Weyl Quantization on Pseudo-Riemannian Manifolds

• Weyl quantization on pseudo-Riemannian manifolds is a method that extends flat-space quantization to curved settings using intrinsic geodesic midpoints and the Levi-Civita connection.
• It employs balanced geodesic constructions, incorporating the Van Vleck–Morette determinant and parallel transport to correctly account for curvature effects in operator symbols.
• Extensions include connection-based and graded quantizations that adapt the formalism for vector bundles and spinor fields, enabling applications to Dirac, Maxwell, and Einstein-type operators.

Searching arXiv for the cited works and closely related papers on Weyl quantization on pseudo-Riemannian manifolds. Weyl quantization on pseudo-Riemannian manifolds denotes a family of constructions that generalize the flat-space Weyl correspondence from $\mathbb R^{2d}$ to curved phase spaces built from a manifold $M$ endowed with a pseudo-Riemannian metric $g$. In the most direct pseudodifferential form, the relevant phase space is $T^*M$, and the quantization is defined intrinsically from the Levi-Civita connection, geodesic midpoint geometry, parallel transport, and the Van Vleck–Morette determinant; in neighboring formulations, the same problem is recast through connection-based quantization of tensorial symbols, through graded symplectic manifolds, or through supercotangent bundles adapted to spin degrees of freedom (Dereziński et al., 2018, Andersson et al., 16 Jul 2025, Muñoz-Díaz et al., 2018, Michel, 2010, Grützmann et al., 2014).

1. Scope of the subject

The expression “Weyl quantization on pseudo-Riemannian manifolds” does not designate a single universally adopted construction. The literature represented here separates into at least four technically distinct strands.

One strand gives a direct pseudodifferential Weyl calculus on $T^*M$. The central construction in this direction is the balanced geodesic Weyl quantization, defined on a pseudo-Riemannian manifold by using geodesic midpoints, half-densities, and $\Delta(x,y)^{1/2}$, and designed to retain the midpoint symmetry and symbol-operator properties of ordinary Weyl quantization (Dereziński et al., 2018). A later extension allows operators on sections of vector bundles with arbitrary metric-compatible bundle connections, and develops the associated star product, Wigner function, self-adjointness correspondence, and examples for Dirac, Maxwell, Yang–Mills, and linearized Einstein operators (Andersson et al., 16 Jul 2025).

A second strand is a curved-configuration-space Wigner–Weyl–Moyal formalism based on chosen canonical coordinates $x^i$ and canonically conjugate momenta $p_i$. This construction is explicit and technically workable, but it is formulated for an $n$-dimensional Riemannian manifold rather than for a general pseudo-Riemannian manifold, and it is chart-based rather than coordinate-free (Gneiting et al., 2013).

A third strand is connection-based quantization of polynomial symbols coming from tensor fields. Here the quantization depends on a symmetric linear connection and can be reformulated through the exponential map. The framework is coordinate-free at the definition level, allows pseudo-Riemannian metrics of arbitrary signature when a metric is used, and gives a dequantization theorem for differential operators, but it does not construct the full modern Weyl pseudodifferential calculus (Muñoz-Díaz et al., 2018).

A fourth strand enlarges the phase space. In one case, the ordinary cotangent bundle is replaced by the supercotangent bundle $\mathcal M=T^*M\oplus \Pi TM$, so that odd variables quantize to Clifford generators and classical spin observables become spinor differential operators (Michel, 2010). In another, one quantizes the degree-$M$0 symplectic graded manifold canonically associated with a pseudo-Euclidean bundle, $M$1, combining ordinary Weyl quantization on the cotangent sector with Clifford quantization on the degree-$M$2 sector (Grützmann et al., 2014).

2. Coordinate-based curved configuration-space Weyl theory

A concrete prototype is the phase-space formalism for curved configuration spaces developed for an $M$3-dimensional Riemannian manifold with metric tensor $M$4 (Gneiting et al., 2013). The phase space is built from chosen coordinates $M$5 and canonically conjugate momenta $M$6, with canonical brackets

$M$7

The Hilbert-space structure uses the invariant density $M$8, so that

$M$9

In this setting the momentum operators are chosen, following DeWitt, as

$g$0

with the contracted Christoffel symbol

$g$1

Thus the metric enters the canonical momentum operator through the connection term. The corresponding Stratonovich–Weyl kernel is

$g$2

and for an operator $g$3 the Weyl symbol is

$g$4

The formal Weyl rule survives for the canonical pairs. In particular,

$g$5

The star product also keeps the standard canonical form once symbols are defined relative to the curved quantizer: $g$6 Curvature therefore enters not through an explicit deformation of the Moyal kernel, but through the quantizer, the momentum operators, and the Weyl symbols of geometric Hamiltonians.

For the classical Hamiltonian

$g$7

the paper adopts

$g$8

and the Weyl symbol becomes

$g$9

with

$T^*M$0

The exact quantum Liouville equation reduces in the semiclassical limit to the classical Liouville equation on $T^*M$1. The limitation is explicit: the framework is formulated for Riemannian configuration manifolds, not for general pseudo-Riemannian or Lorentzian spacetime quantization (Gneiting et al., 2013).

3. Balanced geodesic Weyl quantization on $T^*M$2

The intrinsic pseudodifferential Weyl calculus on a pseudo-Riemannian manifold is the balanced geodesic Weyl quantization (Dereziński et al., 2018). The construction assumes a pseudo-Riemannian manifold $T^*M$3, its Levi-Civita connection, geodesics, the exponential map, Synge’s world function, and the Van Vleck–Morette determinant

$T^*M$4

It is defined on a geodesically convex neighborhood $T^*M$5, so that every $T^*M$6 is connected by a unique distinguished geodesic.

For the $T^*M$7-quantization one sets $T^*M$8 and $T^*M$9, and defines

$T^*M$0

The Weyl case is $T^*M$1. If $T^*M$2 with flat metric, then $T^*M$3, $T^*M$4, geodesics are straight lines, and the formula reduces exactly to the standard flat $T^*M$5-quantization (Dereziński et al., 2018).

The balanced factors are not decorative. They yield the exact Hilbert–Schmidt identity

$T^*M$6

which is the curved analogue of the flat Weyl isometry between $T^*M$7-symbols and Hilbert–Schmidt operators. They also underlie the parity theorem: if $T^*M$8 is an even polynomial in $T^*M$9, then $\Delta(x,y)^{1/2}$0 has only even-degree derivatives, and if $\Delta(x,y)^{1/2}$1 is an odd polynomial, then $\Delta(x,y)^{1/2}$2 has only odd-degree derivatives (Dereziński et al., 2018).

For quadratic symbols the calculus gives a distinguished curvature correction. Specializing to the inverse metric,

$\Delta(x,y)^{1/2}$3

independent of $\Delta(x,y)^{1/2}$4. Thus the balanced quantization of the kinetic energy yields the Laplace-Beltrami or d’Alembertian operator together with the familiar $\Delta(x,y)^{1/2}$5 term (Dereziński et al., 2018).

The star product is defined by $\Delta(x,y)^{1/2}$6. Its asymptotic expansion begins with

$\Delta(x,y)^{1/2}$7

and from order $\Delta(x,y)^{1/2}$8 onward contains genuine manifold corrections involving the Riemann tensor, Ricci tensor, covariant derivatives of curvature, and quadratic curvature terms. The zeroth and first terms are exactly the flat Moyal terms; curvature deforms the product starting at second order (Dereziński et al., 2018).

4. Connection-based polynomial quantization and the exponential map

A distinct but closely related approach quantizes polynomial symbols arising from tensor fields by using a symmetric linear connection $\Delta(x,y)^{1/2}$9 (Muñoz-Díaz et al., 2018). For a symmetric contravariant tensor field $x^i$0 of order $x^i$1, the quantized operator $x^i$2 acts by contraction of $x^i$3 with the symmetrized $x^i$4-fold covariant derivative of a function, multiplied by $x^i$5. The construction is coordinate-free at the definition level and depends only on the connection.

This quantization admits a dequantization theorem: every linear differential operator of order $x^i$6 has a unique decomposition into quantized symmetric contravariant tensors of orders $x^i$7. Through the usual identification of symmetric contravariant tensors with fiberwise polynomial functions on $x^i$8, the dequantized tensor is also the Hamiltonian of the differential operator (Muñoz-Díaz et al., 2018).

The same paper introduces a second quantization method built from the exponential map of the connection. For a symmetric covariant tensor field $x^i$9 of order $p_i$0, one pulls back a function $p_i$1 by the exponential map and evaluates the ordinary $p_i$2-th differential along each tangent fiber at the origin. The key result is Theorem 5.1: $p_i$3 In words, the $p_i$4-th ordinary differential of $p_i$5 at the origin of $p_i$6 equals the symmetrized $p_i$7-fold covariant derivative of $p_i$8 at $p_i$9. This identity explains why the connection-based quantization and the exponential-map quantization coincide (Muñoz-Díaz et al., 2018).

The pseudo-Riemannian relevance is explicit. The framework is formulated for an arbitrary symmetric linear connection, and when a metric is used it may be pseudo-Riemannian of arbitrary signature. In particular, the contravariant metric tensor quantizes to

$n$0

where $n$1 is the Laplacian associated with the metric; in pseudo-Riemannian signature this is the Laplace-Beltrami or d’Alembert-type operator (Muñoz-Díaz et al., 2018).

The extension beyond polynomial symbols proceeds through fields of distributions on tangent fibers. Assuming geodesic completeness, the paper states that functions in $n$2 satisfying the Paley–Wiener–Schwartz conditions in the fibers are quantizable by means of the exponential associated with $n$3, and that those quantized as differential operators are precisely the Hamiltonians (Muñoz-Díaz et al., 2018). This suggests a route toward a pseudodifferential theory, but the paper does not develop a full midpoint-symmetric Weyl calculus.

5. Spinorial and graded extensions

A major neighboring framework replaces the scalar cotangent phase space by phase spaces carrying odd or graded directions. For spinning particles on a pseudo-Riemannian spin manifold $n$4 of signature $n$5, the natural classical phase space is the supercotangent bundle

$n$6

with even coordinates $n$7 and odd coordinates $n$8 encoding spin degrees of freedom (Michel, 2010). The canonical $n$9-form is

$\mathcal M=T^*M\oplus \Pi TM$0

and $\mathcal M=T^*M\oplus \Pi TM$1 is the super-symplectic form. The odd observables

$\mathcal M=T^*M\oplus \Pi TM$2

generate a Lie algebra isomorphic to $\mathcal M=T^*M\oplus \Pi TM$3.

Geometric quantization of $\mathcal M=T^*M\oplus \Pi TM$4, after a suitable choice of polarization, reconstructs the standard spin-geometry objects on $\mathcal M=T^*M\oplus \Pi TM$5. In Darboux coordinates, the quantization map satisfies

$\mathcal M=T^*M\oplus \Pi TM$6

The even momenta therefore become differential operators as in ordinary Weyl quantization, while the odd variables quantize to Clifford generators (Michel, 2010).

This supercotangent formalism also supports a conformally equivariant quantization on conformally flat pseudo-Riemannian spin manifolds. The paper compares tensorial symbols, Hamiltonian symbols on $\mathcal M=T^*M\oplus \Pi TM$7, and spinor differential operators, and constructs a superization map together with a quantization map preserving principal symbols. The Dirac operator appears as the quantization of the basic odd linear symbol $\mathcal M=T^*M\oplus \Pi TM$8, and degree-one superization is related to Killing–Yano tensors and hidden symmetries of spinning particles (Michel, 2010).

A related graded-manifold construction begins from an even-rank pseudo-Euclidean vector bundle $\mathcal M=T^*M\oplus \Pi TM$9 with spinor bundle $M$00. The canonical degree-$M$01 symplectic graded manifold is

$M$02

and the main structural theorem identifies the associated graded algebra of a new filtration on $M$03 with $M$04 (Grützmann et al., 2014). The filtration assigns degree $M$05 to Clifford generators and degree $M$06 to covariant derivatives. The Weyl quantization

$M$07

combines ordinary Weyl quantization on $M$08 with Clifford quantization on $M$09. For $M$10,

$M$11

and for $M$12,

$M$13

The quantization satisfies

$M$14

and defines a symmetric star product on $M$15 (Grützmann et al., 2014). Its main applications are to Courant algebroids and Dirac generating operators rather than to scalar pseudodifferential analysis on $M$16.

6. Vector-bundle-valued Weyl calculus and physically important operators

The vector-bundle extension develops an intrinsic Weyl pseudodifferential calculus on a smooth oriented pseudo-Riemannian manifold $M$17, with operators acting on sections of complex vector bundles endowed with pseudo-Hermitian fiber metrics and compatible connections (Andersson et al., 16 Jul 2025). Symbols are sections of pullback bundles $M$18 over $M$19, and the construction depends on the Levi-Civita connection, the bundle connections, and a real parameter $M$20 controlling the power of the Van Vleck–Morette determinant.

For $M$21 and $M$22, the Wigner function is a bundle-valued symbol

$M$23

defined by a geodesic midpoint formula with parallel transport and the factor $M$24. The corresponding Weyl operator is specified by dual pairing with this Wigner function, and its Schwartz kernel is

$M$25

where $M$26 is the geodesic midpoint and $M$27 is the corresponding relative tangent vector. The inverse kernel-to-symbol map holds modulo smoothing symbols (Andersson et al., 16 Jul 2025).

A central theorem is the curved bundle-valued analogue of the flat Weyl self-adjointness property: $M$28 For trivial line bundles this reduces to the statement that real Weyl symbols correspond to formally self-adjoint operators modulo smoothing terms (Andersson et al., 16 Jul 2025).

The induced star product $M$29 has an exact oscillatory formula involving geodesic triangles, a scalar geometric factor $M$30, and bundle holonomy $M$31. Its semiclassical expansion through third order is

$M$32

with

$M$33

At second and third order the expansion contains Ricci terms, Riemann terms involving $M$34, covariant derivatives of curvature, and bundle-curvature terms $M$35 and $M$36. In flat space with trivial bundles and $M$37, these corrections vanish and the usual Moyal expansion is recovered (Andersson et al., 16 Jul 2025).

This framework also gives explicit Weyl symbols for geometrically significant operators. For the scalar wave operator,

$M$38

For the Dirac operator $M$39, the Weyl symbol is

$M$40

The paper likewise computes symbols for the Maxwell operator, the linearized Yang–Mills operator, and the linearized Einstein operator around a vacuum Einstein background with cosmological constant $M$41 (Andersson et al., 16 Jul 2025).

For $M$42, the Wigner function satisfies the corresponding Moyal equation. If $M$43 and $M$44 is the Weyl symbol of $M$45, then

$M$46

and on globally geodesically convex manifolds, or in Minkowski space, this strengthens to

$M$47

The resulting calculus is intended for quantum field theory on curved spacetimes, semiclassical propagation, and kinetic theory, but it remains local near the diagonal in general (Andersson et al., 16 Jul 2025).

7. Locality, pseudo-Riemannian specificity, and common misconceptions

The pseudo-Riemannian character of the subject is not incidental. It enters through indefinite metric tensors, Levi-Civita connections, geodesic midpoint constructions, the density factors $M$48 or $M$49, the Van Vleck–Morette determinant, and curvature corrections involving $M$50 and $M$51 (Dereziński et al., 2018, Andersson et al., 16 Jul 2025). It also changes the natural operators of interest: the kinetic symbol $M$52 quantizes to a Laplace-Beltrami or d’Alembertian type operator rather than only to an elliptic Laplacian (Muñoz-Díaz et al., 2018).

At the same time, most intrinsic constructions are local. The balanced geodesic calculus is canonical only on a geodesically convex neighborhood of the diagonal and uses a cutoff $M$53; many equivalences are asserted modulo smoothing terms $M$54, $M$55, or $M$56 (Dereziński et al., 2018, Andersson et al., 16 Jul 2025). This locality is especially important in pseudo-Riemannian geometry, where multiple geodesics, conjugate points, or the absence of a connecting geodesic are generic away from the diagonal (Dereziński et al., 2018).

A common misconception is to identify all curved-manifold symbol calculi with a direct Weyl pseudodifferential calculus on $M$57. The literature here shows otherwise. The curved configuration-space Wigner formalism is explicitly Riemannian and chart-based (Gneiting et al., 2013). The connection-and-exponential-map approach gives a geometric quantization of polynomial symbols and a distributional extension, but not a full modern Weyl calculus (Muñoz-Díaz et al., 2018). The supercotangent and degree-$M$58 graded-manifold constructions are highly relevant to spin and Clifford geometry, yet they quantize enlarged phase spaces and target spinor differential operators rather than scalar pseudodifferential operators on $M$59 alone (Michel, 2010, Grützmann et al., 2014).

This suggests a precise taxonomy. The core Weyl theory on pseudo-Riemannian manifolds is the intrinsic midpoint quantization on $M$60 built from geodesic geometry and Van Vleck balancing (Dereziński et al., 2018), together with its extension to bundle-valued symbols and arbitrary compatible bundle connections (Andersson et al., 16 Jul 2025). Around that core lie connection-based polynomial quantizations and super or graded extensions that preserve the Weyl idea of symmetric symbol-operator correspondence while adapting it to tensorial, spinorial, or generalized-geometric settings (Muñoz-Díaz et al., 2018, Michel, 2010, Grützmann et al., 2014).