{"id":"https://openalex.org/W2185108727","doi":null,"title":"Underdeterminacy and Redundance in Maxwell's Equations. I. The Origin of Gauge Freedom","display_name":"Underdeterminacy and Redundance in Maxwell's Equations. I. The Origin of Gauge Freedom","publication_year":2009,"publication_date":"2009-01-01","ids":{"openalex":"https://openalex.org/W2185108727","mag":"2185108727"},"language":"en","primary_location":{"id":"mag:2185108727","doi":null,"is_oa":false,"landing_page_url":"http://www.ejtp.com/articles/ejtpv6i21p39.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null},"volume":null,"type":"article","is_traditional":true,"indexed_in":[],"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/A5048099768","display_name":"P. Enders","orcid":"https://orcid.org/0000-0001-7499-3058","observed_orcids":["https://orcid.org/0000-0001-7499-3058"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"P. Enders","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-7499-3058","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5048099768"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11683","display_name":"Quantum and Classical Electrodynamics","score":0.9994999766349792,"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/T11683","display_name":"Quantum and Classical Electrodynamics","score":0.9994999766349792,"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/T11960","display_name":"Relativity and Gravitational Theory","score":0.0003000000142492354,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"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/T11263","display_name":"Electromagnetic Simulation and Numerical Methods","score":0.0,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/gauge-freedom","display_name":"gauge freedom","score":0.8870000243186951},{"id":"https://openalex.org/keywords/maxwell-s-equations","display_name":"Maxwell's equations","score":0.9020000100135803},{"id":"https://openalex.org/keywords/underdetermination","display_name":"underdetermination","score":0.6729999780654907},{"id":"https://openalex.org/keywords/helmholtz-decomposition","display_name":"Helmholtz decomposition","score":0.7429999709129333},{"id":"https://openalex.org/keywords/longitudinal-electric-field","display_name":"longitudinal electric field","score":0.7860000133514404},{"id":"https://openalex.org/keywords/charge-conservation","display_name":"charge conservation","score":0.7979999780654907},{"id":"https://openalex.org/keywords/transverse-electromagnetic-waves","display_name":"transverse electromagnetic waves","score":0.7289999723434448},{"id":"https://openalex.org/keywords/magnetic-vector-potential","display_name":"vector potential","score":0.7599999904632568},{"id":"https://openalex.org/keywords/aharonov-bohm-effect","display_name":"Aharonov-Bohm effect","score":0.9850000143051147},{"id":"https://openalex.org/keywords/gauge-invariance","display_name":"gauge invariance","score":0.8970000147819519}],"concepts":[{"id":"https://openalex.org/C59282198","wikidata":"https://www.wikidata.org/wiki/Q51501","display_name":"Maxwell's equations","level":2,"score":0.7341498136520386},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.658207893371582},{"id":"https://openalex.org/C76969448","wikidata":"https://www.wikidata.org/wiki/Q6033855","display_name":"Inhomogeneous electromagnetic wave equation","level":4,"score":0.6349220275878906},{"id":"https://openalex.org/C28843909","wikidata":"https://www.wikidata.org/wiki/Q177625","display_name":"Electromagnetic field","level":2,"score":0.6260839700698853},{"id":"https://openalex.org/C65211518","wikidata":"https://www.wikidata.org/wiki/Q16743426","display_name":"Introduction to gauge theory","level":3,"score":0.5905822515487671},{"id":"https://openalex.org/C43498160","wikidata":"https://www.wikidata.org/wiki/Q1076013","display_name":"Electromagnetic tensor","level":3,"score":0.581586480140686},{"id":"https://openalex.org/C2027888","wikidata":"https://www.wikidata.org/wiki/Q521126","display_name":"Vector potential","level":3,"score":0.5662073493003845},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.5396729707717896},{"id":"https://openalex.org/C181830111","wikidata":"https://www.wikidata.org/wiki/Q214850","display_name":"Gauge theory","level":2,"score":0.5321487784385681},{"id":"https://openalex.org/C31593994","wikidata":"https://www.wikidata.org/wiki/Q832289","display_name":"Gauge fixing","level":4,"score":0.5252558588981628},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.4714787006378174},{"id":"https://openalex.org/C196258919","wikidata":"https://www.wikidata.org/wiki/Q1203816","display_name":"Lorenz gauge condition","level":5,"score":0.4704381823539734},{"id":"https://openalex.org/C208081375","wikidata":"https://www.wikidata.org/wiki/Q274502","display_name":"Degrees of freedom (physics and chemistry)","level":2,"score":0.41369712352752686},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.4125611186027527},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.33336734771728516},{"id":"https://openalex.org/C154452299","wikidata":"https://www.wikidata.org/wiki/Q7098853","display_name":"Optical field","level":3,"score":0.2717183530330658},{"id":"https://openalex.org/C115260700","wikidata":"https://www.wikidata.org/wiki/Q11408","display_name":"Magnetic field","level":2,"score":0.25562992691993713},{"id":"https://openalex.org/C200161520","wikidata":"https://www.wikidata.org/wiki/Q5527630","display_name":"Gauge anomaly","level":4,"score":0.2522358000278473},{"id":"https://openalex.org/C40263606","wikidata":"https://www.wikidata.org/wiki/Q105580","display_name":"Gauge boson","level":3,"score":0.12277063727378845}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:2185108727","doi":null,"is_oa":false,"landing_page_url":"http://www.ejtp.com/articles/ejtpv6i21p39.pdf","pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W179485435","https://openalex.org/W1504284124","https://openalex.org/W1523306063","https://openalex.org/W1524052396","https://openalex.org/W1533883395","https://openalex.org/W1538625961","https://openalex.org/W1590648858","https://openalex.org/W1841034626","https://openalex.org/W1933484298","https://openalex.org/W1997169313","https://openalex.org/W2019164214","https://openalex.org/W2019214475","https://openalex.org/W2024637952","https://openalex.org/W2024667731","https://openalex.org/W2028948400","https://openalex.org/W2056936728","https://openalex.org/W2064415053","https://openalex.org/W2101731201","https://openalex.org/W2105076606","https://openalex.org/W2141832944","https://openalex.org/W2322653853","https://openalex.org/W2325435413","https://openalex.org/W2506759652","https://openalex.org/W2593053380","https://openalex.org/W3025285286"],"related_works":["https://openalex.org/W7098747692","https://openalex.org/W7096092045","https://openalex.org/W2897148928","https://openalex.org/W2108393716","https://openalex.org/W7098366219","https://openalex.org/W2016028610","https://openalex.org/W1987724726","https://openalex.org/W2945477584","https://openalex.org/W3085653537","https://openalex.org/W7204895123"],"abstract_inverted_index":{"The":[0],"gauge":[1,116],"freedom":[2],"in":[3,10,18,67,98,110],"the":[4,36,48,53,60,71,79,82,87],"electromagnetic":[5,57],"potentials":[6],"indicates":[7],"an":[8],"underdeterminacy":[9,14],"Maxwell's":[11,19,94,121],"theory.":[12],"This":[13],"will":[15,64,107],"be":[16,65,108],"found":[17],"(1864)":[20],"original":[21],"set":[22],"of":[23,27,55,73,81,86,120],"equations":[24],"by":[25,93],"means":[26],"Helmholtz's":[28],"(1858)":[29],"decomposition":[30],"theorem.":[31],"Since":[32],"it":[33,40],"concerns":[34],"only":[35],"longitudinal":[37,84],"electric":[38],"field,":[39],"is":[41,103],"intimately":[42],"related":[43],"to":[44,52,130],"charge":[45],"conservation,":[46],"on":[47,59],"one":[49],"hand,":[50],"and":[51,75],"transversality":[54],"free":[56],"waves,":[58],"other":[61],"hand":[62],"(as":[63],"discussed":[66],"Pt.":[68],"II).":[69],"Exploiting":[70],"concept":[72],"Newtonian":[74],"Laplacian":[76],"vector":[77,88],"fields,":[78],"role":[80],"static":[83],"component":[85],"potential":[89],"being":[90],"not":[91],"determined":[92],"equations,":[95],"but":[96],"important":[97],"quantum":[99],"mechanics":[100],"(Aharonov-Bohm":[101],"effect)":[102],"elucidated.":[104],"These":[105],"results":[106],"exploited":[109],"Pt.III":[111],"for":[112,125],"formulating":[113],"a":[114],"manifest":[115],"invariant":[117],"canonical":[118],"formulation":[119],"theory":[122],"as":[123],"input":[124],"developing":[126],"Dirac's":[127],"(1949)":[128],"approach":[129],"special-relativistic":[131],"dynamics.":[132],"c":[133]},"counts_by_year":[],"updated_date":"2026-09-26T08:19:03.415552","created_date":"2016-06-24T00:00:00"}