Twisted Product of Functions

• Twisted product of functions is a noncommutative associative convolution that deforms pointwise multiplication using a real antisymmetric matrix θ.
• It employs advanced microlocal techniques, such as the G-wavefront set, to rigorously control the existence and regularity of generalized functions.
• This framework underpins deformation quantization, quantum field theory, and pseudodifferential analysis through its connection with the Moyal star product.

The twisted product of functions, also called the twisted convolution or θ-twisted convolution, is a noncommutative associative product central in noncommutative analysis, deformation quantization, and quantum field theory on noncommutative spaces. For suitable pairs of functions or distributions, it implements a deformation of pointwise multiplication parameterized by a real antisymmetric matrix θ. The twisted product generalizes classical convolution and, via Fourier transform, connects directly to the Moyal–Groenewold star product central to Weyl quantization and noncommutative geometry. Its rigorous definition and analysis require advanced microlocal techniques, especially the G-wavefront set, to control the existence and properties of the product for singular elements such as tempered distributions, ultradistributions, or generalized functions (Bahns et al., 2019, Soloviev, 2012).

1. Formal Definition and Fundamental Properties

Given θ, a real antisymmetric n×n matrix, and tempered distributions $u,v\in \mathcal S'(\mathbb R^n)$, the θ–twisted convolution $u*_\theta v$ is formulated via the Fourier transform $\mathcal F$ and a quadratic "chirp" phase: $$u*_\theta v = \operatorname{diag}^* \Bigl( \mathcal{F}^{-1} \bigl[ (\mathcal{F} u)\otimes(\mathcal{F} v)\,\chi_\theta \bigr] \Bigr)$$ where

$$\chi_\theta(k,p) = e^{-\frac{i}{2}k^T\theta p}, \quad k,p\in\mathbb R^n$$

and $$\operatorname{diag}: \mathbb R^n\to\mathbb R^n\times\mathbb R^n$$ does the pullback $x\mapsto(x,x)$. In classical integral form, when all expressions are defined,

$$(u*_\theta v)(x) = \frac{1}{(2\pi)^n} \int_{\mathbb R^n}\int_{\mathbb R^n} \widehat u(k)\,\widehat v(p)\,e^{-\frac{i}{2}k^T\theta p}\,e^{i(k+p)\cdot x} \,dk\,dp$$

This product is associative and noncommutative except for θ=0, when it coincides with standard convolution (Bahns et al., 2019). On suitable function spaces $E$ with dual $E'$ and a nondegenerate skew-symmetric form σ, one defines

$$(f\,\mathbf{®}\,u)(x) = \int f(y)\,u(x-y)\,e^{2\pi i\,\sigma(y, x-y)}\,dy$$

for $f\in E, u\in E'$ (Soloviev, 2012).

2. Microlocal Conditions for Existence: The G-Wavefront Set

For general distributions, the twisted product is not always defined. Its existence and the regularity of the result are determined by the microlocal singularity structure encoded in the G-wavefront set $WF^G$. The G-wavefront set tracks both local singularities and global growth/oscillation and is characterized via the short-time Fourier transform: $$V_\psi u(x,\xi) = \langle\,u,\; y\mapsto e^{-i y\cdot\xi}\,\overline{\psi(y-x)}\,\rangle$$

A sufficient condition for $u*_\theta v\in \mathcal S'(\mathbb R^n)$ is:

Theorem (Bahns–Schulz, 2019):

Let $(x,\xi)\in WF^G(u), (x,-\xi)\in WF^G(v)$. If the equation $x-\frac{1}{2}\theta\xi=0$ has no such solutions, then $u*_\theta v$ is a well-defined tempered distribution (Bahns et al., 2019).

The wavefront set of the product satisfies

$$WF^G(u*_\theta v) \subset \{ (x+\frac{1}{2}\theta(\xi+\eta),\,\xi+\eta) \mid (x,\xi)\in WF^G(u)\cup\{0\},\ (y,\eta)\in WF^G(v)\cup\{0\},\ y=x+\frac{1}{2}\theta(\xi+\eta) \} \setminus \{0\}$$

This approach yields sequential continuity on subspaces of $\mathcal S'$ with fixed G-wavefront cones.

3. Algebraic Structures and Subspace Algebras

Specific conic subspaces of $\mathcal S'(\mathbb R^n)$ are stable, forming algebras under both pointwise multiplication and $*_\theta$. Two primary families are:

1. Conic cones $I\subset\mathbb R^{2n}\setminus\{0\}$ such that $(x,\xi)\in I\Rightarrow (x,-\xi)\notin I$. The corresponding space

$$\mathcal S'_I(\mathbb R^n) := \{u\in\mathcal S'(\mathbb R^n)\mid WF^G(u)\subset I\}$$

is an algebra.

2. $T_1\subset\mathbb R^n_x\setminus\{0\}$, $T_2\subset\mathbb R^n_\xi\setminus\{0\}$ (closed, conic, $T_2$ stable under addition), with the “θ–twist” condition: $x+\frac{1}{2}\theta\xi\in T_1$ for all $x\in T_1, \xi\in T_2$. Then $\mathcal S'_{T_1\times T_2}(\mathbb R^n)$ is an algebra under $*_\theta$ (Bahns et al., 2019).

Beyond $\mathcal S'$, the framework of twisted convolution and its multiplier algebras extends to Gelfand–Shilov, weighted exponential, and analytic function spaces where the Weyl–Heisenberg group acts continuously (Soloviev, 2012). On such spaces, all ultradistributions with sub-Gaussian growth are convolution multipliers.

4. Connections with the Moyal Star Product and Weyl Correspondence

Via the Fourier transform, the θ–twisted convolution corresponds precisely to the Moyal star product (⋆ₘ) on Schwartz functions: $$(f\star_M g)(x) = \frac{1}{(2\pi)^{2m}} \iint \widehat f(k)\,\widehat g(p)\,e^{-\frac{i}{2}k^T\theta p} \,e^{i(k+p)\cdot x} \,dk\,dp$$ which coincides with $f *_\theta g$ on $\mathcal S(\mathbb R^{2m})$ (Bahns et al., 2019). The Moyal product can also be written as

$$(f\star_M g)(x) = f\,\exp\left[ \frac{i}{2} \overleftarrow{\partial}_i\theta^{ij}\overrightarrow{\partial}_j \right]\,g$$

expanding as a convergent formal series in specific function spaces. In the Weyl quantization scheme, composition of Weyl pseudodifferential operators with symbols $a(x,\xi), b(x,\xi)$ corresponds to $a\star_M b$.

On the level of multiplier algebras, the Fourier image $\mathcal A(\widehat E) = \mathcal F(M(E))$ defines a unital involutive algebra under $\star_M$; it coincides with the intersection of left and right Moyal multipliers on $\widehat E$, and, crucially, this algebraic structure transfers to highly singular distributional and ultradistributional settings (Soloviev, 2012).

5. Functional Analytic Extensions and Decay Conditions

Twisted convolution and the associated Moyal product extend well beyond Schwartz and tempered distribution theory, encompassing a wide hierarchy of nuclear function spaces $E$ and their duals $E'$. For "sufficiently rapidly decreasing" ultradistributions (e.g., with growth slower than a Gaussian), the twisted convolution makes sense and defines algebraic multiplier structures (Soloviev, 2012). Explicitly, in Gelfand–Shilov spaces $S^\beta_\alpha(\mathbb R^d)$ ($\alpha+\beta\geq1$), weighted Beurling or Roumieu symbol spaces, and analytic test function spaces, the full calculus and oscillator product structure remains valid and is governed by convergence criteria derived from the nuclearity and Fourier invariance of the test function space.

6. Applications in Quantum Theory and Pseudodifferential Analysis

The twisted product is fundamental in noncommutative quantum field theory, where it encodes nonlocal interactions and a "fundamental length" via the deformation matrix θ. The formal series

$$(f\star_M g)(x) = \sum_{n=0}^{\infty} \frac{1}{n!} \left(\frac{i}{2}\theta^{\mu\nu}\partial_{x^\mu}\partial_{y^\nu}\right)^n [f(x)g(y)]_{y=x}$$

converges in certain test function spaces (e.g., $P_{1/2}, S^{1/2}_{1/2}$), providing a genuine nonlocal product beyond formal deformation (Soloviev, 2012).

Beyond quantum field theory, the extended twisted convolution product allows for a rigorous pseudodifferential calculus on spaces of ultradistributions, defining operator algebras with explicit asymptotic expansions and calculable products—crucial for analysing differential equations and symbolic structures outside classical regularity frameworks.

7. Summary Table of Core Formulas

Operation	Definition	Domain
θ–Twisted convolution $*_\theta$	$$u*_\theta v = \operatorname{diag}^*( \mathcal{F}^{-1}[ (\mathcal F u)\otimes(\mathcal F v)\chi_\theta ] )$$	$u,v\in\mathcal S'(\mathbb R^n)$, with G-wavefront restriction
Moyal star product $\star_M$	$$(f\star_M g)(x)=\frac{1}{(2\pi)^{2m}}\iint \widehat f(k)\widehat g(p) e^{-\frac{i}{2}k^T\theta p} e^{i(k+p)\cdot x} dk dp$$	$f,g\in\mathcal S(\mathbb R^{2m})$
Oscillatory integral form	$$(f\,\mathbf{®}\,u)(x) = \int f(y)\,u(x-y)\,e^{2\pi i\,\sigma(y,x-y)}\,dy$$	$f\in E$, $u\in E'$

The theory of the twisted product of functions thus provides a robust analytic and microlocal framework for extending noncommutative products to broad classes of generalized functions, with precise control over existence, regularity, and algebraic structure (Bahns et al., 2019, Soloviev, 2012).