An introduction to transposable Schwartz families

  • David CARFI’ University of California Riverside, USA

Abstract




In this paper we start to construct some fundamental features of Dirac Calculus, specifically, we go inside the theory of Heisenberg continuous matrices, which, in our Schwartz Linear Algebra, are represented by Schwartz families. We distinguish the important subclass of transposable continuous matrices and give some basic and very important examples in Quantum Mechanics. So we define transposable Schwartz families and their transpose families, we prove the transposability of Dirac families and Fourier families. We find the transpose of regular-distribution families in a much general case. We define symmetric families, the analogous of symmetric ma- trices in the continuous case. We prove the symmetry of Dirac families and of Fourier families. We define Hermitian families, the analogous of Hermitian matrices in the continuous case. We prove the Hermitianity of Dirac families and of Fourier families. We define unitary families, the analogous of unitary matrices in the continuous case. We prove the unitarity of Dirac families and of the fundamental normalized de Broglie family. Then, we use the transpose of a family to find the components of the superpositions of transposable families, we give a general result and we apply this result to the Dirac families and the eigenfamilies of the vector-wave operator. We shall use the transposable families in next chapters to define the Dirac product in distribution spaces, basic product for the entire foundation of Dirac Calculus and Quantum Mechanics formalism.




References

[1]. I. Antoniou and I. Prigogine. Intrinsic irreversibility and integrability of dynamics. Physica, A192:443-464, 1993.
[2]. R. Balescu. Equilibrium and Nonequilibrium Statistical Mechanics. Wiley & Sons, New York, 1975.
[3]. J. Barros-Neto. An Introduction to the theory of distributions. Marcel Dekker, NewYork, 1973.
[4]. N. Boccara. Functional Analysis: An Introduction for Physicists. Academic Press, Boston, 1990.
[5]. N. Bourbaki. Topologie Générale. (Fascicule de Résultats). Hermann, Paris, 1953.
[6]. N. Bourbaki. Topological Vector Spaces. Hermann, Paris, 1955.
[7]. N. Bourbaki. Intégration. Chapitre 1 a 4. Hermann, Paris, 1965.
[8]. D. Carfì. Superpositions in Prigogine's approach to irreversibility for physical and financial applications. AAPP | Physical, Mathematical, and Natural Sciences, 86(S1):1-13, 2008. https://dx.doi.org/10.1478/C1S0801005.
[9]. D. Carfì. S-Ultralinear Algebra in the space of tempered distributions. AAPP | Physical, Mathematical, and Natural Sciences, 78-79(1):105-130, 2001. http://cab.unime.it/mus/664/.
[10]. D. Carfì. SL-ultradifferentiable Manifolds. Analele Universitatii Bucuresti -Seria Informatica, 50:21-31, 2001. Proceedings of the Centellian of Vranceanu.
[11]. D. Carfì. Dirac-orthogonality in the space of tempered distributions. Journal of Computational and Applied Mathematics, 153(1-2):99-107, 2003. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00634-9.
[12]. D. Carfì. S-ultralinear operators in Quantum Mechanics. In M. Moreau, E. Hideg, K. Martinas, and D. Meyer, editors, Complex systems in natural and social sciences. (Proceedings of the 7th Workshop on Complex Systems in Natural and Social Sciences, Matrafuured, Hungary, September 26-29, 2002), pages 33-46. ELFT, Budapest, 2003.
[13]. D. Carfì. S-linear operators in quantum Mechanics and in Economics. Applied Sciences (APPS), 6(1):7-20, 2004. http://www.mathem.pub.ro/apps/v06/A06.htm.
[14]. D. Carfì. Tangent spaces on S-manifolds. Differential Geometry Dynamical Systems, 6:1-13, 2004. http://www.mathem.pub.ro/dgds/v06/D06-CAR3.pdf.
[15]. D. Carfì. Quantum statistical systems with a continuous range of states. In M. Primicerio, R. Spigler, and V. Valente, editors, 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, pages 189-200. World Scientific, 2005. http://dx.doi.org/10.1142/9789812701817_0018.
[16]. D. Carfì. S-diagonalizable operators in Quantum Mechanics. Glasnik Mathematicki, 40(2):261-301, 2005. http://dx.doi.org/10.3336/gm.40.2.08.
[17]. D. Carfì. An S-Linear State Preference Model. In Communications to SIMAI, volume 1, pages 1-4, 2006. https://dx.doi.org/10.1685/CSC06037.
[18]. D. Carfì. S-convexity in the space of Schwartz distributions and applications. Rendiconti del Circolo Matematico di Palermo, 77(series II):107-122, 2006.
[19]. D. Carfì. Dyson formulas for Financial and Physical evolutions in S’n. Communications to SIMAI Congress, 2:1-10, 2007. https://dx.doi.org/10.1685/CSC06156.
[20]. D. Carfì. Feynman's transition amplitudes in the space S’n. AAPP | Physical, Mathematical, and Natural Sciences, 85(1):1-10, 2007. http://dx.doi.org/10.1478/C1A0701007.
[21]. D. Carfì. S-Linear Algebra in Economics and Physics. Applied Sciences (APPS), 9:48-66, 2007. http://www.mathem.pub.ro/apps/v09/A09-CA.pdf.
[22]. D. Carfì. Topological characterizations of S-linearity. AAPP | Physical, Mathematical, and Natural Sciences, 85(2):1-16, 2007. http://dx.doi.org/10.1478/C1A0702005.
[23]. D. Carfì. Foundations of Superposition Theory, volume 1. Il Gabbiano, 2010. ISBN: 978-88-96293-11-9.
[24]. D. Carfì. The pointwise Hellmann-Feynman theorem. AAPP | Physical, Mathematical, and Natural Sciences, 88(1):1-14, 2010. http://dx.doi.org/10.1478/C1A1001004.
[25]. D. Carfì. Multiplicative operators in the spaces of Schwartz families. ArXiv Paper, pages 1-15, 2011. http://arxiv.org/abs/1104.3908.
[26]. D. Carfì. S-Bases in S-Linear Algebra. ArXiv Paper, pages 1-11, 2011. http://arxiv.org/abs/1104.3324.
[27]. D. Carfì. Schwartz families in tempered distribution spaces. ArXiv Paper, pages 1-15, 2011. http://arxiv.org/abs/1104.4651.
[28]. D. Carfì. Schwartz Linear operators in distribution spaces. ArXiv Paper, pages 1-14, 2011. http://arxiv.org/abs/1104.3380.
[29]. D. Carfì. Spectral expansion of Schwartz linear operators. ArXiv Paper, pages 1-23, 2011. http://arxiv.org/abs/1104.3647.
[30]. D. Carfì. Summable families in tempered distribution spaces. ArXiv Paper, pages 1-7, 2011. http://arxiv.org/abs/1104.4660.
[31]. D. Carfì. 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, 2014. https://dx.doi.org/10.13140/2.1.4959.1360.
[32]. D. Carfì. Spectral expansion of Schwartz linear operators. Researchgate Paper, pages 1-23, 2015. https://dx.doi.org/10.13140/RG.2.1.3688.7762.
[33]. D. Carfì. Differential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms. Journal of Mathematical Economics and Finance, 2(1(2)):85-127, 2016. https://doi.org/10.14505/jmef.v2.1(2).05.
[34]. D. Carfì. Motivations and origins of Schwartz Linear Algebra in Quantum Mechanics. Journal of Mathematical Economics and Finance, 2(2(3)):67-76, 2016. https://doi.org/10.14505/jmef.v2.2(3).04.
[35]. D. Carfì. Filters and limits in filtered spaces. Journal of Mathematical Economics and Finance, 3(2(5)):33-81, 2017.
[36]. D. Carfì. Critical notes about Noether's theorem (Part I). Journal of Mathematical Economics and Finance, 4(2(7)):69-78, 2018. https://doi.org/10.14505/jmef.v4.2(7).04.
[37]. D. Carfì. Position operator. Journal of Mathematical Economics and Finance, 4(1(6)):79-87, 2018. https://doi.org/10.14505/jmef.v4.1(6).05.
[38]. D. Carfì. An introduction to S-Bases in S-Linear Algebra. Journal of Mathematical Economics and Finance, 5(2(9)):35-49, 2019. https://doi.org/10.14505/jmef.v5.2(9).03.
[39]. D. Carfì. Critical notes of Quantum Mechanics: Momentum Operators. Journal of Mathematical Economics and Finance, 5(1(8)):27-45, 2019. https://doi.org/10.14505/jmef.v5.1(8).03.
[40]. D. Carfì. Diracian structures, ket spaces and bras. Journal of Mathematical Economics and Finance, 6(2(11)):41-59, 2020. https://doi.org/10.14505/jmef.v6.2(11).03.
[41]. D. Carfì. Foundations of Superposition Theory. Extended Euclidean Structures in the Space of Tempered Distributions and Applications to Economics and Physics, volume 2. Il Gabbiano, 2020. ISBN: 978-88-96293-24-9.
[42]. D. Carfì. Lecture notes of Relativistic Mechanics: Worldlines, times and velocities of a particle. Journal of Mathematical Economics and Finance, 6(1(10)):57-73, 2020. https://doi.org/10.14505/jmef.v6.1(10).04.
[43]. D. Carfì and C. Germanà. Some properties of a new product in S’n. Journal of Computational and Applied Mathematics, 153(1-2):109-118, 2003. 6th International Symposium on Orthogonal Polynomials, Special Functions and Applications. Elsevier. http://dx.doi.org/10.1016/S0377-0427(02)00635-0.
[44]. D. Carfì and M. Magaudda. Superpositions in Distributions spaces. AAPP | Physical, Mathematical, and Natural Sciences, 85(2):1-14, 2007. http://dx.doi.org/10.1478/C1A0702006.
[45]. D. Carfì and G. Orlando. Transposable Schwartz families. Researchgate Paper, pages 1-15, 2015. https://dx.doi.org/10.13140/RG.2.1.4561.8643.
[46]. P. Cassam-Chenai. Geometric measure of indistinguishability for groups of identical particles. Physical Review, A77(3), 2008. https://dx.doi.org/10.1103/PhysRevA.77.032103.
[47]. P. Cassam-Chenai. Rayleigh-Schrodinger perturbation theory generalized to eigen-operators in non-commutative rings. Journal of Mathematical Chemistry, 49(4):821-835, 2011. https://dx.doi.org/10.1007/s10910-010-9779-y.
[48]. P. Cassam-Chenai. The generalized mean field configuration interaction method. HAL, 2013.
[49]. P. Cassam-Chenai. Spin contamination and noncollinearity in general complex Hartree-Fock wave functions. Theoretical Chemistry Accounts, 134, 2015. https://dx.doi.org/10.1007/s00214-015-1731-6.
[50]. P. Cassam-Chenai, Y. Bouret, M. Rey, S.A. Tashkun, A.V. Nikitin, and VL.G. Tyuterev. Ab initio effective rotational Hamiltonians: A comparative study. International Journal of Quantum Chemistry, 112(9):2201-2220, 2011. https://dx.doi.org/10.1002/qua.23183.
[51]. P. Cassam-Chenai and A. Ilmane. Frequently Asked Questions on the mean field configuration interaction method. I- Distinguishable degrees of freedom. Journal of Mathematical Chemistry, 50(3):652-667, 2012. https://dx.doi.org/10.1007/s10910-011-9912-6.
[52]. P. Cassam-Chenai and D. Jayatilaka. Contributions of the electronic spin and orbital current to the CoCl24- magnetic field probed in polarised neutron diffraction experiments. The Journal of Chemical Physics, 137(6), 2012. https://dx.doi.org/10.1063/1.4737894.
[53]. P. Cassam-Chenai and J. Lievin. Ab initio calculation of the rotational spectrum of methane vibrational ground state. The Journal of Chemical Physics, 136(17), 2012. https://dx.doi.org/10.1063/1.4705278.
[54]. P. Cassam-Chenai and J. Lievin. An improved third order dipole moment surface for methane. Journal of Molecular Spectroscopy, 291:77-84, 2013. https://dx.doi.org/10.1016/j.jms.2013.07.004.
[55]. P. Cassam-Chenai and V. Rassolov. The Electronic Mean-Field Configuration Interaction method: III- the p-orthogonality constraint. Chemical Physics Letters, 487(1-3):147-152, 2010. https://dx.doi.org/10.1016/j.cplett.2010.01.033.
[56]. P. Cassam-Chenai, Y. Scribano, and J. Lievin. Inuence of Kinetic Coupling in Rectilinear Coordinates on the Vibrational Spectrum of Fluoroform. Chemical Physics Letters, 466(1-3):16-20, 2008. https://dx.doi.org/10.1016/j.cplett.2008.10.025.
[57]. J.A. Dieudonn_e. La dualité dans les espaces vectoriels topologiques. Annales scietifiques de l' Ecole Normale Supérieure 3eserie, 59:107-139, 1942.
[58]. J.A. Dieudonné and L. Schwartz. La dualité dans les espaces (F) and (LF). Annales de l'Institut Fourier, 1:61-101, 1949.
[59]. P.A.M. Dirac. The Principles of Quantum Mechanics. Oxford, the Clarendon Press, 1930.
[60]. J. Horvath. Topological Vector Spaces and Distributions, volume 1. Addison-Wesley Publishing Company, 1966.
[61]. S. Kesavan. Topics in Functional Analysis and Applications. Wiley, New Delhi, 1989.
[62]. R. Penrose. Quantum Mechanics: Foundations. Encyclopedia of Mathematical Physics, pages 260-265, 2006.
[63]. I. Prigogine. Non-Equilibrium Statistical Mechanics. Wiley, New York, 1962.
[64]. I. Prigogine. From Being to Becoming: Time and Complexity in the Physical Sciences. Freeman, San Francisco, 1980.
[65]. I. Prigogine. Le leggi del chaos. Laterza, Roma-Bari, 1993.
[66]. L. Schwartz. Functional Analysis. New York University, Courant Institute of Mathematical Sciences, 1964.
[67]. L. Schwartz. Mathematics for the Physical Sciences. Hermann and Addison- Wesley, 1966.
[68]. L. Schwartz. Théorie des Distributions. Hermann, Paris, 1966.
[69]. L. Schwartz. Application of distributions to the theory of elementary particles in quantum mechanics. Gordon and Breach, New York, 1968.
[70]. L. Schwartz. Analyse Hilbertienne. Hermann, Paris, 1979.
[71]. L. Schwartz. Oeuvres Scientifiques I, II, III. American Mathematical Society, 2011.
[72]. R. Shankar. Principles of Quantum Mechanics. Plenum Press, New York, 1994.
[73]. P. Shields. The Theory of Bernoulli Shifts. In Lectures in Mathematics. University of Chicago Press, Chicago, 1973.
[74]. F. Strati. From the Bochner integral to the superposition integral. MPRA Paper 39615, 2012. https://mpra.ub.uni-muenchen.de/39615/.
[75]. F. Strati. On superpositional filtrations. MPRA Paper 40879, 2012. https://mpra.ub.uni-muenchen.de/40879/.
[76]. F. Strati. On defining S-spaces. AAPP - Physical, Mathematical, and Natural Sciences, 1(S2), 2013. http://dx.doi.org/10.1478/AAPP.91S2B2.
[77]. F. Tréves. Topological Vector Spaces, Distributions and Kernels. Dover Books on Mathematics. Dover Publications, 2006.
[78]. K. Yosida. Functional Analysis (6th ed.). Classics in Mathematics. Springer, 1996.
[79]. E. Zeidler. Applied Functional Analysis, volume 1. Springer Verlag, 1995.
Published
2021-06-30
How to Cite
CARFI’, David. An introduction to transposable Schwartz families. Journal of Mathematical Economics and Finance, [S.l.], v. 7, n. 1, p. 23 - 33, june 2021. ISSN 2458-0813. Available at: <https://journals.aserspublishing.eu/jmef/article/view/6488>. Date accessed: 30 nov. 2021. doi: https://doi.org/10.14505/jmef.v7.1(12).02.