Skip to main content
SpringerLink
Log in

Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

The notion of spectral automorphism of an orthomodular lattice was introduced by Ivanov and Caragheorgheopol (Int. J. Theor. Phys. 49(12):3146–3152, 2010) to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spectral automorphisms finding previously missing characterizations of spectral automorphisms, discussing several examples and the possibility to construct such automorphisms in direct products or horizontal sums of lattices. A factorization of the spectrum of a spectral automorphism is found. The last part of the paper addresses the problem of the unitary time evolution of a system from the point of view of the spectral automorphisms theory. An analogue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Beltrametti, E.G., Casinelli, G.: The Logic of Quantum Mechanics. Addison–Wesley, Reading (1981)

    MATH  Google Scholar 

  2. Halmos, P.R.: Introduction to Hilbert Space and Theory of Spectral Multiplicity. Chelsea, New York (1951)

    MATH  Google Scholar 

  3. Ivanov, A., Caragheorgheopol, D.: Spectral automorphisms in quantum logics. Int. J. Theor. Phys. 49(12), 3146–3152 (2010). doi:10.1007/s10773-010-0369-3

    Article  MathSciNet  MATH  Google Scholar 

  4. Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)

    MATH  Google Scholar 

  5. Piron, C.: Foundations of Quantum Physics. Benjamin, Reading (1976)

    MATH  Google Scholar 

  6. Pták, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. VEDA/Kluwer Academic, Bratislava/Dordrecht (1991)

    MATH  Google Scholar 

  7. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Academic Press, New York (1975)

    MATH  Google Scholar 

  8. Varadarajan, V.S.: Geometry of Quantum Theory, vol. I. van Nostrand, Princeton (1968)

    MATH  Google Scholar 

  9. Wright, R.: The structure of projection-valued states: a generalization of Wigner’s theorem. Int. J. Theor. Phys. 16(8), 567–573 (1977)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, Technical University of Civil Engineering in Bucharest, 124 Lacul Tei blv., 020396, Bucharest, Romania

    Dan Caragheorgheopol

  2. “Ilie Murgulescu” Institute of Physical Chemistry, Romanian Academy, 202 Splaiul Independentei, 060021, Bucharest, Romania

    Dan Caragheorgheopol

  3. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, 166 27, Prague, Czech Republic

    Josef Tkadlec

Authors
  1. Dan Caragheorgheopol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Josef Tkadlec
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dan Caragheorgheopol.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Caragheorgheopol, D., Tkadlec, J. Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices. Int J Theor Phys 50, 3750–3760 (2011). https://doi.org/10.1007/s10773-011-0738-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10773-011-0738-6

Keywords

Search

Navigation