The quantum version of the Pythagorean theorem; what is the geometric quantization?

Abstract: A short overview on Geometric Quantization problem (GQ).

THE ORIGINAL PROBLEM

The basic “quantum data’” in the Kostant-Souriau theory are the following cohomology groups:

It does not matter if one does not know what a cohomology group is; the point is that mathematicians began working on the problem of finding a rigorous (canonical) way to associate a phase space (usually a model of a classical physical situation) with a Hilbert space (the space of square integrable wave functions). In the previous formula M is a symplectic manifold (the phase space), L sheaf of polarized sections (a particular overstructure) and H⁰(M,L) corresponds to the “usual” Hilbert space, also called the space of holomorphic sections, or the Hardy space …

Usually it is necessary to consider the action of a compact Lie group G that acts in Hamiltonian fashion on M and that preserve the polarization. G acts on these spaces, so we have a string of representations of G on abstract vector spaces. As Guillemin and Sternberg declare in their book [“Symplectic techniques in physics”] the theory is far from achieving these goals but much of literature was produced in recent years on the subject.

THE QUANTUM AXIOMS

In his work [“The principles of quantum mechanics”], P. Dirac defines the quantum Poisson bracket [.,.] of any two variables u and v as:

where ℏ is the Plank constant. The formula before is considere as one of basic postulates of quantum mechanics. We can summarize these postulates as follows. To start we fix a symplectic manifold (M,ω) of dimension d, with ω the corresponding symplectic structure and an Hilbert space H. The quantization is a “way” to pass from the classical system to the quantum system. In this case the classical system or phase space is described by the symplectic manifold M and the Poisson algebra of smooth function on M denoted by (C∞(M),{·,·}). The quantum system is described by H. We define quantization a map Q from the subset of the commutative algebra of observables C∞(M) to the space of operators in H. Let f ∈ C∞(M) we have that Q(f) : H → H is the corresponding quantum operator. We can summarize the quantum axioms in this scheme:

Quantum axioms (Dirac)

The last postulate say that in the case we consider a connected Lie group G we say that is a group of symmetries of the physical system if we have the two following irreducible representation: one as symplectomorphisms acting on (M,ω) and an other as unitary transformations acting on H.

The classical example is the Schrödinger quantization:

THE LINE BUNDLE

Following the work of [J.Rawnsley and M.Cahen and S. Gutt “Quantization of Kähler Manifolds I”], the first element necessary to the geometric quantization is the Hermitian complex line bundle (L,∇, h). To do this we need a condition on the symplectic form ω of the symplectic structure (M,ω). The condition for the prequantization is that ω must be integral, or geometrically, that the integral of ω over every closed, oriented 2-surface in M is a multiple of 2π. In fact by this assumption we can construct L. We start from L₀= {(m,z,γ) : m ∈ M, z ∈ ℂ, γ a path from m₀ ∈ M to m ∈ M}. Introducing the equivalence relation ∼: (m₁,z₁,γ₁) ∼ (m₂,z₂,γ₂) if and only if m₁ = m₂ and

Here Σ is an oriented 2-surface with boundary γ₁⁻¹◦γ₂. We can do this assuming that M is simply connected so we obtain L = L₀/∼ that is a line bundle on M. Denoting Lₘ the fiber at m and using the equivalence relation we can continue the process. The resulting line bundle is the disjoint union of the spaces:

The Hermitian structure h is given by h((z₁,γ),(z₂,γ)) = z₁·z₂ and the connection ∇ is defined using the parallel transport. Let σ a path from m₁ to m₂ on M, for (z, γ)∈ Lₘ we consider the lift σ’: [0,1] → L of σ: [0,1] → M such that ∇σ’(t) = 0 (along σ(t)) . This last condition is equivalent to say that the section σ of Lₘ over σ is constant. We define the parallel transport by (z,γ) → (z,σ◦γ) and the parallel transport maps that define ∇ are given by the association (z,γ) = σ’(0) → σ’(1). The connection ∇ leave the hermitian structure invariant and if ξ is a vector field on M and s₁, s₂ are smooth sections of L, this compatibility condition can be expressed as follows:

An other important condition in prequantization is the relation between the symplectic form (Kähler form) ω and the curvature form Θ of the connection ∇. The curvature form is a complex 2-form defined as:

where ξ₁,ξ₂ are two fields of M and s is a smooth section of L. The condition on Θ is that:

We call the triple (L,∇,h) the prequantization bundle over (M,ω). We observe that with the hermitian product h and the volume form:

where

is the dimension of M and the times of the multiplication in the volume form. We can consider the space of smooth holomorphic sections s of M such that:

is finite. The L²-completition of this space is an Hilbert space denoted by H or also H⁰(M,L).

THE KÄHLER POLARIZATION

A Kähler polarization of M is a smooth complex distribution D, that is a map that to each point m ∈ M assigns a linear subspace Dₘ of TₘM, such that:

As in [J.Rawnsley and M.Cahen and S. Gutt “Quantization of Kähler Manifolds I”] we consider Kähler manifolds, we define the almost complex structure J using the polarization: J = −iI and J = iI (both relations restricted to D), and the metric

on M by:

for ξ,η vector fields on M. The metric is Hermitian and when ω is closed (or i.e. ∇J =0) it is a Kähler metric. What remain to define are the operators

for every f ∈C∞(M). We note that for every f ∈ C∞(M) we have an Hamiltonian vector field

and the operator

that acts on

We look that if we define the operator:

quantum operator

it is an Hermitian operator and, when f is constant, is only the multiplication by f. This operator satisfy the Dirac axiom.

THE CIRCLE BUNDLE

Let L an hermitian line bundle, considering the dual:

we have the following definition of circle bundle X:

The circle bundle is the boundary of:

that is a strictly pseudoconvex domain in L. We have an induced norm by h, the Levi form ρ and a circle action:

with infinitesimal generator:

We consider the corrispective holomorphic and antiholomorphic space of the tangent space TD:

so there is the composition given by:

elements are of the form:

The Cauchy Riemann operator denoted by:

the contact form:

and the volume form:

and we have that (X,α) is a contact manifold.

The base manifold (symplectic), a model for the classical mechanics and the circles of the circle bundle X

THE HARDY SPACE AND THE SZEGÖ KERNEL

We define the Hardy space

the space that admits the following decomposition:

On the Hardy spaces we have the hermitian product:

of two sections of

the space of holomorphic sections (with the tensor power of the line bundle insted the line bundle L), it is our “Hilbert space’’ and the hermitian product is the usual norm in the functional set of square integrable functions…

There is an unitary isometry between sections on X and sections on M:

where m = π(x) is the projection from the circle bundle to the base manifold.

We can define the projector:

fromthe total Hilbert space to its isotopic component as the Szegö kernel, and the equivariant Szegö kernel:

This formula is a sort of Pythagorean theorem in the quantum Hilbert space. Sure it is very difficult to proceed in calculations. It is simple is some projective compact space with complex holomorphic monomials but we have the magnificent result due to S.Zelditch that gives an asymptotic expansion of our “Quantum Pythagorean Theorem” in terms of a quantum parameter.

THE ASYMPTOTIC EXPANSION OF THE SZEGÖ KERNEL

the parameter k is the inverse of planck constant and tends to infinite …

The result gives an extimation of the dimension of the Hilbert space of holomorphic sections.

The proof is based on the Boutet de Monvel parametrix and the stationary phase lemma of Hörmander in order to treat the oscillating integrals…

REFERENCES

• L.Boutet de Monvel, J.Sjöstrand “Sur la singularité des noyaux de Bergman et de Szegö” Astérisque 34–35 (1976), 123–164.
• L.Boutet de Monvel, V.Guillemin “The spectral theory of Toeplitz operators”, Annals of Mathematics Studies, 99 (1981), Princeton University Press, Prince ton, NJ; University of Tokyo Press, Tokyo.
• P.A.M.Dirac “The principles of quantum mechanics”, Oxford at the clarendon press (fourth edition 1958).
• V.Guillemin, S.Sternberg “Symplectic techniques in physics”, Cambridge, 183 196.
• V.Guillemin, S.Sternberg “Geometric quantization and multiplicities of group representations”, Inv. Math. 67 (1982), no. 1–2, 515–538.
• S.Zelditch “Szegö kernels and a theorem of Tian”, Internat. Math. Res. Notices 6, 317–331, 1998.