{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T15:15:13Z","timestamp":1760800513997,"version":"build-2065373602"},"reference-count":55,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2021,10,3]],"date-time":"2021-10-03T00:00:00Z","timestamp":1633219200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola\u2013Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov\u2013Milne\u2013Pinney equation. Their mapping \u201cEisenhart\u2013Duval\u201d lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.<\/jats:p>","DOI":"10.3390\/sym13101866","type":"journal-article","created":{"date-parts":[[2021,10,11]],"date-time":"2021-10-11T01:59:47Z","timestamp":1633917587000},"page":"1866","update-policy":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/linproxy.fan.workers.dev:443\/https\/orcid.org\/0000-0001-8010-3955","authenticated-orcid":false,"given":"Qiliang","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China"}]},{"ORCID":"https:\/\/linproxy.fan.workers.dev:443\/https\/orcid.org\/0000-0002-1737-3845","authenticated-orcid":false,"given":"Pengming","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519082, China"}]},{"ORCID":"https:\/\/linproxy.fan.workers.dev:443\/https\/orcid.org\/0000-0002-6337-4494","authenticated-orcid":false,"given":"Peter A.","family":"Horvathy","sequence":"additional","affiliation":[{"name":"Institut Denis Poisson, Tours University-Orl\u00e9ans University, UMR 7013, F-37200 Tours, France"}]}],"member":"1968","published-online":{"date-parts":[[2021,10,3]]},"reference":[{"key":"ref_1","unstructured":"Feynman, R.P., and Hibbs, A.R. (1965). Quantum Mechanics and Path Integrals, McGraw-Hill."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Schulman, L. (1981). Techniques and Applications of Path Integration, Wiley.","DOI":"10.1063\/1.2914703"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Khandekar, D.C., Lawande, S.V., and Bhagwat, K.V. (1993). Path-Integral Methods and Their Applications, World Scientific. [1st ed.].","DOI":"10.1142\/1332"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/0003-4916(76)90041-5","article-title":"The Semiclassical Expansion","volume":"97","year":"1976","journal-title":"Ann. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/0003-4916(77)90269-X","article-title":"A New Approach to Gaussian Path Integrals and the Evaluation of the Semiclassical Propagator","volume":"103","author":"Levit","year":"1977","journal-title":"Ann. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF00671761","article-title":"Extended Feynman Formula for Harmonic Oscillator","volume":"18","author":"Horvathy","year":"1979","journal-title":"Int. J. Theor. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1007\/BF02960144","article-title":"Forze non-conservative nella meccanica quantistica","volume":"18","author":"Caldirola","year":"1941","journal-title":"Nuovo Cim."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1143\/ptp\/3.4.440","article-title":"On the Quantization of the Dissipative Systems","volume":"3","author":"Kanai","year":"1948","journal-title":"Prog. Theor. Phys."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/0370-1573(81)90033-8","article-title":"Classical and quantum mechanics of the damped harmonic oscillator","volume":"80","author":"Dekker","year":"1981","journal-title":"Phys. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/0370-1573(86)90029-3","article-title":"Feynman Path Integrals: Some Exact Results and Applications","volume":"137","author":"Khandekar","year":"1986","journal-title":"Phys. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0370-1573(01)00077-1","article-title":"The Quantum damped harmonic oscillator","volume":"362","author":"Um","year":"2002","journal-title":"Phys. Rep."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0375-9601(85)90122-7","article-title":"Transformation of the free propagator to the quadratic propagator","volume":"110","author":"Junker","year":"1985","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/0375-9601(82)90425-X","article-title":"Jackiw Transformation in Path Integrals","volume":"91","author":"Cai","year":"1982","journal-title":"Phys. Lett. A"},{"key":"ref_14","unstructured":"Cai, P.Y., Cai, J.M., and Inomata, A. (1989). A. A time-dependent conformal transformation in Feynman\u2019s path integral. Path integrals from meV to MeV, World Scientific."},{"key":"ref_15","unstructured":"Liang, J.Q., Wang, M., Qiao, S.N., and Su, D.C. (1992, January 12\u201316). Time-dependent conformal transformation in quantum mechanics. Proceedings of the ISATQP-Shanxi 1992, Taiyuan, China."},{"key":"ref_16","first-page":"191","article-title":"The maximal kinematical invariance group of the harmonic oscillator","volume":"46","author":"Niederer","year":"1973","journal-title":"Helv. Phys. Acta"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"591","DOI":"10.2307\/1968307","article-title":"Dynamical trajectories and geodesics","volume":"30","author":"Eisenhart","year":"1928","journal-title":"Ann. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1841","DOI":"10.1103\/PhysRevD.31.1841","article-title":"Bargmann Structures and Newton-cartan Theory","volume":"31","author":"Duval","year":"1985","journal-title":"Phys. Rev. D"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1007\/BF00420564","article-title":"Time Dependent Quantum Systems and Chronoprojective Geometry","volume":"10","author":"Burdet","year":"1985","journal-title":"Lett. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3907","DOI":"10.1103\/PhysRevD.43.3907","article-title":"Celestial mechanics, conformal structures and gravitational waves","volume":"43","author":"Duval","year":"1991","journal-title":"Phys. Rev. D"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"631","DOI":"10.1016\/j.aop.2016.07.033","article-title":"Eisenhart lifts and symmetries of time-dependent systems","volume":"373","author":"Cariglia","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_22","unstructured":"Arnold, V.I. (1978). Supplementary Chapters to the Theory of Ordinary Differential Equations, Nauka."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Arnold, V.I. (1983). Geometrical Methods in the Theory of Ordinary Differential Equations, Springer. (In English).","DOI":"10.1007\/978-1-4684-0147-9"},{"key":"ref_24","unstructured":"Maslov, V.P., Bouslaev, V.C., and Arnol\u2019d, V.I. (1972). Th\u00e9orie des Perturbations et M\u00e9thodes Asymptotiques, Dunod."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF01075861","article-title":"Characteristic class entering in quantization conditions","volume":"1","author":"Arnold","year":"1967","journal-title":"Funktsional\u2019Nyi Anal. Ego Prilozheniya"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1007\/3-540-07789-8_13","article-title":"Construction explicite de l\u2019indice de Maslov. Applications","volume":"50","author":"Souriau","year":"1976","journal-title":"Lect. Notes Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/BF01614222","article-title":"Generating functions for the affine symplectic group","volume":"58","author":"Burdet","year":"1978","journal-title":"Comm. Math. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"32643269","DOI":"10.1063\/1.526073","article-title":"Quantum Systems with Time Dependent Harmonic Part and the Morse Index","volume":"25","author":"Rezende","year":"1984","journal-title":"J. Math. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"065302","DOI":"10.1088\/1751-8113\/44\/6\/065302","article-title":"The quantum Arnold transformation","volume":"44","author":"Aldaya","year":"2011","journal-title":"J. Phys. A"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"475303","DOI":"10.1088\/1751-8113\/45\/47\/475303","article-title":"Symmetries of the quantum damped harmonic oscillator","volume":"45","author":"Guerrero","year":"2012","journal-title":"J. Phys. A"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1260011","DOI":"10.1142\/S0219887812600110","article-title":"Unfolding the quantum Arnold transformation","volume":"9","author":"Guerrero","year":"2012","journal-title":"Int. J. Geom. Meth. Mod. Phys."},{"key":"ref_32","first-page":"1","article-title":"Second order differential equations. Conditions of complete integrability","volume":"9","author":"Ermakov","year":"1880","journal-title":"Univ. Izv. Kiev Series III"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1103\/PhysRev.35.863","article-title":"The numerical determination of characteristic numbers","volume":"35","author":"Milne","year":"1930","journal-title":"Phys. Rev."},{"key":"ref_34","first-page":"68","article-title":"The nonlinear differential equation y\u2032\u2032+p(x)y+cy3=0","volume":"1","author":"Pinney","year":"1959","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1140\/epjc\/s10052-018-5568-8","article-title":"Geometry of the isotropic oscillator driven by the conformal mode","volume":"78","author":"Galajinsky","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1140\/epjc\/s10052-018-5789-x","article-title":"Cosmological aspects of the Eisenhart\u2013Duval lift","volume":"78","author":"Cariglia","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/0375-9601(85)90166-5","article-title":"Exact propagator for the harmonic oscillator with time dependent mass","volume":"113","author":"Cheng","year":"1985","journal-title":"Phys. Lett. A"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.physletb.2018.04.069","article-title":"Screw-symmetric gravitational waves: A double copy of the vortex","volume":"782","author":"Ilderton","year":"2018","journal-title":"Phys. Lett. B"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"022502","DOI":"10.1063\/1.5136078","article-title":"Scaling and conformal symmetries for plane gravitational waves","volume":"61","author":"Zhang","year":"2020","journal-title":"J. Math. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"012015","DOI":"10.1088\/1742-6596\/538\/1\/012015","article-title":"Generalizations of the Ermakov system through the Quantum Arnold Transformation","volume":"538","author":"Guerrero","year":"2014","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_41","first-page":"6658","article-title":"Conformal properties of Chern-Simons vortices in external fields","volume":"D50","author":"Duval","year":"1994","journal-title":"Phys. Rev."},{"key":"ref_42","unstructured":"Gibbons, G.W. (2014). Dark Energy and the Schwarzian Derivative. arXiv."},{"key":"ref_43","unstructured":"Weisstein, E.W. (2021, August 01). Mathieu Function. Available online: https:\/\/linproxy.fan.workers.dev:443\/https\/mathworld.wolfram.com\/MathieuFunction.html."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1007\/BF00403464","article-title":"Exact propagator for the one-dimensional time-dependent quadratic Lagrangian","volume":"14","author":"Cheng","year":"1987","journal-title":"Lett. Math. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1063\/1.3070673","article-title":"Introducing scale symmetry","volume":"25N1","author":"Jackiw","year":"1972","journal-title":"Phys. Today"},{"key":"ref_46","first-page":"802","article-title":"The maximal kinematical invariance group of the free Schrodinger equation","volume":"45","author":"Niederer","year":"1972","journal-title":"Helv. Phys. Acta"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1103\/PhysRevD.5.377","article-title":"Scale and conformal transformations in galilean-covariant field theory","volume":"5","author":"Hagen","year":"1972","journal-title":"Phys. Rev. D"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/BF02785666","article-title":"Conformal Invariance in Quantum Mechanics","volume":"34","author":"Fubini","year":"1976","journal-title":"Nuovo Cim. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0003-4916(90)90354-Q","article-title":"Dynamical Symmetry of the Magnetic Vortex","volume":"201","author":"Jackiw","year":"1990","journal-title":"Ann. Phys."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0003-4916(80)90295-X","article-title":"Dynamical Symmetry of the Magnetic Monopole","volume":"129","author":"Jackiw","year":"1980","journal-title":"Ann. Phys."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"421","DOI":"10.1016\/j.physletb.2018.05.072","article-title":"Memory effect, conformal symmetry and gravitational plane waves","volume":"782","author":"Andrzejewski","year":"2018","journal-title":"Phys. Lett. B"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"155008","DOI":"10.1088\/1361-6382\/ab2394","article-title":"Niederer\u2019s transformation, time-dependent oscillators and polarized gravitational waves","volume":"36","author":"Andrzejewski","year":"2019","journal-title":"Class. Quantum Gravity"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"105019","DOI":"10.1103\/PhysRevD.101.105019","article-title":"Conformal bridge between asymptotic freedom and confinement","volume":"101","author":"Inzunza","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","unstructured":"Dhasmana, S., Sen, A., and Silagadze, Z.K. (2021). Equivalence of a harmonic oscillator to a free particle and Eisenhart lift. arXiv.","DOI":"10.1016\/j.aop.2021.168623"},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Guha, P., and Garai, S. (2021). Integrable modulation, curl forces and parametric Kapitza equation with trapping and escaping. arXiv.","DOI":"10.1007\/s11071-021-06947-6"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/13\/10\/1866\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:09:13Z","timestamp":1760166553000},"score":1,"resource":{"primary":{"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/13\/10\/1866"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,3]]},"references-count":55,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2021,10]]}},"alternative-id":["sym13101866"],"URL":"https:\/\/linproxy.fan.workers.dev:443\/https\/doi.org\/10.3390\/sym13101866","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,10,3]]}}}