On some consequences of the functional generalization of the parallelogram identity

• Autorzy: Mączyński M. M.
• Języki publikacji: EN

Abstrakty

EN

The aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy–Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.

Słowa kluczowe

EN

parallelogram identity; inner product space; Cauchy-Schwarz inequality; Heisenberg inequality; quantum mechanics

PL

reguła równoległoboku; nierówność Cauchy'ego-Schwarza; zasada nieoznaczoności Heisenberga; mechanika kwantowa

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 2
• Strony: 493--506
• Opis fizyczny: Bibliogr. 6 poz.

Twórcy

autor

Mączyński M. M.

• Faculty of Mathematics and Informatics, University of Białystok, 15-097 Białystok, Poland

Bibliografia

• [1] P. Jordan, J. von Neumann, On inner products in linear metric spaces, Annals of Math. 36 (1935), 719–723.
• [2] M. J. Mączyński, A functional characterization of inner product vector spaces, Demonstratio Math. 16 (1983), 797–803.
• [3] M. J. Mączyński, Orthomodularity in partially ordered vector spaces, Bull. Polish Acad. Sci. Math. 36 (1988), 299–306.
• [4] S. Gudder, Algebraic conditions for a function on an abelian groups, Lett. Math. Phys. 3 (1979), 127–133.
• [5] M. J. Mączyński, An abstract derivation of the inequality related to Heisenberg’s uncertainty principle, Rep. Math. Phys. 21 (1985), 281–289.
• [6] P. J. Lahti, M. J. Mączyński, Heisenberg inequality and the complex field in quantum mechanics, J. Math. Phys. 28 (1987), 1764–1769.