Some Remarks on {$\mathfrak g$}-invariant Fedosov Star Products and Quantum Momentum Mappings

15 Pages·0.25 MB·English

by  Michael Frank Müller

Some Remarks on
g-invariant Fedosov Star Products and
Quantum Momentum Mappings
Michael Frank Mu¨ller ∗
3
0 Fakult¨at fu¨r Mathematik und Informatik
0
Universit¨at Mannheim
2
A5 Geb¨aude C
n
a D-68131 Mannheim
J
Germany

5
1 Nikolai Neumaier §

]
A Fakult¨at fu¨r Mathematik und Physik
Q Universit¨at Freiburg
. Hermann-Herder-Straße 3
h
t D-79104 Freiburg i. Br.
a
m Germany
[

2 January 2003
v
1 FR-THEP-2003/01
0
1 Mannheimer Manuskripte 268
1
0
3
0
/
h Abstract
t
a
In these notes we consider the usual Fedosov star product on a symplectic manifold (M,ω)
m
emanating from the fibrewise Weyl product ◦, a symplectic torsion free connection ∇ on M,
v: a formal series Ω ∈ νZ2 (M)[[ν]] of closed two-forms on M and a certain formal series s of
dR
i symmetric contravariant tensor fields on M. For a given symplectic vector field X on M we
X
derive necessaryandsufficient conditions for the triple (∇,Ω,s) determining the starproduct ∗
r
on which the Lie derivative L with respect to X is a derivation of ∗. Moreover, we also give
a X
additional conditions on which L is even a quasi-inner derivation. Using these results we find
X
necessary and sufficient criteria for a Fedosov star product to be g-invariant and to admit a
quantum Hamiltonian. Finally, supposing the existence of a quantum Hamiltonian, we present
a cohomological condition on Ω that is equivalent to the existence of a quantum momentum
mapping. In particular, our results show that the existence of a classical momentum mapping
in general does not imply the existence of a quantum momentum mapping.
∗[email protected]
§[email protected]
1
Contents
1 Introduction 2
2 Preliminaries 3
3 Symplectic Vector Fields as Derivations of ∗ 7
4 g-invariant Star Products ∗ and Quantum Momentum Mappings 10
1 Introduction
The concept of deformation quantization as introduced in the pioneering articles [3] by Bayen,
Flato, Frønsdal, Lichnerowicz and Sternheimer has proved to be an extremely useful framework for
the problem of quantization: the question of existence of star products ⋆ (i.e. formal, associative
deformations of the classical Poisson algebra of complex-valued functions C∞(M) on a symplectic
or more generally, on a Poisson manifold M, such that in the first order of the formal parameter
ν the commutator of the star product yields the Poisson bracket) has been answered positively by
DeWilde andLecomte [9], Fedosov [11], Omori,MaedaandYoshioka [22]inthecase ofasymplectic
phasespace as well as by Kontsevich [18]in themore general case of a Poisson manifold. Moreover,
star products have been classified up to equivalence in terms of geometrical data of the phase space
by Nest and Tsygan [21], Bertelson, Cahen and Gutt [5], Weinstein and Xu [25] on symplectic
manifolds and the classification on Poisson manifolds is due to Kontsevich [18]. Comparisons
between the different results on classification and reviews can be found in articles of Deligne [8],
Gutt and Rawnsley [14, 15], Neumaier [19] and Dito and Sternheimer [10, 23].
Already at the very beginning of the investigations of deformation quantization various notions
of invariance of star products with respect to Lie group resp. Lie algebra actions were introduced
and discussed by Arnal, Cortet, Molin and Pinczon in [2]. Later on it was Xu who systemati-
cally defined the notion of a quantum momentum mapping for g-invariant star products in the
framework of deformation quantization in [26] that naturally generalizes the concept of the mo-
mentum mapping in Hamiltonian mechanics (cf. [1]) and computed the a priori obstructions for
its existence. Actually the notion of a quantum momentum mapping has proved to be essential
for the formulation of the quantum mechanical analogue of the Marsden-Weinstein reduction in
deformation quantization as it was studied by Fedosov in [13], where it was shown that in some
sense ‘reduction commutes with quantization’. For the application of the BRST quantization in
deformationquantization asitwasintroducedanddiscussedbyBordemann,HerbigandWaldmann
in [7] the existence of a quantum momentum mapping also turned out to be a major ingredient
of the construction. For the more special discussion of the example of reduction of star products
for CPn as it was given by Bordemann, Brischle, Emmrich and Waldmann in [6] and was slightly
generalized by Waldmann in [24] again the use of a quantum momentum mapping the existence of
which can be shown explicitly in this case was the key ingredient of the considerations.
Recently in [17] Hamachi has taken up afresh the question under which preconditions the usual
Fedosov star productadmits aquantummomentum mappingand hehas given acondition in terms
of parts of the Fedosov derivation used to define the star product which is assumed to be invariant
with respect to a symplectic Lie Group action on M.
In the present paperwe want to generalize these results into two directions: Firstly we drop the
assumptionofinvarianceofthestarproductwithrespecttoaLiegroupaction andreplaceitbythe
somewhat weaker invariance with respect to the action of a Lie algebra g. Secondly we make the
2
conditions given in [17] more precise and show that assuming that there is a classical momentum
mapping the question of existence of a quantum momentum mapping relies on two cohomological
conditions on the formal series Ω ∈ νZ2 (M)[[ν]] used to construct the g-invariant star product.
dR
The paper is organized as follows: In Section 2 we collect some notations and give a very
short review of Fedosov’s construction. Here we also prove some technical details that enable us to
describeall derivations of theFedosov star productsin a very convenient way which turnsoutto be
very useful for the further investigations. In Section 3 we consider an arbitrary symplectic vector
field on M and give necessary and sufficient conditions for the Lie derivative with respect to this
vector field to be a derivation of the star product ∗ under consideration. Furthermore we can also
specify additional conditions guaranteeing that this derivation is even quasi-inner. In Section 4 we
recall the definitions of g-invariant star products, quantum Hamiltonians and quantum momentum
mappingsfrom[26]andapplyourresultofSection3togivecriteriafortheg-invarianceofaFedosov
star product. Finally, supposing that the Lie algebra action is Hamiltonian and the Hamiltonian
is equivariant with respect to the coadjoint action of g we moreover find conditions that permit
a decision whether quantum momentum mappings do exist. We conclude the paper with some
remarks on possible generalizations and further investigations.
Conventions: By C∞(M), we denote the complex-valued smooth functions and similarly
Γ∞(T∗M) stands for the complex-valued smooth one-forms et cetera. Moreover, we use Einstein’s
summation convention in local expressions.
2 Preliminaries
In this section we shall briefly recall the essentials of Fedosov’s construction of star products on
a symplectic manifold (M,ω). As we assume the reader to be familiar with this construction we
shall restrict to the very minimum to introduce our notation (For more details we refer the reader
to [11, 12] and [19, Sect. 2], where we even used the same notation). Defining
W⊗Λ:= (X∞ Γ∞( sT∗M ⊗ T∗M))[[ν]]. (1)
s=0
W V
it is obvious that W⊗Λ becomes in a natural way an associative, super-commutative algebra and
the product is denoted by µ(a⊗b) = ab for a,b ∈ W⊗Λ (By W⊗Λk we denote the elements of
anti-symmetric degree k and set W := W⊗Λ0.). Besides this pointwise product the Poisson tensor
Λ corresponding to ω gives rise to another associative product ◦ on W⊗Λ by
ν
a◦b = µ◦exp Λiji (∂ )⊗i (∂ ) (a⊗b), (2)
s i s j
2
(cid:16) (cid:17)
which is a deformation of µ. Here i (Y) denotes the symmetric insertion of a vector field Y ∈
s
Γ∞(TM) and similarly i (Y) shall be used to denote the anti-symmetric insertion of a vector field.
a
We set ad(a)b := [a,b] where the latter denotes the deg -graded super-commutator with respect
a
to ◦. Denoting the obvious degree-maps by deg , deg and deg = ν∂ one observes that they
s a ν ν
all are derivations with respect to µ but deg and deg fail to be derivations with respect to ◦.
s ν
Instead Deg := deg +2deg is a derivation of ◦ and hence (W⊗Λ,◦) is formally Deg-graded and
s ν
the corresponding degree is referred to as the total degree. Sometimes we write W ⊗Λ to denote
k
the elements of total degree ≥ k.
In local coordinates we define the differential δ := (1⊗dxi)i (∂ ) which satisfies δ2 = 0 and is a
s i
super-derivation of ◦. Moreover, there is a homotopy operator δ−1 satisfying δδ−1+δ−1δ+σ = id
where σ : W⊗Λ → C∞(M)[[ν]] denotes the projection onto the part of symmetric and anti-
symmetric degree 0 and δ−1a := 1 (dxi ⊗1)i (∂ )a for deg a = ka, deg a = la with k +l 6= 0
k+l a i s a
3
and δ−1a := 0 else. From a torsion free symplectic connection ∇ on M we obtain a derivation
∇ := (1⊗dxi)∇ of ◦ that satisfies the following identities: [δ,∇] = 0, ∇2 = −1ad(R), where
∂i ν
R := 1ω Rt dxi ∨dxj ⊗dxk ∧dxl ∈ W⊗Λ2 involves the curvature of the connection. Moreover
4 it jkl
we have δR = 0 = ∇R by the Bianchi identities.
Now remember the following facts which are just restatements of Fedosov’s original theorems
in [11, Thm. 3.2, 3.3] resp. [12, Thm. 5.3.3]:
ForallΩ ∈ νZ2 (M)[[ν]]andalls ∈ W withσ(s) = 0thereexistsauniqueelementr ∈ W ⊗Λ1
dR 3 2
such that
1
δr = ∇r− r◦r+R+1⊗Ω and δ−1r = s. (3)
ν
Moreover r satisfies the formula
1
r = δs+δ−1 ∇r− r◦r+R+1⊗Ω (4)
ν
(cid:18) (cid:19)
from which r can be determined recursively. In this case the Fedosov derivation
1
D := −δ+∇− ad(r) (5)
ν
is a super-derivation of anti-symmetric degree 1 and has square zero: D2 = 0. Furthermore observe
that the D-cohomology on elements a with positive anti-symmetric degree is trivial since one has
the following homotopy formula DD−1a+D−1Da = a, where D−1a := −δ−1 1 a
id−[δ−1,∇−1ad(r)]
ν
(cf. [12, Thm. 5.2.5]). (cid:16) (cid:17)
Then for any f ∈ C∞(M)[[ν]] there exists a unique element τ(f) ∈ ker(D) ∩ W such that
σ(τ(f)) = f and τ : C∞(M)[[ν]] → ker(D) ∩ W is C[[ν]]-linear and referred to as the Fedosov-
Taylor series corresponding to D. In addition τ(f) can be obtained recursively for f ∈ C∞(M)
from
1
τ(f) =f +δ−1 ∇τ(f)− ad(r)τ(f) . (6)
ν
(cid:18) (cid:19)
Using D−1 one can also write τ(f)= f −D−1(1⊗df). Since D as constructed above is a ◦-super-
derivation ker(D)∩W is a ◦-sub-algebra and a new associative product ∗ for C∞(M)[[ν]], which
turns out to be a star product, is defined by pull-back of ◦ via τ.
Observe that in (3) we allowed for an arbitrary element s ∈ W with σ(s) = 0 that contains no
terms of total degree lower than 3, as normalization condition for r, i.e. δ−1r = s instead of the
usual equation δ−1r = 0. In the following we shall refer to the associative product ∗ defined above
as the Fedosov star product (corresponding to (∇,Ω,s)).
Now we shall give a very convenient description of all derivations of the star product ∗ that
will prove very useful for our further considerations. To this end we consider appropriate fibrewise
quasi-inner derivations of the shape
1
D = − ad(h), (7)
h
ν
where h ∈ W and without loss of generality we assume σ(h) = 0. Our aim is to define C[[ν]]-linear
derivations of ∗byC∞(M)[[ν]] ∋ f 7→ σ(D τ(f))butforan arbitraryelement h ∈ W with σ(h) = 0
h
this mapping fails to be a derivation as D does not map elements of ker(D)∩W to elements of
h
ker(D)∩W. In order to achieve this one must have that D and D super-commute. As D is a
h
C[[ν]]-linear ◦-super-derivation we obviously have
1
[D,D ] = − ad(Dh)
h
ν
4
andhenceobviouslyDhmustbecentral,i.e. Dhhastobeoftheshape1⊗AwithA∈ Γ∞(T∗M)[[ν]]
to have [D,D ] = 0. From D2 = 0 we get that the necessary condition for the solvability of the
h
equation Dh = 1⊗A is the closedness of A since D(1⊗A) = 1⊗dA. But as the D-cohomology
is trivial on elements with positive anti-symmetric degree this condition is also sufficient for the
solvability of the equation Dh= 1⊗A and we get the following statement.
Lemma 2.1 i.) For all formal series A ∈ Γ∞(T∗M)[[ν]] of closed one-forms on M there is a
uniquely determined element h ∈ W such that Dh = 1⊗A and σ(h ) = 0. Moreover h
A A A A
is explicitly given by
h = D−1(1⊗A). (8)
A
ii.) For all A∈ Z1 (M)[[ν]] the mapping D : C∞(M)[[ν]] → C∞(M)[[ν]], where
dR A
1
D f := σ(D τ(f)) = σ − ad(h )τ(f) (9)
A hA ν A
(cid:18) (cid:19)
for f ∈ C∞(M)[[ν]] defines a C[[ν]]-linear derivation of ∗ and hence this construction yields
a mapping Zd1R(M)[[ν]] ∋ A7→ DA ∈ DerC[[ν]](C∞(M)[[ν]],∗).
Proof: The fact that h = D−1(1⊗A) satisfies Dh = 1⊗A is obvious from the homotopy formula for
A A
D and the closedness of A. In addition we have σ(h ) = 0 since D−1 raises the symmetric degree at least
A
by 1. For the uniqueness of h let h be another solution of the equations above, then we obviously have
A A
∞
D(h −h ) = 0 and hence h −h = τ(ϕ) for some ϕ ∈ C (M)[[ν]]. Applying σ to this equation one
A A A A
gets ϕ = 0, since σ(h ) = σ(h ) =f0 and σ(τ(ϕ)) = ϕ, and hence h = h proving that h is uniquely
A A A A A
determinfedbytheaboveequations.fFortheproofofii.) wejustobservethattheequation[D,D ]=0which
hA
is fulfilled according to i.) impflies that D τ(f)=τ(D f) for all f ∈C∞(Mf)[[ν]]. Using this equation and
hA A
the obvious fact that D is a derivation of ◦ it is straightforwardto see using the very definition of ∗ that
hA
D as defined above is a derivation of ∗. The C[[ν]]-linearity of D is also evident from the C[[ν]]-linearity
A A
of τ. (cid:3)
FurthermorewenowareinthepositiontoshowthatoneevenobtainsallC[[ν]]-linearderivations
of ∗ by varying A in the derivations D constructed above.
A
Proposition 2.2 The mapping
Zd1R(M)[[ν]] ∋ A7→ DA ∈ DerC[[ν]](C∞(M)[[ν]],∗)
defined in Lemma 2.1 is a bijection. Moreover, D is a quasi-inner derivation for all f ∈
df
C∞(M)[[ν]], i.e. D = 1ad (f) and the induced mapping [A] 7→ [D ] from H1 (M)[[ν]] ∼=
df ν ∗ A dR
Zd1R(M)[[ν]]/Bd1R(M)[[ν]] to DerC[[ν]](C∞(M)[[ν]],∗)/DerqCi[[ν]](C∞(M)[[ν]],∗) the space of C[[ν]]-lin-
ear derivations of ∗ modulo the quasi-inner derivations, also is bijective.
Proof: First we prove the injectivity of the mapping A 7→ DA. To this end let DA = DA′ then we
get from DhAτ(f) = τ(DAf) and from the analogous equation for A′ that ad(hA −hA′)τ(f) = 0 for all
f ∈C∞(M)[[ν]] and hence hA−hA′ must be central (since it commutes with all Fedosov-Taylorseries), i.e.
we have hA −hA′ = gA,A′ ∈ C∞(M)[[ν]]. But with σ(hA) = σ(hA′) = 0 this implies gA,A′ = 0 and hence
hA = hA′ such that we get 1⊗A = DhA = DhA′ = 1⊗A′ proving the injectivity. For the surjectivity we
start with an arbitrary derivation D of ∗ and want to find closed one-forms A such that D = ∞ νiD
i i=0 Ai
inductively. Assume that we have found such one-forms for 0 ≤ i ≤ k−1 such that D′ = D− k−1νiD
Pi=0 Ai
whichobviouslyisagainaderivationof∗isoftheshapeD′ = ∞ νiD′. Thekthorderinν oftheequation
i=k i P
D′(f∗g)=(D′f)∗g+f∗(D′g)forf,g ∈C∞(M)yieldsthatD′ isavectorfieldX ∈Γ∞(TM). Considering
Pk k
5
the anti-symmetricpartofD′(f∗g)=(D′f)∗g+f∗(D′g)at orderk+1 of ν we get thatthis vectorfield is
symplectic, i.e. L ω =0 and because of the CartanformulaA :=−i ω defines a closedone-formon M.
Xk k Xk
Considering the derivation D it is a straightforwardcomputation using the explicit constructionabove to
Ak
show that D f = X (f)+O(ν) for all f ∈ C∞(M). But then D′−νkD is again a derivation of ∗ that
Ak k Ak
starts in order k+1 of ν and hence the surjectivity follows by induction. The fact that Ddf = ν1ad∗(f) for
all f ∈C∞(M)[[ν]] is obvious from the observation that τ(f) =f −D−1(1⊗df) and the obvious fact that
ad(f)=0. From the above, the well-definedness of the mapping [A]7→[D ] follows and the bijectivity is a
A
direct consequence of the bijectivity of the mapping A7→D . (cid:3)
A
Remark 2.3 Actuallyitiswell-knownthat foranarbitrary star product ⋆on asymplectic manifold
the space of C[[ν]]-linear derivations is in bijection with Z1 (M)[[ν]] and that the quotient space of
dR
these derivations modulo the quasi-inner derivations is in bijection with H1 (M)[[ν]] (cf. [5, Thm.
dR
4.2], observe that the proof given above is just an adaption of the idea of the general proof to our
special situation) but the remarkable thing about Fedosov star products is that these bijections can
be explicitly expressed in terms of D resp. D−1 in a very lucid way which will be useful in the
following.
Toconcludethissectionweshallremovesomeredundancyinthedescriptionofthestarproducts
∗by(∇,Ω,s). Thiswilleasethemoredetailedanalysisinthefollowingsection. Tothisendweshall
recall some well-known facts about symplectic torsion free connections on (M,ω). Given two such
connections say ∇ and ∇′ it is obvious that S∇−∇′(X,Y) := ∇ Y −∇′ Y where X,Y ∈Γ∞(TM)
X X
defines a symmetric tensor field S∇−∇′ ∈ Γ∞( 2T∗M ⊗TM) on M. Defining σ∇−∇′(X,Y,Z) :=
ω(S∇−∇′(X,Y),Z) it is easy to see that σ∇−∇′ ∈ Γ∞( 3T∗M) is a totally symmetric tensor field.
Vice versa given an arbitrary element σ ∈ Γ∞W( 3T∗M) and a symplectic torsion free connection
∇ and defining Sσ ∈ Γ∞( 2T∗M ⊗TM) by σ(X,YW,Z) = ω(Sσ(X,Y),Z) then ∇σ defined by
W
∇σ Y := ∇ Y −Sσ(X,Y) again is a symplectic torsion free connection and all such connections
X X W
can be obtained this way by varying σ. Using these relations we shall compare the corresponding
mappings ∇ and ∇′ on W⊗Λ in the following lemma.
Lemma 2.4 With the notations from above we have
1
∇−∇′ = −(dxj ⊗dxi)i (S∇−∇′(∂ ,∂ )) = ad(T∇−∇′), (10)
s i j
ν
where T∇−∇′ ∈ Γ∞( 2T∗M ⊗T∗M) ⊆ W⊗Λ1 is defined by T∇−∇′(Z,Y;X) := σ∇−∇′(X,Y,Z) =
ω(S∇−∇′(X,Y),Z). Moreover T∇−∇′ satisfies the equations
W
δT∇−∇′ = 0 and ∇T∇−∇′ = R′−R+ ν1T∇−∇′ ◦T∇−∇′ (11)
∇′T∇−∇′ = R′−R− 1T∇−∇′ ◦T∇−∇′,
ν
where R = 1ω Rt dxi ∨ dxj ⊗ dxk ∧ dxl and R′ = 1ω R′t dxi ∨ dxj ⊗ dxk ∧ dxl denote the
4 it jkl 4 it jkl
corresponding elements of W⊗Λ2 that are built from the curvature tensors of ∇ and ∇′.
Proof: The proofof (10) is a straightforwardcomputation using the very definitions fromabove. The first
identity in (11) directly followsfrom(10)and [δ,∇]=[δ,∇′]=0. The other identities in (11) arealsoeasily
obtained squaring equation (10). (cid:3)
Now we are in the position to compare two Fedosov derivations D and D′ resp. the induced
star products ∗ and ∗′ obtained from (∇,Ω,s) and (∇′,Ω′,s′).
6
Proposition 2.5 The Fedosov derivations D and D′ coincide if and only if T∇−∇′−r+r′ = 1⊗ϑ
where ϑ ∈νΓ∞(T∗M)[[ν]] which is equivalent to
σ∇−∇′ ⊗1−s+s′ = ϑ⊗1 and Ω−Ω′ = dϑ. (12)
Proof: Writing down the definitions of D and D′ using equation (10) the first equivalence is obvious since
T∇−∇′ −r+r′ is central in (W⊗Λ,◦) if and only if D = D′. For the proof of the second equivalence first
assume that we have T∇−∇′ −r+r′ = 1⊗ϑ. Applying δ−1 to this equation and using the normalization
condition on r and r′ we obtain the first equation in (12) since δ−1T∇−∇′ =σ∇−∇′ ⊗1. In order to obtain
the second equation in (12) we apply δ to T∇−∇′ −r+r′ =1⊗ϑ and a straightforwardcomputation using
the equations for r and r′ together with the identities from (11) yields the stated result. To prove that the
converseisalsotrue assumethattheequationsin(12)aresatisfiedanddefineB :=r−r′−T∇−∇′+1⊗ϑ∈
W2⊗Λ1. Then again a straightforward computation yields that B satisfies DB = −ν1B◦B and δ−1B = 0
such that the homotopy formula for δ together with σ(B) = 0 implies that B is the unique fixed point of
the mapping W2⊗Λ1 ∋ a 7→ δ−1 ∇a− ν1ad(r)a+ ν1a◦a ∈ W2⊗Λ1. But 0 trivially is a fixed point of
this mapping and hence uniqueness implies that B = 0 proving the other direction of the second stated
(cid:0) (cid:1)
equivalence. (cid:3)
As an important direct consequence of this proposition we get:
Deduction 2.6 For every Fedosov star product ∗ obtained from (∇,Ω,s) with s ∈ W there is
3
a connection ∇′, a formal series Ω′ of closed two-forms and an element s′ ∈ W without terms
4
of symmetric degree 1 such that the star product obtained from (∇′,Ω′,s′) coincides with ∗, and
hence we may without loss of generality restrict to such normalization conditions when varying the
connection and the formal series of closed two-forms arbitrarily.
Proof: We write s = s′ + σ ⊗ 1 − ϑ ⊗ 1 and the preceding proposition states that D coincides with
D′ (and hence the corresponding star products coincide) where D′ is obtained from Ω′ = Ω − dϑ and
∇′ =∇− 1ad(δ(σ⊗1)). (cid:3)
ν
3 Symplectic Vector Fields as Derivations of ∗
Throughout this and the following section let ∗ denote the Fedosov star product obtained from
(∇,Ω,s) as in Section 2 where in view of Deduction 2.6 we may assume that s ∈ W contains no
4
part of symmetric degree 1. Furthermore X ∈ Γ∞(TM) shall always denote a symplectic vector
field on (M,ω) and the space of all these vector fields shall be denoted by Γ∞ (TM) := {Y ∈
symp
Γ∞(TM)|L ω = 0}. It seems to be folklore and actually is not very hard to prove that the
Y
conditions [L ,∇] = 0, L Ω = 0 = L s are sufficient to guarantee that the Lie derivative with
X X X
respect to X is a derivation of ∗. Besides providing a very simple proof of this fact, our aim in this
section is to prove that the converse is also true, i.e. the conditions given above are also necessary
to have that X defines a derivation of ∗. Moreover, we find an additional cohomological condition
involving ω, Ω and X that is equivalent to L being even a quasi-inner derivation.
X
As an important tool we need the deformed Cartan formula (cf. [19, Appx. A]) that relates the
Lie derivative with respect to a symplectic vector field X with the Fedosov derivation D.
Lemma 3.1 For all X ∈ Γ∞ (TM) the Lie derivative L can be expressed in the following
symp X
manner:
1 1
L = Di (X)+i (X)D− ad θ ⊗1+ Dθ ⊗1−i (X)r , (13)
X a a X X a
ν 2
(cid:18) (cid:19)
where D := dxi∨∇ denotes the operator of symmetric covariant derivation and the closed one-
∂i
form θ is defined by θ := i ω.
X X X
7
Proof: SincetheLiederivativeisalocaloperatoritsufficestoprovetheaboveidentityoveranycontractible
open subset U of M. But as X is symplectic it is locally Hamiltonian, i.e. over U there is a function
f ∈ C∞(U) such that X| = X resp. df = θ | . For Hamiltonian vector fields the Cartan formula
U f X U
as above was proved in [19, Prop. 5] and hence equation (13) is valid for all symplectic vector fields
X ∈Γ∞ (TM). (cid:3)
symp
As an immediate consequence of the preceding lemma we have:
Lemma 3.2 For X ∈ Γ∞ (TM) the Lie derivative L is a derivation with respect to ◦. In
symp X
addition we have [δ,L ]= [δ−1,L ] = 0.
X X
Proof: Thefirststatementofthelemmaisobviousfromequation(13)andthecommutationrelationsfollow
from the fact that L is compatible with contractions and preserves the symmetric and the anti-symmetric
X
degree. (cid:3)
After these rather technical preparations we get:
Proposition 3.3 Let X ∈ Γ∞ (TM) then L is a derivation of ∗ if and only if [L ,D] = 0
symp X X
which is equivalent to the existence of a formal series A ∈ Γ∞(T∗M)[[ν]] of closed one-forms such
X
that D θ ⊗1+ 1Dθ ⊗1−i (X)r = 1⊗A .
X 2 X a X
Proof:(cid:0)First let us assume that [L ,D(cid:1)] = 0 then the obvious equation L ◦σ = σ ◦L implies that
X X X
L τ(f) = τ(L f) for all f ∈ C∞(M)[[ν]]. But with this equation and the fact that L is a deriva-
X X X
tion of ◦ it is straightforward to prove that L is a derivation of ∗. Assuming that L is a deriva-
X X
tion of ∗ Proposition 2.2 implies that there is a formal series A of closed one-forms on M such that
X
L f = σ −1ad(D−1(1⊗A ))τ(f) but on the other hand the deformed Cartan formula yields L f =
X ν X X
σ −1ad θ ⊗1+ 1Dθ ⊗1−i (X)r τ(f) andhenceD−1(1⊗A )− θ ⊗1+ 1Dθ ⊗1−i (X)r has
ν (cid:0)X 2 X a (cid:1) X X 2 X a
tobecentral,i.e. aformalfunction. ObservingthatD−1 raisesthesymmetricdegreeatleastby1andthatr
(cid:0) (cid:0) (cid:1) (cid:1) (cid:0) (cid:1)
containsno partofsymmetric degree 0 whichis due to the special shape ofthe normalizationconditionthis
impliesD−1(1⊗A )= θ ⊗1+ 1Dθ ⊗1−i (X)r . ApplyingDtothisequationandusingthehomotopy
X X 2 X a
formula for D together with the fact that A is closed we get D θ ⊗1+ 1Dθ ⊗1−i (X)r =1⊗A .
(cid:0) X (cid:1) X 2 X a X
Assumingfinallythatthis equationisfulfilled,the deformedCartanformulatogetherwithD2 =0obviously
(cid:0) (cid:1)
implies [L ,D]=0 since 1⊗A is central and hence the proposition is proved. (cid:3)
X X
We shall now go on by analysing the condition
1
D θ ⊗1+ Dθ ⊗1−i (X)r = 1⊗A , where dA = 0 (14)
X X a X X
2
(cid:18) (cid:19)
in more detail in order to find out whether it gives rise to conditions on (∇,Ω,s) and X.
Lemma 3.4 For all symplectic vector fields X ∈Γ∞ (TM) we have
symp
1 1
D θ ⊗1+ Dθ ⊗1−i (X)r =−1⊗θ +∇ Dθ ⊗1 −L r−i (X)R−1⊗i Ω. (15)
X X a X X X a X
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
Proof: The proof of this equation is a straightforwardcomputation using the equation that is solved by r
and the deformed Cartan formula (13) once again. (cid:3)
Next we shall need some detailed formulas that describe [∇,L ] in order to simplify the result
X
of the above Lemma. Theproofs of the following two lemmas are justslight variations of theproofs
of [19, Lemma 3 and Lemma 4].
Lemma 3.5 For all X ∈ Γ∞ (TM) the mapping [∇,L ] enjoys the following properties:
symp X
8
i.) In local coordinates one has
[∇,L ] = (dxj ⊗dxi)i ((L ∇) ∂ )= (dxj ⊗dxi)i (S (∂ ,∂ )), (16)
X s X ∂i j s X i j
where the tensor field S ∈ Γ∞(T∗M ⊗T∗M ⊗TM) is defined by
X
(2)
S (∂ ,∂ )= (L ∇) ∂ := L ∇ ∂ −∇ L ∂ −∇ ∂ = R(X,∂ )∂ +∇ X. (17)
X i j X ∂i j X ∂i j ∂i X j LX∂i j i j (∂i,∂j)
ii.) S as defined above is symmetric, i.e. S ∈Γ∞( 2T∗M ⊗TM).
X X
iii.) For all U,V,W ∈ Γ∞(TM) we have ω(W,S (U,VW)) = −ω(S (U,W),V).
X X
Now the tensor field S naturally gives rise to an element T ∈ Γ∞( 2T∗M ⊗T∗M) of W⊗Λ1
X X
of symmetric degree 2 and anti-symmetric degree 1 by
W
T (W,U;V) := ω(W,S (V,U)) (18)
X X
and we have:
Lemma 3.6 The tensor field T as defined in (18) satisfies the following equations:
X
i.) 1ad(T ) =[∇,L ],
ν X X
ii.) T = i (X)R−∇ 1Dθ ⊗1 ,
X a 2 X
iii.) δT = 0 and ∇(cid:0)T = L R(cid:1).
X X X
From the preceding lemma we find that the result of Lemma 3.4 simplifies to
1
D θ ⊗1+ Dθ ⊗1−i (X)r = −1⊗θ −T −L r−1⊗i Ω. (19)
X X a X X X X
2
(cid:18) (cid:19)
Finally we have to find equations that determine L r in order to analyse equation (14).
X
Lemma 3.7 Let X denote a symplectic vector field then L r satisfies the equations
X
1 1
δL r = ∇L r− ad(r)L r− ad(T )r+L R+1⊗di Ω and δ−1L r = L s (20)
X X X X X X X X
ν ν
from which L r is uniquely determined and can be computed recursively from
X
1 1
L r = δL s+δ−1 ∇L r− ad(r)L r− ad(T )r+L R+1⊗di Ω .
X X X X X X X
ν ν
(cid:18) (cid:19)
Proof: For the proof of (20) one just has to apply L to the equations that determine r and to use
X
the commutation relations of the involved mappings. From these equations it is straightforward to find
the recursion formula for L using the homotopy formula for δ. Using statement iii.) of Lemma 3.6 the
X
argument for the uniqueness of the solution of these equations is completely analogous to the one used to
prove the uniqueness of r and hence we leave it to the reader. (cid:3)
After all these preparations we are in the position to formulate the main results of this section.
Theorem 3.8 LetX beasymplectic vectorfieldand let∗be the Fedosov star product corresponding
to (∇,Ω,s), where s ∈ W contains no part of symmetric degree 1. Then, L is a derivation of ∗
4 X
if and only if T = 0, L Ω = 0 and L s = 0, i.e. if and only if X is affine with respect to ∇ and
X X X
s and Ω are invariant with respect to X.
9
Proof: First let T = 0 = L Ω = L s then we have L R = ∇T = 0 and di Ω = 0 and hence
X X X X X X
L r = δ−1 ∇L r− 1ad(r)L r . But this implies L r = 0 and then obviously [D,L ] = 1ad(T +
X X ν X X X ν X
L r) = 0 such that Proposition 3.3 implies that L is a derivation of ∗. To prove the converse we again
X X
(cid:0) (cid:1)
use Proposition 3.3 which says that in case L is a derivation of ∗ there is a formal series A of closed
X X
one-forms on M such that D θ ⊗1+ 1Dθ ⊗1−i (X)r = 1⊗A . Together with equation (19) this
X 2 X a X
yields L r =−(1⊗(θ +A +i Ω)+T ). Applying δ−1 to this equation and using the second equation
X X X X X
(cid:0) (cid:1)
in (20) we get
L s=−(θ +A +i Ω)⊗1−δ−1T .
X X X X X
Now s and hence LXs is in W4 and has no part of symmetric degree 1 such that this equation implies
L s = 0, θ +A +i Ω = 0 and δ−1T = 0. Since θ and A are closed the second of these equations
X X X X X X X
implies 0 = di Ω = L Ω and using the homotopy formula for δ together with δT = 0 the last equation
X X X
yields T =0 which is equivalent to X being affine with respect to ∇ accordingto the Lemmas 3.5 and3.6.
X
Finally one can insert the above expression for L r into the first equation in (20) which turns out to be
X
satisfied identically, which is just a check for consistency. (cid:3)
Finally we can give an additional condition for L to be even a quasi-inner derivation of ∗
X
which is originally due to Gutt [16].
Proposition 3.9 Let X be a symplectic vector field such that L is a derivation of ∗ then L is
X X
even quasi-inner if and only if there is a formal function f ∈C∞(M)[[ν]] such that
df = θ +i Ω = i (ω+Ω) (21)
X X X
and then L = L = −1ad (f), where we have written f = f + f with f ∈ C∞(M) and
f ∈ νC∞(MX)[[ν]].Xf0 ν ∗ 0 + 0
+
Proof: From equation (13) it is obvious that L is quasi-inner if and only if there is a formal function
X
f ∈ C∞(M)[[ν]] such that τ(f) = f +θ ⊗1+ 1Dθ ⊗1−i (X)r but using equation (19) together with
X 2 X a
T = 0, L r = 0 and Df = 1⊗df this is equivalent to (21). In fact the necessary condition for the
X X
solvability of this equation is fulfilled since i Ω is closed according to Theorem 3.8 and θ is closed as X
X X
is symplectic. Moreover, observe that the zeroth order in ν of (21) just means that X is Hamiltonian with
Hamiltonian function f0 and hence the second statement of the Proposition is immediate. (cid:3)
4 g-invariant Star Products ∗ and Quantum Momentum Mappings
In this section we shall usethe results of Theorem 3.8 to findnecessary and sufficient conditions for
the star product ∗ to be invariant with respect to a Lie algebra action. Furthermore Proposition
3.9 gives criteria for the existence of a quantum Hamiltonian and with some little more effort we
shall find a last condition which is necessary and sufficient for this quantum Hamiltonian to define
a quantum momentum mapping for ∗.
Firstletusrecallsomedefinitionsfrom[26]. Letusconsiderafinitedimensionalrealorcomplex
Lie algebra g and let X : g → Γ∞ (TM) : ξ 7→ X denote a Lie algebra anti-homomorphism, i.e.
· symp ξ
[X ,X ] = −X for all ξ,η ∈ g. Then obviously ̺(ξ)f := −L f defines a Lie algebra action of
ξ η [ξ,η] Xξ
g on C∞(M) that naturally extends to a Lie algebra action on C∞(M)[[ν]].
Definition 4.1 With the notations from above a star product ⋆ is called g-invariant in case ̺(ξ)
is a derivation of ⋆ for all ξ ∈ g.
From Theorem 3.8 we obviously get:
10