{"id":"https://openalex.org/W1978607448","doi":"https://doi.org/10.1103/physrev.172.350","title":"Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes","display_name":"Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes","publication_year":1968,"publication_date":"1968-08-10","ids":{"openalex":"https://openalex.org/W1978607448","doi":"https://doi.org/10.1103/physrev.172.350","mag":"1978607448"},"language":"en","primary_location":{"id":"doi:10.1103/physrev.172.350","doi":"https://doi.org/10.1103/physrev.172.350","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physrev.172.350","pdf_url":null,"source":{"id":"https://openalex.org/S54862371","display_name":"Physical Review","issn_l":"0031-899X","issn":["0031-899X","1536-6065"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review","raw_type":"journal-article"},"parent_works":[],"lineage":["https://openalex.org/W1978607448"],"child_works_count":0,"type":"article","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5111733701","display_name":"M. Lax","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Melvin Lax","raw_affiliation_strings":["Bell Telephone Laboratories, Murray Hill, New Jersey"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Bell Telephone Laboratories, Murray Hill, New Jersey","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5111733701"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.5797,"has_fulltext":false,"cited_by_count":443,"citation_normalized_percentile":{"value":0.97333333,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"172","issue":"2","first_page":"350","last_page":"361"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.948199987411499,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.948199987411499,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11261","display_name":"Quantum chaos and dynamical systems","score":0.010300000198185444,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11520","display_name":"Advanced Thermodynamics and Statistical Mechanics","score":0.008100000210106373,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/quantum-noise","display_name":"quantum noise","score":0.8240000009536743},{"id":"https://openalex.org/keywords/quasi-probability-distributions","display_name":"quasiprobability distributions","score":0.7770000100135803},{"id":"https://openalex.org/keywords/quantum-classical-correspondence","display_name":"quantum-classical correspondence","score":0.6959999799728394},{"id":"https://openalex.org/keywords/markov-processes","display_name":"Markov processes","score":0.671999990940094},{"id":"https://openalex.org/keywords/quantum-regression-theorem","display_name":"quantum regression theorem","score":0.9010000228881836},{"id":"https://openalex.org/keywords/normal-ordering","display_name":"normal ordering","score":0.6869999766349792},{"id":"https://openalex.org/keywords/field-operators","display_name":"field operators","score":0.6179999709129333},{"id":"https://openalex.org/keywords/chapman-kolmogorov-equation","display_name":"Chapman-Kolmogorov equation","score":0.8579999804496765},{"id":"https://openalex.org/keywords/commutation-relations","display_name":"commutation relations","score":0.5370000004768372},{"id":"https://openalex.org/keywords/wigner-function","display_name":"Wigner function","score":0.4950000047683716}],"concepts":[{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.6565492153167725},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.5104402899742126},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.47414419054985046},{"id":"https://openalex.org/C84114770","wikidata":"https://www.wikidata.org/wiki/Q46344","display_name":"Quantum","level":2,"score":0.4165956974029541},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.4139522910118103},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3920469880104065},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.22244369983673096},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1103/physrev.172.350","doi":"https://doi.org/10.1103/physrev.172.350","is_oa":false,"landing_page_url":"https://doi.org/10.1103/physrev.172.350","pdf_url":null,"source":{"id":"https://openalex.org/S54862371","display_name":"Physical Review","issn_l":"0031-899X","issn":["0031-899X","1536-6065"],"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/P4310320261","host_organization_name":"American Physical Society","host_organization_lineage":["https://openalex.org/P4310320261"],"host_organization_lineage_names":["American Physical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physical Review","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":34,"referenced_works":["https://openalex.org/W1549773254","https://openalex.org/W1603491459","https://openalex.org/W1968611775","https://openalex.org/W1975422612","https://openalex.org/W1975448072","https://openalex.org/W1984210561","https://openalex.org/W1988147940","https://openalex.org/W1999899718","https://openalex.org/W2006110212","https://openalex.org/W2009229255","https://openalex.org/W2011327948","https://openalex.org/W2014242096","https://openalex.org/W2014299730","https://openalex.org/W2015314874","https://openalex.org/W2025257408","https://openalex.org/W2036465358","https://openalex.org/W2042623116","https://openalex.org/W2043302663","https://openalex.org/W2047046496","https://openalex.org/W2047759905","https://openalex.org/W2053730839","https://openalex.org/W2056471637","https://openalex.org/W2070761172","https://openalex.org/W2074005215","https://openalex.org/W2077847363","https://openalex.org/W2136460417","https://openalex.org/W2161426305","https://openalex.org/W2166370501","https://openalex.org/W2170649178","https://openalex.org/W2228259595","https://openalex.org/W2324014000","https://openalex.org/W3007330514","https://openalex.org/W3136463145","https://openalex.org/W3165898261"],"related_works":["https://openalex.org/W2064407009","https://openalex.org/W2074005215","https://openalex.org/W2065051669","https://openalex.org/W2966247945","https://openalex.org/W7015680052","https://openalex.org/W4224927266","https://openalex.org/W1991373591","https://openalex.org/W2046267337","https://openalex.org/W1581324086","https://openalex.org/W2973051576"],"abstract_inverted_index":{"A":[0,44],"correspondence":[1],"${a}_{j}\\ensuremath{\\leftrightarrow}{\\ensuremath{\\alpha}}_{j}$":[2],"between":[3,32],"operators":[4,106,136,145],"$\\mathrm{a}=[{a}_{1},":[5],"{a}_{2},":[6],"\\ensuremath{\\cdots}{a}_{f}]$":[7],"and":[8,37,121,215],"$c$":[9,40],"numbers":[10],"$\\ensuremath{\\alpha}=[{\\ensuremath{\\alpha}}_{1},":[11],"{\\ensuremath{\\alpha}}_{2},":[12],"\\ensuremath{\\cdots}{\\ensuremath{\\alpha}}_{f}]$":[13],"together":[14],"with":[15,119],"an":[16,29,38,61,207],"arbitrary":[17],"ordering":[18],"rule":[19],"$\\mathcal{C}$":[20],"(e.g.,":[21],"in":[22,112,138,146],"sequence":[23],"from":[24],"1":[25],"to":[26,97,129,165,193,198,206,216],"$f$)":[27],"permit":[28],"association":[30],"$M(\\mathrm{a})=\\mathcal{C}{M}^{(c)}(\\ensuremath{\\alpha})$":[31],"a":[33,53,76,147,211,217],"general":[34,54],"operator":[35],"$M(\\mathrm{a})$":[36],"associated":[39],"number":[41],"function":[42],"${M}^{(c)}(\\ensuremath{\\alpha})$.":[43],"quasiprobability":[45],"$P(\\ensuremath{\\alpha},":[46],"t)$":[47],"is":[48,85,127,179,191],"then":[49],"defined":[50],"so":[51],"that":[52,72,108,118,157,188,203],"ensemble":[55],"average":[56,126,131],"can":[57,91],"be":[58,92,199],"written":[59,137],"as":[60],"ordinary":[62],"integration:":[63],"$\u3008M(\\mathrm{a}(t))\u3009=\\ensuremath{\\int}{M}^{(c)}(\\ensuremath{\\alpha})d\\ensuremath{\\alpha}P(\\ensuremath{\\alpha},":[64],"t)$.":[65],"The":[66],"equation":[67],"for":[68],"$\\frac{\\ensuremath{\\partial}P(\\ensuremath{\\alpha},":[69],"t)}{\\ensuremath{\\partial}t}$":[70],"suggests":[71],"the":[73,102,123,130,133,143,152,158,182,195,221],"$\\ensuremath{\\alpha}$":[74],"obeys":[75],"classical":[77,82,89,124],"Markoff":[78,83],"process.":[79],"If":[80],"this":[81,189],"process":[84],"taken":[86],"literally,":[87],"multitime":[88,125,170,177,212],"averages":[90,171,178],"computed.":[93],"Do":[94],"these":[95],"correspond":[96],"appropriate":[98,208],"quantum":[99,135,183,196],"averages?":[100],"For":[101,142],"case":[103],"of":[104,132,176,210],"field":[105],"such":[107],"$[b,":[109],"{b}^{\\ifmmode\\dagger\\else\\textdagger\\fi{}}]=1$,":[110],"important":[111],"discussing":[113],"laser":[114,148],"statistics,":[115],"we":[116,150],"show":[117,187],"${a}_{1}={b}^{\\ifmmode\\dagger\\else\\textdagger\\fi{}}$":[120],"${a}_{f}=b$,":[122],"equivalent":[128,192],"corresponding":[134],"time-ordered,":[139],"normal-ordered":[140],"sequence.":[141],"atomic":[144],"problem,":[149],"obtain":[151],"desired":[153],"correspondence,":[154],"but":[155],"find":[156],"more":[159],"complicated":[160],"commutation":[161],"rules":[162],"necessarily":[163],"lead":[164],"derivative":[166],"correction":[167],"terms":[168],"when":[169],"are":[172],"taken.":[173],"Our":[174],"derivation":[175],"based":[180],"on":[181,220],"regression":[184],"theorem.":[185],"We":[186],"theorem":[190],"assuming":[194],"system":[197],"Markoffian,":[200],"by":[201],"showing":[202],"it":[204],"leads":[205],"factorization":[209],"density":[213,223],"matrix":[214],"Chapman-Kolmogoroff-like":[218],"condition":[219],"conditional":[222],"matrix.":[224]},"counts_by_year":[{"year":2026,"cited_by_count":4},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":4},{"year":2021,"cited_by_count":6},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":3},{"year":2018,"cited_by_count":9},{"year":2017,"cited_by_count":6},{"year":2016,"cited_by_count":12},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":6},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":10}],"updated_date":"2026-11-14T11:04:58.165337","created_date":"2016-06-24T00:00:00"}