Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics
Abstract
We propose, by Schwartz Linear Algebra, a significant development of Laurent Schwartz Distribution Theory. The study is conducted by following the (natu- ral and straightforward) way of Weak Duality among topological vector spaces, aiming at the construction of a feasible, rigorous, quite elementary and manageable frame- work for Quantum Mechanics. It turns out that distribution spaces reveal themselves an environment more capable to help in Quantum Mechanics than previously thought. The goal of the research, introduced here, consists in showing that the most natural state-spaces of a quantum system, in the infinite dimensional case, are indeed dis- tribution spaces. Moreover, we show new, natural and straightforward mathematical structures that reproduce very closely several physical objects and many operational procedures required in Quantum Mechanics, systematizing the algorithms and nota- tions of Dirac Calculus, in such a way that it becomes a more versatile and more powerful tool, than we are used to think of.
References
Antoniou, I. and I. Prigogine (1993). Intrinsic irreversibility and integrability of dynamics. Physica A192, 443–464.
Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics. Wiley & Sons, New York.
Barros-Neto, J. (1973). An Introduction to the theory of distributions. Marcel Dekker, NewYork.
Boccara, N. (1990). Functional Analysis: An Introduction for Physicists. Academic Press, Boston.
Bourbaki, N. (1953). Topologie Générale. (Fascicule de Résultats). Hermann, Paris.
Bourbaki, N. (1955). Topological Vector Spaces. Hermann, Paris.
Bourbaki, N. (1965). Intégration. Chapitre 1 à 4. Hermann, Paris.
Carfì, D. (1996). Quantum operators and their action on tempered distributions. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 10, 1–20. Available as Researchgate Paper at https://www.researchgate.net/publication/210189140_Quantum_operators_and_their_action_on_tempered_distributions
Carfì, D. (1997). Principles of a generalization of the linear algebra in S’n . Booklets
of the Mathematics Institute of the Faculty of Economics, University of Messina 4, 1–14. Available as Researchgate Paper at https://www.researchgate.net/publication/210189138_Principles_of_a_generalization_of_the_linear_algebra_in_the_spaces_of_tempered_distributions
Carfì, D. (1998). SL-ultralinear operators and some their applications. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina, 1–13. Available as Researchgate Paper at https://www.researchgate.net/publication/210189135_SL-ultralinear_operators_and_some_their_applications
Carfì, D. (2000). SL-ultralinear operators. Annals of Economic Faculty, University of Messina 38, 195–214. Available as Researchgate Paper at https://www.researchgate.net/publication/210189128_SL-ultralinear_operators
Carfì, D. (2001a). S-Ultralinear Algebra in the space of tempered distributions. AAPP | Physical, Mathematical, and Natural Sciences 78-79 (1), 105–130. http://cab.unime.it/mus/664/
Carfì, D. (2001b). SL-ultradifferentiable Manifolds. Analele Universitatii Bucuresti - Seria Informatica 50, 21–31. Proceedings of the Centellian of Vranceanu. Available as Researchgate Paper at https://www.researchgate.net/publication/210189123_SL-ultradifferentiable_Manifolds
Carfì, D. (2002a). S-families, S-bases and the Fourier expansion theorem. Annals of Economic Faculty, University of Messina 40, 117–131. Available as Researchgate Paper at https://www.researchgate.net/publication/210189117_S-families_S-bases_and_the_Fourier_expansion_theorem
Carfì, D. (2002b). On the Schrödinger’s equation associated with an operator admitting a continuous eigenbasis. Rendiconti del Seminario Matematico di Messina 8 (series II), 221–232. Available as Researchgate Paper at https://www.researchgate.net/publication/210189120_On_the_Schredingers_equation_associated_with_an_operator_admitting_a_continouos_eigenbasis
Carfì, D. (2003a). Dirac-orthogonality in the space of tempered distributions. Journal of Computational and Applied Mathematics 153 (1-2), 99–107. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00634-9
Carfì, D. (2003b). S-ultralinear operators in Quantum Mechanics. In M. Moreau, E. Hideg, K. Martinàs, and D. Meyer (Eds.), Complex systems in natural and social sciences. (Proceedings of the 7th Workshop on Complex Systems in Natural and Social Sciences, Màtrafüred, Hungary, September 26-29, 2002), pp. 33–46. ELFT, Budapest. http://www.oszk.hu/mnbwww/K/0723/S.HTML#010. Also available as Researchgate Paper at https://dx.doi.org/10.13140/RG.2.1.2609.0965
Carfì, D. (2004a). S-bases and applications to Physics and Economics. Annals of Economic Faculty, University of Messina, 165–190. Available as Researchgate Paper at https://www.researchgate.net/publication/210189106_S-bases_and_applications_to_physics_and_economics
Carfì, D. (2004b). S-linear operators in quantum Mechanics and in Economics. Applied Sciences (APPS) 6 (1), 7–20. http://www.mathem.pub.ro/apps/v06/A06.htm
Carfì, D. (2004c). Tangent spaces on S-manifolds. Differential Geometry Dynamical Systems 6, 1–13. http://www.mathem.pub.ro/dgds/v06/D06-CAR3.pdf
Carfì, D. (2005a). S-diagonalizable operators in Quantum Mechanics. Glasnik Mathematicki 40 (2), 261–301. http://dx.doi.org/10.3336/gm.40.2.08
Carfì, D. (2005b). Quantum statistical systems with a continuous range of states. In M. Primicerio, R. Spigler, and V. Valante (Eds.), Applied and Industrial Mathematics in Italy (Proceedings of the 7th Conference,Venice, Italy, 20 – 24 September 2004), Volume 69 of Series on Advances in
Mathematics for Applied Sciences, pp. 189–200. World Scientific. http://dx.doi.org/10.1142/9789812701817_0018. Also available as Researchgate Paper at https://www.researchgate.net/publication/210189113_Quantum_statistical_systems_with_a_continuous_range_of_states
Carfì, D. (2006a). An S-Linear State Preference Model. In Communications to SIMAI, Volume 1, pp. 1–4. https://dx.doi.org/10.1685/CSC06037. Also available, in extended version, as Researchgate Paper at https://dx.doi.org/10.13140/RG.2.1.1006.2800
Carfì, D. (2006b). An S-Linear State Preference Model. Researchgate Paper, 1–10. https://dx.doi.org/10.13140/RG.2.1.1006.2800
Carfì, D. (2006c). S-convexity in the space of Schwartz distributions and applications. Rendiconti del Circolo Matematico di Palermo 77 (series II), 107–122. Available as Researchgate Paper at https://www.researchgate.net/publication/210189098_S-convexity_in_the_space_of_Schwartz_distributions_and_applications
Carfì, D. (2007a). Dyson formulas for Financial and Physical evolutions in S’n. Communications to SIMAI Congress 2, 1–10. https://dx.doi.org/10.1685/CSC06156
Carfì, D. (2007b). Feynman’s transition amplitudes in the space S’n. AAPP | Physical, Mathematical, and Natural Sciences 85 (1), 1–10. http://dx.doi.org/10.1478/C1A0701007
Carfì, D. (2007c). S-Linear Algebra in Economics and Physics. Applied Sciences (APPS) 9, 48–66. http://www.mathem.pub.ro/apps/v09/A09-CA.pdf
Carfì, D. (2007d). Topological characterizations of S-linearity. AAPP | Physical, Mathematical, and Natural Sciences 85 (2), 1–16. http://dx.doi.org/10.1478/C1A0702005
Carfì, D. (2008). Superpositions in Prigogine’s approach to irreversibility for physical and financial applications. AAPP | Physical, Mathematical, and Natural Sciences 86 (S1), 1–13. https://dx.doi.org/10.1478/C1S0801005
Carfì, D. (2010a). Foundations of Superposition Theory, Volume 1. Il Gabbiano. ISBN: 978-88-96293-11-9. https://dx.doi.org/10.13140/RG.2.1.3352.2642
Carfì, D. (2010b). The pointwise Hellmann-Feynman theorem. AAPP | Physical, Mathematical, and Natural Sciences 88 (1), 1–14. http://dx.doi.org/10.1478/C1A1001004
Carfì, D. (2011a). S-Bases in S-Linear Algebra. ArXiv Paper, 1–11. http://arxiv.org/abs/1104.3324
Carfì, D. (2011b). Multiplicative operators in the spaces of Schwartz families. ArXiv Paper, 1–15. http://arxiv.org/abs/1104.3908
Carfì, D. (2011c). Schwartz families in tempered distribution spaces. ArXiv Paper, 1–15. http://arxiv.org/abs/1104.4651
Carfì, D. (2011d). Schwartz Linear operators in distribution spaces. ArXiv Paper, 1–14. http://arxiv.org/abs/1104.3380
Carfì, D. (2011e). Spectral expansion of Schwartz linear operators. ArXiv Paper, 1–23. http://arxiv.org/abs/1104.3647
Carfì, D. (2011f). Summable families in tempered distribution spaces. ArXiv Paper, 1–7. http://arxiv.org/abs/1104.4660
Carfì, D. (2014a). Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics. Researchgate Paper, 1–6. https://dx.doi.org/10.13140/2.1.1447.1361
Carfì, D. (2014b). Quantum Mechanics and Dirac Calculus in Schwartz Distribution Spaces, vol. 1. Superpositions in Distribution Spaces, Postulates of Quantum Mechanics in S’n, Schwartz Linear Algebra, S-Representation in Quantum Mechanics, Dirac Orthogonality, S-Linear Quantum Statistics. Il Gabbiano. https://dx.doi.org/10.13140/2.1.4959.1360
Carfì, D. (2015). Spectral expansion of Schwartz linear operators. Researchgate Paper, 1–23. https://dx.doi.org/10.13140/RG.2.1.3688.7762
Carfì, D. (2016a). Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms. Journal of Mathematical Economics and Finance 2 (1(2)), 85–127. http://journals.aserspublishing.eu/jmef/article/view/590
Carfì, D. (2016b). Differential Geometry and Relativity Theories. Vol.1. Tangent vectors, tangent maps, paths, 1-forms. Il Gabbiano. ISBN: 978-88-96293-22-5.
Carfì, D. (2017). Differential Geometry and Relativity Theories vol 1 – Tangent vectors, derivatives, paths, 1-forms, vector fields. Lambert Academic Publishing. ISBN: 978-3-330-02885-2. https://www.lap-publishing.com/catalog/details/store/tr/book/978-3-330-02885-2/differential-geometry-and-relativity-theories-vol-1?search=978-3-330-02885-2
Carfì, D., Caterino, A., and Ceppitelli, R. (2016). State preference models and jointly continuous
utilities. In APLIMAT 2016 - 15th Conference on Applied Mathematics 2016, Proceedings, Volume 1, pp. 163–176. Slovak University of Technology in Bratislava. http://www.proceedings.com/29878.html
Carfì, D. and C. Germanà (1999a). S-nets in the space of tempered distributions and generated operators. Rendiconti del Seminario Matematico di Messina 6 (series II), 113–124. Available as Researchgate Paper at https://www.researchgate.net/publication/267462823_S-nets_in_the_space_of_tempered_distributions_and_generated_operators
Carfì, D. and C. Germanà (1999b). The space of multipliers of S0 and the S-families of tempered distributions. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 5, 1–15. Available as Researchgate Paper at https://www.researchgate.net/publication/210189132_The_space_of_multipliers_of_S%27_and_the_S-families_of_tempered_distributions
Carfì, D. and C. Germanà (2000a). S’-operators and generated families. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 2, 1–16. Available as Researchgate Paper at https://www.researchgate.net/publication/210189126_S%27-operators_and_generated_families
Carfì, D. and C. Germanà (2000b). On the coordinates in an ultra-linearly independent family. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 1, 1–15. Available as Researchgate Paper at https://www.researchgate.net/publication/210189127_On_the_coordinates_in_an_ultralinearly_independent_family
Carfì, D. and C. Germanà (2000c). The coordinate operator in SL-ultralinear algebra. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 4, 1–13. Available as Researchgate Paper at https://www.researchgate.net/publication/210189125_The_coordinate_operator_in_SL-ultralinear_algebra
Carfì, D. and C. Germanà (2003). Some properties of a new product in S’n. Journal of Computational and Applied Mathematics 153 (1-2), 109–118. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00635-0
Carfì, D. and M. Magaudda (2007). Superpositions in Distributions spaces. AAPP | Physical, Mathematical, and Natural Sciences 85 (2), 1–14. http://dx.doi.org/10.1478/C1A0702006
Carfi, D. and G. Orlando (2015). Transposable Schwartz families. Researchgate Paper, 1–15. https://dx.doi.org/10.13140/RG.2.1.4561.8643
Dieudonné, J. (1942). La dualité dans les espaces vectoriels topologiques. Annales scietifiques de l’École Normale Supérieure 3esérie 59, 107–139.
Dieudonné, J. and L. Schwartz (1949). La dualité dans les espaces (F) and (LF). Annales de l’Institut Fourier 1, 61–101.
Dirac, P. (1930). The Principles of Quantum Mechanics. Oxford, the Clarendon Press.
Horváth, J. (1966). Topological Vector Spaces and Distributions, Volume 1. Addison- Wesley Publishing Company.
Kesavan, S. (1989). Topics in Functional Analysis and Applications. Wiley, New Delhi.
Penrose, R. (2006). Quantum Mechanics: Foundations. Encyclopedia of Mathematical Physics, 260–265.
Prigogine, I. (1962). Non-Equilibrium Statistical Mechanics. Wiley, New York.
Prigogine, I. (1980). From Being to Becoming: Time and Complexity in the Physical Sciences. Freeman, San Francisco.
Prigogine, I. (1993). Le leggi del chaos. Laterza, Roma-Bari.
Schwartz, L. (1964). Functional Analysis. New York University, Courant Institute of Mathematical Sciences.
Schwartz, L. (1966a). Mathematics for the Physical Sciences. Hermann and Addison– Wesley.
Schwartz, L. (1966b). Théorie des Distributions. Hermann, Paris.
Schwartz, L. (1968). Application of distributions to the theory of elementary particles in quantum mechanics. Gordon and Breach, New York.
Schwartz, L. (1979). Analyse Hilbertienne. Hermann, Paris.
Schwartz, L. (2011). Oeuvres Scientifiques I, II, III. American Mathematical Society.
Shankar, R. (1994). Principles of Quantum Mechanics. Plenum Press, New York.
Shields, P. (1973). The Theory of Bernoulli Shifts. In Lectures in Mathematics. University of Chicago Press, Chicago.
Trèves, F. (2006). Topological Vector Spaces, Distributions and Kernels. Dover Books on Mathematics. Dover Publications.
Yosida, K. (1996). Functional Analysis (6th ed.). Classics in Mathematics. Springer.
Zeidler, E. (1995). Applied Functional Analysis, Volume 1. Springer Verlag.
Balescu, R. (1975). Equilibrium and Nonequilibrium Statistical Mechanics. Wiley & Sons, New York.
Barros-Neto, J. (1973). An Introduction to the theory of distributions. Marcel Dekker, NewYork.
Boccara, N. (1990). Functional Analysis: An Introduction for Physicists. Academic Press, Boston.
Bourbaki, N. (1953). Topologie Générale. (Fascicule de Résultats). Hermann, Paris.
Bourbaki, N. (1955). Topological Vector Spaces. Hermann, Paris.
Bourbaki, N. (1965). Intégration. Chapitre 1 à 4. Hermann, Paris.
Carfì, D. (1996). Quantum operators and their action on tempered distributions. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 10, 1–20. Available as Researchgate Paper at https://www.researchgate.net/publication/210189140_Quantum_operators_and_their_action_on_tempered_distributions
Carfì, D. (1997). Principles of a generalization of the linear algebra in S’n . Booklets
of the Mathematics Institute of the Faculty of Economics, University of Messina 4, 1–14. Available as Researchgate Paper at https://www.researchgate.net/publication/210189138_Principles_of_a_generalization_of_the_linear_algebra_in_the_spaces_of_tempered_distributions
Carfì, D. (1998). SL-ultralinear operators and some their applications. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina, 1–13. Available as Researchgate Paper at https://www.researchgate.net/publication/210189135_SL-ultralinear_operators_and_some_their_applications
Carfì, D. (2000). SL-ultralinear operators. Annals of Economic Faculty, University of Messina 38, 195–214. Available as Researchgate Paper at https://www.researchgate.net/publication/210189128_SL-ultralinear_operators
Carfì, D. (2001a). S-Ultralinear Algebra in the space of tempered distributions. AAPP | Physical, Mathematical, and Natural Sciences 78-79 (1), 105–130. http://cab.unime.it/mus/664/
Carfì, D. (2001b). SL-ultradifferentiable Manifolds. Analele Universitatii Bucuresti - Seria Informatica 50, 21–31. Proceedings of the Centellian of Vranceanu. Available as Researchgate Paper at https://www.researchgate.net/publication/210189123_SL-ultradifferentiable_Manifolds
Carfì, D. (2002a). S-families, S-bases and the Fourier expansion theorem. Annals of Economic Faculty, University of Messina 40, 117–131. Available as Researchgate Paper at https://www.researchgate.net/publication/210189117_S-families_S-bases_and_the_Fourier_expansion_theorem
Carfì, D. (2002b). On the Schrödinger’s equation associated with an operator admitting a continuous eigenbasis. Rendiconti del Seminario Matematico di Messina 8 (series II), 221–232. Available as Researchgate Paper at https://www.researchgate.net/publication/210189120_On_the_Schredingers_equation_associated_with_an_operator_admitting_a_continouos_eigenbasis
Carfì, D. (2003a). Dirac-orthogonality in the space of tempered distributions. Journal of Computational and Applied Mathematics 153 (1-2), 99–107. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00634-9
Carfì, D. (2003b). S-ultralinear operators in Quantum Mechanics. In M. Moreau, E. Hideg, K. Martinàs, and D. Meyer (Eds.), Complex systems in natural and social sciences. (Proceedings of the 7th Workshop on Complex Systems in Natural and Social Sciences, Màtrafüred, Hungary, September 26-29, 2002), pp. 33–46. ELFT, Budapest. http://www.oszk.hu/mnbwww/K/0723/S.HTML#010. Also available as Researchgate Paper at https://dx.doi.org/10.13140/RG.2.1.2609.0965
Carfì, D. (2004a). S-bases and applications to Physics and Economics. Annals of Economic Faculty, University of Messina, 165–190. Available as Researchgate Paper at https://www.researchgate.net/publication/210189106_S-bases_and_applications_to_physics_and_economics
Carfì, D. (2004b). S-linear operators in quantum Mechanics and in Economics. Applied Sciences (APPS) 6 (1), 7–20. http://www.mathem.pub.ro/apps/v06/A06.htm
Carfì, D. (2004c). Tangent spaces on S-manifolds. Differential Geometry Dynamical Systems 6, 1–13. http://www.mathem.pub.ro/dgds/v06/D06-CAR3.pdf
Carfì, D. (2005a). S-diagonalizable operators in Quantum Mechanics. Glasnik Mathematicki 40 (2), 261–301. http://dx.doi.org/10.3336/gm.40.2.08
Carfì, D. (2005b). Quantum statistical systems with a continuous range of states. In M. Primicerio, R. Spigler, and V. Valante (Eds.), Applied and Industrial Mathematics in Italy (Proceedings of the 7th Conference,Venice, Italy, 20 – 24 September 2004), Volume 69 of Series on Advances in
Mathematics for Applied Sciences, pp. 189–200. World Scientific. http://dx.doi.org/10.1142/9789812701817_0018. Also available as Researchgate Paper at https://www.researchgate.net/publication/210189113_Quantum_statistical_systems_with_a_continuous_range_of_states
Carfì, D. (2006a). An S-Linear State Preference Model. In Communications to SIMAI, Volume 1, pp. 1–4. https://dx.doi.org/10.1685/CSC06037. Also available, in extended version, as Researchgate Paper at https://dx.doi.org/10.13140/RG.2.1.1006.2800
Carfì, D. (2006b). An S-Linear State Preference Model. Researchgate Paper, 1–10. https://dx.doi.org/10.13140/RG.2.1.1006.2800
Carfì, D. (2006c). S-convexity in the space of Schwartz distributions and applications. Rendiconti del Circolo Matematico di Palermo 77 (series II), 107–122. Available as Researchgate Paper at https://www.researchgate.net/publication/210189098_S-convexity_in_the_space_of_Schwartz_distributions_and_applications
Carfì, D. (2007a). Dyson formulas for Financial and Physical evolutions in S’n. Communications to SIMAI Congress 2, 1–10. https://dx.doi.org/10.1685/CSC06156
Carfì, D. (2007b). Feynman’s transition amplitudes in the space S’n. AAPP | Physical, Mathematical, and Natural Sciences 85 (1), 1–10. http://dx.doi.org/10.1478/C1A0701007
Carfì, D. (2007c). S-Linear Algebra in Economics and Physics. Applied Sciences (APPS) 9, 48–66. http://www.mathem.pub.ro/apps/v09/A09-CA.pdf
Carfì, D. (2007d). Topological characterizations of S-linearity. AAPP | Physical, Mathematical, and Natural Sciences 85 (2), 1–16. http://dx.doi.org/10.1478/C1A0702005
Carfì, D. (2008). Superpositions in Prigogine’s approach to irreversibility for physical and financial applications. AAPP | Physical, Mathematical, and Natural Sciences 86 (S1), 1–13. https://dx.doi.org/10.1478/C1S0801005
Carfì, D. (2010a). Foundations of Superposition Theory, Volume 1. Il Gabbiano. ISBN: 978-88-96293-11-9. https://dx.doi.org/10.13140/RG.2.1.3352.2642
Carfì, D. (2010b). The pointwise Hellmann-Feynman theorem. AAPP | Physical, Mathematical, and Natural Sciences 88 (1), 1–14. http://dx.doi.org/10.1478/C1A1001004
Carfì, D. (2011a). S-Bases in S-Linear Algebra. ArXiv Paper, 1–11. http://arxiv.org/abs/1104.3324
Carfì, D. (2011b). Multiplicative operators in the spaces of Schwartz families. ArXiv Paper, 1–15. http://arxiv.org/abs/1104.3908
Carfì, D. (2011c). Schwartz families in tempered distribution spaces. ArXiv Paper, 1–15. http://arxiv.org/abs/1104.4651
Carfì, D. (2011d). Schwartz Linear operators in distribution spaces. ArXiv Paper, 1–14. http://arxiv.org/abs/1104.3380
Carfì, D. (2011e). Spectral expansion of Schwartz linear operators. ArXiv Paper, 1–23. http://arxiv.org/abs/1104.3647
Carfì, D. (2011f). Summable families in tempered distribution spaces. ArXiv Paper, 1–7. http://arxiv.org/abs/1104.4660
Carfì, D. (2014a). Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics. Researchgate Paper, 1–6. https://dx.doi.org/10.13140/2.1.1447.1361
Carfì, D. (2014b). Quantum Mechanics and Dirac Calculus in Schwartz Distribution Spaces, vol. 1. Superpositions in Distribution Spaces, Postulates of Quantum Mechanics in S’n, Schwartz Linear Algebra, S-Representation in Quantum Mechanics, Dirac Orthogonality, S-Linear Quantum Statistics. Il Gabbiano. https://dx.doi.org/10.13140/2.1.4959.1360
Carfì, D. (2015). Spectral expansion of Schwartz linear operators. Researchgate Paper, 1–23. https://dx.doi.org/10.13140/RG.2.1.3688.7762
Carfì, D. (2016a). Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms. Journal of Mathematical Economics and Finance 2 (1(2)), 85–127. http://journals.aserspublishing.eu/jmef/article/view/590
Carfì, D. (2016b). Differential Geometry and Relativity Theories. Vol.1. Tangent vectors, tangent maps, paths, 1-forms. Il Gabbiano. ISBN: 978-88-96293-22-5.
Carfì, D. (2017). Differential Geometry and Relativity Theories vol 1 – Tangent vectors, derivatives, paths, 1-forms, vector fields. Lambert Academic Publishing. ISBN: 978-3-330-02885-2. https://www.lap-publishing.com/catalog/details/store/tr/book/978-3-330-02885-2/differential-geometry-and-relativity-theories-vol-1?search=978-3-330-02885-2
Carfì, D., Caterino, A., and Ceppitelli, R. (2016). State preference models and jointly continuous
utilities. In APLIMAT 2016 - 15th Conference on Applied Mathematics 2016, Proceedings, Volume 1, pp. 163–176. Slovak University of Technology in Bratislava. http://www.proceedings.com/29878.html
Carfì, D. and C. Germanà (1999a). S-nets in the space of tempered distributions and generated operators. Rendiconti del Seminario Matematico di Messina 6 (series II), 113–124. Available as Researchgate Paper at https://www.researchgate.net/publication/267462823_S-nets_in_the_space_of_tempered_distributions_and_generated_operators
Carfì, D. and C. Germanà (1999b). The space of multipliers of S0 and the S-families of tempered distributions. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 5, 1–15. Available as Researchgate Paper at https://www.researchgate.net/publication/210189132_The_space_of_multipliers_of_S%27_and_the_S-families_of_tempered_distributions
Carfì, D. and C. Germanà (2000a). S’-operators and generated families. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 2, 1–16. Available as Researchgate Paper at https://www.researchgate.net/publication/210189126_S%27-operators_and_generated_families
Carfì, D. and C. Germanà (2000b). On the coordinates in an ultra-linearly independent family. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 1, 1–15. Available as Researchgate Paper at https://www.researchgate.net/publication/210189127_On_the_coordinates_in_an_ultralinearly_independent_family
Carfì, D. and C. Germanà (2000c). The coordinate operator in SL-ultralinear algebra. Booklets of the Mathematics Institute of the Faculty of Economics, University of Messina 4, 1–13. Available as Researchgate Paper at https://www.researchgate.net/publication/210189125_The_coordinate_operator_in_SL-ultralinear_algebra
Carfì, D. and C. Germanà (2003). Some properties of a new product in S’n. Journal of Computational and Applied Mathematics 153 (1-2), 109–118. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00635-0
Carfì, D. and M. Magaudda (2007). Superpositions in Distributions spaces. AAPP | Physical, Mathematical, and Natural Sciences 85 (2), 1–14. http://dx.doi.org/10.1478/C1A0702006
Carfi, D. and G. Orlando (2015). Transposable Schwartz families. Researchgate Paper, 1–15. https://dx.doi.org/10.13140/RG.2.1.4561.8643
Dieudonné, J. (1942). La dualité dans les espaces vectoriels topologiques. Annales scietifiques de l’École Normale Supérieure 3esérie 59, 107–139.
Dieudonné, J. and L. Schwartz (1949). La dualité dans les espaces (F) and (LF). Annales de l’Institut Fourier 1, 61–101.
Dirac, P. (1930). The Principles of Quantum Mechanics. Oxford, the Clarendon Press.
Horváth, J. (1966). Topological Vector Spaces and Distributions, Volume 1. Addison- Wesley Publishing Company.
Kesavan, S. (1989). Topics in Functional Analysis and Applications. Wiley, New Delhi.
Penrose, R. (2006). Quantum Mechanics: Foundations. Encyclopedia of Mathematical Physics, 260–265.
Prigogine, I. (1962). Non-Equilibrium Statistical Mechanics. Wiley, New York.
Prigogine, I. (1980). From Being to Becoming: Time and Complexity in the Physical Sciences. Freeman, San Francisco.
Prigogine, I. (1993). Le leggi del chaos. Laterza, Roma-Bari.
Schwartz, L. (1964). Functional Analysis. New York University, Courant Institute of Mathematical Sciences.
Schwartz, L. (1966a). Mathematics for the Physical Sciences. Hermann and Addison– Wesley.
Schwartz, L. (1966b). Théorie des Distributions. Hermann, Paris.
Schwartz, L. (1968). Application of distributions to the theory of elementary particles in quantum mechanics. Gordon and Breach, New York.
Schwartz, L. (1979). Analyse Hilbertienne. Hermann, Paris.
Schwartz, L. (2011). Oeuvres Scientifiques I, II, III. American Mathematical Society.
Shankar, R. (1994). Principles of Quantum Mechanics. Plenum Press, New York.
Shields, P. (1973). The Theory of Bernoulli Shifts. In Lectures in Mathematics. University of Chicago Press, Chicago.
Trèves, F. (2006). Topological Vector Spaces, Distributions and Kernels. Dover Books on Mathematics. Dover Publications.
Yosida, K. (1996). Functional Analysis (6th ed.). Classics in Mathematics. Springer.
Zeidler, E. (1995). Applied Functional Analysis, Volume 1. Springer Verlag.
Published
2017-04-02
How to Cite
CARFI, David.
Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics.
Journal of Mathematical Economics and Finance, [S.l.], v. 2, n. 2, p. 67-76, apr. 2017.
ISSN 2458-0813.
Available at: <https://journals.aserspublishing.eu/jmef/article/view/923>. Date accessed: 03 dec. 2024.
doi: https://doi.org/10.14505//jmef.v2.2(3).04.
Section
Articles
The Copyright Transfer Form to ASERS Publishing (The Publisher)
This form refers to the manuscript, which an author(s) was accepted for publication and was signed by all the authors.
The undersigned Author(s) of the above-mentioned Paper here transfer any and all copyright-rights in and to The Paper to The Publisher. The Author(s) warrants that The Paper is based on their original work and that the undersigned has the power and authority to make and execute this assignment. It is the author's responsibility to obtain written permission to quote material that has been previously published in any form. The Publisher recognizes the retained rights noted below and grants to the above authors and employers for whom the work performed royalty-free permission to reuse their materials below. Authors may reuse all or portions of the above Paper in other works, excepting the publication of the paper in the same form. Authors may reproduce or authorize others to reproduce the above Paper for the Author's personal use or for internal company use, provided that the source and The Publisher copyright notice are mentioned, that the copies are not used in any way that implies The Publisher endorsement of a product or service of an employer, and that the copies are not offered for sale as such. Authors are permitted to grant third party requests for reprinting, republishing or other types of reuse. The Authors may make limited distribution of all or portions of the above Paper prior to publication if they inform The Publisher of the nature and extent of such limited distribution prior there to. Authors retain all proprietary rights in any process, procedure, or article of manufacture described in The Paper. This agreement becomes null and void if and only if the above paper is not accepted and published by The Publisher, or is with drawn by the author(s) before acceptance by the Publisher.