Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem
The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The com...
Uloženo v:
| Vydáno v: | Symmetry (Basel) Ročník 12; číslo 1; s. 39 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Basel
MDPI AG
01.01.2020
|
| Témata: | |
| ISSN: | 2073-8994, 2073-8994 |
| On-line přístup: | Získat plný text |
| Tagy: |
Přidat tag
Žádné tagy, Buďte první, kdo vytvoří štítek k tomuto záznamu!
|
| Abstract | The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunctions and their inner product integrals free from any divergence problem and with Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered, although not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture, including the states with a non-positive norm, may give a consistent physics picture as a version of Lorentz covariant quantum mechanics. The probability interpretation for the usual von Neumann measurements is not a problem, as all wavefunctions restricted to a definite value for the `time’ variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed. |
|---|---|
| AbstractList | The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the full Fock space, and divergence issues with their functional representations. We present here a full solution avoiding those problems. The complete set of Fock states is obtained, together with the corresponding explicit wavefunctions and their inner product integrals free from any divergence problem and with Lorentz symmetry fully maintained without additional constraints imposed. By a simple choice of the pseudo-unitary representation of the underlying symmetry group, motivated from the perspective of the Minkowski spacetime as a representation for the Lorentz group, we obtain the natural non-unitary Fock space picture commonly considered, although not formulated and presented in the careful details given here. From a direct derivation of the appropriate basis state wavefunctions of the finite-dimensional irreducible representations of the Lorentz symmetry, the relation between the latter and the Fock state wavefunctions is also explicitly shown. Moreover, the full picture, including the states with a non-positive norm, may give a consistent physics picture as a version of Lorentz covariant quantum mechanics. The probability interpretation for the usual von Neumann measurements is not a problem, as all wavefunctions restricted to a definite value for the ‘time’ variable are just like those of the usual time independent quantum mechanics. A further understanding from a perspective of the dynamics from the symplectic geometry of the phase space is shortly discussed. |
| Author | Kong, Otto Bedić, Suzana |
| Author_xml | – sequence: 1 givenname: Suzana surname: Bedić fullname: Bedić, Suzana – sequence: 2 givenname: Otto orcidid: 0000-0003-3832-648X surname: Kong fullname: Kong, Otto |
| BookMark | eNptkEFLwzAUx4NMcM6d_AIBj1JNk6ZJj2NsThgoTPFYXtOUZbZNl2TTfXs75mGI7_J_h9__wftdo0FrW43QbUweGMvIoz80MSUxISy7QENKBItkliWDs_0Kjb3fkH444UlKhmg7aaE-eOOxbfHUNl2tg8YrHbCt8NyqT7wKELTHXyas8ey7q40yAX_AXle7VgVjW48r63BY676_B2egDXgBrrGtUfjFK1PXEHri1dmi1s0Nuqyg9nr8myP0Pp-9TRfR8uXpeTpZRopmMkSMSSWVAKmSpNQ65RUVtCozKAshJSMFl1QQIBqk5LRKSSEZ8BI4FWnJMsFG6O50t3N2u9M-5Bu7c_23PqecEypYxpKeuj9Rylnvna7yzpkG3CGPSX7Ump9p7en4D93LgKOE4MDU_3Z-ADsHfY4 |
| CitedBy_id | crossref_primary_10_1007_s11005_025_01927_y crossref_primary_10_1016_j_geomphys_2024_105368 crossref_primary_10_3390_sym13010022 |
| Cites_doi | 10.1016/j.rinp.2020.103636 10.1016/0034-4877(96)84075-4 10.1016/0375-9601(93)90652-G 10.1007/978-94-009-4558-6 10.4236/jhepgc.2019.53031 10.1016/0375-9601(91)90711-G 10.1017/CBO9780511755804 10.1103/PhysRevD.79.045009 10.1007/BF02787240 10.1103/RevModPhys.75.715 10.1063/1.528941 10.4236/jhepgc.2020.61005 10.1063/1.1704991 10.1142/9789812813459 10.1142/7293 10.1088/0305-4470/22/17/002 10.1103/PhysRev.181.1755 10.1103/PhysRev.80.440 10.1063/1.5091245 10.1007/978-94-011-1944-3 10.1007/s10773-019-04075-x 10.1007/3-540-51916-5 |
| ContentType | Journal Article |
| Copyright | 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| Copyright_xml | – notice: 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License. |
| DBID | AAYXX CITATION 7SC 7SR 7U5 8BQ 8FD 8FE 8FG ABJCF ABUWG AFKRA AZQEC BENPR BGLVJ CCPQU DWQXO H8D HCIFZ JG9 JQ2 L6V L7M L~C L~D M7S PHGZM PHGZT PIMPY PKEHL PQEST PQGLB PQQKQ PQUKI PTHSS |
| DOI | 10.3390/sym12010039 |
| DatabaseName | CrossRef Computer and Information Systems Abstracts Engineered Materials Abstracts Solid State and Superconductivity Abstracts METADEX Technology Research Database ProQuest SciTech Collection ProQuest Technology Collection Materials Science & Engineering Collection ProQuest Central (Alumni) ProQuest Central UK/Ireland ProQuest Central Essentials ProQuest Central ProQuest Technology Collection ProQuest One ProQuest Central Korea Aerospace Database SciTech Collection (ProQuest) Materials Research Database ProQuest Computer Science Collection ProQuest Engineering Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional Engineering Database ProQuest Central Premium ProQuest One Academic (New) Publicly Available Content Database ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition Engineering Collection |
| DatabaseTitle | CrossRef Publicly Available Content Database Materials Research Database Technology Collection Technology Research Database Computer and Information Systems Abstracts – Academic ProQuest One Academic Middle East (New) ProQuest Central Essentials ProQuest Computer Science Collection Computer and Information Systems Abstracts ProQuest Central (Alumni Edition) SciTech Premium Collection ProQuest One Community College ProQuest Central ProQuest One Applied & Life Sciences Aerospace Database Engineered Materials Abstracts ProQuest Engineering Collection ProQuest Central Korea ProQuest Central (New) Advanced Technologies Database with Aerospace Engineering Collection Engineering Database ProQuest One Academic Eastern Edition ProQuest Technology Collection ProQuest SciTech Collection METADEX Computer and Information Systems Abstracts Professional ProQuest One Academic UKI Edition Materials Science & Engineering Collection Solid State and Superconductivity Abstracts ProQuest One Academic ProQuest One Academic (New) |
| DatabaseTitleList | Publicly Available Content Database CrossRef |
| Database_xml | – sequence: 1 dbid: PIMPY name: ProQuest - Publicly Available Content Database url: http://search.proquest.com/publiccontent sourceTypes: Aggregation Database |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Sciences (General) |
| EISSN | 2073-8994 |
| ExternalDocumentID | 10_3390_sym12010039 |
| GroupedDBID | 5VS 8FE 8FG AADQD AAYXX ABDBF ABJCF ACUHS ADBBV ADMLS AFFHD AFKRA AFZYC ALMA_UNASSIGNED_HOLDINGS AMVHM BCNDV BENPR BGLVJ CCPQU CITATION E3Z ESX GX1 HCIFZ IAO ITC J9A KQ8 L6V M7S MODMG M~E OK1 PHGZM PHGZT PIMPY PQGLB PROAC PTHSS TR2 TUS 7SC 7SR 7U5 8BQ 8FD ABUWG AZQEC DWQXO H8D JG9 JQ2 L7M L~C L~D PKEHL PQEST PQQKQ PQUKI |
| ID | FETCH-LOGICAL-c298t-338c8c7a8c44dee65f272fd9adb78830b58270a0ea8852f60b83a5da5276d3973 |
| IEDL.DBID | M7S |
| ISICitedReferencesCount | 3 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000516823700039&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 2073-8994 |
| IngestDate | Fri Jul 25 12:00:28 EDT 2025 Tue Nov 18 22:07:15 EST 2025 Sat Nov 29 07:17:48 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c298t-338c8c7a8c44dee65f272fd9adb78830b58270a0ea8852f60b83a5da5276d3973 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0003-3832-648X |
| OpenAccessLink | https://www.proquest.com/docview/2550273934?pq-origsite=%requestingapplication% |
| PQID | 2550273934 |
| PQPubID | 2032326 |
| ParticipantIDs | proquest_journals_2550273934 crossref_primary_10_3390_sym12010039 crossref_citationtrail_10_3390_sym12010039 |
| PublicationCentury | 2000 |
| PublicationDate | 2020-01-01 |
| PublicationDateYYYYMMDD | 2020-01-01 |
| PublicationDate_xml | – month: 01 year: 2020 text: 2020-01-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Basel |
| PublicationPlace_xml | – name: Basel |
| PublicationTitle | Symmetry (Basel) |
| PublicationYear | 2020 |
| Publisher | MDPI AG |
| Publisher_xml | – name: MDPI AG |
| References | Kong (ref_22) 2019; 58 ref_12 ref_11 Feynman (ref_26) 1950; 80 Chew (ref_25) 2019; 5 ref_19 ref_18 ref_17 Kong (ref_10) 2019; 2075 ref_16 Bruce (ref_3) 1993; 106A ref_15 Gazeau (ref_5) 1993; 179 Zmuidizinas (ref_13) 1966; 7 Moshinsky (ref_2) 1989; 22 Johnson (ref_14) 1969; 181 ref_21 Aldaya (ref_4) 1991; 156 ref_20 Bars (ref_7) 2009; 79 ref_1 Cirelli (ref_23) 1990; 31 Aldaya (ref_6) 1996; 37 ref_9 Zurek (ref_24) 2003; 75 ref_8 |
| References_xml | – ident: ref_9 doi: 10.1016/j.rinp.2020.103636 – volume: 37 start-page: 387 year: 1996 ident: ref_6 article-title: Group-Theoretical Construction of the Quantum Relativistic Harmonic Oscillator publication-title: Rep. Math. Phys. doi: 10.1016/0034-4877(96)84075-4 – volume: 179 start-page: 67 year: 1993 ident: ref_5 article-title: Relativistic Harmonic Oscillator and Space Curvature publication-title: Phys. Lett. A doi: 10.1016/0375-9601(93)90652-G – ident: ref_1 doi: 10.1007/978-94-009-4558-6 – volume: 5 start-page: 553 year: 2019 ident: ref_25 article-title: Observables and Dynamics, Quantum to Classical from a Relativity Symmetry and Noncommutative Geometric Perspective publication-title: J. High Energy Phys. Gravit. Cosmo. doi: 10.4236/jhepgc.2019.53031 – volume: 156 start-page: 381 year: 1991 ident: ref_4 article-title: The Quantum Relativistic Harmonic Oscillator: Generalized Hermite Polynomials publication-title: Phys. Lett. A doi: 10.1016/0375-9601(91)90711-G – ident: ref_18 doi: 10.1017/CBO9780511755804 – volume: 79 start-page: 045009 year: 2009 ident: ref_7 article-title: Relativistic Harmonic Oscillator Revisited publication-title: Phys. Rev. D doi: 10.1103/PhysRevD.79.045009 – volume: 106A start-page: 711 year: 1993 ident: ref_3 article-title: The Klein-Gordon Oscillator publication-title: Il Nuovo C. doi: 10.1007/BF02787240 – volume: 75 start-page: 715 year: 2003 ident: ref_24 article-title: Decoherence, Einselection, and the Quantum Origins of the Classical publication-title: Rev. Mod. Phys. doi: 10.1103/RevModPhys.75.715 – volume: 31 start-page: 2891 year: 1990 ident: ref_23 article-title: Quantum Mechanics as an Infinite-dimensional Hamiltonian System with Uncertainty Structure: Part I publication-title: J. Math. Phys. doi: 10.1063/1.528941 – ident: ref_8 doi: 10.4236/jhepgc.2020.61005 – volume: 7 start-page: 764 year: 1966 ident: ref_13 article-title: Unitary Representations of the Lorentz Group on 4-Vector Manifolds publication-title: J. Math. Phys. doi: 10.1063/1.1704991 – ident: ref_12 doi: 10.1142/9789812813459 – ident: ref_16 doi: 10.1142/7293 – volume: 22 start-page: L817 year: 1989 ident: ref_2 article-title: The Dirac oscillator publication-title: J. Phys. A doi: 10.1088/0305-4470/22/17/002 – volume: 181 start-page: 1755 year: 1969 ident: ref_14 article-title: Position Operators and Proper Time in Relativistic Quantum Mechanics publication-title: Phys. Rev. doi: 10.1103/PhysRev.181.1755 – volume: 80 start-page: 440 year: 1950 ident: ref_26 article-title: Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction publication-title: Phys. Rev. doi: 10.1103/PhysRev.80.440 – ident: ref_15 – ident: ref_17 – ident: ref_19 – ident: ref_20 – volume: 2075 start-page: 100001 year: 2019 ident: ref_10 article-title: Quantum Spacetime Pictures and Particle Dynamics from a Relativity Perspective publication-title: AIP Conf. Proc. doi: 10.1063/1.5091245 – ident: ref_21 doi: 10.1007/978-94-011-1944-3 – volume: 58 start-page: 1803 year: 2019 ident: ref_22 article-title: The First Physics Picture of Contractions from a Fundamental Quantum Relativity Symmetry Including All Known Relativity Symmetries, Classical and Quantum publication-title: Int. J. Theor. Phys. doi: 10.1007/s10773-019-04075-x – ident: ref_11 doi: 10.1007/3-540-51916-5 |
| SSID | ssj0000505460 |
| Score | 2.1901653 |
| Snippet | The earlier treatments of the Lorentz covariant harmonic oscillator have brought to light various difficulties, such as reconciling Lorentz symmetry with the... |
| SourceID | proquest crossref |
| SourceType | Aggregation Database Enrichment Source Index Database |
| StartPage | 39 |
| SubjectTerms | Fock state Harmonic oscillators Hilbert space Minkowski space Norms Quantum field theory Quantum mechanics Quantum physics Representations Spacetime Symmetry Wave functions |
| Title | Analysis on Complete Set of Fock States with Explicit Wavefunctions for the Covariant Harmonic Oscillator Problem |
| URI | https://www.proquest.com/docview/2550273934 |
| Volume | 12 |
| WOSCitedRecordID | wos000516823700039&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVHPJ databaseName: ROAD: Directory of Open Access Scholarly Resources customDbUrl: eissn: 2073-8994 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000505460 issn: 2073-8994 databaseCode: M~E dateStart: 20080101 isFulltext: true titleUrlDefault: https://road.issn.org providerName: ISSN International Centre – providerCode: PRVPQU databaseName: Engineering Database customDbUrl: eissn: 2073-8994 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000505460 issn: 2073-8994 databaseCode: M7S dateStart: 20090301 isFulltext: true titleUrlDefault: http://search.proquest.com providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest - Publicly Available Content Database customDbUrl: eissn: 2073-8994 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000505460 issn: 2073-8994 databaseCode: PIMPY dateStart: 20090301 isFulltext: true titleUrlDefault: http://search.proquest.com/publiccontent providerName: ProQuest – providerCode: PRVPQU databaseName: ProQuest Central customDbUrl: eissn: 2073-8994 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0000505460 issn: 2073-8994 databaseCode: BENPR dateStart: 20090301 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEF60evDiW3wzBw8qBJNN0mxOotKioDX4QD2F3c0GCra1TSx48bc7k2x9gHjxEkIeEPhmZ-abnXzD2F4uY61CxR2llHACz1MOhhHpxFkW0sagMrW6_mXU6YjHxzixBbfCtlVOfGLlqLOBphr5Eaa-JL0S-8Hxy9ChqVG0u2pHaEyzGVJJ8KrWvdvPGgtNaQuabv1bno_s_qh463m0_-vScPDvgeinH66CS3vhv5-1yOZtWgkntR0ssSnTX2ZLduEWsG_VpQ9W2HAiQwKDPpA3QOAM3JoSBjm00TtCnX8CVWiBWvS6ulvCgxwbioGVmQJmuoCZI74_Rq6N4MC5HPVIZReuMaaiaSGVh6QeVrPK7tutu7Nzx85dcDSPRekga9VCR1LoIMiMaYY5j3iexTJTSJh9V4WCR650jRQi5HnTVcKXYSZDHjUzzG_8NdboD_pmnYHIcx4jZxEm9wLl6lhQ13-Ip7j2Pa422OEEhFRbUXKajfGcIjkhxNJviG2wvc-HX2otjt8f255AldoFWaRfOG3-fXuLzXGi1FWVZZs1ytGr2WGzelx2i9EumzltdZKb3crO6PjewmvJxVXy9AFHCOBV |
| linkProvider | ProQuest |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1dS9xAFL1YFdqXVm1Fq7X3QaEtBJNJspl5KKWoy4rrVtBS3-LMZAILNaubdIt_yt_Ye_OhFopvPvQtkEkgmTPnfsydcwG2c62siY3wjDHSi4LAeGRGtKeyLOaNQeMadf1hMhrJ83N1Mge33VkYLqvsOLEm6mxiOUe-S64vS6-oMPpyde1x1yjeXe1aaDSwOHI3vylkKz8f7tP87gjRPzjbG3htVwHPCiUrj2IyK22ipY2izLlenItE5JnSmaFwMPRNLEXia99pKWOR93wjQx1nOhZJLyPrHdJ7n8ECuRFC1aWCp3c5He4KF_X85hhgGCp_t7y5DHi_2edm5A8N39-8Xxuz_qv_7TcswcvWbcavDc6XYc4VK7DcElOJH1r17I-v4bqTWcFJgcx2BEyHp67CSY59Yn9s_GvkDDRyCeLYjiv8oWeObXy9DJE8eSTPmJ6faVqhRYUDPb1kFWH8Rj4DLZ2KRpw0zXjewPcn-fRVmC8mhVsDlHkuFMVk0uVBZHyrJJ9qiOmSuC0QZh0-dZOe2lZ0nXt__Ewp-GKEpA8Qsg7bd4OvGq2Rfw_b7KCRtoRTpve4ePv47ffwfHB2PEyHh6OjDXghOH1QZ5Q2Yb6a_nLvYNHOqnE53aqxjXDx1Cj6A0p4OE4 |
| linkToPdf | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3fa9RAEB7qVcQXa_2B1arzUEGFcMkmuew-lFKsR4_WM6BifYq7m104sLn2Ek_6r_nXOXNJagXxrQ--BbIJJPvNzDezu98A7HitrEmNCIwxMkiiyAQURnSgyjLlhUHjWnX942w6lScnKl-Dn_1ZGN5W2fvElaMu55Zr5EOiviy9ouJk6LttEfnBeO_sPOAOUrzS2rfTaCFy5C5-UPpW704OaK5fCDF--_HNYdB1GAisULIJKD-z0mZa2iQpnRulXmTCl0qXhlLDODSpFFmoQ6elTIUfhUbGOi11KrJRSZE8pvfegHWi5IkYwHo-eZd_uazwcI-4ZBS2hwLjWIXD-uI04tXnkFuTXw2Df0aBVWgbb_zPP-Uu3OkINe63FrAJa666B5udy6rxZaer_eo-nPcCLDivkP0gQdbhB9fg3OOY4gK2zBu5No28OXFmZw1-1kvH0X9loEgcH4kz0_NLTbZbNXioF6esL4zviU2QUTU0Im_b9DyAT9fy6Q9hUM0r9whQei8UZWvS-SgxoVWSzzukdEleLxJmC173AChsJ8fOXUG-FZSWMVqKK2jZgp3LwWetCsnfh233MCk6V1QXvzHy-N-3n8MtAk9xPJkePYHbgusKq1LTNgyaxXf3FG7aZTOrF886oCN8vW4Y_QIfV0KE |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Analysis+on+Complete+Set+of+Fock+States+with+Explicit+Wavefunctions+for+the+Covariant+Harmonic+Oscillator+Problem&rft.jtitle=Symmetry+%28Basel%29&rft.au=Bedi%C4%87%2C+Suzana&rft.au=Kong%2C+Otto&rft.date=2020-01-01&rft.issn=2073-8994&rft.eissn=2073-8994&rft.volume=12&rft.issue=1&rft.spage=39&rft_id=info:doi/10.3390%2Fsym12010039&rft.externalDBID=n%2Fa&rft.externalDocID=10_3390_sym12010039 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=2073-8994&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=2073-8994&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=2073-8994&client=summon |