Infinitesimal and infinite numbers as an approach to quantum mechanics

Vieri Benci1, Lorenzo Luperi Baglini2, and Kyrylo Simonov2

1Dipartimento di Matematica, Università degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy
2Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern Platz 1, 1090 Vienna, Austria

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Non-Archimedean mathematics is an approach based on fields which contain infinitesimal and infinite elements. Within this approach, we construct a space of a particular class of generalized functions, ultrafunctions. The space of ultrafunctions can be used as a richer framework for a description of a physical system in quantum mechanics. In this paper, we provide a discussion of the space of ultrafunctions and its advantages in the applications of quantum mechanics, particularly for the Schrödinger equation for a Hamiltonian with the delta function potential.

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[1] F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics (World Scientific, 2005),​10.1142/​7038.

[2] A. Robinson, Non-standard Analysis (North-Holland, 1974).

[3] L. O. Arkeryd, N. J. Cutland and C. W. Henson (Eds.), Nonstandard Analysis: Theory and Applications (Springer, 1997), DOI:10.1007/​978-94-011-5544-1.

[4] S. Albeverio, in Mathematics+Physics: Lectures on Recent Results, edited by L. Streit (World Scientific, 1986), Vol. 2, pp. 1–49, DOI:10.1142/​9789814503068_0001.

[5] S. Albeverio, in Nonstandard Analysis and its Applications, edited by N. Cutland (Cambridge University Press, 1988), pp. 182–220, DOI:10.1017/​CBO9781139172110.005.

[6] J. Harthong, Adv. Appl. Math. 2, 24 (1981), DOI:10.1016/​0196-8858(81)90038-5.

[7] J. Harthong, Études sur la mécanique quantique (Astérisque, Vol. 111, Société Mathématique de France, 1984), pp. 20–25.

[8] M. O. Farrukh, J. Math. Phys. 16, 177 (1975), DOI:10.1063/​1.522525.

[9] S. Albeverio, J. E. Fenstad and R. Høegh-Krohn, Trans. Amer. Math. Soc. 252, 275 (1979), DOI:10.1090/​S0002-9947-1979-0534122-5.

[10] F. Bagarello and S. Valenti, Int. J. Theor. Phys. 27, 557 (1988), DOI:10.1007/​BF00668838.

[11] S. Albeverio, F. Gesztesy, R. Høegh-Krohn and H. Holden, Solvable Models in Quantum Mechanics (Springer, 1988), DOI:10.1007/​978-3-642-88201-2.

[12] A. Raab, J. Math. Phys. 45, 47 (2004), DOI:10.1063/​1.1812358.

[13] V. Benci, Adv. Nonlinear Stud. 13, 461 (2013), DOI:10.1515/​ans-2013-0212.

[14] V. Benci and L. Luperi Baglini, Discrete Contin. Dyn. Syst. Ser. S 7, 593 (2014), DOI:10.3934/​dcdss.2014.7.593.

[15] V. Benci and L. Luperi Baglini, Monatsh. Math. 176, 503 (2014), DOI:10.1007/​s00605-014-0647-x.

[16] V. Benci and L. Luperi Baglini, in Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems, Flagstaff, Arizona, USA, 2012, edited by J. M. Neuberger, M. Chherti, P. Girg and P. Takac, Electron. J. Diff. Eqns., Conference 21 (2014), pp. 11–21.

[17] V. Benci and L. Luperi Baglini, in Analysis and Topology in Nonlinear Differential Equations, edited by D. G. Figueiredo, J. M. do Ó and C. Tomei (Birkhäuser, 2014), Vol. 85, pp. 61–86, DOI:10.1007/​978-3-319-04214-5_4.

[18] V. Benci, L. Luperi Baglini and M. Squassina, Adv. Nonlinear Anal. 9, 124 (2018), DOI:10.1515/​anona-2018-0146.

[19] V. Benci and L. Luperi Baglini, Arab. J. Math. 4, 231 (2015), DOI:10.1007/​s40065-014-0114-5.

[20] P. Ehrlich, Arch. Hist. Exact Sci. 60, 1 (2006), DOI:10.1007/​s00407-005-0102-4.

[21] R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis (Springer, 1998), DOI:10.1007/​978-1-4612-0615-6.

[22] P. Fletcher, K. Hrbacek, V. Kanovei, M. G. Katz, C. Lobry, and S. Sanders, Real Anal. Exch., 42, 193 (2017), DOI:10.14321/​realanalexch.42.2.0193.

[23] V. Benci, M. Di Nasso and M. Forti, in Nonstandard Methods and Applications in Mathematics, edited by N. J. Cutland, M. Di Nasso and D. A. Ross (A K Peters/​CRC Press, 2006), pp. 3–44, DOI:10.1017/​9781316755761.002.

[24] V. Benci, in Calculus of Variations and Partial Differential Equations, edited by G. Buttazzo, A. Marino and M. K. V. Murthy (Springer, 2000), pp. 285–326, DOI:10.1007/​978-3-642-57186-2_12.

[25] V. Benci, MatematicaMente 218–222 (2016–2017).

[26] V. Benci, Alla scoperta dei numeri infinitesimi: Lezioni di analisi matematica esposte in un campo non-archimedeo (Aracne editrice, 2018).

[27] V. Benci, I numeri e gli insiemi etichettati (Conferenze del seminario di matematica dell'Università di Bari, Vol. 261, Laterza, 1995).

[28] V. Benci and M. Di Nasso, Adv. Math. 173, 50 (2003), DOI:10.1016/​S0001-8708(02)00012-9.

[29] V. Benci, M. Di Nasso and M. Forti, Ann. Pure Appl. Logic 143, 43 (2006), DOI:10.1016/​j.apal.2006.01.008.

[30] V. Benci and M. Forti, The Euclidean numbers, in preparation.

[31] V. Benci, An improved setting for generalized functions: robust ultrafunctions, in preparation.

[32] L. Schwartz, C. R. Acad. Sci. Paris 239, 847 (1954).

[33] J. F. Colombeau, Elementary Introduction to New Generalized Functions (North Holland, 1985).

[34] V. Benci, L. Horsten and S. Wenmackers, Milan J. Math. 81, 121 (2013), DOI:10.1007/​s00032-012-0191-x.

[35] V. Benci, L. Horsten and S. Wenmackers, Brit. J. Phil. Sci. 69, 509 (2018), DOI:10.1093/​bjps/​axw013.

[36] F. Gieres, Rep. Prog. Phys. 63, 1893 (2000), DOI:10.1088/​0034-4885/​63/​12/​201.

[37] S. Flügge, Practical Quantum Mechanics (Springer, 1999), pp. 35–40, DOI:10.1007/​978-3-642-61995-3.

[38] M. Belloni and R. W. Robinett, Phys. Rep. 540, 25 (2014), DOI:10.1016/​j.physrep.2014.02.005.

[39] I. Mitra, A. DasGupta and B. Dutta-Roy, Am. J. Phys. 66, 1101 (1998), DOI:10.1119/​1.19051.

[40] M. de Llano, A. Salazar and M. A. Solís, Rev. Mex. Phys. 51, 626 (2005).

[41] S. Geltman, J. Atom. Mol. Opt. Phys. 2011, 573179 (2011), DOI:10.1155/​2011/​573179.

[42] A. Farrell and B. P. van Zyl, Can. J. Phys. 88, 817 (2010), DOI:10.1139/​P10-061.

[43] R. Jackiw, in M. A. B. Bég Memorial Volume, edited by A. Ali and P. Hoodboy (World Scientific, 1991), pp. 25–42, DOI:10.1142/​1447.

[44] S. Albeverio, Z. Brzeźniak and L. D\c abrowski, J. Funct. Anal. 130, 220 (1995), DOI:10.1006/​jfan.1995.1068.

[45] S. Albeverio and L. Nizhnik, Ukr. Math. J. 52, 664 (2000), DOI:10.1007/​BF02487279.

[46] G. Dell'Antonio, A. Michelangeli, R. Scandone and K. Yajima, Ann. Henri Poincaré 19, 283 (2018), DOI:10.1007/​s00023-017-0628-4.

[47] R. Scandone, arXiv:1901.02449 (2019).

[48] R. M. Cavalcanti, Rev. Bras. Ensino Fis. 21, 336 (1999).

[49] S.-L. Nyeo, Am. J. Phys. 68, 571 (2000), DOI:10.1119/​1.19485.

Cited by

[1] Raffaele Scandone, Lorenzo Luperi Baglini, and Kyrylo Simonov, "A characterization of singular Schrödinger operators on the half-line", Canadian Mathematical Bulletin 1 (2020).

[2] Vieri Benci, "An improved setting for generalized functions: fine ultrafunctions", arXiv:2105.01490.

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