--- _id: '455' abstract: - lang: eng text: The derivation of effective evolution equations is central to the study of non-stationary quantum many-body systems, and widely used in contexts such as superconductivity, nuclear physics, Bose–Einstein condensation and quantum chemistry. We reformulate the Dirac–Frenkel approximation principle in terms of reduced density matrices and apply it to fermionic and bosonic many-body systems. We obtain the Bogoliubov–de Gennes and Hartree–Fock–Bogoliubov equations, respectively. While we do not prove quantitative error estimates, our formulation does show that the approximation is optimal within the class of quasifree states. Furthermore, we prove well-posedness of the Bogoliubov–de Gennes equations in energy space and discuss conserved quantities acknowledgement: Open access funding provided by Institute of Science and Technology (IST Austria). The authors acknowledge support by ERC Advanced Grant 321029 and by VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). The authors would like to thank Sébastien Breteaux, Enno Lenzmann, Mathieu Lewin and Jochen Schmid for comments and discussions about well-posedness of the Bogoliubov–de Gennes equations. alternative_title: - Annales Henri Poincare article_processing_charge: No author: - first_name: Niels P full_name: Benedikter, Niels P id: 3DE6C32A-F248-11E8-B48F-1D18A9856A87 last_name: Benedikter orcid: 0000-0002-1071-6091 - first_name: Jérémy full_name: Sok, Jérémy last_name: Sok - first_name: Jan full_name: Solovej, Jan last_name: Solovej citation: ama: Benedikter NP, Sok J, Solovej J. The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations. Annales Henri Poincare. 2018;19(4):1167-1214. doi:10.1007/s00023-018-0644-z apa: Benedikter, N. P., Sok, J., & Solovej, J. (2018). The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations. Annales Henri Poincare. Birkhäuser. https://doi.org/10.1007/s00023-018-0644-z chicago: Benedikter, Niels P, Jérémy Sok, and Jan Solovej. “The Dirac–Frenkel Principle for Reduced Density Matrices and the Bogoliubov–de Gennes Equations.” Annales Henri Poincare. Birkhäuser, 2018. https://doi.org/10.1007/s00023-018-0644-z. ieee: N. P. Benedikter, J. Sok, and J. Solovej, “The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations,” Annales Henri Poincare, vol. 19, no. 4. Birkhäuser, pp. 1167–1214, 2018. ista: Benedikter NP, Sok J, Solovej J. 2018. The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations. Annales Henri Poincare. 19(4), 1167–1214. mla: Benedikter, Niels P., et al. “The Dirac–Frenkel Principle for Reduced Density Matrices and the Bogoliubov–de Gennes Equations.” Annales Henri Poincare, vol. 19, no. 4, Birkhäuser, 2018, pp. 1167–214, doi:10.1007/s00023-018-0644-z. short: N.P. Benedikter, J. Sok, J. Solovej, Annales Henri Poincare 19 (2018) 1167–1214. date_created: 2018-12-11T11:46:34Z date_published: 2018-04-01T00:00:00Z date_updated: 2023-09-19T10:07:41Z day: '01' ddc: - '510' - '539' department: - _id: RoSe doi: 10.1007/s00023-018-0644-z external_id: isi: - '000427578900006' file: - access_level: open_access checksum: 883eeccba8384ad7fcaa28761d99a0fa content_type: application/pdf creator: system date_created: 2018-12-12T10:11:57Z date_updated: 2020-07-14T12:46:31Z file_id: '4914' file_name: IST-2018-993-v1+1_2018_Benedikter_Dirac.pdf file_size: 923252 relation: main_file file_date_updated: 2020-07-14T12:46:31Z has_accepted_license: '1' intvolume: ' 19' isi: 1 issue: '4' language: - iso: eng month: '04' oa: 1 oa_version: Published Version page: 1167 - 1214 publication: Annales Henri Poincare publication_status: published publisher: Birkhäuser publist_id: '7367' pubrep_id: '993' quality_controlled: '1' scopus_import: '1' status: public title: The Dirac–Frenkel principle for reduced density matrices and the Bogoliubov–de Gennes equations tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 19 year: '2018' ...