Bopp Pseudodifferential Operators

• Bopp pseudodifferential operators are a phase-space formulation of quantum mechanics that recast canonical quantization into a pseudodifferential calculus linked to the Moyal star product.
• They utilize Bopp shifts and a harmonic representation to preserve canonical commutation relations and mirror the properties of the Weyl calculus within a symplectically covariant framework.
• Their formalism unifies deformation quantization and the analysis of mixed quantum states by providing a consistent method to construct phase-space density operators.

Bopp pseudodifferential operators constitute a phase-space formulation of quantum mechanics, originating from Fritz Bopp's 1956 construction. These operators promote the canonical quantization rules to a pseudodifferential calculus on phase-space functions, establishing a direct link to the Moyal star product and giving a symplectically covariant framework that unifies deformation quantization with the analysis of mixed quantum states via density operators. Their formalism and properties mirror those of the Weyl calculus but operate intrinsically on functions over phase space rather than configuration space (Gosson, 21 Nov 2024).

1. Bopp Shifts and Canonical Quantization in Phase Space

The essential innovation of Bopp operators is the use of shifted action on phase-space functions $Y(z)$ with $z = (x, p) \in \mathbb{R}^{2n}$. In contrast to the standard Schrödinger representation,

$$x_j \mapsto x_j,\qquad p_j \mapsto -i\hbar\partial_{x_j},$$

the Bopp shifts assign

$$x^B_j := x_j + \frac{i\hbar}{2}\partial_{p_j},\qquad p^B_j := p_j - \frac{i\hbar}{2}\partial_{x_j},$$

when applied to phase-space functions. The commutation relation

$[x^B_j,\, p^B_k] = i\hbar\, \delta_{jk}$

demonstrates that the canonical structure is preserved, so the Bopp operators provide a direct phase-space realization of the Heisenberg algebra. This makes possible a full translation of standard operator methods to the phase-space setting.

2. The Bopp Pseudodifferential Calculus and Harmonic Representation

Given a phase-space symbol $a(z)$, the corresponding Bopp operator acts by functional calculus: $\OpB(a) = a(x^B, p^B) = a\left(x + \frac{i\hbar}{2}\partial_p,\; p - \frac{i\hbar}{2}\partial_x\right).$ This operator may be realized equivalently via the harmonic (or integral) representation: $\OpB(a)Y(z) = (2\pi\hbar)^{-n} \int_{\mathbb R^{2n}} \F_\sigma a(z_0)\; T(z_0)Y(z)\;dz_0,$ where $\F_\sigma a$ is the symplectic Fourier transform of $a$: $\F_\sigma a(z_0) = \int e^{-\frac{i}{\hbar} \sigma(z, z_0)} a(z)\, dz,$ and $T(z_0)$ implements the Bopp displacement: $$T(z_0)Y(z) = e^{\frac{i}{\hbar} \sigma(z, z_0)} Y(z - z_0).$$ With $\sigma((x, p), (x', p')) = p \cdot x' - x \cdot p'$ as the symplectic form, this framework directly parallels the integral kernel structure of the Weyl quantization, but is adapted to the $2n$-dimensional phase space.

The operator $\OpB(a)$ can itself be interpreted as a Weyl operator on phase space with symbol $\tilde{a}(z, \zeta) = a(z - \tfrac{1}{2} J\zeta)$, where $$J = \left(\begin{array}{cc} 0 & I \ -I & 0 \end{array}\right)$$.

3. Composition, Moyal Star Product, and Wavepacket Transform

A crucial structural property is the unitary equivalence of Bopp and configuration-space Weyl operators, established via the wavepacket (cross-Wigner) transform: $$U_\phi \psi(z) = (2\pi\hbar)^{-n/2} \int_{\mathbb R^n} e^{-\frac{i}{\hbar} p \cdot y} \psi\left(x + \frac{y}{2}\right) \overline{\phi\left(x - \frac{y}{2}\right)}\, dy.$$ For any symbol $a$,

$\OpB(a)\, U_\phi = U_\phi \, \OpW(a),$

where $\OpW(a)$ denotes Weyl quantization on configuration space. This intertwining leads to algebraic mirroring of composition: $\OpB(a)\OpB(b) = \OpB(a *_\hbar b),$ with $a *_\hbar b$ the Moyal product given by: $$(a *_\hbar b)(z) = (2\pi\hbar)^{-2n} \iint e^{\frac{2i}{\hbar}\sigma(u, v)} a(z + u)\, b(z + v)\, du\, dv,$$ and, by its power-series expansion,

$$a *_\hbar b = \sum_{r=0}^\infty \frac{1}{r!}\left(\frac{i\hbar}{2}\right)^r P_r(a, b),$$

where $P_1(a, b) = \{a, b\}$ is the Poisson bracket and $P_r$ its $r$th iterated power. Notably,

$\OpB(a) Y = a *_\hbar Y,$

so Bopp operators act as convolution with respect to the Moyal star product.

4. Density Operators in the Bopp Formalism

In quantum mechanics, trace-class density operators $\rho$ on $L^2(\mathbb R^n)$ possess Weyl symbols

$\rho = \OpW\big( (2\pi\hbar)^{-n} \, \varrho(z) \big),\quad \varrho(z) = \sum_j \lambda_j W\psi_j(z),$

with $\{ \psi_j \}$ an orthonormal set and $W\psi_j$ the associated Wigner functions. The Bopp pseudodifferential operator associated to the same symbol is

$\rho^B := \OpB\bigl( (2\pi\hbar)^{-n} \varrho \bigr).$

While $\rho^B$ is not of trace class on all of $L^2(\mathbb R^{2n})$, its restriction to the phase-space image $\mathcal H_\phi = \Im U_\phi$ is a bona fide density operator: $$\rho^B_{\phi} = \sum_j \lambda_j | U_\phi \psi_j \rangle \langle U_\phi \psi_j |,$$ with positivity, trace one, and spectral equivalence to $\rho$ preserved. This establishes a bridge between phase-space quantizations and the operator framework for mixed quantum states.

5. Deformation Quantization and Mixed-State Evolution

Bopp operators realize deformation quantization directly at the level of density matrices and their action on phase-space distributions. When acting on a test function $Y = U_\phi \chi$, the operator $\rho^B$ produces a deformation-quantized Liouville evolution: $$\rho^B Y = \sum_{r=0}^\infty \frac{1}{r!} P_r(\varrho, W(\chi, \phi)),$$ where $W(\chi,\phi)$ is the cross-Wigner distribution and $P_r$ denotes the $r$-fold iterated Poisson bracket. Thus, the entire dynamics of mixed quantum states is expressed in terms of phase-space Poisson algebra and deformation quantization, demonstrating the full structural unity of Bopp calculus, star products, and mixed-state quantum theory.

6. Structural Summary and Mathematical Significance

Bopp pseudodifferential operators supply the phase-space counterpart to the Weyl quantization, distinguished algebraic properties through the Moyal star product, and a unitarily intertwined structure via the wavepacket transform. Their application to density operators yields positive, normalized, and spectrally equivalent phase-space density matrices under restriction, faithfully capturing the quantum statistical structure. This formalism thus unifies Weyl calculus, Moyal star products, deformation quantization, and the analysis of mixed states in a single symplectically covariant framework (Gosson, 21 Nov 2024).