Polarization-Free Quantization

• Polarization-Free Quantization is a method that avoids choosing a Lagrangian polarization by using non-perturbative path-integrals and abstract algebraic constructions to quantize symplectic and Poisson manifolds.
• It recovers standard schemes such as Weyl–Moyal, Berezin–Toeplitz, and Kontsevich quantization, providing a unified framework across geometric and deformation paradigms.
• For linear field theories, the approach unifies Fock state constructions via a projective-limit method, ensuring the resultant quantum algebra is independent of any polarization choice.

Polarization-free quantization denotes a class of quantization procedures for classical symplectic or Poisson manifolds—and for linear field theories—that circumvent the necessity of choosing a Lagrangian polarization or a decomposition into positive and negative frequency modes. Polarizations, while essential to canonical geometric quantization and traditional Fock space constructions, are known to introduce ambiguities and obstruct invariance under automorphism groups, particularly in cases lacking a canonical splitting, such as quantum field theories on curved backgrounds or general symplectic manifolds. Recent developments provide path-integral and projective-limit constructions that unify or exhaust all possible quantization schemes, ensuring the resultant quantum theory is independent of arbitrary polarization choices (Lackman, 2024, Lanéry, 2016).

1. Fundamental Structures in Polarization-Free Quantization

On a symplectic manifold $(M,\omega)$ with the prequantum integrality condition $[\frac{1}{\hbar}\omega] \in 2\pi H^2(M;\mathbb{Z})$, the traditional geometric quantization procedure begins with the construction of the prequantum line bundle and proceeds through the selection of a Lagrangian polarization to yield a physical Hilbert space. Polarization-free quantization reconstructs this process by deploying either non-perturbative path-integrals over field-space (e.g., via the Poisson sigma model on the disk) or, for linear field theories, by leveraging the abstract Weyl $\mathrm{C}^*$-algebra and its projective limit state space, entirely avoiding any polarization or choice of mode decomposition (Lackman, 2024, Lanéry, 2016).

In the case of geometric quantization, the polarization-free quantization map

$Q:C_c^\infty(M) \longrightarrow \mathcal{A}_\hbar$

assigns to functions on $M$ operators in a noncommutative $\mathrm{C}^*$-algebra $\mathcal{A}_\hbar$, constructed explicitly via boundary path integrals of the Poisson sigma model. This map is defined on the groupoid of homotopy classes of loops $\Pi_1(M)$, making use of a canonical multiplicative prequantum line bundle $L_1 \rightarrow \Pi_1(M)$ (Lackman, 2024).

For linear field theories, the full space $S$ of (sufficiently regular) classical solutions, equipped with a symplectic form $\Omega$, is quantized by constructing the abstract Weyl algebra generated by operators $W(f)$ satisfying the canonical commutation relations, and the universal projective limit state space

$$S = \varprojlim\{S_\lambda,\, \operatorname{Tr}_{\lambda_2 \to \lambda_1}\}$$

where each $S_\lambda$ is the space of trace-class operators on a finite-mode Fock Hilbert space associated to a finite symplectic family $\lambda$, and the transitions are defined via partial traces (Lanéry, 2016).

2. Explicit Formulation of the Polarization-Free Quantization Map

The polarization-free scheme in geometric settings employs the Poisson sigma model (PSM) to define the quantization map as follows: $$Q:\ C_c^\infty(M) \to \Gamma(L_1 \to \Pi_1(M)), \qquad Q(f)[\gamma] = \left( \gamma,\ \int_{X: S^1\to M,\, X|_{S^1_-} = \gamma,\, X|_{S^1_+} = \gamma} f(\gamma(1/2))\, e^{\frac{i}{\hbar}\int_D X^*\omega}\, DX \right)$$ Here, $\gamma \in \Pi_1(M)$ denotes a homotopy class of loops, and the integral is over fields on the disk matching $\gamma$ on specified boundary segments. This operator construction, applied to sections of $L_1$, produces operators $Q(f)$ on $L^2(M,L)$ with integral kernels given in local Darboux coordinates by

$$K_f(m',m) = \frac{1}{(2\pi\hbar)^n} \int_{X(0)=m,\, X(1)=m'} \exp\left(\frac{i}{\hbar}\int_D X^*\omega\right)\, DX$$

for $n = \frac{1}{2}\dim M$. This construction totally avoids any reference to or choice of Lagrangian polarization (Lackman, 2024).

In linear theories, the polarization-free approach works at the level of abstract states rather than explicit operator representatives in a chosen Fock space. The key objects are the inductive limit $\mathrm{C}^*$-algebra $\overline{B}$ and its projective-limit state space, containing all possible Fock states arising from different (e.g., complex structure-induced) splittings of the phase space (Lanéry, 2016).

3. Recovery and Unification of Standard Quantization Schemes

Polarization-free quantization frameworks are structured to recover traditional quantization schemes as special cases or limits:

• Weyl–Moyal quantization: For $M = T^*\mathbb{R}^n$ equipped with its flat Kähler connection, the PSM path integral yields the Moyal star-product and, upon representation, produces the standard Weyl quantization on $L^2(\mathbb{R}^n)$ (Lackman, 2024).
• Berezin–Toeplitz and Fedosov quantization: On compact Kähler manifolds, a specific choice of symplectic connection identifies a cocycle in the lattice PSM formulation, which, in the limit as the mesh vanishes, recovers Fedosov’s star-product. The Berezin–Toeplitz quantization is reconstructed on the holomorphic subspace of $L^2(M,L)$ with a semi-classical expansion matching the known star-products (Lackman, 2024).
• Kontsevich deformation quantization: Perturbatively, the disk amplitude of the PSM produces the Kontsevich formula for deformation quantization of arbitrary Poisson manifolds, thus unifying the geometric and deformation quantization paradigms (Lackman, 2024).

For linear field theories, the projective construction encompasses all Fock states corresponding to any reasonable choice of polarization, and does so simultaneously and constructively. The resulting state space admits automorphisms induced by the full group of bounded symplectomorphisms of the phase space (Lanéry, 2016).

4. Invariance under Change of Polarization and the Role of Schur’s Lemma

Traditional geometric quantization’s dependence on the choice of polarization leads to ambiguities, especially when comparing representations associated to different polarizations. In the polarization-free approach, the full $\mathrm{C}^*$-algebra $\mathcal{A}_\hbar$ constructed acts on the entire prequantum Hilbert space $\mathcal{H} = L^2(M, L)$; each subspace defined by a covariantly constant polarization $P$ is an invariant subspace of $\mathcal{A}_\hbar$. If two polarized subspaces $\mathcal{H}_{P_1}$ and $\mathcal{H}_{P_2}$ are both irreducible representations, Schur’s lemma ensures their commutants are scalar, and they are equivalent up to a unique projective unitary intertwiner, provided their intersection is nontrivial, e.g., via the Blattner–Kostant–Sternberg pairing (Lackman, 2024).

In linear field theory, the universal projective state space is designed so that any automorphism of the phase space lifts to a $*$-automorphism of the quantum algebra, acting bijectively on the state space. The resulting quantum theory is thus manifestly independent of any a priori polarization choice (Lanéry, 2016).

5. Polarization-Free Quantization of the Torus and the Noncommutative Torus

The quantization of the symplectic 2-torus $T^2_{(p,q)} = \mathbb{R}^2 / 2\mathbb{Z}^2$ with symplectic form $\omega = n\,dp \wedge dq$ provides a concrete illustration. The polarization-free scheme yields:

• The symplectic groupoid $$\Pi_1(T^2) \cong (\mathbb{R}^2 \times T^2)/2 \mathbb{Z}^2$$ with the standard source and target maps.
• The PSM path integral produces a Moyal-type star-product:

$$(f \star g)(p,q) = \int_{T^2 \times T^2} f(p_1, q_1) g(p_2, q_2) e^{i n [(p_2 - p)(q_1 - q) - (q_2 - q)(p_1 - p)]} dp_1 dq_1 dp_2 dq_2$$

• The corresponding quantum algebra is the standard noncommutative torus $C^*(\mathbb{Z}^2, \theta = n^{-1})$, generated by unitaries $U = e^{ip}$, $V = e^{iq}$ with $UV = e^{2\pi i / n} VU$.
• The holomorphically polarized subspace is $n$-dimensional and is spanned by theta functions $\{\vartheta_j\}_{j=0}^{n-1}$, obeying

$$Q(U)\vartheta_j = \vartheta_{j+1}, \qquad Q(V)\vartheta_j = e^{2\pi i j / n} \vartheta_j$$

yielding the canonical irreducible representation of the noncommutative torus algebra (Lackman, 2024).

6. Polarization-Free Quantization for Linear Field Theories

For real linear field theories, the polarization-free quantization constructs the quantum theory via the abstract Weyl $\mathrm{C}^*$-algebra:

• The phase space $S$ (e.g., initial data for the Klein–Gordon equation) is endowed with a symplectic form $\Omega$.
• The Weyl operators $W(f)$ generate the algebra via