Algebras of distributions suitable for
phase-space quantum mechanics. I
January 13, 2026
The twisted product of functions on \(\mathbb{R}^{2N}\) is extended to a \(*\)-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.
-3.25ex 1.5ex Introduction
This is the first of two papers whose aim is to give a rigorous formulation to the Weyl–Wigner–Groenewold–Moyal or phase-space approach to quantum mechanics of spinless, nonrelativistic particles. (In a future article, we will show how spin may be incorporated also in this formalism.) In recent years, this approach has received increasing attention [1]–[5]. However, much remains to be done to unify its different strands. On the one hand, much useful quantum physics can be done using the distribution functions in the sense of Wigner [4]. On the other hand, most mathematical attention has centred on the Weyl operator calculus [6]–[9]. As Groenewold [10] and Moyal [11] have shown, one can work with functions on the classical phase space only, in a self-contained way, using the “twisted product” concept. Similarly, the practitioners of “deformation theory” [2] have given a promising axiomatic basis for quantum mechanics, but with mathematical tools rather different from the usual functional-analytic methods of quantum theory.
We attempt here to establish a mathematically rigorous and physically manageable formulation for quantum mechanics in phase space. To obtain the right mathematical context, we must, for example, specify those pairs of functions whose twisted product may be formed; and a suitable function space should include as many observables of physical interest as possible. To include the basic observables of position and momentum, we must leave aside the algebra of bounded operators on a Hilbert space: in rigorizing the phase-space approach, one soon finds that it is useful to work with locally convex topological vector spaces which are not necessarily Banach spaces.
The paper is organized as follows. In Sec. [sec:sec:Schwartz-algebras], we review the properties of the twisted product and convolution in the Schwartz space \(\mathcal{S}(\mathbb{R}^{2N})\). In Sec. [sec:sec:Moyal-algebra], we dualize these notions to the case where one or both factors are tempered distributions, and identify the Moyal algebra \(\mathcal{M}\), that is, the largest \(*\)-algebra of distributions where these operations are defined and associative. We show that \(\mathcal{M}\) is invariant under Fourier transformations. In Sec. [sec:sec:Moyal-regularity], we consider the regularity properties of the twisted product and convolution, and show that distributions of compact support belong to the Moyal algebra. In Sec. [sec:sec:matrix-basis], we construct an orthonormal basis in \(\mathcal{S}(\mathbb{R}^{2N})\); using this basis, we show that the twisted product may be presented as a matrix product of double sequences. As a consequence, we construct a net of Sobolev-like spaces of tempered distributions, some of which are Banach algebras with respect to the twisted product; these permit a more detailed examination of the Moyal algebra \(\mathcal{M}\).
-3.25ex 1.5ex The algebras (\(\mathcal{S}_2,\times\)) and (\(\mathcal{S}_2,\diamond\))
Throughout this paper, we work with certain spaces of functions and distributions over \(\mathbb{R}^{2N}\), regarded as the phase space \(T^*(\mathbb{R}^N)\). For \(u,v \in \mathbb{R}^{2N}\), we write \(u'v\) and \(u'Jv\) for the ordinary and symplectic scalar products of \(u\) and \(v\). Choosing and fixing an orthonormal symplectic basis for \(\mathbb{R}^{2N}\), we write \[u = (u_1,u_2,\dots,u_{2N}) = (q_1,\dots,q_N,p_1,\dots,p_N)\] and \[v = (v_1,v_2,\dots,v_{2N}) = (\tilde{q}_1,\dots,\tilde{q}_N,\tilde{p}_1,\dots,\tilde{p}_N),\] where explicitly \[u'v := \sum_{i=1}^N (q_i\tilde{q}_i + p_i\tilde{p}_i), \qquad u'Jv := \sum_{i=1}^N (q_i\tilde{p}_i - p_i\tilde{q}_i),\] where \(J\) is the matrix \(\begin{pmatrix} 0 & 1_N \\ -1_N & 0 \end{pmatrix}\) in the chosen basis. Note that \(v'Ju = -u'Jv\) and \(u'Ju = 0\).
We define \(\mathcal{S}_2 := \mathcal{S}(\mathbb{R}^{2N})\) as the Schwartz space of smooth rapidly decreasing functions on \(\mathbb{R}^{2N}\). If \(f \in \mathcal{S}_2\), \(s \in \mathbb{R}^{2N}\) and \(1 \leqslant j \leqslant 2N\), we define \(f^*(u) := \overline{f(u)}\), \(\check f(u) := f(-u)\), and also: \[\begin{align} {2} (\mu_j f)(u) &:= u_j f(u), &\qquad \partial_j f &:= \frac{\partial f}{\partial u_j}, \\ (\tau_sf)(u) &:= f(u - s), &\qquad (\varepsilon_sf)(u) &:= e^{is'Ju}f(u), \end{align}\] and \[\hat{\partial}_j f := \begin{cases} \partial_{j+N} f & \text{if 1 \leqslant j \leqslant N,} \\ -\partial_{j-N} f & \text{if N < j \leqslant 2N.} \end{cases}\]
We make three normalizations which are a little unconventional. First, for integrals over \(\mathbb{R}^{2N}\) we use the Haar measure \(dx := (2\pi)^{-N}\,d^{2N}x\) where \(d^{2N}x\) is Lebesgue measure. (This gets rid of powers of \(2\pi\) in Fourier transforms [12]). In particular, \(\int e^{-x^2/2} \,dx = 1\). Secondly, we use the bilinear form \[\langle f,g\rangle := \int f(x) g(x) \,dx\] and the sesquilinear form \[(f\mathbin|g) := 2^{-N} \langle f^*,g\rangle = 2^{-N} \int \overline{f(x)} g(x) \,dx\] whenever the integrals converge. For \(f \in L^2(\mathbb{R}^{2N})\), we will use the norm \(\|f\| := (f\mathbin|f)^{1/2}\). Thirdly, for Planck’s constant we take \(\hbar = 2\) (rather than the usual \(\hbar = 1\)).
We define an ordinary Fourier transform \(\mathcal{F}\) and two symplectic Fourier transforms \(F\) and \(\widetilde{F}\) [13] by \[(\mathcal{F}f)(u) := \int f(t) e^{-it'u} \,dt, \quad (Ff)(u) := \int f(t) e^{-it'Ju} \,dt, \quad (\widetilde{F}f)(u) := \int f(t) e^{it'Ju} \,dt.\] The transforms \(\mathcal{F}\), \(F\) and \(\widetilde{F}\) are commuting isomorphisms (of Fréchet spaces) of \(\mathcal{S}_2\) onto \(\mathcal{S}_2\), and satisfy the following formulas: \[\begin{align} {2} Ff(u) &= \mathcal{F}f(Ju), &\quad \widetilde{F}f(u) &= \mathcal{F}f(-Ju), \\ F^2 &= \widetilde{F}^2 = \text{Id}, &\quad \widetilde{F}f &= (Ff)^\vee = F(\check f), \quad (Ff)^* = \widetilde{F}(f^*), \\ F(\tau_s f) &= \varepsilon_{-s} Ff, &\quad F(\varepsilon_s f) &= \tau_{-s} Ff, \\ F(\hat{\partial}_j f) &= -i\mu_j Ff, &\quad F(\mu_j f) &= i\hat{\partial}_j Ff, \\ \duo{Ff,g} &= \langle f,\widetilde{F}g,,\rangle &\quad (Ff\mathbin|g) &= (f\mathbin|Fg). \end{align}\]
Definition 1. If \(f,g \in \mathcal{S}_2\), the twisted product \(f \times g\) is defined by \[\begin{align} (f \times g)(u) &:= \iint f(v) g(w) \exp\bigl( i(u'Jv + v'Jw + w'Ju) \bigr) \,dv \,dw \nonumber \\ &= \iint f(u + s) g(u + t) \,e^{is'Jt} \,ds \,dt. \label{eq:twisted-product} \end{align}\tag{1}\] The twisted convolution \(f \diamond g\) is defined by \[(f \diamond g)(u) := \int f(u - t) g(t) \,e^{-iu'Jt} \,dt. \label{eq:twisted-colvoln}\tag{2}\]
Remark 1. It was von Neumann [14] who introduced the twisted convolution (although he gave it no name) in order to establish the uniqueness of the Schroedinger representation. It has been used by Kastler and others [13], [15], [16] to study the canonical commutation relations.
Remark 2. The twisted product is nothing but the Weyl functional calculus [7] seen from another point of view. As in [1] and elsewhere, one may regard it as \(f \times g = \mathcal{F}^{-1}(\mathcal{F}f \diamond \mathcal{F}g)\); but perhaps a more natural motivation is the following. The pointwise product \(f(u)g(u)\) is not suitable for quantum mechanics since the uncertainty principle forbids localization at a point in phase space. Following Sławianowski [17], we seek to replace it by some other product which is translation and symplectic equivariant, associative, and nonlocal. In [3] it is shown that the only integral kernels satisfying translation and symplectic equivariance and associativity are \(a\delta(s)\delta(t)\) – for the pointwise product – and \(be^{ics'Jt}\), where \(a,b,c\) are constants which we may set equal to \(1\).
Proposition 1. If \(f,g \in \mathcal{S}_2\), then \(f \times g \in \mathcal{S}_2\), the map \((f,g) \mapsto f \times g\) is a continuous bilinear operation on \(\mathcal{S}_2\), and \[\begin{align} \partial_j(f \times g) &= \partial_j f \times g + f \times\partial_j g; \nonumber \\ \mu_j(f \times g) &= f \times\mu_j g + i\hat{\partial}_j f \times g = \mu_j f \times g - if \times\hat{\partial}_j g. \label{eq:Leibniz-rules} \end{align}\qquad{(1)}\]
Proof. The Leibniz formula follows by differentiating 1 under the integral sign, and ?? is a straightforward calculation. By induction on these formulas, \(f \times g\) lies in \(\mathcal{S}_2\). If \(\alpha= (\alpha_1,\dots,\alpha_{2N}) \in \mathbb{N}^{2N}\), we write \(\partial^\alpha= \partial_1^{\alpha_1} \cdots \partial_{2N}^{\alpha_{2N}}\) and similarly define \(\mu^\alpha\), \(\hat{\partial}^\alpha\). Then \[\mu^\alpha\,\partial^\gamma(f \times g) = \sum_{\beta\leqslant\alpha} \sum_{\varepsilon\leqslant\gamma} (-i)^{|\beta|} \binom{\alpha}{\beta} \binom{\gamma}{\varepsilon} \mu^{\alpha-\beta} \,\partial^{\gamma-\varepsilon} f \times\hat{\partial}^\beta\,\partial^\varepsilon g.\] From 1 we get \(\|f \times g\|_\infty \leqslant\|f\|_1 \,\|g\|_1\). Since the topology of \(\mathcal{S}_2\) is given by the seminorms \(p_{\alpha\gamma}(f) := \|\mu^\alpha\,\partial^\gamma f\|_\infty\) or by \(q_{\alpha\gamma}(f) := \|\mu^\alpha\,\partial^\gamma f\|_1\), the estimates \[p_{\alpha\gamma}(f \times g) \leqslant\sum_{\beta\leqslant\alpha} \sum_{\varepsilon\leqslant\gamma} \binom{\alpha}{\beta} \binom{\gamma}{\varepsilon} q_{\alpha-\beta,\gamma-\varepsilon}(f) q_{0,\eta+\varepsilon}(g),\] with \(\eta_j = \beta_{j\pm N}\) for all \(j\), show that \((f,g) \mapsto f \times g\) is jointly continuous for the topology of \(\mathcal{S}_2\). ◻
The various Fourier transforms intertwine \(\times\) and \(\diamond\), just as with “ordinary” products and convolutions. In fact, even more is true: by applying a symplectic Fourier transform to one side only, we can interchange the operations \(\times\) and \(\diamond\). This allows us to work with the operation most convenient to any particular calculation, transferring the result to the other one by Fourier-invariance of \(\mathcal{S}_2\). Explicitly, we find: \[f \times g = Ff \diamond g = f \diamond \widetilde{F}g, \qquad f \diamond g = Ff \times g = f \times\widetilde{F}g, \label{eq:Fourier-switch}\tag{3}\] since, for example, \[\begin{align} (f \times g)(u) &= \iint f(v) g(w) \,e^{-iv'J(u-w)} e^{iw'Ju} \,dv \,dw \\ &= \int Ff(u - w) g(w) \,e^{-iu'Jw} \,dw = (Ff \diamond g)(u). \end{align}\] We also find \[\mathcal{F}(f \times g) = \mathcal{F}f \diamond \mathcal{F}g, \qquad \mathcal{F}(f \diamond g) = \mathcal{F}f \times\mathcal{F}g, \label{eq:Fourier-transfer}\tag{4}\] and exactly analogous formulas with \(\mathcal{F}\) replaced by \(F\) or \(\widetilde{F}\). Also, \[(f \times g) \times h = f \times(g \times h), \qquad (f \diamond g) \diamond h = f \diamond (g \diamond h), \label{eq:twisted-assoc}\tag{5}\] since \[\begin{align} ((f \diamond g) \diamond h)(u) &= \iint f(u - t - s) g(s) h(t) \,e^{-i(u'Jt+(u-t)'Js)} \,ds \,dt \\ &= \iint f(u - v) g(v - t) h(t) \,e^{-i(u'Jv-t'Jv)} \,dt \,dv \\ &= (f \diamond (g \diamond h))(u) \end{align}\] and applying 4 yields the associativity of \(\times\). Next, \[(f \times g)^* = g^* \times f^*, \qquad (f \diamond g)^* = g^* \diamond f^*,\] since, for instance, \[\begin{align} (f \times g)^*(u) &= \iint f^*(u + s) g^*(u + t) \,e^{-is'Jt} \,ds \,dt \\ &= \iint g^*(u + t) f^*(u + s) \,e^{it'Js} \,dt \,ds \\ &= (g^* \times f^*)(u). \end{align}\]
A fact of fundamental importance is the following identity.
Proposition 2. If \(f,g \in \mathcal{S}_2\), then \[\int (f \times g)(u) \,du = \int (g \times f)(u) \,du = \int f(u) g(u) \,du. \label{eq:tracial-property}\qquad{(2)}\]
Proof. \[\begin{align} \int (f \times g)(u) \,du &= \mathcal{F}(f \times g)(0) = (\mathcal{F}f \diamond \mathcal{F}g)(0) = \int \mathcal{F}f(-t) \mathcal{F}g(t) \,dt \\ &= (\mathcal{F}f * \mathcal{F}g)(0) = \mathcal{F}(fg)(0) = \int f(u) g(u) \,du \end{align}\] where \(*\) denotes ordinary convolution. ◻
The cyclicity inherent in the tracial identity ?? is what allows us to push through the extension via duality. We note an important consequence of ?? .
Proposition 3. If \(f,g,h \in \mathcal{S}_2\), then \[\begin{align} \langle f \times g,h\rangle &= \langle f,g \times h\rangle = \langle g,h \times f\rangle; \label{eq:product-transfer} \\ \langle f \diamond g,h\rangle &= \langle f,\check g \diamond h\rangle = \langle g,h \diamond \check f\rangle; \label{eq:convol-transfer} \\ (h\mathbin|f \times g) &= (f^* \times h\mathbin|g) = (h \times g^*\mathbin|f). \nonumber \end{align}\] {#eq: sublabel=eq:eq:product-transfer,eq:eq:convol-transfer}
Proof. From ?? and the associativity 5 , we find that all three expressions in ?? are equal to \(\int (f \times g \times h)(u) \,du\). Now ?? follows from 3 , and the third formula is immediate. ◻
-3.25ex 1.5ex Duality and the Moyal algebra
Having established a calculus for functions in \(\mathcal{S}_2\) with twisted product and convolution, we now extend it to a larger algebra of tempered distributions. First we consider the twisted product or convolution of a tempered distribution and a test function.
For \(T \in \mathcal{S}_2'\), \(h \in \mathcal{S}_2\), we write \(\langle T,h\rangle := T(h)\). We also extend our previous notations in the usual way: \[\begin{align} \langle\mathcal{F}T,h\rangle &:= \langle T,\mathcal{F}h\rangle, \quad \langle\check T,h\rangle := \langle T,\check h\rangle, \\ (T\mathbin|h) &:= 2^{-N}\langle T^*,h\rangle := 2^{-N}\langle T,h^*\rangle^*, \\ \langle FT,h\rangle &:= \langle T,\widetilde{F}h\rangle, \quad \langle\widetilde{F}T,h\rangle := \langle T,Fh\rangle, \\ \langle\partial_j T,h\rangle &:= -\langle T,\partial_j h,,\rangle \quad \langle\mu_j T,h\rangle := \langle T,\mu_j h\rangle. \end{align}\]
Definition 2. For \(T \in \mathcal{S}_2'\), \(f,h \in \mathcal{S}_2\), we define \(T \times f\), \(f \times T\), \(T \diamond f\) and \(f \diamond T\) in \(\mathcal{S}_2'\) by \[\begin{align} {2} \langle T \times f,h\rangle &:= \langle T,f \times h\rangle, &\quad \langle f \times T,h\rangle &:= \langle T,h \times f\rangle, \label{eq:transposed-product} \\ \langle T \diamond f,h\rangle &:= \langle T,\check f \diamond h,,\rangle &\quad \langle f \diamond T,h\rangle &:= \langle T,h \diamond \check f\rangle. \nonumber \end{align}\tag{6}\] The continuity of \(\times\) and \(\diamond\) in \(\mathcal{S}_2\) implies that each right hand side is continuous and linear in \(h\) and thus defines an element of \(\mathcal{S}_2'\). By ?? and ?? , these are extensions of the corresponding operations on \(\mathcal{S}_2\).
Throughout this paper, every dual space \(E'\) of a locally convex space \(E\) is topologized by the strong dual topology, that of uniform convergence on bounded subsets of \(E\). Then the four bilinear maps \(\colon \mathcal{S}_2' \times\mathcal{S}_2 \to \mathcal{S}_2'\) defined above are hypocontinuous. Indeed, since \(\mathcal{S}_2\) and \(\mathcal{S}_2'\) are barrelled it suffices [18], [19] to check separate continuity. For example, for fixed \(T\), the map \(f \mapsto \langle T,f \times h\rangle\) is continuous, uniformly so for \(h\) in a bounded subset of \(\mathcal{S}_2\) (by the joint continuity of \(f\) and \(h\)), so that \(f \mapsto T \times f\) is continuous from \(\mathcal{S}_2\) to \(\mathcal{S}_2'\). For fixed \(f\), \(T \mapsto T \times f\) is the transpose of the continuous map \(h \mapsto f \times h\) of \(\mathcal{S}_2\) into \(\mathcal{S}_2\), and as such is continuous from \(\mathcal{S}_2'\) to \(\mathcal{S}_2'\).
All formulas of Sec. [sec:sec:Schwartz-algebras] involving \(f\) and \(g\) extend to analogous formulae for \(T\) and \(f\) (e.g., \(T \times f = FT \diamond f = T \diamond \widetilde{F}f\)). This is easily checked since \(A = B\) in \(\mathcal{S}_2'\) iff \(\langle A,h\rangle = \langle B,h\rangle\) for all \(h \in \mathcal{S}_2\), and we may reduce to the \(\mathcal{S}_2\) case using 6 .
We write \(\mathbb{1}\) for the constant function with value \(1\), and \(\delta\) for the Dirac measure of mass one supported at \(0\). These are the identities for the operations \(\times\) and \(\diamond\): \[\mathbb{1}\times f = f \times\mathbb{1}= f, \qquad \delta\diamond f = f \diamond \delta= f.\] This follows from \[\langle\mathbb{1}\times f,h\rangle = \langle\mathbb{1},f \times h\rangle = \int (f \times h)(u) \,du = \langle f,h\rangle\] by ?? : thus \(\mathbb{1}\times f = f\) as elements of \(\mathcal{S}_2'\). We show below that \(\mathbb{1}\times f\) is continuous, so \(\mathbb{1}\times f = f\) as functions in \(\mathcal{S}_2\). The other half of the equation follows from 4 , since \(F\delta= \widetilde{F}\delta= \mathcal{F}\delta= \mathbb{1}\). From 3 we also obtain the formulas \[\mathbb{1}\diamond f = \delta\times f = \widetilde{F}f, \qquad f \diamond \mathbb{1}= f \times\delta= Ff. \label{eq:Fourier-cross}\tag{7}\] Using the fact that \(u_j = \mu_j\mathbb{1}\), we obtain from ?? the important identities \[u_j \times f = \mu_j f + i\,\hat{\partial}_j f, \qquad f \times u_j = \mu_j f - i\,\hat{\partial}_j f,\] which in the \((q,p)\) notation become \[\begin{align} {2} q_j \times f &= \biggl( q_j + i\frac{\partial}{\partial p_j} \biggr) f, &\qquad p_j \times f &= \biggl( p_j - i\frac{\partial}{\partial q_j} \biggr) f, \nonumber \\[\jot] f \times q_j &= \biggl( q_j - i\frac{\partial}{\partial p_j} \biggr) f, &\qquad f \times p_j &= \biggl( p_j + i\frac{\partial}{\partial q_j} \biggr) f. \label{eq:Weyl-operators} \end{align}\tag{8}\]
If \(T \in \mathcal{S}_2'\) and \(f \in \mathcal{S}_2\), the ordinary convolution \(T*f\) is [18], [20] a smooth function in \(\mathcal{O}_C\), whereas the ordinary product \(Tf\) is a “rapidly decreasing distribution” in \(\mathcal{O}_C'\) but need not be smooth. In contrast, 4 – extended to \(\mathcal{S}_2'\) – shows that the twisted product and twisted convolution have similar properties of smoothness and of growth at infinity. To see this, we first note that 2 may be rewritten as \[(f \diamond g)(u) = \langle\varepsilon_{-u} \tau_u \check f,g\rangle = \langle f,\varepsilon_u \tau_u \check g\rangle.\] Thus, in convolution formulas such as \((T*f)(u) = \langle T,\tau_u \check f\rangle\), the translations \(\tau_u\) are replaced by \(\varepsilon_u \tau_u\) or \(\varepsilon_{-u} \tau_u\).
Theorem 1. If \(T \in \mathcal{S}_2'\), \(f \in \mathcal{S}_2\), then \(T \times f\), \(f \times T\), \(T \diamond f\) and \(f \diamond T\) are smooth functions on \(\mathbb{R}^{2N}\), given by \[\begin{align} (T \times f)(u) &= \langle T,\varepsilon_u \tau_u Ff\rangle, \nonumber \\ (f \times T)(u) &= \langle T,\varepsilon_{-u} \tau_u\widetilde{F}f\rangle \label{eq:product-formula} \\ (T \diamond f)(u) &= \langle T,\varepsilon_u \tau_u\check f\rangle, \nonumber \\ (f \diamond T)(u) &= \langle T,\varepsilon_{-u} \tau_u\check f\rangle. \label{eq:convol-formula} \end{align}\] {#eq: sublabel=eq:eq:product-formula,eq:eq:convol-formula}
Proof. If \(h \in \mathcal{S}_2\), the maps \(u \mapsto \tau_u h\), \(u \mapsto \varepsilon_u h\) are continuous from \(\mathbb{R}^{2N}\) to \(\mathcal{S}_2\), so the right hand sides of these formulas are jointly continuous in \(u\) and \(f\). By transposition, since \(\langle T,\varepsilon_u \tau_u\check f\rangle = \langle\varepsilon_{-u} \tau_u \check T,f\rangle\), they are also continuous in \(T\). These right hand sides define separately continuous extensions to \(\mathcal{S}_2' \times\mathcal{S}_2\) of the twisted product and convolution on the dense subspace \(\mathcal{S}_2 \times\mathcal{S}_2\); since these extensions are necessarily unique, they coincide with \(T \times f\), etc., as defined earlier.
Now \[\partial_j h(u) = \pm \lim_{c\to 0} c^{-1} \bigl( h(u + ce_j) - h(u) \bigr),\] where \(e_k\) is the \(k\)th basis vector in \(\mathbb{R}^{2N}\), whenever this limit exists. By calculation, we find that \[\lim_{c\to 0} c^{-1} \langle T,\varepsilon_{u+ce_j} \tau_{u+ce_j} Ff - \varepsilon_u \tau_u Ff\rangle = (\partial_j T \times f)(u) + (T \times\partial_j f)(u)\] as expected, so by induction \(T \times f\) is infinitely differentiable, with the Leibniz formula \(\partial_j(T \times f) = \partial_j T \times f + T \times\partial_j f\) holding as an equality between smooth functions. The other three cases are similar. ◻
Let \(\mathcal{E}_2\) denote the space of smooth functions on \(\mathbb{R}^{2N}\), with the topology of uniform convergence of all derivatives on compact sets. Ordinary convolution operators are precisely those which commute with translations; we may characterize the twisted convolution operators as those which commute with “twisted translations”.
Theorem 2. Let \(L \colon \mathcal{S}_2 \to \mathcal{E}_2\) be linear and continuous; then there is a unique \(T \in \mathcal{S}_2'\) with \(L(f) = T \times f\) for all \(f \in \mathcal{S}_2\) iff \(L\) commutes with \(\{\,\varepsilon_u \tau_u : u \in \mathbb{R}^{2N}\,\}\).
Proof. From ?? we find that \[\begin{align} \varepsilon_v \tau_v(T \times f)(u) &= e^{iv'Ju} (T \times f)(u - v) = e^{iv'Ju} \langle T,\varepsilon_{u-v} \tau_{u-v} Ff\rangle \\ &= e^{iv'Ju} \langle T,t \mapsto e^{i(u-v)'Jt} Ff(t - u + v)\rangle \\ &= \langle T,t \mapsto e^{i(u'Jt - v'J(t-u))} \varepsilon_{-v} Ff(t - u)\rangle \\ &= \langle T,t \mapsto e^{iu'Jt} \tau_{-v} \varepsilon_{-v} Ff(t - u)\rangle = \langle T,\varepsilon_u \tau_u F(\varepsilon_v \tau_v f)\rangle \end{align}\] so that \(f \mapsto T \times f\) commutes with any \(\varepsilon_v \tau_v\).
On the other hand, given \(L \colon \mathcal{S}_2 \to \mathcal{E}_2\) which commutes with all \(\varepsilon_v \tau_v\), we define \(T \in \mathcal{S}_2'\) by \(\langle T,h\rangle := L(Fh)(0)\). (So \(T\) is unique.) For \(u \in \mathbb{R}^{2N}\), \(f \in \mathcal{S}_2\), we then obtain \[\begin{align} (T \times f)(u) &= \langle T,\varepsilon_u \tau_u Ff,=\rangle \langle T,F(\varepsilon_{-u} \tau_{-u}f)\rangle \\ &= L(\varepsilon_{-u} \tau_{-u}f)(0) = \varepsilon_{-u} \tau_{-u} (Lf)(0) \\ &= \tau_{-u}(Lf)(0) = (Lf)(u). \tag*{\qed} \end{align}\] ◻
We can now define the Moyal \(*\)-algebra \(\mathcal{M}\). We define it as the intersection of two spaces \(\mathcal{M}_L\) and \(\mathcal{M}_R\) which, to the best of our knowledge, were first considered by Antonets [21].
Definition 3.
\(\mathcal{M}_L := \{\,S \in \mathcal{S}_2' : S \times f \in \mathcal{S}_2 \text{ for all } f \in \mathcal{S}_2\,\}\);
\(\mathcal{M}_R := \{\,R \in \mathcal{S}_2' : f \times R \in \mathcal{S}_2 \text{ for all } f \in \mathcal{S}_2\,\}\);
\(\mathcal{M}:= \mathcal{M}_L \cap \mathcal{M}_R\).
Note that \(S \in \mathcal{M}_L\) iff \(S^* \in \mathcal{M}_R\) since \((S \times f)^* = f^* \times S^*\). Since \(\mathcal{S}_2\) is a Fréchet space, the maps \(f \mapsto S \times f\), \(f \mapsto f \times R\) are continuous from \(\mathcal{S}_2\) to \(\mathcal{S}_2\) by the closed graph theorem.
It is clear that \(\mathcal{S}_2 \subset \mathcal{M}\) and that \(\mathbb{1}\in \mathcal{M}\). The formulas 7 show that \(\delta\in \mathcal{M}\). Now, if \(S \in \mathcal{M}_L\) and \(f \in \mathcal{S}_2\), we have \[\begin{align} \partial_j S \times f &= \partial_j(S \times f) - S \times\partial_j f, \\ \mu_j S \times f &= \mu_j(S \times f) + iS \times\hat{\partial}_j f, \end{align}\] and so \(\partial_j S \in \mathcal{M}_L\) and \(\mu_j S \in \mathcal{M}_L\); thus \(\mathcal{M}_L\), and similarly \(\mathcal{M}_R\) and \(\mathcal{M}\), is closed under partial differentation and multiplication by polynomials. Hence, in particular, all polynomials lie in \(\mathcal{M}\).
We now extend the twisted product to the case of one distribution in \(\mathcal{M}\) and one in \(\mathcal{S}_2'\) (so that \(\mathcal{S}_2'\) is an \(\mathcal{M}\)-bimodule).
Definition 4. If \(R \in \mathcal{M}_R\), \(S \in \mathcal{M}_L\), \(T \in \mathcal{S}_2'\), we define \(T \times S\), \(R \times T\) in \(\mathcal{S}_2'\) by \[\langle T \times S,h\rangle := \langle T,S \times h\rangle, \qquad \langle R \times T,h\rangle := \langle T,h \times R\rangle, \label{eq:Moyal-bimodule}\tag{9}\] for all \(h \in \mathcal{S}_2\). Since the right hand sides are continuous in \(h\), \(T \times S\) and \(R \times T\) are defined in \(\mathcal{S}_2'\).
If \(R,S \in \mathcal{M}\), \(T \in \mathcal{S}_2'\) and \(f,g,h \in \mathcal{S}_2\), we may compute: \[\begin{align} \langle(T \times f) \times g,h\rangle &= \langle T \times f,g \times h\rangle = \langle T,f \times g \times h\rangle = \langle(T \times(f \times g),h\rangle, \\ \langle(R \times S) \times f,h\rangle &= \langle R \times S,f \times h\rangle = \langle R,S \times f \times h\rangle = \langle R \times(S \times f),h\rangle. \end{align}\] In particular, \((R \times S) \times f \in \mathcal{S}_2\) for \(f \in \mathcal{S}_2\), so \(R \times S \in \mathcal{M}_L\). Then \[\langle(T \times R) \times S,h\rangle = \langle T \times R,S \times h\rangle = \langle T,R \times S \times h\rangle = \langle T \times(R \times S),h\rangle.\] We conclude that \(\mathcal{M}\) is an associative algebra; in fact, it is a \(*\)-algebra since, for \(R,S \in \mathcal{M}\), \[\begin{align} \langle(R \times S)^*,h\rangle &= \langle R \times S,h^*\rangle^* = \langle R,S \times h^*\rangle^* \\ &= \langle R^*,h \times S^*\rangle = \langle S^* \times R^*,h\rangle. \end{align}\]• We may also note that since \(S \diamond f = S \times\widetilde{F}f\) and \(f \diamond R = Ff \times R\), we have \[\begin{align} \mathcal{M}_L &= \{\,S \in \mathcal{S}_2' : S \diamond f \in \mathcal{S}_2 \text{ for all } f \in \mathcal{S}_2\,\}, \\ \mathcal{M}_R &= \{\,R \in \mathcal{S}_2' : f \diamond R \in \mathcal{S}_2 \text{ for all } f \in \mathcal{S}_2\,\}, \end{align}\] so \(\mathcal{M}\) is also a \(*\)-algebra under \(\diamond\), where we define \(\langle T \diamond S,h\rangle := \langle T,\check S \diamond h\rangle\) and \(\langle R \diamond T,h\rangle := \langle T,h \diamond \check R\rangle\). The invariance of \(\mathcal{M}\) under the several Fourier transforms now follows easily. One easily checks that the formulas of Sec. [sec:sec:Schwartz-algebras] remain valid when \(f,g\) are replaced by \(R,S \in \mathcal{M}\).
Remark 3. We show below that \(\{\,f \times g : f,g \in \mathcal{S}_2\,\}\) equals \(\mathcal{S}_2\). Thus \(\mathcal{M}\) is the maximal \(*\)-algebra which we may define by duality. For, if \(T \in \mathcal{S}_2'\) with \(T \times f, f \times T \in \mathcal{M}\) for all \(f \in \mathcal{S}_2\), then by writing \(f = g \times h\) we see that \(T \times f\) and \(f \times T\) both lie in \(\mathcal{S}_2\), since \(T \times f = (T \times g) \times h\) and \(f \times T = g \times(h \times T)\); hence \(T \in \mathcal{M}\).
Remark 4. \(\mathcal{M}\), \(\mathcal{M}_L\), \(\mathcal{M}_R\) and \(\mathcal{S}_2'\) are distinct spaces of distributions.
-3.25ex 1.5ex Regularity properties
In this section we consider in more detail the growth conditions on resultants of twisted products or convolutions. We identify a space of smooth functions, \(\mathcal{O}_T\), which contains all functions defined by ?? and ?? , and we show that \(\mathcal{O}_T\) is a normal space of distributions. As a consequence, its dual space \(\mathcal{O}'_T\) contains all distributions of compact support and is contained in \(\mathcal{M}\).
If \((E_i)_{i\in I}\) is a collection of locally convex spaces, the projective topology on the intersection \(E := \bigcap_{i\in I} E_i\) is the weakest locally convex topology such that all inclusions \(E \subset E_i\) are continuous. The inductive topology on the union \(F := \bigcup_{i\in I} E_i\) is the strongest locally convex topology such that all inclusions \(E_i \subset F\) are continuous. We will use the projective topology on decreasing intersections, and the inductive topology on increasing unions, without further comment.
For \(f \in C^m(\mathbb{R}^{2N})\), \(k,m \in \mathbb{N}\), let \[p_{k,m}(f) := \sup\{\,(1+u^2)^{-k-|\alpha|/2} |\partial^\alpha f(u)| : u \in \mathbb{R}^{2N},\;|\alpha| \leqslant m\,\} \label{eq:Weyl-seminorms}\tag{10}\] (where \(u^2 = u'u = u_1^2 +\cdots+ u_{2N}^2\)), and let \(\mathcal{V}_k^m\) be the space of all \(f \in C^m\) such that the function \((1 + u^2)^{-k-|\alpha|/2} \,\partial^\alpha f(u)\) vanishes at infinity for all \(|\alpha| \leqslant m\), normed by \(p_{k,m}\). Now let \[\mathcal{V}_k := \bigcap_{m\in\mathbb{N}} \mathcal{V}_k^m, \qquad \mathcal{O}_T := \bigcup_{k\in\mathbb{N}} \mathcal{V}_k. \label{eq:OT-filtration}\tag{11}\]
Proposition 4. \(\mathcal{O}_C \subset \mathcal{O}_T \subset \mathcal{O}_M\) with continuous inclusions.
The proof is easy and will be omitted.
Theorem 3. If \(T \in \mathcal{S}_2'\), \(f \in \mathcal{S}_2\), then \(T \times f\), \(f \times T\), \(T \diamond f\) and \(f \diamond T\) all lie in \(\mathcal{O}_T\). Moreover, these four bilinear maps of \(\mathcal{S}_2' \times\mathcal{S}_2\) into \(\mathcal{O}_T\) are separately continuous.
Proof. It suffices to consider the case of \(T \diamond f\).
Differentiating the equality \(\varepsilon_u \tau_u(T \diamond f) = T \diamond (\varepsilon_u \tau_u f)\), with \(u = tJe_j\), at \(t = 0\), we get \[(\hat{\partial}_j + i\mu_j)(T \diamond f) = T \diamond (\hat{\partial}_j + i\mu_j)f,\] so that \[\hat{\partial}_j(T \diamond f) = T \diamond (\hat{\partial}_j + i\mu_j)f - i\mu_j(T \diamond f),\] and by induction we get, for \(\alpha\in \mathbb{N}^{2N}\), \[\hat{\partial}^\alpha(T \diamond f)(u) = \sum_{\beta\leqslant\alpha} P_\beta(u) (T \diamond f_\beta)(u) \label{eq:Leibniz-again}\tag{12}\] where \(P_\beta(u)\) is a polynomial of degree at most \(|\beta|\), and \(f_\beta\in \mathcal{S}_2\), for \(\beta\leqslant\alpha\). From 10 , we need only show that \(T \diamond f\) is polynomially bounded.
Any \(T \in \mathcal{S}_2'\) can be written [20] as \(T = \hat{\partial}^\gamma Q\), with \(\gamma\in \mathbb{N}^{2N}\), where \(Q\) is a polynomially bounded continuous function on \(\mathbb{R}^{2N}\). Recalling ?? , \(\hat{\partial}_j Q \diamond f = \hat{\partial}_j(Q \diamond f) - iQ \diamond \mu_j f\), and by iteration \[T \diamond f = \hat{\partial}^\gamma Q \diamond f = \sum_{\varepsilon\leqslant\gamma} \hat{\partial}^\varepsilon(Q \diamond g_\varepsilon) \label{eq:convol-expand}\tag{13}\] for certain \(g_\varepsilon\in \mathcal{S}_2\). Combining this with 12 , we need only show that \(Q \diamond f\) is polynomially bounded.
If \(|Q(t)| \leqslant C(1 + t^2)^k\), then \[\begin{align} |(Q \diamond f)(u)| &= |\int Q(t) e^{iu'Jt} f(u - t) \,dt| \leqslant\int C(1 + t^2)^k |f(u - t)| \,dt \\ &\leqslant\int 2^k C(1 + u^2)^k (1 + (u - t)^2)^k |f(u - t)| \,dt = K(1 + u^2)^k, \end{align}\] where \(K = 2^k C \int (1 + s^2)^k |f(s)| \,ds\) is finite since \(f \in \mathcal{S}_2\).
Since \(K\) depends continuously on \(f\), the map \(f \mapsto Q \diamond f\) is continuous from \(\mathcal{S}_2\) into \(\mathcal{V}_{k+1}\). Since each \(g_\varepsilon\) in 13 depends continuously of \(f\), and since \(p_{k,m}(\partial_j f) \leqslant p_{k-1,m+1}(f)\), so that \(\partial_j \colon \mathcal{V}_{k-1} \to \mathcal{V}_k\) is continuous, we conclude that \(f \mapsto T \diamond f\) is continuous from \(\mathcal{S}_2\) into \(\mathcal{V}_{k+|\gamma|+1}\) and hence from \(\mathcal{S}_2\) into \(\mathcal{O}_T\).
Now fix \(f \in \mathcal{S}_2\) and let \(T\) vary in \(\mathcal{S}_2'\). Then \(K\) is a multiple of \(C\), so if \(V\) is a neighbourhood of zero in \(\mathcal{O}_T\), then \(V \cap \mathcal{V}_{k+|\gamma|+1}\) is a zero-neighbourhood in \(\mathcal{V}_{k+|\gamma|+1}\) and thus contains all \(T \diamond f\) with \(T = \hat{\partial}^\gamma Q\) and \(|Q(t)| \leqslant C(1 + t^2)^k\) for \(C \leqslant c_{k\gamma}\) with \(c_{k\gamma}\) small enough. Let \[B := \biggl\{ h \in \mathcal{S}_2 : \int (1 + u^2)^r \,|\partial^\alpha h(u)| \,du \leqslant\frac{1}{c_{r\alpha}} \text{ for all } r \in \mathbb{N},\;\alpha\in \mathbb{N}^{2N} \biggr\}.\] Then \(B\) is bounded in \(\mathcal{S}_2\) and its polar \(B^\circ\) is a neighbourhood of \(0\) in \(\mathcal{S}_2'\) such that \(T \diamond f \in V\) whenever \(T \in B^\circ\). ◻
Remark 5. The fact that \(T \times f \in \mathcal{O}_M\) has been noted in [7].
A normal space of distributions (on \(\mathbb{R}^{2N}\)) [18] is a locally convex space \(\mathcal{R}\) where \(\mathcal{D}\subset \mathcal{R}\subset \mathcal{D}'\) with continuous inclusions and \(\mathcal{D}\) is dense in \(\mathcal{R}\). (Here \(\mathcal{D}\) is the space of test functions of compact support on \(\mathbb{R}^{2N}\).)
Lemma 1. \(\mathcal{V}_k^m\) is a normal space of distributions.
Proof. We adapt the analogous proof of Horváth [18] for \(\mathcal{S}_{-k}^m\). Take \(g \in \mathcal{D}\) with \(g(u) = 1\) for \(u^2 \leqslant 1\) and \(0 \leqslant g(u) \leqslant 1\) for all\(u \in \mathbb{R}^{2N}\). Set \(g_\varepsilon(u) := g(\varepsilon u)\) for \(\varepsilon> 0\). Then for \(f \in \mathcal{V}_k^m\) we have \(fg_\varepsilon\in \mathcal{D}^m\) (the space of \(C^m\) functions of compact support) and from 10 we get \[\begin{align} p_{k,m}(f - fg_\varepsilon) &= \sup \biggl\{ (1 + u^2)^{-k-|\alpha|/2} \sum_{\beta\leqslant\alpha} \binom{\alpha}{\beta} \,|\partial^{\alpha-\beta}(1 - g(\varepsilon u)) \,\partial^\beta f(u)| : |\alpha| \leqslant m, \;u \in \mathbb{R}^{2N} \biggr\} \\ &\leqslant C \sup \biggl\{ (1 + u^2)^{-k-|\alpha|/2} \sum_{\beta\leqslant\alpha} \binom{\alpha}{\beta} \,|\partial^\beta f(u)| : |\alpha| \leqslant m, \;u^2 \geqslant\varepsilon^{-2} \biggr\} \\ &\leqslant 2^m C \sup\{\,(1 + u^2)^{-k-|\beta|/2} \,|\partial^\beta f(u)| : |\beta| \leqslant m, \;u^2 \geqslant\varepsilon^{-2}\,\}, \end{align}\] where we may take \[C = 1 + \sup\{\,|\partial^\gamma g(u)| : |\gamma| \leqslant m,\;u \in \mathbb{R}^{2N}\,\}.\] Thus \(fg_\varepsilon\to f\) in \(\mathcal{V}_k^m\) as \(\varepsilon\to 0\), and so \(\mathcal{D}\) is dense in \(\mathcal{V}_k^m\). Hence \(\mathcal{D}\) is dense in \(\mathcal{V}_k^m\) since it is dense in \(\mathcal{D}^m\). On the other hand, since \(q_{k+m,m}(f) \leqslant p_{k,m}(f) \leqslant q_{k,m}(f)\) for \(f \in \mathbb{C}^m(\mathbb{R}^{2N})\), we get a chain of continuous inclusions: \[\mathcal{D}\subset \mathcal{S}_{-k}^m \subset \mathcal{V}_k^m \subset \mathcal{S}_{-k-m}^m \subset \mathcal{D}'. \eqno \qed\] ◻
Lemma 2. Let \((\mathcal{R}_k)_{k\in\mathbb{N}}\) be a sequence of normal spaces of distributions.
If \(\mathcal{R}_{k+1} \subset \mathcal{R}_k\) with a continuous inclusion for all \(k\), and if \(\mathcal{R}:= \bigcap_{k\in\mathbb{N}} \mathcal{R}_k\) with the projective topology;
or if \(\mathcal{R}_k \subset \mathcal{R}_{k+1}\) with a continuous inclusion for all \(k\), and if \(\mathcal{R}:= \bigcup_{k\in\mathbb{N}} \mathcal{R}_k\) with the inductive topology;
then \(\mathcal{R}\) is a normal space of distributions.
Proof. (1) We have \(\mathcal{D}\subset \mathcal{R}\subset \mathcal{R}_k \subset \mathcal{D}'\) for all \(k \in \mathbb{N}\). The first inclusion is continuous since \(\mathcal{R}\) has the projective topology and each \(\mathcal{D}\subset \mathcal{R}_k\) is continuous; the continuity of the other inclusions is clear. If \(V\) is a neighbourhood of \(0\) in \(\mathcal{R}\) and if \(f \in \mathcal{R}\), then \(V = V_k \cap \mathcal{R}\) where \(V_k\) is a \(0\)-neighbourhood in some \(\mathcal{R}_k\); then \(f + V_k\) contains some \(g \in \mathcal{D}\), and hence \(g \in (f + V)\): so \(\mathcal{D}\) is dense in \(\mathcal{R}\).
(2) We have \(\mathcal{D}\subset \mathcal{R}_k \subset \mathcal{R}\subset \mathcal{D}'\) for all \(k \in \mathbb{N}\). The third inclusion is continuous since \(\mathcal{R}\) has the inductive topology and each \(\mathcal{R}_k \subset \mathcal{D}'\) is continuous; the continuity of the other inclusions is clear. If \(V\) is a neighbourhood of \(0\) in \(\mathcal{R}\) and if \(f \in \mathcal{R}\), then \(f \in \mathcal{R}_k\) for some \(k\) and \(V \cap \mathcal{R}_k\) is a \(0\)-neighbourhood in some \(\mathcal{R}_k\); then \(f + (V \cap \mathcal{R}_k)\) contains some \(g \in \mathcal{D}\), and hence \(g \in (f - V)\): so \(\mathcal{D}\) is dense in \(\mathcal{R}\). ◻
Already in [18], \(\mathcal{O}_C\) has been shown to be a normal space of distributions, where the proof technique is essentially the application of Lemma 2 to the definition ?? of \(\mathcal{O}_C\). From ?? we also obtain the normality of \(\mathcal{O}_M\). Combining the two Lemmas with the definition 11 of \(\mathcal{O}_T\), we get normality of \(\mathcal{O}_T\). Thus each inclusion in the chain \[\mathcal{D}\subset \mathcal{O}_C \subset \mathcal{O}_T \subset \mathcal{O}_M \subset \mathcal{D}'\] is continuous and has dense image (since \(\mathcal{D}\) is dense in all these spaces). Therefore the transposed maps \[\mathcal{D}\subset \mathcal{O}_M' \subset \mathcal{O}'_T \subset \mathcal{O}_C' \subset \mathcal{D}'\] are one-to-one and continuous. We identify each dual space with its image in \(\mathcal{D}'\). Since we can interpolate \(\mathcal{S}_2\) and \(\mathcal{S}_2'\) into both chains, the dual spaces consist of tempered distributions.
The space of distributions of compact support on \(\mathbb{R}^{2N}\) [20] is the dual space \(\mathcal{E}_2'\) of \(\mathcal{E}_2\). We can now show that it is contained in the Moyal algebra.
Theorem 4. The spaces \(\mathcal{E}_2'\), \(\mathcal{O}_M'\) and \(\mathcal{O}'_T\) are contained in \(\mathcal{M}\).
Proof. Transposing \(\mathcal{O}_T \subset \mathcal{O}_M \subset \mathcal{E}_2\), we get \(\mathcal{E}_2' \subset \mathcal{O}_M' \subset \mathcal{O}'_T\), so we need only check that \(\mathcal{O}'_T \subset \mathcal{M}\).
If \(S \in \mathcal{O}'_T\) and \(T \in \mathcal{S}_2'\), we may define \(S \times T\), \(T \times S\) by transposition: \[\langle S \times T,h\rangle := \langle S,T \times h\rangle, \qquad \langle T \times S,h\rangle := \langle S,h \times T\rangle, \label{eq:more-products}\tag{14}\] for \(h \in \mathcal{S}_2\); since the right hand sides are continuous in \(h\), by Theorem 3, \(S \times T\) and \(T \times S\) are defined in \(\mathcal{S}_2'\).
Moreover, for a fixed \(h \in \mathcal{S}_2\), the maps \(T \mapsto T \times h\), \(T \mapsto h \times T\) are continuous \(\colon\mathcal{S}_2'\to\mathcal{O}_T\) by Theorem 3, so they transpose to continuous maps \(S \mapsto h \times S\), \(S \mapsto S \times h\) from \(\mathcal{O}'_T\) into \((\mathcal{S}_2')' = \mathcal{S}_2\), via \[\langle h \times S,T\rangle := \langle S,T \times h\rangle, \qquad \langle S \times h,T\rangle := \langle S,h \times T\rangle. \label{eq:more-transposes}\tag{15}\] Combining 14 and 15 , we get \[\langle T \times S,h\rangle := \langle T,S \times h\rangle, \qquad \langle S \times T,h\rangle := \langle T,h \times S\rangle, \label{eq:more-extensions}\tag{16}\] for \(T \in \mathcal{S}_2'\), \(S \in \mathcal{O}'_T\), \(h \in \mathcal{S}_2\). We have shown that \(S \times h \in \mathcal{S}_2\), \(h \times S \in \mathcal{S}_2\) whenever \(h \in \mathcal{S}_2\), \(S \in \mathcal{O}'_T\); these are resultants of separately continuous extensions to \(\mathcal{O}'_T \times\mathcal{S}_2\) of the twisted product on \(\mathcal{S}_2\), and since \(\mathcal{S}_2\) is dense in \(\mathcal{O}'_T\) these extensions are unique. Since 16 is formally identical with 9 , the twisted products 14 and 15 are consistent with previous ones, and we conclude that \(\mathcal{O}'_T \subset \mathcal{M}\). Since \(\mathbb{1}\notin \mathcal{O}'_T\), we have \(\mathcal{O}'_T \neq \mathcal{M}\). ◻
Corollary 1. The space \(\mathcal{O}_C\) is contained in \(\mathcal{M}\). In particular, the (ordinary) convolution of any function in \(\mathcal{S}_2\) with any tempered distribution belongs to \(\mathcal{M}\).
Proof. Since \(\mathcal{O}_C\) is reflexive [22], it is enough to note that \(\mathcal{F}(\mathcal{O}_M') = \mathcal{O}_C\). Then use Theorem 4. Also [18], \(T * f\) belongs to \(\mathcal{O}_C\) if \(f \in \mathcal{S}_2\), \(T \in \mathcal{S}_2'\). ◻
If \(R,S \in \mathcal{S}_2'\), their tensor product \(R \otimes S \in \mathcal{S}'(\mathbb{R}^{4N})\) is given by \(\langle R \otimes S,f \otimes g\rangle := \langle R,f\rangle\,\langle S,g\rangle\). If \(R,S \in \mathcal{M}\), we have \[\begin{align} \langle R \diamond S,h\rangle &= \langle R,\check S \diamond h\rangle = \langle R,u \mapsto \langle S,\varepsilon_{-u} \tau_{-u} h\rangle\rangle \\ &= \langle R \otimes S,(u,v) \mapsto e^{-iu'Jv} h(u + v)\rangle. \end{align}\] Writing \(h_2(u,v) := e^{-iu'Jv} h(u + v)\), we find that \[\partial_u^\alpha\partial_v^\gamma h_2(u,v) = \sum_{\beta\leqslant\alpha} \sum_{\varepsilon\leqslant\gamma} i^{|\varepsilon|-|\beta|} \binom{\alpha}{\beta} \binom{\gamma}{\varepsilon} (Jv)^\beta(Ju)^\varepsilon\,e^{-iu'Jv} (\partial_u^{\alpha-\beta} \partial_v^{\gamma-\varepsilon} h)(u + v). \label{eq:deriv-expansion}\tag{17}\] Thus \(h \mapsto h_2\) is continuous as a map \(\colon \mathcal{E}_2 \to \mathcal{E}(\mathbb{R}^{4N})\). Since \((R,S) \mapsto R \otimes S\) is jointly continuous \(\colon \mathcal{E}'_2 \times\mathcal{E}'_2 \to \mathcal{E}'(\mathbb{R}^{4N})\) and \(\langle R \diamond S,h\rangle = \langle R \otimes S,h_2\rangle\), we find that \((R,S) \mapsto R \diamond S\) is jointly continuous on \(\mathcal{E}'_2\).
By the Paley–Wiener theorem, \(F(\mathcal{E}'_2) = \mathcal{F}(\mathcal{E}'_2) =: \mathcal{O}_\mathrm{exp}\) is the space of functions in \(\mathcal{O}_M\) which extend to analytic functions of exponential type, and by Theorem 4 and the Fourier-invariance of \(\mathcal{M}\), \(\mathcal{O}_\mathrm{exp}\) is contained in \(\mathcal{M}\). (\(\mathcal{O}_\mathrm{exp}\) carries the topology induced by \(\mathcal{F}\) from \(\mathcal{E}'_2\).)
For convenience, we write \(\hat{\mu}_j f := \mu_{j+N}f\) if \(j \leqslant N\), \(\hat{\mu}_j f := -\mu_{j-N}f\) if \(j > N\). If \(h \in \mathcal{E}_2\), the expansion \[e^{-iu'Jv} h(u + v) = \sum_{k=0}^\infty \frac{(-i)^k}{k!}\, (u'Jv)^k h(u + v)\] converges uniformly on compact subsets of \(\mathbb{R}^{2N}\), together with all derivatives on account of 17 , and this convergence is uniform on bounded subsets of \(\mathcal{E}_2\). Thus \[\langle S \diamond T,h\rangle = \sum_{k=0}^\infty \frac{(-i)^k}{k!}\, \langle S \otimes T,(u,v) \mapsto (u'Jv)^k h(u + v)\rangle.\] Since \((u'Jv)^k = \sum_{|\alpha|=k} \frac{k!}{\alpha!}\, u^{\alpha\prime} (Jv)^\alpha\), and \((\hat{\mu}_j f)(v) = (Jv)_j f(v)\) for all \(f \in \mathcal{S}_2\) and each \(j\), we derive \[\begin{align} \langle S \diamond T,h\rangle &= \sum_{k=0}^\infty \sum_{|\alpha|=k} \frac{(-i)^k}{\alpha!}\, \langle S \otimes T,(u,v) \mapsto u^{\alpha\prime} (Jv)^\alpha h(u + v)\rangle \nonumber \\ &= \sum_{\alpha\in\mathbb{N}^{2N}} \frac{(-i)^{|\alpha|}}{\alpha!}\, \langle\mu^\alpha S \otimes\hat{\mu}^\alpha T,(u,v) \mapsto h(u + v)\rangle \nonumber \\ &= \sum_{\alpha\in\mathbb{N}^{2N}} \frac{(-i)^{|\alpha|}}{\alpha!}\, \langle\mu^\alpha S * \hat{\mu}^\alpha T,h\rangle \label{eq:convol-expansion} \end{align}\tag{18}\] where the series converges uniformly for \(h\) in bounded subsets of \(\mathcal{E}_2\). We are now able to expand the twisted product as a series of products of derivatives.
Theorem 5. If \(S,T \in \mathcal{O}_\mathrm{exp}\), then \[S \times T = \sum_{\alpha\in\mathbb{N}^{2N}} \frac{i^{|\alpha|}}{\alpha!} \,(\partial^\alpha S)\,(\hat{\partial}^\alpha T)\] with convergence in the topology of \(\mathcal{O}_\mathrm{exp}\).
Proof. We apply the Fourier transform \(\mathcal{F}\) to 18 , replacing \(S,T\) by \(\mathcal{F}^{-1}S,\mathcal{F}^{-1}T\). By the continuity of \(\mathcal{F}\) we get \[\begin{align} S \times T &= \sum_\alpha\frac{(-i)^{|\alpha|}}{\alpha!}\, \mathcal{F}(\mu^\alpha\mathcal{F}^{-1}S * \hat{\mu}^\alpha\mathcal{F}^{-1}T) \\ &= \sum_\alpha\frac{(-i)^{|\alpha|}}{\alpha!}\, i^{2|\alpha|} \,(\partial^\alpha S)(\hat{\partial}^\alpha T) = \sum_\alpha\frac{i^{|\alpha|}}{\alpha!} \,(\partial^\alpha S)\,(\hat{\partial}^\alpha T). \tag*{\qed} \end{align}\] ◻
Corollary 2. If \(S,T \in \mathcal{O}_\mathrm{exp}\), then \[S \times T - T \times S = 2i \sum_{r=0}^\infty \sum_{|\alpha|=2r+1} \frac{(-1)^r}{\alpha!} \,(\partial^\alpha S)\,(\hat{\partial}^\alpha T). \label{eq:Moyal-bracket}\qquad{(3)}\]
Remark 6. The restriction to \(\mathcal{E}'_2\) or \(\mathcal{O}_\mathrm{exp}\) is only needed to guarantee convergence of the series indexed by \(\alpha\). If either \(S\) or \(T\) is a polynomial, these series are finite sums, and the expansions are valid.
Remark 7. For \(N = 1\), the leading term in ?? is the ordinary Poisson bracket \(\partial_q S \partial_p T - \partial_p S \partial_q T\). As a differential operator on \(S \otimes T\), we may formally write \[\begin{align} S \times T - T \times S &= 2i \biggl( \sum_{r=0}^\infty \frac{(-1)^r}{(2r+1)!}\, (\partial_q \otimes\partial_p - \partial_p \otimes\partial_q)^{2r+1} \biggr) (S \otimes T) \\ &= 2i \bigl( \sin(\partial_q \otimes\partial_p - \partial_p \otimes\partial_q) \bigr) (S \otimes T), \end{align}\] an expression first derived by Moyal [11] and called the “Moyal bracket”.
-3.25ex 1.5ex The matricial form of the twisted product
By working in the Schwartz space \(\mathcal{S}_2\), we avoid the usual continuity problems for the creation and annihilation operators for the harmonic oscillator, as has been observed before [23]. In the present context, these operators are represented by first-degree polynomials in \(\mathcal{M}\). To avoid notational clutter, we will take \(N = 1\) in this section and the next; but the results go through in the general case with the systematic use of multi-indices. We write \(u = (q,p)\) and use \(q,p,\partial_q,\partial_p\) in place of \(\mu_1,\mu_2,\partial_1,\partial_2\) respectively. We introduce the notations: \[\begin{align} a &:= \frac{q + ip}{\sqrt{2}}, \quad \bar a:= \frac{q - ip}{\sqrt{2}}\,, \\ \frac{\partial}{\partial a} &:= \frac{\partial_q - i\partial_p}{\sqrt{2}}, \quad \frac{\partial}{\partial\bar a} := \frac{\partial_q + i\partial_p}{\sqrt{2}}\,, \\ H &:= a\bar a= {\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}(q^2 + p^2) = {\mathchoice{\tfrac{1}{2}}{\tfrac{1}{2}}{{\scriptstyle\frac{1}{2}}}{{\scriptstyle\frac{1}{2}}}}u^2, \quad f_0 = 2 e^{-\bar aa} = 2 e^{-H}. \end{align}\]
From 8 we obtain the formulas \[\begin{align} {2} a \times f &= af + \frac{\partial f}{\partial\bar a}\,, &\qquad f \times a &= af - \frac{\partial f}{\partial\bar a}\,, \nonumber \\ \bar a\times f &= \bar af - \frac{\partial f}{\partial a}\,, &\qquad f \times\bar a&= \bar af + \frac{\partial f}{\partial a}\,. \label{eq:one-step} \end{align}\tag{19}\]
We get at once the following equalities in \(\mathcal{M}\): \[\bar a\times a = H - 1, \quad a \times\bar a= H + 1; \quad a \times\bar a- \bar a\times a = 2. \label{eq:two-step}\tag{20}\]
The third equality is of course the canonical commutation relation for \(a\) and \(\bar a\): recall that we have taken units in which \(\hbar = 2\). We note also that \(a \times a \times\cdots\times a \text{ (n~times) } = a^n\).
The Gaussian function \(f_0\) has several nice properties: it is (pointwise) positive, it is a fixed point for the various Fourier transforms, and it is an idempotent in \(\mathcal{S}_2\) for both the twisted product and the twisted convolution. Moreover, it is a unit vector in \(L^2(\mathbb{R}^2)\), because of our choices of normalization.
Notice that if \(g \in \mathcal{O}_M\), 19 implies that \[a \times(gf_0) = \Bigl( \frac{\partial g}{\partial\bar a}\Bigr) f_0, \quad \bar a\times(gf_0) = \Bigl( 2\bar ag - \frac{\partial g}{\partial a} \Bigr) f_0. \label{eq:basic-steps}\tag{21}\] Taking \(g = 2^m \bar a^m\), we find that \[\bar a\times(2^m \bar a^m f_0) = \Bigl( 2^{m+1}\bar a^{m+1} - 2^m \frac{\partial\bar a^m}{\partial a} \Bigr) f_0 = 2^{m+1} \bar a^{m+1} f_0,\] so we get by induction that \(\bar a^m \times f_0 = 2^m \bar a^m f_0\) if \(m \in \mathbb{N}\). If \(n > m\), this gives \[a^n \times\bar a^m \times f_0 = a^n \times(2^m \bar a^m f_0) = 2^m \,\frac{\partial^n}{\partial\bar a^n} (\bar a^m) f_0 = 0,\] and if \(n < m\), then \[f_0 \times a^n \times\bar a^m = (a^m \times\bar a^n \times f_0)^* = 0.\] Also, \[\begin{align} f_0 \times a^n \times\bar a^n \times f_0 &= f_0 \times a^n \times(2^n \bar a^n f_0) = f_0 \times 2^n\, \frac{\partial^n}{\partial\bar a^n} (\bar a^n) f_0 \\ &= 2^n n!\, f_0 \times f_0 = 2^n n!\, f_0. \end{align}\] To summarize: \[f_0 \times a^n \times\bar a^m \times f_0 = \delta_{mn} 2^n n!\, f_0 \quad\text{for}\quad m,n \in \mathbb{N}. \label{eq:vev}\tag{22}\]
We now introduce an orthonormal basis for \(L^2(\mathbb{R}^2)\), which we declare as a doubly indexed family of functions in \(\mathcal{S}_2\), since this family forms a system of matrix units with respect to the twisted product.
Definition 5. For \(m,n \in \mathbb{N}\), we define \(f_{mn} \in \mathcal{S}_2\) by \[f_{mn} := \frac{1}{\sqrt{2^{m+n}\,m!\,n!}}\, \bar a^m \times f_0 \times a^n. \label{eq:matrix-basis}\tag{23}\]
From 22 we get directly \[\begin{align} f_{mn} \times f_{kl} &= (2^{m+n+k+l}\,m!\,n!\,k!\,l!)^{-1/2} \bar a^m \times f_0 \times a^n \times\bar a^k \times f_0 \times a^l \nonumber \\ &= \frac{\delta_{nk}}{\sqrt{2^{m+l}\,m!\,l!}}\, \bar a^m \times f_0 \times a^l = \delta_{nk} f_{ml}. \label{eq:matrix-units} \end{align}\tag{24}\] This implies that \[\begin{align} 2(f_{mn}\mathbin|f_{kl}) &= \langle f_{nm},f_{kl}\rangle = \int (f_{nm} \times f_{kl})(u) \,du = \delta_{mk} \int f_{nl}(u) \,du \\ &= \frac{\delta_{mk}}{\sqrt{2^{n+l}\,n!\,l!}} \int (\bar a^n \times f_0 \times f_0 \times a^l)(u) \,du \\ &= \frac{\delta_{mk}}{\sqrt{2^{n+l}\,n!\,l!}} \int (f_0 \times a^l \times\bar a^n \times f_0)(u) \,du \\ &= \delta_{mk} \,\delta_{nl} \int f_0(u) \,du = 2 \,\delta_{mk} \,\delta_{nl} \end{align}\] so that \(\{\,f_{mn} : m,n \in \mathbb{N}\,\}\) is orthonormal in \(L^2(\mathbb{R}^2)\). This family is complete since – see 28 below – all the Hermite functions on \(\mathbb{R}^2\) are linear combinations of the \(f_{mn}\).
Remark 8. The basis \((f_{mn})\) lies in \(\mathcal{S}_2\), as is clear from 23 since \(a,\bar a\in \mathcal{M}\). It has also the important property of diagonalizing the Fourier transforms; one readily checks that \[\mathcal{F}(f_{mn}) = (-i)^{m+n} f_{mn}, \quad F(f_{mn}) = (-1)^n f_{mn}, \quad \widetilde{F}(f_{mn}) = (-1)^m f_{mn}, \label{eq:Fourier-eigenbasis}\tag{25}\] and hence also \(\check f_{mn} = (-1)^{m+n} f_{mn}\).
To present \(f_{mn}\) explicitly, we use polar coordinates \(q + ip =: \rho e^{i\alpha}\); note that \(\rho^2 = q^2 + p^2 = u^2\).
By induction, applying 21 to 23 , we derive \[f_{mn} = \frac{1}{\sqrt{2^{m+n}\,m!\,n!}} \sum_{k=0}^n (-1)^k \binom{m}{k} \binom{n}{k}\, k!\, 2^{m+n-k} \bar a^{m-k} a^{n-k} f_0\] and, since \(a = (1/\sqrt{2})\,\rho e^{i\alpha}\), \(\bar a= (1/\sqrt{2})\,\rho e^{-i\alpha}\), we get, after some rearrangement, \[f_{mn}(\rho,\alpha) = 2(-1)^n \sqrt{\frac{n!}{m!}}\, e^{-i\alpha(m-n)} \rho^{m-n} L_n^{m-n}(\rho^2) \,e^{-\rho^2/2} \label{eq:Laguerre-basis}\tag{26}\] and in particular \[f_{nn}(\rho,\alpha) = 2(-1)^n L_n(\rho^2) \,e^{-\rho^2/2} \label{eq:diagonal-basis}\tag{27}\] where \(L_n\) and \(L_n^{m-n}\) are the usual Laguerre polynomials of order \(n\).
Remark 9. Equation 27 agrees with the Wigner function of the \(n\)th energy level of the harmonic oscillator, obtained in [10]; and 26 agrees with the “transition” between levels of the harmonic oscillator, first derived in [24]; see also [3] and [25].
Now we represent \(\mathcal{S}_2\), \(\mathcal{S}_2'\) and \(L^2(\mathbb{R}^2)\) as sequence spaces of coefficients after expansion in the twisted Hermite basis \((f_{mn})\). Our treatment is in the spirit of Simon’s work [26] with the ordinary Hermite basis. Since here the coefficients form a doubly indexed family, we may consider their matrix product, which turns out to correspond to the twisted product of the associated functions or distributions.
The fundamental fact which underlies the sequence constructions is that the twisted Hermite basis “diagonalizes” the oscillator Hamiltonian \(H\) (its eigenvalues are odd integers rather than half-integers due to our convention that \(\hbar = 2\)).
Proposition 5. If \(m,n \in \mathbb{N}\), then \[H \times f_{mn} = (2m + 1) f_{mn}, \qquad f_{mn} \times H = (2n + 1) f_{mn}.\]
Proof. From 23 we get at once \[\begin{align} {2} a \times f_{mn} &= \sqrt{2m}\, f_{m-1,n}, &\qquad f_{mn} \times a &= \sqrt{2n + 2}\, f_{m,n+1} \\ \bar a\times f_{mn} &= \sqrt{2m + 2}\, f_{m+1,n}, &\qquad f_{mn} \times\bar a&= \sqrt{2n}\, f_{m,n-1}, \end{align}\] (with \(f_{mn} = 0\) if \(m\) or \(n\) is \(-1\)). The result follows from 20 . ◻
Let us write \(A(f) := H \times f \times H\). We could consider \(A\) as an operator on \(L^2(\mathbb{R}^2)\) with domain \(\mathcal{S}_2\); as such, \(A\) is symmetric and closable, and clearly unbounded. Indeed, \((A \pm iI)f_{mn} = ((2m+1)(2n+1) \pm i)f_{mn}\) and hence \(A \pm iI\) has dense range: thus \(A\) is essentially self-adjoint. Moreover, \(A^{-1}\) has finite-dimensional eigenspaces and \[\sum_{m,n=0}^\infty (2m+1)^{-2} (2n+1)^{-2} = (\pi^2/8)^2\] is finite, so \(A^{-1}\) is a Hilbert–Schmidt operator. It is not hard to check that the seminorms \(f \mapsto \|A^k f\|\), for \(k \in \mathbb{N}\), generate the topology of \(\mathcal{S}_2\).
Remark 10. Let \(B = u^2 - \Delta\) be the usual Hermite operator on \(L^2(\mathbb{R}^2)\). We have \(Bf = H \times f + f \times H\), so \(Bf_{mn} = 2(m+n+1)f_{mn}\). If \(h_k(x) := (2^{k-1}\,k!)^{-1/2} H_k(x) e^{-x^2/2}\) is the usual Hermite function of degree \(k\), we conclude that \[f_{mn} = \sum_{k+l=m+n} c_{mn}^{kl} h_k \otimes h_l, \qquad h_k \otimes h_l = \sum_{m+n=k+l} b_{kl}^{mn} f_{mn}, \label{eq:basis-change}\tag{28}\] for some constants \(c_{mn}^{kl}\), \(b_{kl}^{mn}\). In fact, we may compute that \[c_{mn}^{kl} = 2^{(m-n)/2} i^{2m+l} {\binom{m+n}{l}}^{1/2} {\binom{m+n}{m}}^{-1/2} P_m^{l-m,k-m}(0)\] where \(P_m^{l-m,k-m}\) is the usual Jacobi polynomial. (This takes care of the completeness argument for the \(f_{mn}\).)
We can now characterize \(\mathcal{S}_2\) and \(\mathcal{S}_2'\) as sequence spaces.
Theorem 6. Let \(\sss\) be the Fréchet space of rapidly decreasing double sequences \(c = (c_{mn})\) such that \[r_k(c) := \biggl[ \sum_{m,n=0}^\infty (2m+1)^{2k} (2n+1)^{2k} |c_{mn}|^2 \biggr]^{1/2}\] is finite for all \(k \in \mathbb{N}\), topologized by the seminorms \((r_k)_{k\in\mathbb{N}}\). For \(f \in \mathcal{S}_2\), let \(c\) be the sequence of coefficients in the expansion \(f = \sum_{m,n=0}^\infty c_{mn} f_{mn}\). Then \(f \mapsto c\) is an isomorphism of Fréchet spaces from \(\mathcal{S}_2\) onto \(\sss\).
Proof. If \(f \in \mathcal{S}_2\), then \(\|A^k f\| < \infty\) for all \(k \in \mathbb{N}\), so that \(r_k(c) = \|A^k f\|\) is finite for all \(k\). It follows that \(f \mapsto c\) is a one-to-one topological isomorphism of \(\mathcal{S}_2\) into \(\sss\).
Given any \(c \in \sss\), for \(M,N \in \mathbb{N}\) let \(c^{MN}\) be the double sequence defined by \(c_{mn}^{MN} := c_{mn}\) if \(m \leqslant M\), \(n \leqslant N\) and \(c_{mn}^{MN} := 0\) otherwise. Then \(r_k(c^{MN} - c) \to 0\) as \(M,N\to\infty\), for each \(k\), so that the functions \(\sum_{m=0}^M \sum_{n=0}^N c_{mn} f_{mn}\) form a Cauchy sequence in \(\mathcal{S}_2\) and hence converge to a function \(f\) which maps onto \(c\). ◻
For \(T \in \mathcal{S}_2'\), \(m,n \in \mathbb{N}\), define \(b_{mn} := \langle T,f_{nm}\rangle\). Then \[\langle T,f,=\rangle \sum_{m,n=0}^\infty c_{nm} \langle T,f_{nm}\rangle = \sum_{m,n=0}^\infty c_{nm} b_{mn} \label{eq:matrix-expansion}\tag{29}\] where the series converges absolutely, for each \(f = \sum_{m,n=0}^\infty c_{mn}f_{mn} \in \mathcal{S}_2\). Since \(T \in \mathcal{S}_2'\), there exist \(k \in \mathbb{N}\), \(K > 0\) such that \[|\langle T,f,|\rangle \leqslant K \|A^k f\| = K\,r_k(c) \label{eq:sequence-estimate}\tag{30}\] for all \(f \in \mathcal{S}_2\), and since \[\langle T,f,=\rangle \sum_{m,n=0}^\infty (2m + 1)^{-k} (2n + 1)^{-k} b_{mn} (2n + 1)^k (2m + 1)^k c_{nm}\] the Schwarz inequality gives \(r_{-k}(b) \leqslant K\). Thus, whenever \(b_{mn}\) is a double sequence with \(r_{-k}(b)\) finite for some \(k\), the series \(\sum_{m,n=0}^\infty b_{mn}f_{mn}\) converges weakly to \(T\) in \(\mathcal{S}_2'\), and 30 shows that the convergence is uniform on bounded subsets of \(\mathcal{S}_2\), so the series converges to \(T\) in the strong dual topology of \(\mathcal{S}_2'\).
The main result of this section is now easy.
Theorem 7. If \(a,b \in \sss\) correspond respectively to \(f,g \in \mathcal{S}_2\) as coefficient sequences in the twisted Hermite basis, then the sequence corresponding to the twisted product \(f \times g\) is the matrix product \(ab\), where \[(ab)_{mn} := \sum_{k=0}^\infty a_{mk} b_{kn}. \label{eq:matrix-product}\qquad{(4)}\]
Proof. From 24 and the continuity of \(\times\) in \(\mathcal{S}_2\), we get \[\begin{align} f \times g &= \biggl(\sum_{m,k} a_{mk} f_{mk} \biggr) \times\biggl(\sum_{r,n} b_{rn} f_{rn} \biggr) \\ &= \sum_{m,k,r,n} a_{mk} b_{rn}\, f_{mk} \times f_{rn} = \sum_{m,k,n} a_{mk} b_{kn}\, f_{mn}. \tag*{\qed} \end{align}\] ◻
Corollary 3. \(\{\,f \times g : f,g \in \mathcal{S}_2\,\} = \mathcal{S}_2\).
Proof. It suffices to show that any \(c \in \sss\) can be written as the matrix product of two sequences in \(\sss\). We use Howe’s argument [25] to show this.
Set \(d_m := \bigl( \sup\{\,|c_{jr}| : j \in \mathbb{N},\;r \geqslant m\,\} \bigr)^{1/2}\) for \(m \in \mathbb{N}\), and let \(d\) be the “diagonal” sequence with entries \(d_m\delta_{mn}\). Then one verifies that \(r_k(d)^2 \leqslant C_k r_{2k+2}(c)\) for some constants \(C_k\), so that \(d \in \sss\). Now if we set \(b_{mn} := c_{mn}/d_n\), we get \(|b_{mn}| = |c_{mn}|/d_n \leqslant d_n^2/d_n = d_n\) and thus \(b \in \sss\) also. Clearly \(bd = c\). ◻
Remark 11. The sequence \(a \diamond b\) corresponding to \(f \diamond g\) is \[(a \diamond b)_{mn} := \sum_{k=0}^\infty (-1)^k a_{mk} b_{kn}.\] Since, by 3 and 7 , \(f \diamond g = f \times\delta\times g\), it suffices to show that \(\delta\) is represented by the diagonal matrix with entries \((-1)^m \delta_{mn}\); this follows from 25 , since \(\mathcal{F}\mathbb{1}= \delta\). Thus the entire theory of the twisted product and convolution could be developed, at least formally, in the matrix language and without mention of the symplectic Fourier transforms; the basic transformation formulas – see 3 – are: \[R \diamond S = R \times\delta\times S, \qquad R \times S = R \diamond \mathbb{1}\diamond S.\]
We now show that ?? gives a second way of defining the twisted product for many pairs of distributions, which lie in spaces of Sobelev type.
Definition 6. For \(s,t \in \mathbb{R}\), we denote by \(\mathcal{G}_{s,t}\) the Hilbert space obtained by completing \(\mathcal{S}_2\) with respect to the norm \[\|f\|_{s,t} := \biggl( \sum_{m,n=0}^\infty (2m + 1)^s (2n + 1)^t |c_{mn}|^2 \biggr)^{1/2}. \label{eq:st-norms}\tag{31}\]
Observe that \(\mathcal{G}_{0,0} = L^2(\mathbb{R}^2)\) with \(\|f\|_{0,0} = \|f\|\). An orthonormal basis for \(\mathcal{G}_{s,t}\) is given by the functions \((2m + 1)^{-s/2} (2n + 1)^{-t/2} f_{mn}\), and thus \[f = \sum_{m,n=0}^\infty c_{mn} f_{mn}\] with convergence in the \((s,t)\)-norm, for all \(f \in \mathcal{G}_{s,t}\). Note that \(\mathcal{S}_2 = \bigcap_{s,t\in\mathbb{R}} \mathcal{G}_{s,t}\) topologically. Since \(\mathcal{S}_2 \subset \mathcal{G}_{s,t}\) is a continuous inclusion with dense image, \(\mathcal{G}_{s,t}\) is a normal space of tempered distributions, and the transpose of \(\mathcal{S}_2 \subset \mathcal{G}_{s,t}\) is the inclusion \(\mathcal{G}_{-t,-s} \subset \mathcal{S}_2'\). Also, from 30 , \(\mathcal{S}_2' = \bigcup_{s,t\in\mathbb{R}} \mathcal{G}_{s,t}\) (topologically). Furthermore, \(\mathcal{G}_{s,t} \subset \mathcal{G}_{q,r}\) with a continuous inclusion iff \(s \geqslant q\) and \(t \geqslant r\). Note also that \(f^* \in \mathcal{G}_{t,s}\) whenever \(f \in \mathcal{G}_{s,t}\).
If \(g = \sum_{m,n=0}^\infty b_{mn} f_{mn} \in \mathcal{G}_{q,r}\), we define formally \[f \times g := \sum_{m,n=0}^\infty \biggl( \sum_{k=0}^\infty c_{mk} b_{kn} \biggr) f_{mn}. \label{eq:matricial-product}\tag{32}\]
Theorem 8.
The serie 32 converges in \(\mathcal{G}_{s,r}\) if \(t + q \geqslant 0\), and in that case \[\|f \times g\|_{s,r} \leqslant\|f\|_{s,t}\, \|g\|_{q,r}\,.\]
\(\mathcal{G}_{s,t}\) is a Banach algebra under the twisted product 32 whenever \(s + t \geqslant 0\); for \(s \geqslant 0\), \(\mathcal{G}_{s,s}\) is a Banach \(*\)-algebra.
The Fourier transforms \(F,\widetilde{F},\mathcal{F}\) are unitary isometries of each \(\mathcal{G}_{s,t}\) onto itself.
The twisted product 32 is consistent with previous definitions.
Proof. (1) From the Schwarz inequality we get \[\begin{align} \|f \times g\|_{s,r}^2 &\leqslant\sum_{m,n=0}^\infty (2m + 1)^s \biggl( \sum_{k=0}^\infty |c_{mk} b_{kn}| \biggr)^2 (2n + 1)^r \\ &= \sum_{m,n=0}^\infty (2m + 1)^s \biggl( \sum_{k=0}^\infty |c_{mk}| (2k + 1)^{t/2} (2k + 1)^{-t/2} |b_{kn}| \biggr)^2 (2n + 1)^r \\ &\leqslant\sum_{m,k=0}^\infty (2m + 1)^s |c_{mk}|^2 (2k + 1)^t \sum_{l,n=0}^\infty (2l + 1)^{-t} |b_{ln}|^2 (2n + 1)^r \\ &= \|f\|_{s,t}^2 \,\|g\|_{-t,r}^2 \leqslant\|f\|_{s,t}^2 \,\|g\|_{q,r}^2 \end{align}\] whenever \(q \geqslant-t\), which yields convergence of 32 in this case.
(2) follows by taking \(q = s\), \(r = t\).
(3) The Fourier invariance of \(\mathcal{G}_{s,t}\) is evident from 31 and 25 .
(4) If \(T = \sum_{m,n=0}^\infty d_{mn} f_{mn} \in \mathcal{S}_2'\), then \[\begin{align} \langle T,f \times g\rangle &= \sum_{m,n=0}^\infty d_{mn} \biggl( \sum_{k=0}^\infty c_{mk} b_{kn} \biggr) \nonumber \\ &= \sum_{k,n=0}^\infty \biggl( \sum_{m=0}^\infty d_{nm} c_{mk} \biggr) b_{kn} = \langle T \times f,g\rangle, \label{eq:matricial-transfer} \end{align}\tag{33}\] where the convergence of the double sums is absolute by 29 and that of the simple sums is also absolute, in the second case because \(T \times f\) lies in some \(\mathcal{G}_{s,t}\) and \(g\) lies in \(\mathcal{S}_2 \subset \mathcal{G}_{-t,-s}\); thus we may interchange the summations, obtaining \[T \times f = \sum_{k,n=0}^\infty \bigl( \sum_{m=0}^\infty d_{nm} c_{mk} \bigr) f_{nk}.\]
If \(f \in \mathcal{M}_L\), \(g \in \mathcal{S}_2\), then by Definition 4, \[\langle T \times f,g\rangle := \langle T,f \times g\rangle = \sum_{m,n=0}^\infty d_{nm} \biggl( \sum_{k=0}^\infty c_{mk} b_{kn} \biggr);\] the double sum converges absolutely since \(f \times g \in \mathcal{S}_2\), so we may interchange the order of summation to recover 33 . ◻
Remark 12. We see that if \(f,g \in L^2(\mathbb{R}^2)\), then \(f \times g \in L^2(\mathbb{R}^2)\) and \(\|f \times g\| \leqslant\|f\|\,\|g\|\). Moreover, \(f \times g\) lies in \(C_0(\mathbb{R}^2)\): the continuity follows by adapting the analogous argument for (ordinary) convolution.
Remark 13. Notice from 31 that \(A(\mathcal{G}_{s,t}) = \mathcal{G}_{s-2,t-2}\), where we write \(A(T) := H \times T \times H\) for any \(T \in \mathcal{S}_2'\), which makes sense since \(H \in \mathcal{M}\). Clearly \(A(\mathcal{M}) \subset \mathcal{M}\), so that if \(\mathcal{M}\) were to contain any \(\mathcal{G}_{s,t}\), it would contain them all. However, \(\mathcal{M}\neq \mathcal{S}_2'\), and thus \(\mathcal{M}\) contains no \(\mathcal{G}_{s,t}\); in particular, \(L^2(\mathbb{R}^{2N}) \nsubseteq \mathcal{M}\). Hence \(\mathcal{M}\) really provides a different extension of \(\times\) on \(\mathcal{S}_2\) from the Banach algebras \(\mathcal{G}_{s,t}\) \((s + t \geqslant 0)\).
-3.25ex 1.5ex Conclusion and outlook
We have been led to define the twisted product of a pair of distributions and to introduce the Moyal \(*\)-algebra \(\mathcal{M}\) under the twisted product. A rigorous formulation of the phase-space approach to quantum theory, in the arena given by \(\mathcal{M}\), should address the following problems:
find an equivalent of the spectral theorem;
solve the dynamical equations using such a spectral theorem;
describe the algebraic structure of the state space corresponding to \(\mathcal{M}\).
We plan to take up these problems. Our first task is to give \(\mathcal{M}\) an appropriate topology: this we do in the following paper [27].
-3.25ex 1.5ex *Acknowledgements
We are grateful for helpful correspondence from John Horváth and Peter Wagner. We would like to thank Prof. Abdus Salam and the International Centre for Theoretical Physics, for their hospitality during a stay in which this work was completed. We gratefully acknowledge support from the Vicerrectoría de Investigación of the Universidad de Costa Rica.