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In studying quantum field theory on such a simple noncommutative spacetime, the question arose how to extend the above products (viz. the Weyl calculus) to (tempered) distributions. This question, which is interesting in its own right, had been raised and partially answered in different contexts, e.g. by Maillard [13] as well as Gracia-Bond´ıaand Varilly [7, 8] (in the context of quantum mechanics), by Dubois-Violette, Kriegl, Maeda and Michor [5] as well as Lechner and Waldmann [12] (in the context of noncommutative geometry) and by others, among them Zahn [21] (in the context of quantum field theory on noncommutative spaces, see also [2]). More generally, the question can be raised in the context of the Weyl calculus (see e.g. Folland’s [6]): Every continuous operator from S (Rn) → S ′(Rn) has a Weyl symbol in S ′(R2n), and hence the composability of two operators is equivalent to the existence of the Weyl product of their symbols. Using the global or Gabor wavefront set, which is a microlocal concept adapted to tempered distributions and in particular measures the behaviour at infinity (ignored by the classical wavefront set), we give a sufficient criterion that guarantees the existence of the twisted convolution product that was first published in the doctoral thesis [18]. To give this result a broader audience and to embed it into context, we present this result and its proof in more detail here, give examples of topological algebras of tempered distributions with respect to the twisted convolution and the twisted convolution product and compare our result to the literature.

2 The G-wavefront set

Originally introduced by H¨ormander [10] to study the propagation of of singularities of quadratic hyperbolic operators, the G-wavefront set WFG (G for global or Gabor) is a microlocal tool that takes into account the behaviour of a (tempered) distribution at infinity. While the classical wavefont set measures the behaviour of a distribution in cones in the cotangent space T ∗Rn, the G-wavefront set is characterized by open subsets in the boundary S2n−1 ≃ (R2n \{0}/ ∼), with ∼ given by the dilation, of the compactification2 B2n of the cotangent space T ∗Rn ≃ R2n. For the reader’s convenience, we recall here the basic definition and properties. We choose the characterization of WFG in terms of short term Fourier transform, instead of the microlocal one, as it requires the least prior knowledge. Its equivalence to H¨ormander’s definition was proved in [17]. Let ψ ∈ S (Rn) \{0} be a window function, e.g. the (normalized) Gaussian. Then the short term Fourier transform of f ∈ S (Rn) is

1 −iξ·x V (f)(x, ξ)= 2 f(y)ψ(y − x) e dy (2.1) ψ (2π)n/ kψk2 Rn Z n 2n ′ n This gives a continuous map Vψ : S (R ) → S (R ) which extends to a map Vψ : S (R ) → S ′(R2n) by 1 V (u)(x, ξ)= 2 (u, M τ ψ) ψ (2π)n/ kψk2 ξ x iξ·x where Mξg(x)= e g(x) and τx denotes the translation and where the pairing is (u,g)= ′ u(g). Observe that for u ∈ S , (x, ξ) 7→ Vψ(u)(x, ξ) is a polynomially bounded continuous function. 2Taking the compactification Bn × Bn of T ∗Rn ≃ Rn × Rn, one derives the so-called SG-wavefront set. A detailed comparison and analysis of these wavefront sets can be found in [18] where it is also proved that the G-wavefront set coincides with Nakamura’s homogeneous wavefront set.

2 ′ n 2n Definition 2.1 Let u ∈ S (R ), let (x0, ξ0) ∈ R \{0}. Then (x0, ξ0) is not in the wavefront set WFG(u) of u if and only if for one (hence for all) ψ ∈ S \{0}, there is an 2n open cone Γ0 ⊂ R \{0} containing (x0, ξ0) such that

2 k (1 + k(x, ξ)k ) |Vψu(x, ξ)| is bounded on Γ0 for all k =0, 1, 2,... .

2n Observe that WFG is a closed conic (w.r.t. variable and covariable jointly) subset of R \{0}. Since the cones in the above definition are localized “at infinity” (i.e. in the compactifica- tion’s boundary), all local information is lost, e.g. the δ-distribution at a ∈ Rn always has n wavefront set WFG(δa) = {0}× (R \{0}) for any a. In fact, lack of smoothness at finite points leads to a contribution {0}× (Rn \{0}) to the wavefront set, lack of decay and linear oscillations at infinity are encoded in (Rn \{0}) ×{0}, while the rest of R2n \{0} encodes higher than linear oscillations at infinity (see [17, 18] and the discussion below).

′ n Proposition 2.2 (Properties of WFG [H¨ormander]) Let u ∈ S (R ). We have

′ (i) Regularity: u ∈ S ⊂ S ⇔ WFG(u)= ∅.

(ii) Symmetry under Fourier transform: (x, ξ) ∈ WFG(u) ⇔ (ξ, −x) ∈ WFG(F u) (iii) Multiplication by a chirp: For A a real, symmetric n × n matrix, we have

i T 2 x (Ax) (x, ξ) ∈ WFG(u) ⇔ (x, ξ + Ax) ∈ WFG(e u)

n m T (iv) Pullback: Let A : R → R be linear. If WFG ∩{(0, ξ)|A ξ = 0} = ∅, the pullback A∗u of u along A is uniquely defined as a tempered distribution and

∗ T WFG(A u) ⊂{(x, A ξ) | (Ax, ξ) ∈ WFG(u)} ∪ ker A ×{0}

Proof: Statement (i) is proved in [10], statements (ii) and (iii) in [11, Thm 18.5.9]. State- ment (iv) is proved in [10] (Proposition 2.3 and 2.9). Observe, however, that in the second of these propositions, the contribution from the kernel of A was accidentally omitted.

As an easy corollary observe that from (ii) above and the wavefront set of δa it follows that ia· n n WFG(e ) = (R \{0}) ×{0} for any a ∈ R . The G-wavefront set is appropriate to test whether the pointwise product of two tenpered distributions exists and is again temepered:

n Theorem 2.3 Let u, v ∈ S (R ) be such that (0, ξ) ∈ WFG(u) implies that (0, −ξ) 6∈ WFG(v). Then the pointwise product of u and v is uniquely defined as the extension of the pointwise product of two Schwartz functions and it is again tempered. It is given by the pullback diag∗(u ⊗ v) of the tensor product along the diagonal map diag(x) = (x, x).

The proof was not explicitly given in [10] which concentrated instead on pairings of distri- butions. It follows immediately from our more general theorem below.

3 3 The twisted convolution product on S ′

To extend the twisted convolution product to tempered distributions, we rewrite the product of f,g ∈ S (R2m) in terms of the tensor product and the pullback along the diagonal map,

i T F −1 F F − 2 k (θp) f ∗ g(x) = (k,p)→(x,y) (f)(k) ⊗ (g)(p) e x=y   i T   = diag∗ F −1 F (f)(k) ⊗ F (g)(p) e− 2 k (θp) (x) ∗  −1    = diag F ((F (f) ⊗ F (g)) · χθ ) (x) where the equality of the first line with (1.2) follows with a simple
variable transform (k → 2m 2m k − p) and where χθ is the chirp on R × R ,

0 θ χ (K) = exp(− i KT (ΘK)) , Θ= 1 θ 2 2 −θ 0   k T 1 T T T since for K = p , we have K (ΘK)= 2 (k θp − p θk)= k θp. Observe that indeed, we have ΘT = Θ.   We now prove our main result.

Theorem 3.1 Let θ be an antisymmetric n × n matrix, and let u, v ∈ S ′(Rn) be such that

1 x − 2 θξ = 0 (3.1) has no solution (x, ξ) with (x, ξ) ∈ WFG(u) and (x, −ξ) ∈ WFG(v). Then the twisted convolution product of u and v is defined as

∗ −1 u ∗θ v = diag F ((F (u) ⊗ F (v)) χθ ) and its G-wavefront set WFG(u ∗θ v) is contained in

1 1 (x + 2 θη,ξ + η) (x, ξ) ∈ WFG(u) ∪{0} , (y, η) ∈ WFG(v) ∪{0} ,y = x + 2 θ(η + ξ) \{0} n o It yields a sequentially continuous map S ′ × S ′ → S ′ . WFG(u) WFG(v) Γ

R2n S ′ Rn Here, for any closed cone Γ ⊂ \{0}, Γ( ) is the space of all tempered distributions with WFG contained in Γ, endowed with the H¨ormander topology.

Proof: From Proposition 2.2 (ii), we deduce that the wavefront set of F u⊗F v is contained in (ξ, η; −x, −y) | (x, ξ) ∈ WFG(u) ∪{0} and (y, η) ∈ WFG(v) ∪{0} \{0} and from Proposition 2.2 (iii), it follows that the wavefront set of (F u⊗F v) · χθ is contained in ξ x ξ ; − − Θ | (x, ξ) ∈ WF (u) ∪{0} and (y, η) ∈ WF (v) ∪{0} \{0} η y η G G         −x − 1 θη = 2 | {z 1 } −y + 2 θξ !

Taking the inverse Fourier transform then yields (again by Proposition 2.2 (ii))

1 1 x + 2 θη,y − 2 θξ; ξ, η | (x, ξ) ∈ WFG(u) ∪{0} and (y, η) ∈ WFG(v) ∪{0} \{0} 
4 −1 for the G-wavefront set of F ((F u ⊗ F v) · χθ). By Proposition 2.2 (iv), the pullback onto the diagonal is well-defined if the above has empty intersection with

{(0; ξ, η) | ξ + η =0} .

1 This is guaranteed by our assumption on the wavefront sets of u and v: If x + 2 θη =0= 1 y − 2 θξ and η = −ξ, then 1 x − 2 θξ =0 , (y, η) = (x, −ξ) and by assumption, the first equation does not have a solution (x, ξ) ∈ WFG(u) such that (x, −ξ) ∈ WFG(v). By Proposition 2.2 (iv), its wavefront set is contained in

−1 {(x, ξ + η) | (x, x; ξ, η) ∈ WFG(F ((F u ⊗ F v) · χθ))} ∪ ker(diag) ×{0} which in turn is equal to

1 1 (x + 2 θη,ξ + η) (x, ξ) ∈ WFG(u) ∪{0} , (y, η) ∈ WFG(v) ∪{0} ,y = x + 2 θ(η + ξ) \{0} n o R2n where we got rid of the total zero ker(diag) ×{0} = {0} ∈ , as it is by definition not contained in the wavefront set. The continuity statement follows from the sequential continuity of the elementary operations above which constitute the twisted convolution product. 

The statement in Theorem 2.3 for the pointwise product now follows automatically, by setting θ = 0. For the other extreme, where θ is invertible (hence n must be even), the condition on the wavefront sets in the theorem reads

−1 −1 (x, 2 θ x) ∈ WFG(u) implies (x, −2 θ x) 6∈ WFG(v) .

Observe that H¨ormander’s classical wavefront set would be unsuitable for this investigation, as – but for homogeneous distributions – it lacks the symmetric behaviour under Fourier transform which WFG possesses.

4 Algebras in S ′

An immediate consequence of our main theorem is the following:

Corollary 4.1 Let θ be an antisymmetric n × n matrix. Let Γ be a conic subset of R2n \ 0 such that (x, ξ) ∈ Γ implies that (x, −ξ) 6∈ Γ, then pointwise multiplication and twisted S ′ Rn convolution product of any two u, v ∈ Γ( ) yield well-defined tempered distributions.

Observe however, that the wavefront set of the resulting distributions will in general no S ′ longer be an element of Γ.

n Corollary 4.2 Let θ be an antisymmetric n × n matrix. Let Γ2 be a conic subset of R \ 0 n that is closed under addition and let Γ1 be a conic subset of R such that

1 x ∈ Γ1, ξ ∈ Γ2 implies x + 2 θξ ∈ Γ1 (4.1) S ′ Rn Then Γ1×Γ2 ( ) is an algebra under pointwise multiplication and under the twisted con- volution product.

5 Proof: Since Γ2 is closed under addition, but does not contain 0, condition (3.1) in the theorem is trivially satisfied. The pointwise product has wavefront set contained in Γ1 × Γ2, since Γ2 is closed under addition. This is also true for the twisted convolution product (with 1  θ non trivial) since by definition, x + 2 θη ∈ Γ1 for any η ∈ Γ2 and x ∈ Γ1.

0 −1 Example Let n = 2 and let θ = . LetΓ be the positive lightcone {(ξ , ξ )|ξ2 ≤ 1 0 2 0 1 0 2   ξ1 , ξ0 ≥ 0}. Then Γ2 is closed under addition, and ξ ∈ Γ2 implies −ξ 6∈ Γ2. Setting Γ1 equal to the left half space will satisfy the condition (4.1).

Obserse that these algebras are non-empty since for any closed closed conic set Γ in R2n \{0}, S ′ Rn there is a distribution u ∈ Γ( ), see e.g. [19]. We now consider some older results from the literature in view of our framework. To this end, we first observe that for f,g ∈ S , the twisted convolution 1.1 can be written as

−1 −1 f⋆g(y)= Fx→y((F f ∗ F g)(x)) (4.2) in terms of the twisted convolution product. We deduce:

S ′ Rn Corollary 4.3 The space {0}×Rn\{0}( ) forms an algebra under the twisted convolu- tion ⋆.

S ′ Rn To see this, let u, v ∈ {0}×Rn\{0}( ). We conclude from our theorem and the behaviour of WFG under Fourier transform (Proposition 2.2 (ii)) that the twisted convolution product of F −1u and F −1v is well-defined, since (3.1) has no solution at all with (x, ξ) ∈ Rn \{0}×{0}. We conclude moreover, that the wavefront set of F −1u∗F −1v is contained in Rn \{0}×{0}, and hence, the wavefront set we obtain by taking the additional Fourier transform (as prescribed by (4.2)) is again {0}× Rn \{0}.

′ Rn Consider in this context the following result from [5]: Let OM ( ) denote the space of n speedily decreasing distributions, i.e. the dual space of OM (R ), ∞ Rn Nn Z 2 k α {f ∈ C ( )| for any α ∈ 0 there is k ∈ s.t. (1 + |x| ) ∂ f is bbd} , ′ R2m endowed with the topology given by all these seminorms. Then for u, v ∈ OM ( ), the ′ R2m twisted convolution (1.1) u ⋆ v with the standard symplectic θ is again in OM ( ). To see the relation with Corollary 4.3, we first observe with [5] that S S ′ S ′ ′ S ′ ⊂ OC ⊂ OM ⊂ and ⊂ OM ⊂ OC ⊂

n 2 k α where OC(R ) is the space of smooth functions such that there is k ∈ Z s.t. (1+|x| ) ∂ f is ′ Rn Rn bounded for all multindices α, and that moreover, by Fourier transform, OM ( ) ≃ OC ( ). n It is easy to see that the G-wavefront set (in the sense of definition 2.1) of a ∈ OC (R ) is contained in Rn \{0}×{0}, as the polynomial growth in k(x, ξ)k can (for x 6= 0) be estimated by a linear combination of derivatives (w.r.t. x) of ψ(y − x) eixξ, and rewriting

β −iξx −iξy β −iξ(x−y) ∂x ψ(y − x) e = e ∂x ψ(y − x) e ,  
−iξy n the claim follows by integration by parts, using that y 7→ a(y) e is in OC (R ) for a ∈ Rn ′ Rn Rn OC ( ). From the isomorphism OM ( ) ≃ OC ( ) and the behaviour of WFG under ′ Fourier transform (Proposition 2.2(ii)) we conclude that the wavefront set of u ∈ OM is contained in {0}× Rn \{0}.

6 ′ Hence, we now obtain from corollary 4.3, that the twisted convolution of u, v ∈ OM is well n defined, and its G-wavefront set is again contained in {0}× R \{0}. Observe that if OC ′ n n were equal to {u ∈ S (R ) | WFG(u) ⊂ R \{0}×{0}}, then we would get exactly the ′ result of [5] that the twisted convolution u ⋆ v is again in OM . On the other hand, we cannot mimick the construction from [7, 8] in our context. There, the so-called Moyal algebra was given as the intersection of

′ ′ ML = {S ∈ S |S×f ∈ S for all f ∈ S } and MR = {S ∈ S |f×S ∈ S for all f ∈ S } with yet another (related) twisted product × given in terms of the standard symplectic form θ. If, in our context, we were to replace this product by the twisted convolution product and ′ ′ to consider ML,∗ := {u ∈ S |u∗f ∈ S for all f ∈ S }, then we would find that any u ∈ S has a well-defined twisted convolution product with any f ∈ S . However, the inclusion we have for the G-wavefront set of u∗f is too crude to say when exactly the twisted convolution product is a Schwartz function: Asking

1 1 (x + 2 θη,ξ + η) (x, ξ) ∈ WFG(u) ∪{0} , (y, η) ∈ WFG(f) ∪{0} ,y = x + 2 θ(η + ξ) \{0} n o S for f ∈ , i.e. WF G(f)= ∅, to be empty is too strong, as it amounts to requiring u to be Schwartz itself.

Nevertheless, the global wavefront set is a tool that provides a sufficent criterion for the existence and makes it possible to set up spaces of tempered distributions that form algebras under pointwise multiplication, ⋆ or ∗. It remains an interesting open problem to investigate other (exact) Rieffel deformations in this context, e.g. the ones studied in [12, 3].

References

[1] D. Bahns, S. Doplicher, K. Fredenhagen and G. Piacitelli, On the unitarity prob- lem in space/time noncommutative theories, Phys. Lett. B 533 (2002) pp. 178–181. doi:10.1016/S0370-2693(02)01563-0 [2] D. Bahns, The ultraviolet infrared mixing problem on the noncommutative Moyal space, arXiv:1012.3707, accepted for publication in AHP. [3] Pierre Bieliavsky, Axel de Goursac, Florian Spinnler, Non-formal deformation quantiza- tion and star-exponential of the Poincare Group, Proceedings of the conference ”XXXI Workshop on Geometric Methods in Physics”, Bialowieza, June 2012 [4] S. Doplicher, K. Fredenhagen and J. Roberts, The quantum structure of spacetime at the Planck scale and quantum fields, Comm. Math. Phys. 172 (1995), no. 1, 187–220. projecteuclid.org/euclid.cmp/1104273963 [5] M. Dubois-Violette, A. Kriegl, Y. Maeda, P. Michor, Smooth *-algebras, Progress of Theoretical Physics Supplement Number 144 (2001), 54-78. doi:10.1143/PTPS.144.54 [6] G. Folland, Harmonic Analysis in Phase Space. (AM-122), Volume 122, Princeton Uni- versity Press, 1989

[7] J. V´arilly and J. Gracia-Bond´ıa, Algebras of distributions suitable for phasespace quan- tum mechanics. I, J. Math. Phys. 29, 869 (1988). doi:10.1063/1.528200

7 [8] J. V´arilly and J. Gracia-Bond´ıa, Algebras of distributions suitable for phasespace quan- tum mechanics. II. Topologies on the Moyal algebra J. Math. Phys. 29, 880 (1988). doi:10.1063/1.527984 [9] H. J. Groenewold, On the Principles of elementary quantum mechanics, Physica 12 (1946) pp. 405 – 460. doi:10.1016/S0031-8914(46)80059-4. [10] L. H¨ormander, Quadratic hyperbolic operators, Microlocal Analysis and Applications, Lecture Notes in Mathematics vol. 1495, L. Cattabriga, L. Rodino (Eds.), 1991, 118– 160. [11] L. H¨ormander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer, 1994. [12] G. Lechner and S. Waldmann, Strict deformation quantization of locally convex algebras and modules, Journal of Geometry and Physics 99 (2016) pp. 111–144 doi:10.1016/j.geomphys.2015.09.013 [13] J. M. Maillard, On the twisted convolution product and the Weyl trans- formation of tempered distributions, J. Geom. Phys. 3 (1986), 231–261. doi:10.1016/0393-0440(86)90021-5

[14] J. E. Moyal, Quantum mechanics as a statistical theory, Proceedings of the Cambridge Philosophical Society, 45 (1949) pp. 99–124. doi:10.1017/S0305004100000487. [15] J. v. Neumann, Die Eindeutigkeit der Schr¨odingerschen Operatoren, Math. Ann. 104, (1931) pp. 570–578, doi:10.1007/BF01457956 [16] M. Rieffel, On the operator algebra for the space-time uncertainty relations, in Operator algebras and quantum field theory (Rome, 1996), 374–382, Internat. Press, Cambridge, MA, 1997. [17] L. Rodino and P. Wahlberg, The Gabor wave front set, Monatsh. Math. 173 (2014), 625-655, doi:10.1007/s00605-013-0592-0 [18] R. Schulz, Microlocal Analysis of Tempered Distributions Schulz, Doctoral thesis, G¨ottingen, 2014. hdl.handle.net/11858/00-1735-0000-0022-5F76-E [19] R. Schulz and P. Wahlberg, Equality of the homogeneous and the Gabor wave front set, Commun. Partial Differential Equations, 42:5, (2017) pp. 703–730, doi:10.1080/03605302.2017.1300173 [20] E.P. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) pp. 749–759. doi:10.1103/PhysRev.40.749 [21] J. Zahn, The wave front set of oscillatory integrals with inhomogeneous phase function, Journal of Pseudo-Differential Operators and Applications 2:101-113,2011, doi:10.1007/s11868-011-0024-7

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