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Orthomodularity and the incompatibility of relativity and quantum mechanics

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Abstract

We show that orthomodularity in general and non-existence of isotropic vectors in particular decisively yield the geometry of quantum mechanics and that a fundamental reason why quantum mechanics and relativity cannot be unified is because of the non-existence of isotropic vectors.

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Notes

  1. Non-zero vectors with zero norm.

  2. Let E be a precompact subset (in the norm topology) of an infinite dimensional, separable Hilbert space. Then there exists \((e_{n}) \) such that \(\sum \nolimits _{n=1}^{\infty }\left\langle x,e_{n}\right\rangle <\infty \) for all \(x\in E\).

  3. Not to be confused with a split space in theory of Quadratic Forms, a completely opposite concept.

References

  1. Ardnt, M., Hornberger, K.: Testing the limits of quantum mechanical superpositions. Nat. Phys. 10, 271–277 (2014)

    Article  Google Scholar 

  2. d Muynck, W.M.: Foundations of Quantum Mechanics: An Empiricist Approach. Kluwer Academic Publishers, Dordrecht. ISBN 1-4020-0932-1 (2002)

  3. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47(10), 777–780 (1935)

    Article  MATH  Google Scholar 

  4. Colbeck, R., Renner, R.: Is a system’s wave function in one-to-one correspondence with its elements of reality? Phys. Rev. Lett. 108, 150402 (2012)

    Article  Google Scholar 

  5. Lawden, D.F.: The Mathematical Principles of Quantum Mechanics. Dover Publications. ISBN: 978-0-48-644223-5 (2005)

  6. Efinger, H.J.: A nonlinear unitary framework for quantum state reduction. Department of Scientific Computing Technical Report Series (2005). https://www.cosy.sbg.ac.at/research/tr/2005-03_Efinger.pdf

  7. Rédei, M. (Editor) John von Neumann: Selected Letters, 27: History of Mathematics, Rhode Island, Am. Math. Soc. and Lon. Math. Soc. (2005)

  8. Bhatt, S.J.: A Seminorm with square property on a Banach Algebra is submultiplicative. Proc. Am. Math. Soc. 117(2), 435–438 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Piziak, R.: Mackey closure operator. J. Lond. Math. Soc. 2, 33–38 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  10. Piziak, R.: Sesquilinear forms in infinite dimensions. Pac. J. Math. 43(2), 475–481 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  11. Solèr, M.P.: Characterisation of Hilbert spaces by orthomodular spaces. Commun. Algebra 23(1), 219–243 (1995)

    Article  MATH  Google Scholar 

  12. Alvarez, J.A.: \(C\ast \)-algebras of operators in non-archimedean Hilbert spaces. Comment. Math. Univ. Carolin. 33(4), 573–580 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Shirbisheh, V.: Lectures on \(C\ast \) -algebras. arXiv:1211.3404 (2012, preprint)

  14. Malik, A.N., Kamran, T.: Non-isometric involutive anti-automorphisms. arXiv:1603.08498v1 (2016)

  15. Khrennikov, A.Y.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Springer, The Netherlands (1997)

    Book  MATH  Google Scholar 

  16. Hazewinkel, M. (ed.): Basis. Encyclopedia of Mathematics. Springer, Berlin (2001)

    Google Scholar 

  17. Brunet, O.: Orthogonality and dimensionality. Axioms 2, 477–489 (2013)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Department, Quaid-e-Azam University, Islamabad, Pakistan

    Abdullah Naeem Malik & Tayyab Kamran

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  1. Abdullah Naeem Malik
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  2. Tayyab Kamran
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Correspondence to Abdullah Naeem Malik.

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Malik, A.N., Kamran, T. Orthomodularity and the incompatibility of relativity and quantum mechanics. Quantum Stud.: Math. Found. 4, 171–179 (2017). https://doi.org/10.1007/s40509-016-0092-8

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  • DOI: https://doi.org/10.1007/s40509-016-0092-8

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