The origins of the trace formula View Full Text


Ontology type: schema:Chapter     


Chapter Info

DATE

1997

AUTHORS

Martin C. Gutzwiller

ABSTRACT

The trace formula establishes a mathematical relation between the spectrum of energy levels of a dynamical system in quantum mechanics on one hand, and the set of periodic orbits in the corresponding system in classical mechanics on the other hand. Such a formula was first derived by Atle Selberg in the context of geodesics on a surfaces of constant negative curvature for the purposes of number theory, while other mathematicians and some physicists were mostly inspired by Weyl to get the asymptotic distribution of eigenvalues of differential operators. Eventually, the author obtained a more general trace formula on the basis of Feynman’s path integral, and a similar result was obtained by Colin de Verdiere shortly thereafter on more geometric grounds. The use of various zetafunctions in differential geometry and statistical physics is mentioned at the end. More... »

PAGES

8-28

Book

TITLE

Classical, Semiclassical and Quantum Dynamics in Atoms

ISBN

978-3-540-63004-3
978-3-540-69055-9

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/bfb0105967

DOI

http://dx.doi.org/10.1007/bfb0105967

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1025182508


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/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "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"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Thomas J Watson Research Lab, IBM, Yorktown Heights", 
          "id": "http://www.grid.ac/institutes/grid.410484.d", 
          "name": [
            "Thomas J Watson Research Lab, IBM, Yorktown Heights"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Gutzwiller", 
        "givenName": "Martin C.", 
        "id": "sg:person.07737020712.55", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07737020712.55"
        ], 
        "type": "Person"
      }
    ], 
    "datePublished": "1997", 
    "datePublishedReg": "1997-01-01", 
    "description": "The trace formula establishes a mathematical relation between the spectrum of energy levels of a dynamical system in quantum mechanics on one hand, and the set of periodic orbits in the corresponding system in classical mechanics on the other hand. Such a formula was first derived by Atle Selberg in the context of geodesics on a surfaces of constant negative curvature for the purposes of number theory, while other mathematicians and some physicists were mostly inspired by Weyl to get the asymptotic distribution of eigenvalues of differential operators. Eventually, the author obtained a more general trace formula on the basis of Feynman\u2019s path integral, and a similar result was obtained by Colin de Verdiere shortly thereafter on more geometric grounds. The use of various zetafunctions in differential geometry and statistical physics is mentioned at the end.", 
    "editor": [
      {
        "familyName": "Friedrich", 
        "givenName": "Harald", 
        "type": "Person"
      }, 
      {
        "familyName": "Eckhardt", 
        "givenName": "Bruno", 
        "type": "Person"
      }
    ], 
    "genre": "chapter", 
    "id": "sg:pub.10.1007/bfb0105967", 
    "isAccessibleForFree": false, 
    "isPartOf": {
      "isbn": [
        "978-3-540-63004-3", 
        "978-3-540-69055-9"
      ], 
      "name": "Classical, Semiclassical and Quantum Dynamics in Atoms", 
      "type": "Book"
    }, 
    "keywords": [
      "trace formula", 
      "path integral", 
      "Feynman path integral", 
      "constant negative curvature", 
      "general trace formula", 
      "statistical physics", 
      "dynamical systems", 
      "differential geometry", 
      "number theory", 
      "differential operators", 
      "classical mechanics", 
      "quantum mechanics", 
      "asymptotic distribution", 
      "periodic orbits", 
      "mathematical relations", 
      "geometric grounds", 
      "Atle Selberg", 
      "negative curvature", 
      "formula", 
      "mechanics", 
      "mathematicians", 
      "Weyl", 
      "eigenvalues", 
      "geodesics", 
      "Selberg", 
      "physics", 
      "integrals", 
      "energy levels", 
      "operators", 
      "physicists", 
      "orbit", 
      "geometry", 
      "theory", 
      "curvature", 
      "Colin", 
      "system", 
      "set", 
      "distribution", 
      "similar results", 
      "spectra", 
      "results", 
      "relation", 
      "surface", 
      "basis", 
      "hand", 
      "authors", 
      "ground", 
      "end", 
      "origin", 
      "purpose", 
      "use", 
      "context", 
      "levels"
    ], 
    "name": "The origins of the trace formula", 
    "pagination": "8-28", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1025182508"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/bfb0105967"
        ]
      }
    ], 
    "publisher": {
      "name": "Springer Nature", 
      "type": "Organisation"
    }, 
    "sameAs": [
      "https://doi.org/10.1007/bfb0105967", 
      "https://app.dimensions.ai/details/publication/pub.1025182508"
    ], 
    "sdDataset": "chapters", 
    "sdDatePublished": "2022-12-01T06:51", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20221201/entities/gbq_results/chapter/chapter_340.jsonl", 
    "type": "Chapter", 
    "url": "https://doi.org/10.1007/bfb0105967"
  }
]
 

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/bfb0105967'

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/bfb0105967'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bfb0105967'

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

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bfb0105967'


 

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

117 TRIPLES      22 PREDICATES      78 URIs      71 LITERALS      7 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/bfb0105967 schema:about anzsrc-for:01
2 anzsrc-for:0101
3 schema:author N08e9791e4ec24cf290502228cd49f52d
4 schema:datePublished 1997
5 schema:datePublishedReg 1997-01-01
6 schema:description The trace formula establishes a mathematical relation between the spectrum of energy levels of a dynamical system in quantum mechanics on one hand, and the set of periodic orbits in the corresponding system in classical mechanics on the other hand. Such a formula was first derived by Atle Selberg in the context of geodesics on a surfaces of constant negative curvature for the purposes of number theory, while other mathematicians and some physicists were mostly inspired by Weyl to get the asymptotic distribution of eigenvalues of differential operators. Eventually, the author obtained a more general trace formula on the basis of Feynman’s path integral, and a similar result was obtained by Colin de Verdiere shortly thereafter on more geometric grounds. The use of various zetafunctions in differential geometry and statistical physics is mentioned at the end.
7 schema:editor N8d2c37383de54ef0a33da6a2849701cd
8 schema:genre chapter
9 schema:isAccessibleForFree false
10 schema:isPartOf N1e91ab8143b04f87aa0274c313ced504
11 schema:keywords Atle Selberg
12 Colin
13 Feynman path integral
14 Selberg
15 Weyl
16 asymptotic distribution
17 authors
18 basis
19 classical mechanics
20 constant negative curvature
21 context
22 curvature
23 differential geometry
24 differential operators
25 distribution
26 dynamical systems
27 eigenvalues
28 end
29 energy levels
30 formula
31 general trace formula
32 geodesics
33 geometric grounds
34 geometry
35 ground
36 hand
37 integrals
38 levels
39 mathematical relations
40 mathematicians
41 mechanics
42 negative curvature
43 number theory
44 operators
45 orbit
46 origin
47 path integral
48 periodic orbits
49 physicists
50 physics
51 purpose
52 quantum mechanics
53 relation
54 results
55 set
56 similar results
57 spectra
58 statistical physics
59 surface
60 system
61 theory
62 trace formula
63 use
64 schema:name The origins of the trace formula
65 schema:pagination 8-28
66 schema:productId N1791ba89d5c442388a00430a4e38dd6a
67 N8eafea5d1ca7425997e22028f0785136
68 schema:publisher N2d1d53928c66410e85bb11faf762ce21
69 schema:sameAs https://app.dimensions.ai/details/publication/pub.1025182508
70 https://doi.org/10.1007/bfb0105967
71 schema:sdDatePublished 2022-12-01T06:51
72 schema:sdLicense https://scigraph.springernature.com/explorer/license/
73 schema:sdPublisher Nef3aaf8c673d4d3cb8cd55be07f952d9
74 schema:url https://doi.org/10.1007/bfb0105967
75 sgo:license sg:explorer/license/
76 sgo:sdDataset chapters
77 rdf:type schema:Chapter
78 N08e9791e4ec24cf290502228cd49f52d rdf:first sg:person.07737020712.55
79 rdf:rest rdf:nil
80 N1791ba89d5c442388a00430a4e38dd6a schema:name dimensions_id
81 schema:value pub.1025182508
82 rdf:type schema:PropertyValue
83 N1e91ab8143b04f87aa0274c313ced504 schema:isbn 978-3-540-63004-3
84 978-3-540-69055-9
85 schema:name Classical, Semiclassical and Quantum Dynamics in Atoms
86 rdf:type schema:Book
87 N24435b39a61341269dcc9be57bc6b385 schema:familyName Eckhardt
88 schema:givenName Bruno
89 rdf:type schema:Person
90 N2d1d53928c66410e85bb11faf762ce21 schema:name Springer Nature
91 rdf:type schema:Organisation
92 N3c49f2ed19c5424ba28381933716f218 schema:familyName Friedrich
93 schema:givenName Harald
94 rdf:type schema:Person
95 N8d2c37383de54ef0a33da6a2849701cd rdf:first N3c49f2ed19c5424ba28381933716f218
96 rdf:rest Nfd94518b86874a03ab091c1b782bca3b
97 N8eafea5d1ca7425997e22028f0785136 schema:name doi
98 schema:value 10.1007/bfb0105967
99 rdf:type schema:PropertyValue
100 Nef3aaf8c673d4d3cb8cd55be07f952d9 schema:name Springer Nature - SN SciGraph project
101 rdf:type schema:Organization
102 Nfd94518b86874a03ab091c1b782bca3b rdf:first N24435b39a61341269dcc9be57bc6b385
103 rdf:rest rdf:nil
104 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
105 schema:name Mathematical Sciences
106 rdf:type schema:DefinedTerm
107 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
108 schema:name Pure Mathematics
109 rdf:type schema:DefinedTerm
110 sg:person.07737020712.55 schema:affiliation grid-institutes:grid.410484.d
111 schema:familyName Gutzwiller
112 schema:givenName Martin C.
113 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.07737020712.55
114 rdf:type schema:Person
115 grid-institutes:grid.410484.d schema:alternateName Thomas J Watson Research Lab, IBM, Yorktown Heights
116 schema:name Thomas J Watson Research Lab, IBM, Yorktown Heights
117 rdf:type schema:Organization
 




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


...