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Wigner–Weyl transform
Wigner–Weyl transform
Wigner–Weyl transform
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In quantum mechanics, the Wigner–Weyl transform or Weyl–Wigner transform (after Hermann Weyl and Eugene Wigner) is the invertible mapping between functions in the quantum phase space formulation and Hilbert space operators in the Schrödinger picture.

Often the mapping from functions on phase space to operators is called the Weyl transform or Weyl quantization, whereas the inverse mapping, from operators to functions on phase space, is called the Wigner transform. This mapping was originally devised by Hermann Weyl in 1927 in an attempt to map symmetrized classical phase space functions to operators, a procedure known as Weyl quantization.[1] It is now understood that Weyl quantization does not satisfy all the properties one would require for consistent quantization and therefore sometimes yields unphysical answers. On the other hand, some of the nice properties described below suggest that if one seeks a single consistent procedure mapping functions on the classical phase space to operators, the Weyl quantization is the best option: a sort of normal coordinates of such maps. (Groenewold's theorem asserts that no such map can have all the ideal properties one would desire.)

Regardless, the Weyl–Wigner transform is a well-defined integral transform between the phase-space and operator representations, and yields insight into the workings of quantum mechanics. Most importantly, the Wigner quasi-probability distribution is the Wigner transform of the quantum density matrix, and, conversely, the density matrix is the Weyl transform of the Wigner function.

In contrast to Weyl's original intentions in seeking a consistent quantization scheme, this map merely amounts to a change of representation within quantum mechanics; it need not connect "classical" with "quantum" quantities. For example, the phase-space function may depend explicitly on the reduced Planck constant ħ, as it does in some familiar cases involving angular momentum. This invertible representation change then allows one to express quantum mechanics in phase space, as was appreciated in the 1940s by Hilbrand J. Groenewold[2] and José Enrique Moyal.[3][4]

In more generality, Weyl quantization is studied in cases where the phase space is a symplectic manifold, or possibly a Poisson manifold. Related structures include the Poisson–Lie groups and Kac–Moody algebras.

Definition of the Weyl quantization of a general observable

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The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be (q,p), and let f be a function defined everywhere on phase space. In what follows, we fix operators P and Q satisfying the canonical commutation relations, such as the usual position and momentum operators in the Schrödinger representation. We assume that the exponentiated operators and constitute an irreducible representation of the Weyl relations, so that the Stone–von Neumann theorem (guaranteeing uniqueness of the canonical commutation relations) holds.

Basic formula

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The Weyl transform (or Weyl quantization) of the function f is given by the following operator in Hilbert space,[5][6]

Throughout, ħ is the reduced Planck constant.

It is instructive to perform the p and q integrals in the above formula first, which has the effect of computing the ordinary Fourier transform of the function f, while leaving the operator . In that case, the Weyl transform can be written as[7]

.

We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function , but then when applying the Fourier inversion formula, we substitute the quantum operators and for the original classical variables p and q, thus obtaining a "quantum version of f."

A less symmetric form, but handy for applications, is the following,

In the position representation

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The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,[8]

Inverse map

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The inverse of the above Weyl map is the Wigner map (or Wigner transform), which was introduced by Eugene Wigner,[9] which takes the operator Φ back to the original phase-space kernel function f,

For example, the Wigner map of the oscillator thermal distribution operator is[6]

If one replaces in the above expression with an arbitrary operator, the resulting function f may depend on the reduced Planck constant ħ, and may well describe quantum-mechanical processes, provided it is properly composed through the star product, below.[10] In turn, the Weyl map of the Wigner map is summarized by Groenewold's formula,[6]

Weyl quantization of polynomial observables

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While the above formulas give a nice understanding of the Weyl quantization of a very general observable on phase space, they are not very convenient for computing on simple observables, such as those that are polynomials in and . In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators and . For example, the Wigner map of the quantum angular-momentum-squared operator L2 is not just the classical angular momentum squared, but it further contains an offset term −3ħ2/2, which accounts for the nonvanishing angular momentum of the ground-state Bohr orbit.

Properties

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Weyl quantization of polynomials

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The action of the Weyl quantization on polynomial functions of and is completely determined by the following symmetric formula:[11]

for all complex numbers and . From this formula, it is not hard to show that the Weyl quantization on a function of the form gives the average of all possible orderings of factors of and factors of :where , and is the set of permutations on N elements.

For example, we have

While this result is conceptually natural, it is not convenient for computations when and are large. In such cases, we can use instead McCoy's formula[12]

This expression gives an apparently different answer for the case of from the totally symmetric expression above. There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator. (The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of in terms of the operators , , and and verify the first expression in McCoy's formula with .)

It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side. (An exact correspondence is impossible, in light of Groenewold's theorem.) For example, Moyal showed the

Theorem: If is a polynomial of degree at most 2 and is an arbitrary polynomial, then we have .

Weyl quantization of general functions

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See also

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References

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Further reading

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Wigner–Weyl transform

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The Wigner–Weyl transform, also known as the Weyl–Wigner transform, is an invertible mathematical mapping in that associates operators acting on a with functions defined on classical , thereby providing a phase-space formulation of quantum theory that is fully equivalent to the standard Schrödinger representation. This transform enables the representation of quantum states and observables in terms of quasiprobability distributions, such as the Wigner function, which reveal both quantum interference effects and classical-like behavior in phase space. Introduced independently by physicists and in the early 1930s, the transform emerged during the foundational development of as a means to bridge classical and quantum descriptions. proposed the in 1932 to derive quantum corrections to classical , defining it as a of the density matrix off-diagonal elements in position space. , in his 1931 English translation of his 1927 work on and , outlined a quantization rule that maps classical symbols (functions of position and ) to non-commuting quantum operators via a symmetric ordering prescription, known as Weyl quantization. The combined framework, often termed the Weyl–Wigner–Moyal formalism, incorporates the for the evolution of phase-space functions, analogous to the in but with quantum corrections via the star product. Key properties of the Wigner–Weyl transform include its covariance under unitary transformations, such as and transformations, which preserve the phase-space structure, and its ability to handle both pure and mixed quantum states through the density operator. Unlike true probability distributions, the Wigner function can take negative values, highlighting non-classical quantum features like interference, though it marginalizes correctly to position and probability densities. In the (ℏ→0\hbar \to 0), the transform recovers the standard commutative phase-space mechanics, making it invaluable for semiclassical approximations. The transform has found wide applications across quantum physics, including for describing coherent and squeezed states, for time-frequency analysis, and for studying electron dynamics in solids. It also underpins modern developments in theory, such as entanglement detection via phase-space methods, and numerical simulations like the Wigner approach for solving quantum transport problems. Despite challenges like the non-positivity of quasidistributions, extensions to discrete systems, relativistic fields, and curved phase spaces continue to expand its utility.

Introduction

Overview and Motivation

The Wigner–Weyl transform establishes a bidirectional, invertible mapping Φ\Phi between self-adjoint operators A^\hat{A} on the L2(Rn)L^2(\mathbb{R}^n) and real-valued functions f(q,p)f(q,p) defined on the classical R2n\mathbb{R}^{2n}, where qq and pp represent position and coordinates, respectively. This correspondence assumes familiarity with the canonical quantum operators Q^\hat{Q} (position) and P^\hat{P} (), satisfying the commutation relation [Q^,P^]=iℏI^[\hat{Q}, \hat{P}] = i\hbar \hat{I} with ℏ\hbar denoting the reduced Planck's constant, and the underlying phase-space structure inherited from classical . While the transform preserves key algebraic structures such as traces and expectations, it does not yield a fully commutative due to the inherent non-commutativity of quantum operators, highlighting the irreducible quantum features even in phase-space representations. The primary motivation for the Wigner–Weyl transform lies in its provision of a phase-space formulation of , which facilitates the analysis of using tools analogous to those in classical , such as for evolution. This approach enables the interpretation of quantum states via quasi-probability distributions on , bridging the gap between wave mechanics and classical trajectories without relying on coordinate or representations alone. Originally developed to compute quantum corrections to in classical phase-space integrals, it has become essential for revealing non-classical effects like interference and in a geometrically intuitive framework. A representative example is the Wigner function W(q,p)W(q,p), obtained by applying the transform to a operator ρ\rho: W(q,p)=1(2πℏ)n∫Rndy ⟨q+y/2∣ρ∣q−y/2⟩e−ip⋅y/ℏ,W(q,p) = \frac{1}{(2\pi\hbar)^n} \int_{\mathbb{R}^n} dy \, \langle q + y/2 | \rho | q - y/2 \rangle e^{-i p \cdot y / \hbar}, which serves as a quasi-probability that can exhibit negative values, thereby capturing quantum interference phenomena absent in classical probability distributions.

Historical Background

The Wigner–Weyl transform originated with Hermann Weyl's efforts to establish a systematic correspondence between classical phase-space functions and quantum operators, known as Weyl quantization. In his 1927 paper, Weyl proposed a rule for quantizing polynomials in position and by symmetrizing the operator ordering, aiming to provide a consistent mapping that aligned with the emerging formalism of . This approach was part of the broader push in the 1920s to axiomatize physical theories, reflecting influences from Hilbert's sixth problem on formalizing the mathematical foundations of physics. Eugene Wigner extended this framework in 1932 by developing the inverse transform, now called the Wigner function, which maps quantum density operators to phase-space distributions. Published in the context of , Wigner's work allowed for a probabilistic interpretation of quantum states in , facilitating calculations of thermodynamic properties like equilibrium corrections. This contribution bridged Weyl's quantization with applications in statistical ensembles, highlighting the transform's utility for semiclassical approximations. Post-World War II advancements clarified the limitations and refined the theoretical structure of the transform. In 1946, Hilbrand Groenewold proved a theorem demonstrating that no exact, invertible exists between all classical functions and quantum operators under any quantization scheme, including Weyl's, due to inconsistencies in higher-order terms. Building on this, José Moyal formulated a comprehensive phase-space in 1949, introducing the star product () as a non-commutative multiplication on phase-space functions that preserves the of quantum operators. These developments solidified the Wigner–Weyl transform as a cornerstone of deformation quantization, emphasizing its role in reconciling classical and quantum descriptions despite inherent non-equivalences.

Mathematical Definition

Wigner Transform

The Wigner transform provides a mapping from a quantum mechanical operator A^\hat{A} acting on the of square-integrable functions to a function WA^(q,p)W_{\hat{A}}(q, p) defined on classical phase space, where q∈Rnq \in \mathbb{R}^n represents position coordinates and p∈Rnp \in \mathbb{R}^n represents momentum coordinates. This transform is central to the phase-space formulation of quantum mechanics, allowing operators to be represented as functions that resemble classical observables while capturing quantum effects. The explicit formula for the Wigner transform in nn dimensions is WA^(q,p)=1(2πℏ)n∫Rndy ⟨q+y/2∣A^∣q−y/2⟩e−ip⋅y/ℏ,W_{\hat{A}}(q, p) = \frac{1}{(2\pi \hbar)^n} \int_{\mathbb{R}^n} \mathrm{d}y \, \langle q + y/2 | \hat{A} | q - y/2 \rangle e^{-i p \cdot y / \hbar}, where the integration is over the displacement vector yy, ℏ\hbar is the reduced Planck's constant, and ⟨⋅∣A^∣⋅⟩\langle \cdot | \hat{A} | \cdot \rangle denotes the matrix element in the position basis. This expression was originally introduced by Wigner in 1932 for the specific case of the density operator to derive quantum corrections to classical thermodynamic equilibrium distributions. The general extension to arbitrary operators followed in the development of the Weyl-Wigner correspondence, as formalized by Groenewold in 1946.80084-1) The resulting phase-space function WA^(q,p)W_{\hat{A}}(q, p) is interpreted as a quasi-probability distribution, meaning it can take negative values, which highlights inherently quantum features such as interference and oscillations that have no classical analog. For a pure state with density operator ρ^=∣ψ⟩⟨ψ∣\hat{\rho} = |\psi\rangle\langle\psi|, the Wigner transform reduces to the Wigner function Wψ(q,p)W_\psi(q, p), providing a visual representation of the quantum state in phase space where regions of negativity indicate non-classical behavior. This quasi-probabilistic nature arises directly from the off-diagonal matrix elements in the integral, distinguishing it from true probability densities.80084-1) Key properties of the Wigner transform include its preservation of the Hilbert-Schmidt inner product under the mapping, such that the trace of the product of two operators equals the phase-space integral of their transforms: Tr⁡(A^B^)=∫dq dp WA^(q,p)WB^(q,p)\operatorname{Tr}(\hat{A} \hat{B}) = \int \mathrm{d}q \, \mathrm{d}p \, W_{\hat{A}}(q, p) W_{\hat{B}}(q, p). This orthogonality relation ensures that the transform is invertible and maintains the algebraic structure of quantum mechanics in phase space. Additionally, the marginals of the Wigner function for a density operator recover the standard probability distributions: integrating over momentum gives the position probability density, ∫dp Wρ^(q,p)=∣ψ(q)∣2\int \mathrm{d}p \, W_{\hat{\rho}}(q, p) = |\psi(q)|^2, while integrating over position yields the momentum probability density, ∫dq Wρ^(q,p)=∣ψ~(p)∣2\int \mathrm{d}q \, W_{\hat{\rho}}(q, p) = |\tilde{\psi}(p)|^2, where ψ~(p)\tilde{\psi}(p) is the Fourier transform of the position wave function ψ(q)\psi(q). These marginal properties link the phase-space representation back to the traditional wave-function formulation.80084-1) As a simple example, consider the Q^\hat{Q}, whose Wigner transform is WQ^(q,p)=qW_{\hat{Q}}(q, p) = q, independent of the variable pp. This reflects the classical correspondence, where the position is simply the coordinate qq in , and it serves as a basis for quantizing in higher dimensions.80084-1)

Weyl Quantization

The Weyl quantization maps a phase-space function f(q,p)f(q, p), known as the , to a corresponding quantum operator A^=Opw(f)\hat{A} = \mathrm{Op}^w(f) on the of square-integrable wave functions, providing the quantization rule that inverts the dequantization of the Wigner transform. This mapping ensures that classical are promoted to self-adjoint operators while preserving key symmetries of the . Originally introduced by for in 1927, the general form was developed by Groenewold in 1946 to handle arbitrary smooth functions. The basic formula for the Weyl quantization in nn dimensions is given by A^=Opw(f)=(1(2πℏ)n)∫dnq dnp f(q,p)∫dna dnb ei[a⋅(P^−p)+b⋅(Q^−q)]/ℏ,\hat{A} = \mathrm{Op}^w(f) = \left( \frac{1}{(2\pi \hbar)^n} \right) \int \mathrm{d}^n q \, \mathrm{d}^n p \, f(q, p) \int \mathrm{d}^n a \, \mathrm{d}^n b \, e^{i [a \cdot (\hat{P} - p) + b \cdot (\hat{Q} - q)] / \hbar}, where Q^\hat{Q} and P^\hat{P} are the position and momentum operators satisfying [Q^j,P^k]=iℏδjk[\hat{Q}_j, \hat{P}_k] = i \hbar \delta_{jk}, and the inner integral represents a Fourier decomposition over translations in phase space. Equivalently, in one dimension or using complex coordinates in quantum optics contexts, this can be expressed as A^=∫dq dp2πℏ f(q,p) D^(q,p),\hat{A} = \int \frac{\mathrm{d} q \, \mathrm{d} p}{2\pi \hbar} \, f(q, p) \, \hat{D}(q, p), where D^(q,p)=exp⁡[i(pQ^−qP^)/ℏ]\hat{D}(q, p) = \exp\left[ i (p \hat{Q} - q \hat{P}) / \hbar \right] is the displacement (or Weyl-Heisenberg) operator, which generates unitary translations in phase space. This formulation highlights the role of the symplectic Fourier transform, as the quantization kernel arises from the Fourier transform of ff with respect to the symplectic form ω=dq∧dp\omega = \mathrm{d} q \wedge \mathrm{d} p, ensuring the mapping is linear and bijective on suitable function spaces like the Schwartz class. The Weyl quantization is distinguished by its uniqueness among ordering prescriptions, as it guarantees covariance under phase-space translations: if f′(q,p)=f(q−δq,p−δp)f'(q, p) = f(q - \delta q, p - \delta p), then Opw(f′)=T^(δq,δp) Opw(f) T^†(δq,δp)\mathrm{Op}^w(f') = \hat{T}(\delta q, \delta p) \, \mathrm{Op}^w(f) \, \hat{T}^\dagger(\delta q, \delta p), where T^\hat{T} is the unitary translation operator, preserving the symplectic structure without introducing ℏ\hbar-dependent anomalies. For illustration, consider the monomial symbol f(q,p)=qpf(q, p) = q p; under Weyl quantization, it maps to the symmetrized operator A^=12(Q^P^+P^Q^)\hat{A} = \frac{1}{2} (\hat{Q} \hat{P} + \hat{P} \hat{Q}), reflecting the balanced ordering that resolves the classical ambiguity in operator products. This example underscores how Weyl quantization systematically handles bilinear terms, with the inverse recovered via the Wigner transform.

Position Representation and Inverse Mapping

The position representation of a Weyl-quantized operator A^\hat{A} is given by its kernel in the position basis, which connects the phase-space symbol f(q,p)f(q, p) to the matrix elements ⟨x∣A^∣y⟩\langle x | \hat{A} | y \rangle. Specifically, for an nn-dimensional system, ⟨x∣A^∣y⟩=1(2πℏ)n∫dnp eip⋅(x−y)/ℏf(x+y2,p),\langle x | \hat{A} | y \rangle = \frac{1}{(2\pi \hbar)^n} \int d^n p \, e^{i p \cdot (x - y)/\hbar} f\left( \frac{x + y}{2}, p \right), where ℏ\hbar is the reduced Planck's . This expression arises from the Weyl ordering rule, which symmetrizes the classical function f(q,p)f(q, p) in the operator correspondence, ensuring the kernel captures the evaluation in position space. The inverse mapping recovers the phase-space symbol from the operator via the Wigner transform, confirming the bijectivity of the correspondence between symbols and operators on suitable function spaces. The formula is f(q,p)=1(2πℏ)n∫dnx ⟨q+x/2∣A^∣q−x/2⟩e−ip⋅x/ℏ,f(q, p) = \frac{1}{(2\pi \hbar)^n} \int d^n x \, \langle q + x/2 | \hat{A} | q - x/2 \rangle e^{-i p \cdot x /\hbar}, which integrates the off-diagonal kernel elements centered at position qq and Fourier-transforms them with respect to the relative coordinate xx. This relation demonstrates that the Wigner–Weyl transform is invertible, mapping operators to unique symbols and vice versa, provided ff is sufficiently smooth (e.g., in the Schwartz space). In one dimension, the normalization often uses h=2πℏh = 2\pi \hbar to reflect the elementary phase-space volume. The Stratonovich–Weyl correspondence generalizes this framework beyond the flat phase space of R2n\mathbb{R}^{2n} to arbitrary symplectic manifolds or non-Cartesian bases, employing coherent states to define operator kernels. In this extension, the kernel is constructed using an overcomplete set of coherent states ∣ϕz⟩|\phi_z\rangle, where zz parameterizes the , leading to a symbol f(z)=∫⟨ϕz∣A^∣ϕz⟩K(z)dμ(z)f(z) = \int \langle \phi_z | \hat{A} | \phi_z \rangle K(z) d\mu(z) with a reproducing kernel K(z)K(z) that ensures unitarity and trace preservation. This approach maintains the bijective mapping while accommodating curved geometries, such as those in groups or settings.

Key Properties

Quantization of Polynomials

The Weyl quantization maps classical polynomials on to quantum operators via symmetric ordering, ensuring a balanced treatment of non-commuting position QQ and PP operators satisfying [Q,P]=iℏ[Q, P] = i\hbar. This approach, introduced by in 1927 as a means to associate phase-space functions to operators, resolves the operator-ordering ambiguity inherent in quantizing monomials like qmpnq^m p^n by averaging over all possible permutations of the factors. For a qmpnq^m p^n, the Weyl quantization Opw(qmpn)\mathrm{Op}^w(q^m p^n) is defined as the symmetrized product: Opw(qmpn)=1(m+n)!∑σ∈Sm+nQmσPnσ,\mathrm{Op}^w(q^m p^n) = \frac{1}{(m+n)!} \sum_{\sigma \in S_{m+n}} Q^{m_\sigma} P^{n_\sigma}, where the sum runs over all (m+n)!(m+n)! permutations σ\sigma of mm position factors and nn factors, though the multinomial coefficient (m+nm)\binom{m+n}{m} more precisely normalizes the average over the distinct orderings. This symmetric rule produces a unique operator for each and preserves hermiticity when the classical symbol is real-valued. An equivalent explicit expansion, known as McCoy's , expresses the quantization in terms of nested commutators and binomial sums: Opw(qmpn)=∑k=0min⁡(m,n)m! n!(m−k)! (n−k)! 2kk! qm−kpn−k(iℏ)k,\mathrm{Op}^w(q^m p^n) = \sum_{k=0}^{\min(m,n)} \frac{m! \, n! }{ (m-k)! \, (n-k)! \, 2^k k! } \, q^{m-k} p^{n-k} (i \hbar)^k, providing a recursive way to compute the operator by incorporating the iteratively; this was derived by N. R. McCoy in as a correspondence between classical and quantum functions. A key result concerning quantization is Moyal's theorem, which establishes an exact correspondence between classical s and quantum commutators for polynomials of total degree at most 2 under Weyl ordering. Specifically, if ff and gg are such polynomials, then [Opw(f),Opw(g)]=iℏOpw({f,g}),[\mathrm{Op}^w(f), \mathrm{Op}^w(g)] = i\hbar \mathrm{Op}^w(\{f, g\}), where {f,g}\{f, g\} is the ; this holds because higher-order terms in the expansion vanish for low-degree polynomials. For higher degrees, ordering ambiguities arise, as the Weyl map introduces non-trivial corrections from commutators, though the correspondence remains asymptotic in ℏ\hbar. This theorem, articulated in J. E. Moyal's formulation of phase-space , is particularly relevant for quadratic Hamiltonians like p2/2mp^2/2m plus potential V(q)V(q), where Opw(p2/2m+V(q))=P2/2m+V(Q)\mathrm{Op}^w(p^2/2m + V(q)) = P^2/2m + V(Q) exactly, facilitating Ehrenfest's theorem. As an illustrative example, consider the monomial q2pq^2 p: its Weyl quantization is the average of the three possible orderings, Opw(q2p)=13(Q2P+QPQ+PQ2)=Q2P−iℏQ,\mathrm{Op}^w(q^2 p) = \frac{1}{3} (Q^2 P + Q P Q + P Q^2) = Q^2 P - i\hbar Q, where the commutators [Q,P]=iℏ[Q, P] = i\hbar shift the pure symmetric form, demonstrating how non-commutativity modifies the classical expression. This explicit computation highlights the systematic nature of symmetric ordering for practical applications in quantum Hamiltonians.

Quantization of General Functions

The Weyl quantization scheme, initially developed for polynomial symbols, extends naturally to more general phase-space functions through an operator-valued integral representation. For a symbol f(q,p)f(q, p) belonging to the L2(R2n)L^2(\mathbb{R}^{2n}) space or the S(R2n)\mathcal{S}(\mathbb{R}^{2n}) of rapidly decreasing smooth functions, the corresponding quantum operator A^\hat{A} is defined as A^=1(2πℏ)n∫R2nf(q,p) Δ(q,p) dq dp,\hat{A} = \frac{1}{(2\pi \hbar)^n} \int_{\mathbb{R}^{2n}} f(q, p) \, \Delta(q, p) \, dq \, dp, where Δ(q,p)=ei(pQ^−qP^)/ℏ\Delta(q, p) = e^{i (p \hat{Q} - q \hat{P})/\hbar} denotes the unitary associated with the Heisenberg-Weyl group, acting on the L2(Rn)L^2(\mathbb{R}^n). This formulation provides a rigorous mapping from classical observables to self-adjoint operators, preserving key structures like the trace and preserving the L2L^2 norm in the sense that the Weyl map is unitary between L2(R2n)L^2(\mathbb{R}^{2n}) and the Hilbert-Schmidt operators on L2(Rn)L^2(\mathbb{R}^n). In the framework of pseudodifferential operator theory, the Weyl quantization corresponds to a specific ordering prescription in the symbol calculus, where the symbol f(q,p)f(q, p) is interpreted as the Weyl symbol of A^\hat{A}, distinct from other orderings like normal or antinormal forms. This interpretation arises from the oscillatory integral kernel of the operator, given by (A^ψ)(x)=1(2πℏ)n∫R2nei(x−y)⋅η/ℏf(x+y2,η)ψ(y) dy dη,(\hat{A} \psi)(x) = \frac{1}{(2\pi \hbar)^n} \int_{\mathbb{R}^{2n}} e^{i (x - y) \cdot \eta / \hbar} f\left( \frac{x + y}{2}, \eta \right) \psi(y) \, dy \, d\eta, which embodies the midpoint (Weyl) ordering and facilitates analysis of operator composition via the in . Convergence of the requires the ff to be bounded or sufficiently rapidly decreasing to ensure the operator is well-defined and bounded on L2(Rn)L^2(\mathbb{R}^n); for symbols in the , the resulting operator inherits similar decay properties, mapping S\mathcal{S} isomorphically onto the space of Schwartz-class pseudodifferential operators. For broader classes, such as tempered distributions S′(R2n)\mathcal{S}'(\mathbb{R}^{2n}), the quantization extends continuously, yielding unbounded operators that are still densely defined, though careful regularization is needed to handle oscillatory behavior and ensure trace-class properties when applicable. A representative example is the Gaussian symbol f(q,p)=1πe−(q2+p2)f(q, p) = \frac{1}{\pi} e^{-(q^2 + p^2)} (in units where ℏ=1\hbar = 1), which quantizes to the rank-one onto a Gaussian wavepacket state, specifically the operator ∣ψ0⟩⟨ψ0∣|\psi_0\rangle\langle\psi_0| for the of the , ψ0(x)=π−1/4e−x2/2\psi_0(x) = \pi^{-1/4} e^{-x^2/2}. This illustrates how smooth, positive-definite symbols yield positive operators, aligning with physical interpretations in and coherent state theory.

Moyal Product and Bracket

The Moyal star product, also known as the , arises naturally in the Wigner–Weyl formalism as the phase-space representation of the operator product under Weyl quantization. It provides a non-commutative multiplication on functions over that corresponds to the composition of the associated quantum operators. For smooth functions f(q,p)f(q, p) and g(q,p)g(q, p) on the , the is defined as f⋆g(q,p)=f(q,p)exp⁡(iℏ2(∂q←∂p→−∂p←∂q→))g(q,p),f \star g (q, p) = f(q, p) \exp\left( \frac{i \hbar}{2} \left( \overleftarrow{\partial_q} \overrightarrow{\partial_p} - \overleftarrow{\partial_p} \overrightarrow{\partial_q} \right) \right) g(q, p),
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