Infinitesimal and infinite numbers as an approach to quantum mechanics

Vieri Benci; Lorenzo Luperi Baglini; Kyrylo Simonov

doi:10.22331/q-2019-05-03-137

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

Published:	2019-05-03, volume 3, page 137
Eprint:	arXiv:1901.10945v3
Doi:	https://doi.org/10.22331/q-2019-05-03-137
Citation:	Quantum 3, 137 (2019).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

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.

► BibTeX data

@article{Benci2019infinitesimal,
  doi = {10.22331/q-2019-05-03-137},
  url = {https://doi.org/10.22331/q-2019-05-03-137},
  title = {Infinitesimal and infinite numbers as an approach to quantum mechanics},
  author = {Benci, Vieri and Baglini, Lorenzo Luperi and Simonov, Kyrylo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {137},
  month = may,
  year = {2019}
}

► References

[1] F. Strocchi, An Introduction to the Mathematical Structure of Quantum Mechanics (World Scientific, 2005), DOI:doi.org/10.1142/7038.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1017/CBO9781139172110.005

[6] J. Harthong, Adv. Appl. Math. 2, 24 (1981), DOI:10.1016/0196-8858(81)90038-5.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1090/S0002-9947-1979-0534122-5

[10] F. Bagarello and S. Valenti, Int. J. Theor. Phys. 27, 557 (1988), DOI:10.1007/BF00668838.
https://doi.org/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.
https://doi.org/10.1007/978-3-642-88201-2

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

[13] V. Benci, Adv. Nonlinear Stud. 13, 461 (2013), DOI:10.1515/ans-2013-0212.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://ejde.math.txstate.edu/conf-proc/21/b1/benci.pdf

[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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1007/s40065-014-0114-5

[20] P. Ehrlich, Arch. Hist. Exact Sci. 60, 1 (2006), DOI:10.1007/s00407-005-0102-4.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1007/978-3-642-57186-2_12

[25] V. Benci, MatematicaMente 218–222 (2016–2017).
http://www.mathesis.verona.it/matematicamente/archivio-matematicamente/

[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.
https://doi.org/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.
https://doi.org/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).
http://sites.mathdoc.fr/OCLS/pdf/OCLS_1954__21__1_0.pdf

[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.
https://doi.org/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.
https://doi.org/10.1093/bjps/axw013

[36] F. Gieres, Rep. Prog. Phys. 63, 1893 (2000), DOI:10.1088/0034-4885/63/12/201.
https://doi.org/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.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1119/1.19051

[40] M. de Llano, A. Salazar and M. A. Solís, Rev. Mex. Phys. 51, 626 (2005).
https://rmf.smf.mx/pdf/rmf/51/6/51_6_626.pdf

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

[42] A. Farrell and B. P. van Zyl, Can. J. Phys. 88, 817 (2010), DOI:10.1139/P10-061.
https://doi.org/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.
https://doi.org/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.
https://doi.org/10.1006/jfan.1995.1068

[45] S. Albeverio and L. Nizhnik, Ukr. Math. J. 52, 664 (2000), DOI:10.1007/BF02487279.
https://doi.org/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.
https://doi.org/10.1007/s00023-017-0628-4

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

[48] R. M. Cavalcanti, Rev. Bras. Ensino Fis. 21, 336 (1999).
http://www.sbfisica.org.br/rbef/pdf/v21_336.pdf

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

Cited by

[1] Mirte van der Eyden, Tim Netzer, and Gemma De las Cuevas, "Halos and undecidability of tensor stable positive maps", Journal of Physics A: Mathematical and Theoretical 55 26, 264006 (2022).

[2] Raffaele Scandone, Lorenzo Luperi Baglini, and Kyrylo Simonov, "A characterization of singular Schrödinger operators on the half-line", Canadian Mathematical Bulletin 64 4, 923 (2021).

[3] Jeffrey Barrett and Isaac Goldbring, "Everettian Mechanics with Hyperfinitely Many Worlds", Erkenntnis 89 4, 1367 (2024).

[4] Vieri Benci, "An improved setting for generalized functions: fine ultrafunctions", arXiv:2105.01490, (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 13:59:55) and SAO/NASA ADS (last updated successfully 2026-08-09 13:59:56). The list may be incomplete as not all publishers provide suitable and complete citation data.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.