Geometry of quantum states View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1968-03

AUTHORS

Bogdan Mielnik

ABSTRACT

In the first part of this work, an attempt of a realistic interpretation ofquantum logic is presented. Propositions of quantum logic are interpreted as corresponding to certain macroscopic objects called filters; these objects are used to select beams of particles. The problem of representing the propositions as projectors in a Hilbert space is considered and the classical approach to this question due to Birkhoff and von Neumann is criticized as neglecting certain physically important properties of filters. A new approach to this problem is proposed. The second part of the paper contains a revision of the concept of a state in quantum mechanics. The set of all states of a physical system is considered as an abstract space with a geometry determined by the transition probabilities. The existence of a representation of states by vectors in a Hilbert space is shown to impose strong limitations on the geometric structure of the space of states. Spaces for which this representation does not exist are called non-Hilbertian. Simple examples of non-Hilbertian spaces are given and their possible physical meaning is discussed. The difference between Hilbertian and non-Hilbertian spaces is characterized in terms of measurable quantities. More... »

PAGES

55-80

References to SciGraph publications

  • 1967-12. On the algebraic structure of quantum mechanics in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf01654032

    DOI

    http://dx.doi.org/10.1007/bf01654032

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1019198240


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "University of Warsaw", 
              "id": "https://www.grid.ac/institutes/grid.12847.38", 
              "name": [
                "Institute of Theoretical Physics, Warsaw University, Poland"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Mielnik", 
            "givenName": "Bogdan", 
            "id": "sg:person.014150321311.19", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014150321311.19"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf01646019", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1027252280", 
              "https://doi.org/10.1007/bf01646019"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01646019", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1027252280", 
              "https://doi.org/10.1007/bf01646019"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1063/1.1703794", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1057773740"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2307/1968621", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1069674038"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.2307/1968656", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1069674071"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "1968-03", 
        "datePublishedReg": "1968-03-01", 
        "description": "In the first part of this work, an attempt of a realistic interpretation ofquantum logic is presented. Propositions of quantum logic are interpreted as corresponding to certain macroscopic objects called filters; these objects are used to select beams of particles. The problem of representing the propositions as projectors in a Hilbert space is considered and the classical approach to this question due to Birkhoff and von Neumann is criticized as neglecting certain physically important properties of filters. A new approach to this problem is proposed. The second part of the paper contains a revision of the concept of a state in quantum mechanics. The set of all states of a physical system is considered as an abstract space with a geometry determined by the transition probabilities. The existence of a representation of states by vectors in a Hilbert space is shown to impose strong limitations on the geometric structure of the space of states. Spaces for which this representation does not exist are called non-Hilbertian. Simple examples of non-Hilbertian spaces are given and their possible physical meaning is discussed. The difference between Hilbertian and non-Hilbertian spaces is characterized in terms of measurable quantities.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1007/bf01654032", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1136216", 
            "issn": [
              "0010-3616", 
              "1432-0916"
            ], 
            "name": "Communications in Mathematical Physics", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "9"
          }
        ], 
        "name": "Geometry of quantum states", 
        "pagination": "55-80", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "5b7903979da87dbf8e8a319f8565bcaf2397b79ad59d55534de18aa7a7baf41c"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/bf01654032"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1019198240"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/bf01654032", 
          "https://app.dimensions.ai/details/publication/pub.1019198240"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-11T09:11", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000338_0000000338/records_47976_00000000.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1007/BF01654032"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf01654032'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf01654032'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf01654032'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf01654032'


     

    This table displays all metadata directly associated to this object as RDF triples.

    74 TRIPLES      21 PREDICATES      31 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/bf01654032 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N5bfb7b99acbf48638f89c3f955b60f3c
    4 schema:citation sg:pub.10.1007/bf01646019
    5 https://doi.org/10.1063/1.1703794
    6 https://doi.org/10.2307/1968621
    7 https://doi.org/10.2307/1968656
    8 schema:datePublished 1968-03
    9 schema:datePublishedReg 1968-03-01
    10 schema:description In the first part of this work, an attempt of a realistic interpretation ofquantum logic is presented. Propositions of quantum logic are interpreted as corresponding to certain macroscopic objects called filters; these objects are used to select beams of particles. The problem of representing the propositions as projectors in a Hilbert space is considered and the classical approach to this question due to Birkhoff and von Neumann is criticized as neglecting certain physically important properties of filters. A new approach to this problem is proposed. The second part of the paper contains a revision of the concept of a state in quantum mechanics. The set of all states of a physical system is considered as an abstract space with a geometry determined by the transition probabilities. The existence of a representation of states by vectors in a Hilbert space is shown to impose strong limitations on the geometric structure of the space of states. Spaces for which this representation does not exist are called non-Hilbertian. Simple examples of non-Hilbertian spaces are given and their possible physical meaning is discussed. The difference between Hilbertian and non-Hilbertian spaces is characterized in terms of measurable quantities.
    11 schema:genre research_article
    12 schema:inLanguage en
    13 schema:isAccessibleForFree false
    14 schema:isPartOf N2086c341854e46338a2bfb421b9628e0
    15 N32744bf2a91f4eac9ab824fedc4607a5
    16 sg:journal.1136216
    17 schema:name Geometry of quantum states
    18 schema:pagination 55-80
    19 schema:productId N291362ec7a044f7eb463e453674df9fe
    20 N40f5446ed6844d0f9d3098b7021cd036
    21 Nbcc46005d3714260b71d67c52c78e2d9
    22 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019198240
    23 https://doi.org/10.1007/bf01654032
    24 schema:sdDatePublished 2019-04-11T09:11
    25 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    26 schema:sdPublisher N42e9795d87b24a99abf100d96a468dcc
    27 schema:url http://link.springer.com/10.1007/BF01654032
    28 sgo:license sg:explorer/license/
    29 sgo:sdDataset articles
    30 rdf:type schema:ScholarlyArticle
    31 N2086c341854e46338a2bfb421b9628e0 schema:volumeNumber 9
    32 rdf:type schema:PublicationVolume
    33 N291362ec7a044f7eb463e453674df9fe schema:name doi
    34 schema:value 10.1007/bf01654032
    35 rdf:type schema:PropertyValue
    36 N32744bf2a91f4eac9ab824fedc4607a5 schema:issueNumber 1
    37 rdf:type schema:PublicationIssue
    38 N40f5446ed6844d0f9d3098b7021cd036 schema:name readcube_id
    39 schema:value 5b7903979da87dbf8e8a319f8565bcaf2397b79ad59d55534de18aa7a7baf41c
    40 rdf:type schema:PropertyValue
    41 N42e9795d87b24a99abf100d96a468dcc schema:name Springer Nature - SN SciGraph project
    42 rdf:type schema:Organization
    43 N5bfb7b99acbf48638f89c3f955b60f3c rdf:first sg:person.014150321311.19
    44 rdf:rest rdf:nil
    45 Nbcc46005d3714260b71d67c52c78e2d9 schema:name dimensions_id
    46 schema:value pub.1019198240
    47 rdf:type schema:PropertyValue
    48 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    49 schema:name Mathematical Sciences
    50 rdf:type schema:DefinedTerm
    51 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    52 schema:name Pure Mathematics
    53 rdf:type schema:DefinedTerm
    54 sg:journal.1136216 schema:issn 0010-3616
    55 1432-0916
    56 schema:name Communications in Mathematical Physics
    57 rdf:type schema:Periodical
    58 sg:person.014150321311.19 schema:affiliation https://www.grid.ac/institutes/grid.12847.38
    59 schema:familyName Mielnik
    60 schema:givenName Bogdan
    61 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014150321311.19
    62 rdf:type schema:Person
    63 sg:pub.10.1007/bf01646019 schema:sameAs https://app.dimensions.ai/details/publication/pub.1027252280
    64 https://doi.org/10.1007/bf01646019
    65 rdf:type schema:CreativeWork
    66 https://doi.org/10.1063/1.1703794 schema:sameAs https://app.dimensions.ai/details/publication/pub.1057773740
    67 rdf:type schema:CreativeWork
    68 https://doi.org/10.2307/1968621 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069674038
    69 rdf:type schema:CreativeWork
    70 https://doi.org/10.2307/1968656 schema:sameAs https://app.dimensions.ai/details/publication/pub.1069674071
    71 rdf:type schema:CreativeWork
    72 https://www.grid.ac/institutes/grid.12847.38 schema:alternateName University of Warsaw
    73 schema:name Institute of Theoretical Physics, Warsaw University, Poland
    74 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...