• Skip to content (access key 1)
  • Skip to search (access key 7)
FWF — Austrian Science Fund
  • Go to overview page Discover

    • Research Radar
    • Discoveries
      • Emmanuelle Charpentier
      • Adrian Constantin
      • Monika Henzinger
      • Ferenc Krausz
      • Wolfgang Lutz
      • Walter Pohl
      • Christa Schleper
      • Anton Zeilinger
    • scilog Magazine
    • Austrian Science Awards
      • FWF Wittgenstein Awards
      • FWF START Awards
    • excellent=austria
      • Clusters of Excellence
      • Emerging Fields
    • In the Spotlight
      • 40 Years of Erwin Schrödinger Fellowships
      • Quantum Austria
    • Dialogs and Talks
      • think.beyond Summit
    • E-Book Library
  • Go to overview page Funding

    • Portfolio
      • excellent=austria
        • Clusters of Excellence
        • Emerging Fields
      • Projects
        • Principal Investigator Projects
        • Principal Investigator Projects International
        • Clinical Research
        • 1000 Ideas
        • Arts-Based Research
        • FWF Wittgenstein Award
      • Careers
        • ESPRIT
        • FWF ASTRA Awards
        • Erwin Schrödinger
        • Elise Richter
        • Elise Richter PEEK
        • doc.funds
        • doc.funds.connect
      • Collaborations
        • Specialized Research Groups
        • Special Research Areas
        • Research Groups
        • International – Multilateral Initiatives
        • #ConnectingMinds
      • Communication
        • Top Citizen Science
        • Science Communication
        • Book Publications
        • Digital Publications
        • Open-Access Block Grant
      • Subject-Specific Funding
        • AI Mission Austria
        • Belmont Forum
        • ERA-NET HERA
        • ERA-NET NORFACE
        • ERA-NET QuantERA
        • ERA-NET TRANSCAN
        • Alternative Methods to Animal Testing
        • European Partnership Biodiversa+
        • European Partnership ERA4Health
        • European Partnership ERDERA
        • European Partnership EUPAHW
        • European Partnership FutureFoodS
        • European Partnership OHAMR
        • European Partnership PerMed
        • European Partnership Water4All
        • Gottfried and Vera Weiss Award
        • netidee SCIENCE
        • Herzfelder Foundation Projects
        • Quantum Austria
        • Rückenwind Funding Bonus
        • WE&ME Award
        • Zero Emissions Award
      • International Collaborations
        • Belgium/Flanders
        • Germany
        • France
        • Italy/South Tyrol
        • Japan
        • Luxembourg
        • Poland
        • Switzerland
        • Slovenia
        • Taiwan
        • Tyrol–South Tyrol–Trentino
        • Czech Republic
        • Hungary
    • Step by Step
      • Find Funding
      • Submitting Your Application
      • International Peer Review
      • Funding Decisions
      • Carrying out Your Project
      • Closing Your Project
      • Further Information
        • Integrity and Ethics
        • Inclusion
        • Applying from Abroad
        • Personnel Costs
        • PROFI
        • Final Project Reports
        • Final Project Report Survey
    • FAQ
      • Project Phase PROFI
        • Accounting for Approved Funds
        • Labor and Social Law
        • Project Management
      • Project Phase Ad Personam
        • Accounting for Approved Funds
        • Labor and Social Law
        • Project Management
      • Expiring Programs
        • FWF START Awards
  • Go to overview page About Us

    • Mission Statement
    • FWF Video
    • Values
    • Facts and Figures
    • Annual Report
    • What We Do
      • Research Funding
        • Matching Funds Initiative
      • International Collaborations
      • Studies and Publications
      • Equal Opportunities and Diversity
        • Objectives and Principles
        • Measures
        • Creating Awareness of Bias in the Review Process
        • Terms and Definitions
        • Your Career in Cutting-Edge Research
      • Open Science
        • Open-Access Policy
          • Open-Access Policy for Peer-Reviewed Publications
          • Open-Access Policy for Peer-Reviewed Book Publications
          • Open-Access Policy for Research Data
        • Research Data Management
        • Citizen Science
        • Open Science Infrastructures
        • Open Science Funding
      • Evaluations and Quality Assurance
      • Academic Integrity
      • Science Communication
      • Philanthropy
      • Sustainability
    • History
    • Legal Basis
    • Organization
      • Executive Bodies
        • Executive Board
        • Supervisory Board
        • Assembly of Delegates
        • Scientific Board
        • Juries
      • FWF Office
    • Jobs at FWF
  • Go to overview page News

    • News
    • Press
      • Logos
    • Calendar
      • Post an Event
      • FWF Informational Events
    • Job Openings
      • Enter Job Opening
    • Newsletter
  • Discovering
    what
    matters.

    FWF-Newsletter Press-Newsletter Calendar-Newsletter Job-Newsletter scilog-Newsletter

    SOCIAL MEDIA

    • LinkedIn, external URL, opens in a new window
    • Twitter, external URL, opens in a new window
    • Facebook, external URL, opens in a new window
    • Instagram, external URL, opens in a new window
    • YouTube, external URL, opens in a new window

    SCILOG

    • Scilog — The science magazine of the Austrian Science Fund (FWF)
  • elane login, external URL, opens in a new window
  • Scilog external URL, opens in a new window
  • de Wechsle zu Deutsch

  

Orthogonality and Symmetry

Orthogonality and Symmetry

Thomas Vetterlein (ORCID: 0000-0003-0571-9551)
  • Grant DOI 10.55776/PIN5424624
  • Funding program Principal Investigator Projects International
  • Status ongoing
  • Start April 1, 2025
  • End March 31, 2028
  • Funding amount € 370,556
  • Project website
  • E-mail

Weave: Österreich - Belgien - Deutschland - Luxemburg - Polen - Schweiz - Slowenien - Tschechien

Disciplines

Mathematics (80%); Physics, Astronomy (20%)

Keywords

    Orthoset, Orthomodular Lattice, Dagger Category, Mathematical Foundations Of Quantum Mechanics, Hilbert lattice

Abstract

Quantum physics is a theory that takes some getting used to and its use may be accompanied by a certain discomfort until it has become routine. Over the years, countless approaches have been proposed to justify the peculiar formalism, but the discussion on the foundations is still ongoing. Our project work is intended to be a further contribution to this issue. We start at the lowest possible level. We are less concerned with physics itself than with the most important structures employed in quantum physics and we adopt a mathematical perspective. The basic model of quantum physics is the complex Hilbert space or, more generally, a C*-algebra. The definitions of both these structures are not well comprehensible without a firm mathematical background and the question arises as to whether the structures cannot be reduced to simpler ones. We take up the long-standing efforts that were once initiated by a work of Birkhoff and von Neumann, who proposed to describe the Hilbert space by a means of certain algebra that emerges from the set of its subspaces. Our work circles around the notion of orthogonality. The focus is thus on a concept that occurs in mathematics at numerous places. Although the term is linked to a clear geometric concept two vectors that form a right angle are orthogonal , its role in mathematics is not so easy to grasp. In quantum physics, the (pure) states of a physical system are described by vectors of a Hilbert space and orthogonality means that the transition probability from one state to the other as a result of a measurement is zero. Remarkably, this binary relation determines the structure; in a certain sense, the Hilbert space is reducible to the concept of orthogonality. Equipped with the orthogonality relation alone, a Hilbert space forms a so-called orthoset and everything else can be reconstructed from it. Part of the project is to describe the relevant type of orthosets in the simplest possible way. Similarly, a C*-algebra can also be assigned an orthoset, although in this case the orthoset alone is not sufficient for a description. A further part of the project is to find ways to extend the description without great complications. What kind of properties are eligible in order to specify a certain orthoset? This is where a further keyword comes into play: symmetry. Structure-preserving self-mappings are of central importance in physics and likewise in this project. The notion of an orthoset is as general as that of an undirected graph; the notion of symmetry refers to the vast theory of groups; both together, however, seem to offer an extraordinary potential when it comes to the characterisation of standard structures used in physics.

Research institution(s)
  • Universität Linz - 79%
  • Technische Universität Wien - 21%
Project participants
  • Bert Lindenhovius, Slovak Academy of Sciences , national collaboration partner
  • Helmut Länger, Technische Universität Wien , national collaboration partner
  • Karl Svozil, Technische Universität Wien , national collaboration partner
  • Mike Behrisch, Technische Universität Wien , associated research partner
International project participants
  • Andre Kornell, Dalhousie University - Canada
  • Jan Kühr, Palacky University - Czechia
  • Ivan Chajda, Palacky University - Czechia
  • Dominik Lachman, Palacky University - Czechia
  • Milan Matoušek - Czechia
  • David Kruml, Masarykova Univerzita - Czechia
  • Mirko Navara, Czech Technical University - Czechia, international project partner
  • Mirko Navara, Czech Technical University - Czechia
  • Jan Paseka, Masarykova Univerzita - Czechia
  • Isar Stubbe, Universite du Littoral Cote d Opale - France
  • Daniele Mundici, University of Florence - Italy
  • Antonio Ledda, Università degli Studi di Cagliari - Italy
  • Antonio Di Nola, Università degli Studi di Salerno - Italy
  • Giuseppina Barbieri, Università degli Studi di Salerno - Italy
  • Anatolij Dvurecenskij, Slovak Academy of Sciences - Slovakia
  • Bert Lindenhovius, Slovak Academy of Sciences - Slovakia
  • John Harding, New Mexico State University - USA
  • Chris Heunen, University of Edinburgh - United Kingdom

Research Output

  • 5 Citations
  • 8 Publications
Publications
  • 2025
    Title Foulis Quantales and Complete Orthomodular Lattices
    DOI 10.1007/978-3-031-97225-6_25
    Type Book Chapter
    Author Botur M
    Publisher Springer Nature
    Pages 309-321
  • 2025
    Title Adjointable maps between linear orthosets
    DOI 10.1016/j.jmaa.2025.129494
    Type Journal Article
    Author Paseka J
    Journal Journal of Mathematical Analysis and Applications
    Pages 129494
  • 2025
    Title Induced Orthogonality in Semilattices with 0 and in Pseudocomplemented Lattices and Posets
    DOI 10.1007/s11083-025-09696-y
    Type Journal Article
    Author Chajda I
    Journal Order
    Pages 1-16
    Link Publication
  • 2025
    Title Categories of Orthosets and Adjointable Maps
    DOI 10.1007/s10773-025-06031-4
    Type Journal Article
    Author Paseka J
    Journal International Journal of Theoretical Physics
    Pages 164
    Link Publication
  • 2025
    Title Foulis m-semilattices and their modules
    DOI 10.1109/ismvl64713.2025.00044
    Type Conference Proceeding Abstract
    Author Botur M
    Pages 196-201
  • 2025
    Title All minimal clones generated by {0, 1}-valued majority operations on a five-element set
    DOI 10.1109/ismvl64713.2025.00042
    Type Conference Proceeding Abstract
    Author Behrisch M
    Pages 184-189
  • 2025
    Title Many-valued aspects of tense and related operators
    DOI 10.1016/j.fss.2025.109509
    Type Journal Article
    Author Botur M
    Journal Fuzzy Sets and Systems
    Pages 109509
  • 2025
    Title A Dagger Kernel Category of Complete Orthomodular Lattices
    DOI 10.1007/s10773-025-05965-z
    Type Journal Article
    Author Botur M
    Journal International Journal of Theoretical Physics
    Pages 111

Discovering
what
matters.

Newsletter

FWF-Newsletter Press-Newsletter Calendar-Newsletter Job-Newsletter scilog-Newsletter

Contact

Austrian Science Fund (FWF)
Georg-Coch-Platz 2
(Entrance Wiesingerstraße 4)
1010 Vienna

office(at)fwf.ac.at
+43 1 505 67 40

General information

  • Job Openings
  • Jobs at FWF
  • Press
  • Philanthropy
  • scilog
  • FWF Office
  • Social Media Directory
  • LinkedIn, external URL, opens in a new window
  • Twitter, external URL, opens in a new window
  • Facebook, external URL, opens in a new window
  • Instagram, external URL, opens in a new window
  • YouTube, external URL, opens in a new window
  • Cookies
  • Whistleblowing/Complaints Management
  • Accessibility Statement
  • Data Protection
  • Acknowledgements
  • Social Media Directory
  • © Österreichischer Wissenschaftsfonds FWF
© Österreichischer Wissenschaftsfonds FWF