equations (??) or equations (??) in terms of the hab , gab =: e2πihab . Consequently, every set
of functions satisfying these conditions can be viewed as transition functions of some line bundle.
Conditions for prequantization : A manifold (M, ω) is then said quantizable if one can build a prequantization, i.e. if one can choose αi and fk j (see equations (??) and (??)) so that
(i)
ω|U j = dα j ,
(3.27)
(ii)
αi − α j|U jk = d fk j ,
(3.28)
(iii)
e ~ f jk =: l jk define transition functions for a line bundle over M .
i
(3.29)
The condition (iii) implies that Ci jk defined by the following equation
hi j + h jk + hki =
1
( fi j + f jk + fki ) = Ci jk
2π~
23
(3.30)
is a Z–valued cocycle (see section ??). We see that the existence of a line bundle can be rephrased in terms
p
of the Cech cohomology HCech
(M, K), with K = R or C. It is related to the fact that the class c, [c], in
2
HCech
(M, K) is integral (or not), i.e. the fact that it has a representative which takes only integral values.
But we also mentioned the existence of an isomorphism between Cech and de Rham cohomology classes
(in section ??). Without entering the details, this isomorphism allows one to rephrase the existence of a line
bundle satisfying the required conditions (i), (ii) and (iii) here above (i.e. the possibility to construct a prequantization) directly in terms of the symplectic form ω.
The statement is that one can build a prequantization of the classical system (M, ω) iff
h i
ω
2
2π~ ∈ HdR (M, Z) ,
i.e. the cohomology class of
ω
2π~
has to be integral, or equivalently its integral on any
closed 2–surface has to be an integer.
Now suppose that this condition is satisfied. We arrived at the conclusion that wavefunctions are given by
section Γ(L) of a line bundle (L, π, M) whose transition functions are given by (??) and ??, and that the
operators O f associated with classical observables f are
:=2πihα̃a ,X f i
z }| {
i
O f = −i~(X f − hα, X f i + f ,
~
(3.31)
with ω|Ua = dαa . But, from the definition of a connexion on a line bundle, see equation (??), this is nothing
other than
O f = −i~∇ f + f ,
(3.32)
where the connexion one–form of the connexion ∇ is {(Ua , α̃a )}. The curvature of this connexion, see equation
(??), is thus,
Ω|Ua = −dα̃a = −
1
1
dαa = −
ω|U .
2π~
2π~ a
(3.33)
A related theorem is that a 2–form Ω is the curvature 2–form of a line bundle iff Ω is integral ([?, ?, ?] and
references therein).
The last ingredient to get a Hilbert space is to endow the space of sections with a scalar product, which is
done by asking that the line bundle possesses a hermitian structure, see (??). It defines a scalar product on
the space of compactly supported sections,
hs1 , s2 i := (
1 n
)
2π~
Z
H x (s1 (x), s2 (x)) µ .
M
24
(3.34)
[one can show, see [?, ?], that every line bundle can be endowed with a hermitian structure]. Moreover, if the
connexion is hermitian (see (??)), then the operators are symplectic :
hO f s1 , s2 i = hs1 , O f s2 i .
(3.35)
It can be shown that, given a line bundle with a connexion one–form {(Ua , θa )}, a compatible hermitian structrue H(., .) exists if and only if θa − θ̄a is exact, or, because Ω|Ua = dθa , if the curvature Ω is real [?, ?].
Summary of the prequantization procedure: Let us consider a classical system (M, ω) satisfying the preh i
ω
2
∈ HdR
(M, Z). The prequantization procedure consists in (i) assigning to the
quantization condition, i.e. 2π~
classical states, i.e. points on the manifold M, quantum states which are sections on the line bundle (L, π, M)
whose transition functions are given by lk j (see equations (??) and (??)), and (ii) assigning to the classical
observables, i.e. functions f on the manifold M, quantum observables which are given by O f = −i~∇ f + f .
Moreover, with the scalar product given in equation (??), the space of sections on the line bundle (L, π, M)
becomes an Hilbert space.
So, we are getting close. But we are not yet done, as we can illustrate on a very simple example. Consider
Q = Rn and phase space M = T ? Q = R2n , ω = d p j ∧ dq j with j = 1, ..., n. M admits an action of G = R2n by
symplectomorphisms. (a, b) ∈ G = R2n acts on (p, q) ∈ M = R2n by
σ(a,b) (p, q) = (p + a, q + b) .
(3.36)
The fundamental vector fields of this action are ∂ pi and ∂q j , and one can show that this action is almost
hamiltonian [e.g. i(−∂ pi ω = −dqi +cst. ]. Actually, it is not hamiltonian, but there exists a canonical way to
make a hamiltonian action from an almost one (basically consider G̃ = G × R, the extension of G → then the
action of G̃ = connected and simply connected Lie group with Lie algebra G̃). In this case, the fundamental
vector field of the action of G̃ are unaffected (see [?]). Note that, in this case, since ω is exact globally,
[ω] = 0. Thus the system is clearly quantizable, and the only line bundle up to equivalence is the trivial one
L = R2n × C, whose sections can be identified with functions on R2n . We found that
Op j
=
Oq j
=
∂
∂qi
∂
i~
∂p j
−i~
which are the operators associated with the observables p j and q j corresponding to the hamiltonian fundamental vector fields − ∂q∂ j and
∂
∂p j
of the action of R2n .
If one considers the action of these operators on the subspace Cq (R2n ) ⊂ C(R2n ) of functions depending only
on qi , on sees that
O p j f = −i~
∂f
,
∂q j
Oq j f = q j f ,
f ∈ Cq (R2n ) ,
(3.37)
thus Cq (Rn ) is an invariant subspace, and the representation is not irreducible. One must then find a natural
way to restrict the Hilbert space. (i.e. in this example, go from C(R2n ) to C(Rn )). This leads us to introduce
more ingredients, namely the notion of polarization.
25
3.4
Polarization
We have seen that the prequantization procedure is not sufficient to take into account the irreducibility condition. Remember that the irreducibility condition expresses the fact that if the classical system has an hamiltonian action of a group G, one wants that quantum states to form an irreducible representation of G. To
deal with this condition, we first to need introduce some mathematical definitions. The notion of polarization
will play a central role. To illustrate this purpose, we give an important example of polarization which is
related to Kälher manifolds. Next, we will use the introduced notions to select a subspace of functions on
M to be quantized. Finally we mention very nice theorems about the fact that quantum states now form a
representation of the group G.
3.4.1
Basic definitions
First, we consider complex vector fields on M by extending C–linearly the symplectic form to T M C =
T M ⊗ C = T M ⊕ iT M, i.e. iX+iY ω = i x ω + i iY ω ∀X, Y ∈ T M.
Let ρ be a symplectic form on the vector space V. The symplectic complementary vectorial subspace, W ρ =
{X ∈ V|ρ(X, Y) = 0 , ∀Y ∈ V} of the vectorial subspace W is said to be
- coisotropic if W ρ ⊂ W,
- isotropic if W ρ ⊃ W,
- lagrangian if W ρ = W.
A complex distribution D on M is a sub–fibre of the complex tangent fibre, i.e. to each point m ∈ M is
associated a vectorial sub–space Dm of T m M C which varies smoothly with m. The set of vector fields tangent to D is denoted VD (M). A distribution is said to be involutive if it is closed under the Lie bracket,
i.e. ∀X, Y ∈ VD (M), [X, Y] ∈ VD (M).
A polarization P on a symplectic manifold (M, D) is a complex distribution of T M C such that
P1. Pm is lagrangian for all m ∈ M,
P2. P is involutive,
P3. the dimension of P ∩ P̄ ∩ T M is constant (P̄ denotes the distribution complex conjugate to P).
Remark : The distributions invariant under conjugation are necessarily complexified real distributions. The
complex distributions P ∩ P̄ and P + P̄ are the complexified of the real distributions D = P ∩ P̄ ∩ T M and
E = (P + P̄) ∩ T M. As P is involutive, P̄ is also involutive, as well as D. Since D is a real distribution, the
Frobenius theorem implies that D defines a foliation of M, denoted M/D.
26
A polarization is said to be strongly integrable if E is involutive. Then M/D and M/E are differentiable
manifolds.
If (M, ω) possesses a line bundle (L, π, M) endowed with an hermitian connection ∇, a polarization is said to
be admissible for the connection if, in the neighbourhood of any point m, there exists a symplectic potential
adapted to the connection, i.e. hθ, Xi = 0 for all X ∈ VP (X).
The notion of strongly integrable polarization is important since it can be shown that such a polarization is
always admissible for a given connection.
3.4.2
An example : Kälherian polarization
A Kälherian polarization is a polarization P such that P ∩ P̄ = {0}.
We will see that these polarizations enjoy very nice properties and that the notion of (almost) complex manifold appears naturally in this context.
It is easy to show that P ∩ T M = {0} and P ∩ iT M = {0}, indeed, if X ∈ Pm ∩ T m M, then X̄ = X ∈ P̄m ∩ T m M
thus X ∈ Pm ∩ P¯m ∩ T m M = {0} (and similarly for P ∩ iT M = {0}). Let X and Y belong to Pm , X − Y is
not in T m M /{0} and X − Y is not in iT m M /{0}. Therefore <(X) = <(Y) iff =(X) = =(Y). This implies that
there exists an isomorphism J : T m M → T m M such that each element of Pm is of the form X + iJm (X) with
X ∈ T m M. Let us mention some properties of J,
• As P is complex, we have J 2 = −1 and since Pm varies smoothly with m, Jm form a tensor J of type
(1, 1) on M.
• Since P is lagrangian, we have for all X, Y ∈ VP (M),
O =
=
ω(X + iJ(X), Y + iJ(Y))
[ω(X, Y) − ω(J(X), J(Y))] + i[ω(X, J(Y)) − ω(Y, J(X))].
The real part of the equation implies that J preserves the symplectic form, while the imaginary part
tells us that the tensor g(X, Y) := ω(X, J(Y)) on M is symmetric.
• Moreover, g is non degenerate: if g(X, Y) = 0 ∀Y then iX ω = 0 and therefore X = 0.
• J preserves ω since g(J(X), J(Y)) = ω(J(X), J 2 (X)) = ω(X, J(X)) = g(Y, X) = g(X, Y).
Some definitions
1. Almost complex structure : Let us put names on all these objects. If M is a differentiable manifold, an
almost complex structure is a (1,1)–tensor J on M such that Jm2 = −1Tm M for all m ∈ M. The application
Jm gives T m M a structure of complex vector space by (x + iy)X = xX + yJ(X). A couple (M, J) is an
27
almost complex manifold.
Of course, on T m M, Jm has no proper values. Its C–linear extension T m M C has proper values ±i. We
can therefore decompose T m M C into the direct sum of
T m(1,0) M
=
{X ∈ T m M C |Jm (X) = +iX} ,
T m(0,1) M
=
{X ∈ T m M C |Jm (X) = −iX} .
2. Complex structure : Let F̄ and F denote the complex distribution on M defined by T m(1,0) and T m(0,1) M.
An almost complex structure J on M is a complex structure if the complex distributions F and F̄ are
involutive and of dimension n. It is possible to show that this gives to M the structure of a complex
manifold, i.e. the local charts ψa take their values in a complex vector space and the transition function
ψa ◦ ψ−1
b are holomorphic isomorphisms. This is equivalent to saying that there exist local holomorphic
coordinates (za , z̄a ) such that for all m ∈ M, ( ∂z∂a )m and ( ∂z̄∂a )m are eigenvectors of Jm of eigenvalues
+i and −i, or equivalently that there exist, near any point, real local coordinates (xa , ya ) such that
Jm ( ∂x∂a )m = ( y∂a )m and Jm ( ∂y∂a )m = −( x∂a )m .
3. Kähler and almost Käher manifold : An (almost) Kähler (resp. Kähler) manifold is a triple (M, ω, J)
such that
K1. (M, ω) is a symplectic manifold,
K2. (M, J) is an almost complex (resp. complex) manifold,
K3. J and ω are compatible in the sense that, for all X, Y ∈ V(M),
ω(J(X), J(Y)) = ω(X, Y) .
Note that in both cases, the distributions F and F̄ are lagrangian. On a Kähler manifold, the distributions F
and F̄ are therefore Kählerian polarizations since they are involutive and lagrangian, and that their intersection
is empty. They are called holomorphic and anti–holomorphic polarizations. Note that in local holomorphic
coordinates, these polarizations are generated by respectively by the vector fields
∂
∂z̄a
and
∂
∂za .
Remark: Conversely, as we have seen in the beginning of this section, any Kählerian polarization P over a
symplectic manifold induces a structure of Kähler manifold.
On any Kähler manifold (M, ω, J), the metric tensor g defined by
g(X, Y) := ω(X, J(Y)) ,
(3.38)
is non degenerate, and therefore corresponds to a (pseudo–)riemanian metric. The complex structure J is said
positive if g is positive definite, and if it comes from a polarization, the polarization is also said to be positive.
One can define a (pseudo–)hermitian structure k by
k(X, Y) := g(X, Y) + iω(X, Y) .
28
(3.39)
It is easy to see that k is non degenerate and is compatible with J.
Conversely, on any almost complex manifold (M, J) endowed with a (pseudo–)hermitian scalar product k
compatible with J, one can define a non degenerate 2–form ω, called fundamental form of (M, J), by
ω(X, Y) := =k(X, Y) .
(3.40)
If this form is closed, and therefore symplectic, it is called Kähler form. If J is almost complex or complex,
one gets an almost Kähler manifold or Kähler manifold.
In holomorphic local coordinates, any Kähler form can be written as,
ω=i
∂2 K
¯ ,
dza ∧ dz̄b = i∂∂K
∂za ∂z̄b
(3.41)
where K(z, z̄) is a real function called Kähler potential.
This is interesting because it means that for any Kähler manifold (M, ω, J) endowed with a prequantic line
bundle, in the neighborhood of a point where the Kähler potential is K, the local symplectic potentials are
¯ are adapted to the holomorphic (and anti–holomorphic) polarizations.
given by θ = i~∂K (and θ0 = −i~∂K)
We can conclude that these polarizations are therefore always admissible for any connection.
3.4.3
Polarization of the states and quantizable functions
If L is a line bundle over (M, ω), ∇ a connection having curvature
ω
i~
and P a strongly integrable polarization,
the space of polarized sections is defined by,
ΓP (M, L) := {s ∈ Γ(M, L)|∇X s = 0 ∀X ∈ P} .
Note that on an open set Ui with trivalising section si and a potential θi adapted to the connection, these
conditions read, for a local section s = fi si , as X( fi ) = 0. If there exist locally coordinates xa , yb such that the
polarization is generated by the vector files
∂
∂xa ,
these conditions boil down to the fact that f depends only on
the coordinates y , indeed the conditions X( fi ) = 0 reduce to xa fi = 0.
b
The operators act now on polarized sections, we should therefore keep only the operators that send a polarized
section to a polarized section.There is a Lie subalgebra C P of functions on M such that O f operates on
ΓP (M, L) for every f in C P :
C P := { f ∈ C ∞ (M)| [X f , X] ∈ P ∀X ∈ P} ,
indeed
∇X O f s =
=
=
=
~
∇X ( ∇X f s + f s)
i
~
~
∇[X,X f ] s + X( f )s + R(X, X f )s
i
i
~
∇[X,X f ] s + (d f (X) + ω(X, X f ))s
i
~
∇[X,X f ] s = 0
∀s ∈ ΓF (M, L) .
i
29
Up to now, we have introduced a way of selecting a sub–space of functions on M, namely C P . The idea is
of course that we should quantize only this sub–space of functions. Let us now return to the irreducibility
condition. If a Lie group G has a Hamiltonian action σ on (M, ω) and if {λY , Y ∈ G} are the corresponding functions, one would like that the functions λY are quantizable. This is the case if one can choose a
polarization P which is invariant by G, i.e. σ(g)∗ P x = Pσ(g)x , ∀g ∈ G, ∀x ∈ M. Indeed,
σ(g)∗ P x = Pσ(g)x
⇒ [Ỹ, X] ∈ P x ,
⇒ [XλY , X] ∈ P x ,
∀X ∈ P x , ∀Y ∈ G
∀X ∈ P x , ∀Y ∈ G
⇒ λY ∈ C P , ∀Y ∈ G
(3.42)
If one has a G–invariant polarization on (M, ω) then
Y→
i
Oλ
~ Y
(3.43)
is a homomorphism of G into operators on Γ∞
P (M, L).
It has been proven by Kostant that this exponentiates to a representation of G on polarized section when the
action of G on M is transitive. Moreover when G is either a semi–simple, compact or solvable group, the
representations so–obtained are irreducible.
The application of the geometric quantization program to integral coadjoint orbits of G in G? admitting an
invariant polarization is the starting point of Kirillov’s orbit method [here integral refers to the fact that the
prequantization condition is fulfilled for the orbit]. It states that an irreducible representation of G should
correspond to a coadjoint orbit of G. For some groups (notably simply connected nilpotent Lie groups), there
is a perfect bijection between orbits and irreducible unitary representations. However the general case is far
more complicated (i.e. there exists representations which are not attached to orbits), and we refer the reader
e.g. to [?] for a short discussion of this wide subject.
Acknowledgements
We are grateful to Pierre Bieliavsky for enlightening discussions. SdB thanks the Engineering and Physical
Sciences Research Council for financial support (Hodge fellowship).
30
Appendix A
Conventions and formulas
iX f ω = d f = ω(X f , .)
ω = −dθ
{ f, g} = Xg ( f ) = ω(X f , Xg )
X{ f,g} = −[X f , Xg ]
dα(X, Y) = X(α(Y)) − Y(α(X)) − α([X, Y])
O f = −i~X f − α( f ) + f = −i∇X f + f
α(X f ) = hα, X f i
s = fa sa
∇X s = (X( fa ) − 2πihαa , Xi fa )sa .
Ω=
ω
2π~
∗ denotes the pull–back or push–forward
? denotes the dual
31
Bibliography
[1] S. Gutt, “Symplectic geometry,”. Course given at Université Libre de Bruxelles.
[2] A. Cannas Da Silva, “Symplectic geometry,” in Handbook of Differential Geometry, vol. 2 (F.J.E.
Dillen and L.C.A. Verstraelen, eds.), math.SG/0505366 .
[3] Y. Voglaire, “Quantification de particules topologiques avec symétrie, ” Master’s thesis, Université
catholique de Louvain, 2005.
[4] M. Nakahara, “Geometry, topology and physics,” Bristol, UK: Hilger 505 p. (Graduate student series
in physics), 1990.
[5] A. M. Perelomov, “Integrable systems of classical mechanics and Lie algebras,” Vol.1, Birkhauser
Verlag Basel, 1990.
[6] J. M. Figueroa–O’Farrill, “BRST Cohomology and its Applications to Two Dimensional Conformal
Field Theory,” PhD thesis, [http://www.maths.ed.ac.uk/ jmf/Research/PVBLICATIONS/Thesis.pdf].
[7] N. M. J. Woodhouse, “Geometric Quantization,” second edition, Clarendon Press Oxford, 1991.
[8] S. Detournay, “Deformations of anti-de Sitter black holes, ” hep-th/0611031.
[9] D.A. Vogan, review of “Lectures on the orbit method”, http://www.ams.org/bull/2005-42-04/S02730979-05-01065-7/S0273-0979-05-01065-7.pdf
[10] D. J. Simms, N. M. J. Woodhouse, “Lectures on Geometric Quantization,” Lecture Notes in Physics
53, Springer-Verlag, 1976
[11] A. Weinstein, “Lectures on Symplectic Manifolds,” Conf. Board of the Math. Sciences, Reg. Conf.
Series in Math. 29, AMS, Providence, R.I., 1977
32