In general, an arbitrary symplectic manifold has no associated “configuration space,” and therefore the classical and quantum mechanical viewpoints must be adapted to this new context based on the available structure.
The classical picture
The central objects in the classical picture of mechanics on an arbitrary symplectic manifold (P, ω) are the semi-classical states, represented as before by lagrangian submanifolds of P equipped with half-densities, and the vector space of observablesC∞(P).
With respect to pointwise multiplication, C∞(P) forms a commutative associative alge-bra. Additionally, the symplectic form onP induces a Lie algebra structure onC∞(P) given by the Poisson bracket
{f, g}=Xg·f,
where Xg = ˜ω−1(dg) denotes the hamiltonian vector field associated tog. These structures satisfy the compatibility condition
{f h, g}=f{h, g}+{f, g}h,
and are referred to collectively by calling C∞(P) the Poisson algebraof P.
The classical system evolves along the trajectories of the vector field XH associated to the choice of a hamiltonian H: P → R. If f is any observable, then Hamilton’s equations assume the form
f˙={H, f}.
Note that in local Darboux coordinates, the Poisson bracket is given by {f, g}=X
Setting f =qj or f =pj in the Poisson bracket form of Hamilton’s equations above yields their familiar form, as in Section 3.3.
Two functionsf, g ∈C∞(P) are said to be ininvolutionif {f, g}= 0, in which case the hamiltonian flows of f and g commute. An observable in involution with the hamiltonian H is called a first integral or constant of the motion of the system. A collection fi of functions in involution on P is said to be complete if the vanishing of {fi, g} for all i implies that g is a function of the formg(x) = h(f1(x),· · · , fn(x)).
The quantum mechanical picture
Quantum mechanical observables should be the vector spaceA(HP) of self-adjoint linear op-erators on some complex Hilbert spaceHP associated to P. The structure of a commutative, non-associative algebra is defined on A by Jordan multiplication:
A◦B = 1
2(AB+BA),
representing the quantum analog of pointwise multiplication in the Poisson algebra of P. Similarly, the ~-dependent commutator
[A, B]~ = i
~(AB−BA)
defines a Lie algebra structure on A analogous to the Poisson bracket onC∞(P).
Quantum mechanical states are vectors inHP. The time-evolution of the quantum system is determined by a choice of energy operator ˆH, which acts on states via the Schr¨odinger equation:
ψ˙ = i
~ Hψ.ˆ
A collection {Aj} of quantum observables is said to be complete if any operator B which commutes with each Aj is a multiple of the identity. This condition is equivalent to the irreducibility of {Aj}, in other words, no nontrivial subspace of HP is invariant with respect to each j.
Quantization
Although neither an underlying configuration manifold nor its intrinsic Hilbert space is generally available in the context of arbitrary symplectic manifolds, the basic goal of quan-tization remains the same as in the case of cotangent bundles: starting with a symplectic manifold P, we wish to identify a ∗-algebra A of operators which give the quantum analog of the “system” at hand. Early approaches to this problem were based on the principle that, regardless of how the algebra Ais identified, the correspondence between classical and quantum observables should be described by a linear map from the Poisson algebra C∞(P) toA which satisfies the following criteria known as the Dirac axioms
Definition 5.33 A linear map ρ:C∞(P)→S(HP) is called aquantization provided that it satisfies
1. ρ(1) =identity.
2. ρ({f, g}) = [ρ(f), ρ(g)]~
3. For some complete set of functions f1,· · · , fn in involution, the operators ρ(f1),· · · , ρ(fn) form a complete commuting set.
As was eventually proven by Groenwald and Van Hove (see [1] for a proof), a quantization of all classical observables in this sense does not exist in general.
To make the basic quantization problem more tractable, we first enlarge the class of classical objects to be quantized, but then relax the criteria by which quantum and classical objects are to correspond. Based on an idea of Weyl and von Neumann, we first replace the classical observables by the groups of which they are the infinitesimal generators. The basic classical objects are then symplectomorphisms, which should be represented by unitary operators on quantum Hilbert spaces.
A general formulation of the quantization problem is then to define a “functor” from the symplectic category to the category of (hermitian) linear spaces. This means that to each symplectic manifold P we should try to assign a Hilbert space HP in such a way that HP is dual to HP, and HP×Q is canonically isomorphic to a (completed) tensor product HP⊗Hb Q. However this is accomplished, each canonical relation L ∈ Hom(P, Q) must then be assigned to a linear operator TL ∈Hom(HP,HQ)' HQ⊗H∗P in a way which commutes with compositions, i.e.,
TL◦L˜ =TL˜ ◦TL
for each ˜L∈Hom(Q, R). In these abstract terms, our classical and quantum correspondences can be expressed as follows.
Object Classical version Quantum version
basic spaces symplectic manifold (P, ω) hermitian vector spaceHP
Q×P HQ⊗Hb P
Q×P Hom(HP,HQ)
point ∗ C
state lagrangian submanifold L element ofHP
space of observables Poisson algebraC∞(P) symmetric operators onHP
At this stage, two relevant observations are apparent from our earlier study of WKB quantization. First, in addition to just a lagrangian submanifold L ⊂ P, it may require more data (such as a symbol) to determine an element of HP in a consistent way. Second, the Hilbert space HP may carry some sort of filtration (e.g. by powers of ~ or by degree of smoothness), and the quantization may be “correct” only to within a certain degree of accuracy as measured by the filtration. By “correctness” we mean that composition of canonical relations should correspond to composition of operators. As we have already seen, it is too much to require that this condition be satisfied exactly. The best we can hope for is a functorial relation, rather than a mapping, from the classical to the quantum category.
6 Fourier Integral Operators
For our first examples of a quantization theory which gives a functorial relation between the classical (symplectic) category and the quantum category of hermitian vector spaces, we return to WKB quantization in cotangent bundles. In this context, semi-classical states are represented by pairs (L,s) consisting of an immersed lagrangian submanifoldLinT∗(M×N) equipped with asymbols. Our discussion begins by defining a suitable notion of composition for such states. The WKB quantization I~(L,s) of the state (L,s) is then regarded as the Schwartz kernel of the corresponding operator in Hom(HM,HN). A particular concrete case of this classical-quantum correspondence is given by the symbol calculus of Fourier integral operators.
6.1 Compositions of semi-classical states
In Section 4.4, we defined a semi-classical state in a cotangent bundle T∗M as a pair (L,s) consisting of a quantizable lagrangian submanifold L⊂ T∗M and a symbols on L. In this section, we study certain natural transformations of semi-classical states.
Reduction of semi-classical states
We will say that a reducible pair (C, L) in a symplectic manifold P is properly reducible if the quotient of I =L×P C by its characteristic foliation is a smooth, Hausdorff manifold and the mapI →LC is proper.
Lemma 6.1 If (C, L) is a properly reducible pair in a symplectic manifold P, then there exists a natural linear map
|Ω|1/2L⊗ |Ω|1/2C→ |Ω|1/2LC.
Proof.First note that ifV is a symplectic vector space, together with a lagrangian subspace L and a coisotropic subspace C, then the exact sequence
0→L∩C⊥ →L∩C →LC →0 gives rise to the isomorphism
|Λ|1/2LC⊗ |Λ|1/2(L∩C⊥)' |Λ|1/2(L∩C).
The linear maps v 7→(v,−v) and (x, y)7→x+y define a second exact sequence 0→L∩C →L⊕C →L+C →0,
from which we get
|Λ|1/2LC ⊗ |Λ|1/2(L∩C⊥)⊗ |Λ|1/2(L+C)' |Λ|1/2L⊗ |Λ|1/2C.
Finally, the exact sequence
0→(L+C)⊥ →V →ω˜ (L+C)∗ →0
combined with the half-density on V induced by the symplectic form defines a natural isomorphism
|Λ|1/2(L+C)' |Λ|1/2(L+C)⊥. Since L∩C⊥= (L+C)⊥, we arrive at
|Λ|1/2LC ⊗ |Λ|(L∩C⊥)' |Λ|1/2L⊗ |Λ|1/2C.
Now consider a properly reducible pair (C, L) in a symplectic manifold P. By the pre-ceding computations, there is a linear map
|Ω|1/2L⊗ |Ω|1/2C → |Ω|1/2LC⊗ |Ω|(FI).
Since the quotient mapI →LC is proper, integration over its fibers is well-defined and gives the desired linear map
|Ω|1/2L⊗ |Ω|1/2C→ |Ω|1/2LC.
2 Let M be a smooth manifold and consider a submanifold N ⊂ M equipped with a foliation F such that the leaf space NF is a smooth, Hausdorff manifold. From Section 5.1 we recall that the integrability of F implies that
CN ={(x, p)∈T∗M :x∈N, Fx ⊂ker(p)}
is a coisotropic submanifold of T∗M whose reduced space is the cotangent bundle T∗NF of the leaf space NF. If L is an immersed lagrangian submanifold of T∗M such that (CN, L) form a reducible pair, then we denote by I the fiber product L×T∗M CN of L and CN and consider the following commutative diagram
L −−−→ T∗M
rL
x
ι x
I −−−→rCN CN
π
y
y
pC
LC
−−−→ T∗N
Our goal is describe how a symbol s onL naturally induces a symbol sC on LC. Lemma 6.2 In the notation of the diagram above, there is a natural isomorphism
r∗LΦL,~ →π∗ΦLC,~.
Proof.From the proof of Lemma 5.16 it follows easily that the pull-back toI of the Liouville forms onT∗M andT∗N coincide, and thus, the pull-back toI of the prequantum line bundles over T∗M and T∗N are canonically isomorphic.
Similarly, letEbe the symplectic vector bundle overIgiven by the pull-back ofT(T∗M).
Lagrangian subbundles λ, λ0 ofE are then induced by the immersed lagrangian submanifold L and the vertical subbundle V M of T(T∗M). By definition, the pull-back of the Maslov bundle of Lto I is canonically isomorphic to the bundle Mλ,λ0(E).
Now the tangent bundle of CN induces a coisotropic subbundle C of E, and one must check that the pull-back of the Maslov bundle of LC to I is canonically isomorphic to MλC,λ0
C(C/C⊥). From Theorem 5.22 we therefore obtain a canonical isomorphism of rL∗ML
with π∗MLC. By tensoring this isomorphism with the isomorphism of prequantum bundles described in the preceding paragraph, we arrive at the desired isomorphism of phase bundles.
2 A parallel section s of ΦL,~ pulls back to a parallel section of rL∗ΦLC,~, which, under the isomorphism of Lemma 6.2, identifies with a parallel section of π∗ΦLC,~. Since parallel sections of π∗ΦLC,~ identify naturally with parallel sections of ΦLC,~, we obtain a map
(parallel sections of ΦL,~)→(parallel sections of ΦLC,~).
By tensoring with the map of density spaces given above, we obtain, for each half-densityσ onC, a natural map of symbol spaces
SL→SLC.
We will denote the image of a symbol son L under this map bysC,σ. Composition of semi-classical states
IfL1, L2 are immersed lagrangian submanifolds ofT∗M, T∗N, then their productL2×L1gives a well-defined immersed lagrangian submanifold of T∗(M ×N) via the Schwartz transform SM,N:T∗N×T∗M →T∗(M×N). By the fundamental properties of the Schwartz transform described in Proposition 3.32, it follows easily that the phase bundle ΦL2×L1,~ is canonically isomorphic to the external tensor product ΦL2,~Φ−1L
1,~. From the discussion in Appendix A we also have a linear isomorphism of density bundles |Λ|1/2(L2×L1)→ |Λ|1/2L2|Λ|1/2L1, and thus there is a natural linear map of symbol spaces
SL2⊗ SL1 → SL2×L1
which we denote by (s1,s2)→s2 s∗1.
Definition 6.3 The product of semi-classical states (L1,s1),(L2,s2) in T∗M, T∗N is the semi-classical state (L2×L1,s2s∗1) in T∗(M×N).
2
Turning to compositions, we first note that if M, V are smooth manifolds, then the Schwartz transform SM,V : T∗V ×T∗M → T∗(M ×V) identifies each canonical relation in Hom(T∗M, T∗V) with an immersed lagrangian submanifold inT∗(M×V). Our goal is now to describe how semi-classical states (L,s) in T∗(M ×N) and (L0,s0) in T∗(N ×V) define a semi-classical state in T∗(M × V) when L, L0 are composable as canonical relations in Hom(T∗M, T∗N) and Hom(T∗N, T∗V), respectively.
We begin with the following commutative diagram
T∗V ×T∗N ×T∗N ×T∗M −−−→ T∗(M ×N×N ×V)
y
y T∗V ×T∗M −−−→ T∗(M×V)
where the upper horizontal arrow denotes the compositionSM×N◦(SN,V×SM,N) of Schwartz tranforms, and the lower horizontal arrow equals the Schwartz transform SM,V. The left vertical arrow represents the reduction relation defined by the coisotropic submanifoldC = T∗V ×∆T∗N × T∗M, while the right vertical arrow is the reduction relation defined by the image of C under the Schwartz transform. L0 ◦L ∈ Hom(T∗M, T∗V) is defined as the reduction of L2×L1 by the coisotropic submanifold T∗V ×∆T∗N ×T∗M of T∗V ×T∗N × T∗N ×T∗M. Under the Schwartz transform,
T∗V ×T∗N ×T∗N ×T∗M →T∗(M ×N ×N ×V),
the submanifold T∗V ×∆T∗N ×T∗M maps to the conormal submanifoldCV,N,M ofT∗(V × N×N×M) defined byV×∆N×M ⊂V ×N×N×M and its product foliation by subsets of the formv×∆N×mfor (v, m)∈V ×M. Thus, the Schwartz transform ofL2×L1reduces by CV,N,M to yield the image ofL2◦L1under the Schwartz transformT∗V×T∗M →T∗(V×M).
From Lemma 6.1, it follows that for any properly reducible pair L1, L2, we obtain a natural linear map of symbol spaces
SL2 ⊗ SL1 → SL2◦L1
given by the composition of the product map above and the reduction mapSL2×L1 → SL2◦L1
defined by the natural half-density onCV,N,M induced by the symplectic forms onT∗V, T∗N, and T∗M. We denote the image ofs2s1 under this map by s2 ◦s1.
We will say that semi-classical states (L1,s1) in T∗(M ×N) and (L2,s2) in T∗(N ×V) are composableif the conormal submanifoldCV,N,M and immersed lagrangian submanifold L2×L1 form a properly reducible pair.
Definition 6.4 If a semi-classical state (L1,s1) in T∗(M ×N) is composable with a semi-classical state (L2,s2) in T∗(N ×V), then their composition is defined as the semi-classical state (L2◦L1,s2◦s1) in T∗(M ×V).
2
Example 6.5 Consider a semi-classical state (LF,˜s) inT∗(M×N), whereLF is the Schwartz transform of the graph of a symplectomorphism F: T∗M →T∗N. If L is an immersed la-grangian submanifold of T∗M, then Definition 6.4 provides that for each symbol s ∈ SL, the semi-classical state (L,s) in T∗M is transformed by F into the semi-classical state (LF ◦L,˜s◦s) in T∗N.
To make this correspondence more explicit, let us choose a particular lagrangian immer-sion ι:L→T∗M representing L (see the discussion in Section 4.4). We then note that the natural half-density |ωnM|1/2 onLF ≈T∗M enables us to identify the symbol spaceSLF with the product
ΓLF ⊗CC∞(LF,C).
Given a parallel section ˜s and smooth complex-valued function h onLF, we find by a com-putation that the isomorphism SL→ SLF◦L determined by the symbol ˜s= ˜s⊗h is given by s⊗a7→s0⊗(a·ι∗h), whereais any half-density onL ands0 is the unique parallel section of ΦLF◦L,~ such that ι∗s˜7→s0⊗s−1 under the canonical isomorphism ι∗ΦLF,~ 'ΦLF◦L,~⊗Φ−1L,
~. 4 Example 6.6 As a special case of Example 6.5, note that if the symplectomorphism F : T∗M → T∗N is the cotangent lift of a diffeomorphism M → N, then for each ~ > 0, the phase bundle of of LF admits a canonical parallel section, and so the symbol space SLF identifies naturally with R+×C∞(LF,C). If h ∈C∞(LF,C), then with this identification, the composition of (LF, h) with a semi-classical state (L,s) equals the semi-classical state (LF ◦L,(h◦ι)·s).
Similarly, if β is a closed 1-form on M and the symplectomorphism F : T∗M → T∗M equals fiberwise translation by β, then ~ ∈ R+ is admissible for LF if and only if [β] is
~-integral. For such ~, each parallel section of the phase bundle ΦLF,~ identifies with an oscillatory function of the form ceiS/~ for c ∈ R and some S: M → T~ satisfying S∗dσ = β. A computation then shows that the composition of a semi-classical state of the form (LF, eiS/~⊗h) with (L, ι,s) yields the semi-classical state (LF ◦L, ei(S◦πL)/~(h◦ι)·s).
4