Abstract
The properties of the generalized Gelfand-Shilov spaces \(S_{{b_n}}^{{a_k}}\) are studied from the viewpoint of deformation quantization. We specify the conditions on the defining sequences (ak) and (bn) under which \(S_{{b_n}}^{{a_k}}\) is an algebra with respect to the twisted convolution and, as a consequence, its Fourier transformed space \(S_{{a_k}}^{{b_n}}\) is an algebra with respect to the Moyal star product. We also consider a general family of translation-invariant star products. We define and characterize the corresponding algebras of multipliers and prove the basic inclusion relations between these algebras and the duals of the spaces of ordinary pointwise and convolution multipliers. Analogous relations are proved for the projective counterpart of the Gelfand-Shilov spaces. A key role in our analysis is played by a theorem characterizing those spaces of type S for which the function exp(iQ(x)) is a pointwise multiplier for any real quadratic form Q.
Similar content being viewed by others
References
L. Álvarez-Gaumé and M. A. Vázquez-Mozo, “General properties of non-commutative field theories,” Nucl. Phys. B 668(1–2), 293–321 (2003).
M. A. Antonets, “The classical limit for Weyl quantization,” Lett. Math. Phys. 2(3), 241–245 (1978).
A. G. Athanassoulis, N. J. Mauser, and T. Paul, “Coarse-scale representations and smoothed Wigner transforms,” J. Math. Pures Appl. 91(3), 296–338 (2009).
S. Beiser, H. Römer, and S. Waldmann, “Convergence of the Wick star product,” Commun. Math. Phys. 272(1), 25–52 (2007).
F. A. Berezin and M. A. Shubin, The Schrödinger Equation (Mosk. Gos. Univ., Moscow, 1983; Kluwer, Dordrecht, 1991).
M. Blaszak and Z. Domański, “Phase space quantum mechanics,” Ann. Phys. 327(2), 167–211 (2012).
E. Brüning and S. Nagamachi, “Relativistic quantum field theory with a fundamental length,” J. Math. Phys. 45(6), 2199–2231 (2004).
M. Cappiello and J. Toft, “Pseudo-differential operators in a Gelfand-Shilov setting,” Math. Nachr. 290(5–6), 738–755 (2017).
M. Chaichian, M. N. Mnatsakanova, A. Tureanu, and Yu. Vernov, “Test functions space in noncommutative quantum field theory,” J. High Energy Phys. 2008(09), 125 (2008).
S. Doplicher, K. Fredenhagen, and J. E. Roberts, “The quantum structure of spacetime at the Planck scale and quantum fields,” Commun. Math. Phys. 172(1), 187–220 (1995).
V. Ya. Fainberg and M. A. Soloviev, “Causality, localizability, and holomorphically convex hulls,” Commun. Math. Phys. 57(2), 149–159 (1977).
G. B. Folland, Harmonic Analysis in Phase Space (Princeton Univ. Press, Princeton, NJ, 1989), Ann. Math. Stud. 122.
V. Gayral, J. M. Gracia-Bondía, B. Iochum, T. Schücker, and J. C. Várilly, “Moyal planes are spectral triples,” Commun. Math. Phys. 246(3), 569–623 (2004).
I. M. Gelfand and G. E. Shilov, Spaces of Fundamental and Generalized Functions (Fizmatgiz, Moscow, 1958; Academic, New York, 1968), Generalized Functions 2.
M. de Gosson, Symplectic Geometry and Quantum Mechanics (Birkhüauser, Basel, 2006).
J. M. Gracia-Bondía and J. C. Váarilly, “Algebras of distributions suitable for phase-space quantum mechanics. I,” J. Math. Phys. 29(4), 869–879 (1988).
A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires (Am. Math. Soc., Providence, RI, 1955), Mem. AMS, No. 16.
L. Hörmander, The Analysis of Linear Partial Differential Operators. I: Distribution Theory and Fourier Analysis (Springer, Berlin, 1983).
L. Hörmander, The Analysis of Linear Partial Differential Operators. III: Pseudo-differential Operators (Springer, Berlin, 1985).
H. Komatsu, “Projective and injective limits of weakly compact sequences of locally convex spaces,” J. Math Soc. Japan 19(3), 366–383 (1967).
H. Komatsu, “Ultradistributions. I: Structure theorems and a characterization,” J. Fac. Sci., Univ. Tokyo, Sect. IA 20(1), 25–105 (1973).
G. Köthe, Topological Vector Spaces. II (Springer, New York, 1979).
J. M. Maillard, “On the twisted convolution product and the Weyl transformation of tempered distributions,” J. Geom. Phys. 3(2), 231–261 (1986).
S. Mandelbrojt, “Sur un problème de Gelfand et Šilov,” Ann. Sci. Éc. Norm. Supér., Sér. 3, 77(2), 145–166 (1960).
R. Meise and D. Vogt, Introduction to Functional Analysis (Clarendon, Oxford, 1997).
B. S. Mitjagin, “Nuclearity and other properties of spaces of type S,” Am. Math. Soc. Transl., Ser. 2, 93, 45–59 (1970) [transl. from Tr. Mosk. Mat. Obshch. 9, 317–328 (1960)].
J. E. Moyal, “Quantum mechanics as a statistical theory,” Proc. Camb. Philos. Soc. 45, 99–124 (1949).
G. J. Murphy, C *-Algebras and Operator Theory (Academic, Boston, 1990).
S. Nagamachi and N. Mugibayashi, “Hyperfunction quantum field theory,” Commun. Math. Phys. 46(2), 119–134 (1976).
S. Nagamachi and N. Mugibayashi, “Hyperfunction quantum field theory. II: Euclidean Green’s functions,” Commun. Math. Phys. 49(3), 257–275 (1976).
J. von Neumann, “Die Eindeutigkeit der Schröodingerschen Operatoren,” Math. Ann. 104, 570–578 (1931).
V. P. Palamodov, “Fourier transforms of infinitely differentiable functions of rapid growth,” Tr. Mosk. Mat. Obshch. 11, 309–350 (1962).
S. Pilipović and B. Prangoski, “Anti-Wick and Weyl quantization on ultradistribution spaces,” J. Math. Pures Appl. 103(2), 472–503 (2015).
B. Prangoski, “Pseudodifferential operators of infinite order in spaces of tempered ultradistributions,” J. Pseudo-Diff. Oper. Appl. 4(4), 495–549 (2013).
Quantum Mechanics in Phase Space: An Overview with Selected Papers, Ed. by C. K. Zachos, D. B. Fairlie, and T. L. Curtright (World Scientific, Hackensack, NJ, 2005).
H. H. Schaefer, Topological Vector Spaces (MacMillan, New York, 1966).
N. Seiberg and E. Witten, “String theory and noncommutative geometry,” J. High Energy Phys. 1999(09), 032 (1999).
G. E. Shilov, “On a problem of quasianalyticity,” Dokl. Akad. Nauk SSSR 102(5), 893–895 (1955).
M. A. Shubin, Pseudodifferential Operators and Spectral Theory (Springer, Berlin, 1987), Springer Ser. Sov. Math.
A. G. Smirnov, “On topological tensor products of functional Fréchet and DF spaces,” Integral Transforms Spec. Funct. 20(3–4), 309–318 (2009).
M. A. Solov’ev, “On the Fourier-Laplace transformation of generalized functions,” Theor. Math. Phys. 15(1), 317–328 (1973) [transl. from Teor. Mat. Fiz. 15 (1), 3–19 (1973)].
M. A. Solov’ev, “Spacelike asymptotic behavior of vacuum expectation values in nonlocal field theory,” Theor. Math. Phys. 52(3), 854–862 (1982) [transl. from Teor. Mat. Fiz. 52 (3), 363–374 (1982)].
M. A. Soloviev, “An extension of distribution theory and of the Paley-Wiener-Schwartz theorem related to quantum gauge theory,” Commun. Math. Phys. 184(3), 579–596 (1997).
M. A. Soloviev, “Axiomatic formulations of nonlocal and noncommutative field theories,” Theor. Math. Phys. 147(2), 660–669 (2006) [transl. from Teor. Mat. Fiz. 147 (2), 257–269 (2006)].
M. A. Soloviev, “Star product algebras of test functions,” Theor. Math. Phys. 153(1), 1351–1363 (2007) [transl. from Teor. Mat. Fiz. 153 (1), 3–17 (2007)].
M. A. Soloviev, “Noncommutativity and θ-locality,” J. Phys. A: Math. Theor. 40(48), 14593–14604 (2007).
M. A. Soloviev, “Quantum field theory with a fundamental length: A general mathematical framework,” J. Math. Phys. 50(12), 123519 (2009).
M. A. Soloviev, “Reconstruction in quantum field theory with a fundamental length,” J. Math. Phys. 51(9), 093520 (2010).
M. A. Soloviev, “Moyal multiplier algebras of the test function spaces of type S,” J. Math. Phys. 52(6), 063502 (2011).
M. A. Soloviev, “Twisted convolution and Moyal star product of generalized functions,” Theor. Math. Phys. 172(1), 885–900 (2012) [transl. from Teor. Mat. Fiz. 172 (1), 9–27 (2012)].
M. A. Soloviev, “Generalized Weyl correspondence and Moyal multiplier algebras,” Theor. Math. Phys. 173(1), 1359–1376 (2012) [transl. from Teor. Mat. Fiz. 173 (1), 38–59 (2012)].
M. A. Soloviev, “Algebras with convergent star products and their representations in Hilbert spaces,” J. Math. Phys. 54(7), 073517 (2013).
M. A. Soloviev, “Star products on symplectic vector spaces: Convergence, representations, and extensions,” Theor. Math. Phys. 181(3), 1612–1637 (2014) [transl. from Teor. Mat. Fiz. 181 (3), 568–596 (2014)].
M. A. Soloviev, “Integral representations of the star product corresponding to the s-ordering of the creation and annihilation operators,” Phys. Scr. 90(7), 074008 (2015).
R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That (W. A. Benjamin, New York, 1964).
K. Takahashi, “Distribution functions in classical and quantum mechanics,” Prog. Theor. Phys., Suppl. 98, 109–156 (1989).
J. Toft, “Images of function and distribution spaces under the Bargmann transform,” J. Pseudo-Diff. Oper. Appl. 8(1), 83–139 (2017).
J. C. Váarilly and J. M. Gracia-Bondía, “Algebras of distributions suitable for phase-space quantum mechanics. II: Topologies on the Moyal algebra,” J. Math. Phys. 29(4), 880–887 (1988).
V. S. Vladimirov, Methods of the Theory of Functions of Many Complex Variables (Nauka, Moscow, 1964; M.I.T. Press, Cambridge, MA, 1966).
V. S. Vladimirov, Generalized Functions in Mathematical Physics (Nauka, Moscow, 1976; Mir, Moscow, 1979).
H. Weyl, The Theory of Groups and Quantum Mechanics (Dover Publ., New York, 1931).
M. W. Wong, Weyl Transforms (Springer, New York, 1998).
V. V. Zharinov, “Compact families of locally convex topological vector spaces, Fréchet-Schwartz and dual Fréechet-Schwartz spaces,” Russ. Math. Surv. 34(4), 105–143 (1979) [transl. from Usp. Mat. Nauk 34 (4), 97–131 (1979)].
Author information
Authors and Affiliations
Corresponding author
Additional information
This article was submitted by the author simultaneously in Russian and English
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 306, pp. 235–257.
Rights and permissions
About this article
Cite this article
Soloviev, M.A. Spaces of Type S as Topological Algebras under Twisted Convolution and Star Product. Proc. Steklov Inst. Math. 306, 220–241 (2019). https://doi.org/10.1134/S0081543819050195
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543819050195