Post-Newtonian Description of Quantum Systems in Gravitational Fields
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Post-Newtonian Description of Quantum Systems in Gravitational Fields Von der QUEST-Leibniz-Forschungsschule der Gottfried Wilhelm Leibniz Universität Hannover zur Erlangung des Grades Doktor der Naturwissenschaften Dr. rer. nat. genehmigte Dissertation von M. A. St.Philip Klaus Schwartz, geboren am 12.07.1994 in Langenhagen. 2020 Mitglieder der Promotionskommission: Prof. Dr. Elmar Schrohe (Vorsitzender) Prof. Dr. Domenico Giulini (Betreuer) Prof. Dr. Klemens Hammerer Gutachter: Prof. Dr. Domenico Giulini Prof. Dr. Klemens Hammerer Prof. Dr. Claus Kiefer Tag der Promotion: 18. September 2020 This thesis was typeset with LATEX 2# and the KOMA-Script class scrbook. The main fonts are URW Palladio L and URW Classico by Hermann Zapf, and Pazo Math by Diego Puga. Abstract This thesis deals with the systematic treatment of quantum-mechanical systems situated in post-Newtonian gravitational fields. At first, we develop a framework of geometric background structures that define the notions of a post-Newtonian expansion and of weak gravitational fields. Next, we consider the description of single quantum particles under gravity, before continuing with a simple composite system. Starting from clearly spelled-out assumptions, our systematic approach allows to properly derive the post-Newtonian coupling of quantum-mechanical systems to gravity based on first principles. This sets it apart from other, more heuristic approaches that are commonly employed, for example, in the description of quantum-optical experiments under gravitational influence. Regarding single particles, we compare simple canonical quantisation of a free particle in curved spacetime to formal expansions of the minimally coupled Klein– Gordon equation, which may be motivated from the framework of quantum field theory in curved spacetimes. Specifically, we develop a general WKB-like post-Newtonian expansion of the Klein–Gordon equation to arbitrary order in the inverse of the velocity of light. Furthermore, for stationary spacetimes, we show that the Hamiltonians arising from expansions of the Klein–Gordon equation and from canonical quantisation agree up to linear order in particle momentum, independent of any expansion in the inverse of the velocity of light. Concerning the topic of composite systems, we perform a fully detailed systematic derivation of the first order post-Newtonian quantum Hamiltonian describing the dynamics of an electromagnetically bound two-particle system which is situated in external electromagnetic and gravitational fields. This calculation is based on previous work by Sonnleitner and Barnett, which we significantly extend by the inclusion of a weak gravitational field as described by the Eddington–Robertson parametrised post-Newtonian metric. In the last, independent part of the thesis, we prove two uniqueness results char- acterising the Newton–Wigner position observable for Poincaré-invariant classical Hamiltonian systems: one is a direct classical analogue of the well-known quantum Newton–Wigner theorem, and the other clarifies the geometric interpretation of the Newton–Wigner position as ‘centre of spin’, as proposed by Fleming in 1965. Keywords: quantum systems under gravity, post-Newtonian expansion, post-Newtonian gravity, weak gravity iii Zusammenfassung Diese Arbeit beschäftigt sich mit der systematischen Beschreibung quantenmechani- scher Systeme in post-Newton’schen Gravitationsfeldern. Zunächst entwickeln wir geo- metrische Hintergrundstrukturen, welche die Konzepte einer post-Newton’schen Ent- wicklung und schwacher Gravitationsfelder zu definieren ermöglichen. Anschließend beschäftigen wir uns mit der Beschreibung einzelner Quantenteilchen unter Gravitation und wenden uns schließlich einem einfachen zusammengesetzten System zu. Unsere von klar formulierten Annahmen ausgehende systematische Vorgehensweise ermöglicht es, die post-Newton’sche Kopplung quantenmechanischer Systeme an Gravitation im eigentlichen Sinne herzuleiten. Dies unterscheidet sie von anderen, heuristischeren Herangehensweisen, wie sie beispielsweise oft zur Beschreibung quantenoptischer Experimente unter Gravitation benutzt werden. Für einzelne Teilchen vergleichen wir die einfache kanonische Quantisierung freier Teilchen in gekrümmten Raumzeiten mit formalen Entwicklungen der minimal gekop- pelten Klein-Gordon-Gleichung, welche quantenfeldtheoretisch motiviert werden kön- nen. Konkret entwickeln wir eine allgemeine WKB-artige post-Newton’sche Entwicklung der Klein-Gordon-Gleichung zu beliebiger Ordnung im Inversen der Lichtgeschwindig- keit. Ferner zeigen wir für stationäre Raumzeiten, dass die Hamilton-Operatoren, welche aus Entwicklungen der Klein-Gordon-Gleichung bzw. mit kanonischer Quanti- sierung hergeleitet werden, zu linearer Ordnung im Teilchenimpuls übereinstimmen, unabhängig von jeglicher Entwicklung im Inversen der Lichtgeschwindigkeit. Wir leiten den in erster Ordnung post-Newton’schen Hamiltonoperator vollständig her, der die Dynamik eines elektromagnetisch gebundenen Zwei-Teilchen-Systems beschreibt, das sich in sowohl einem externen elektromagnetischen als auch einem Gravitationsfeld befindet. Diese Rechnung basiert auf einer Arbeit von Sonnleitner und Barnett, die wir durch die Einbeziehung der Gravitation maßgeblich erweitern. Im letzten, unabhängigen Teil der Arbeit beweisen wir zwei Eindeutigkeitsresulta- te über die Newton-Wigner-Ortsobservable für Poincaré-invariante klassische Hamil- ton’sche Systeme. Eines ist ein direktes klassisches Analogon des quantenmechanischen Newton-Wigner-Satzes; das andere gibt eine klare Charakterisierung der geometrischen Interpretation des Newton-Wigner-Orts als „Spin-Zentrum“, die 1965 von Fleming vorgeschlagen wurde. Schlagworte: Quantensysteme unter Gravitation, post-Newton’sche Entwicklung, post-Newton’sche Gravitation, schwache Gravitation v Summarium1 hoc opus est de descriptione sastematica systematium mechanicorum quanticorum in campis gravitalibus post Newtonum. primo recessas structuras geometricas elaborabi- mus, quibus consilia expansionis post Newtonum parvisque campis gravitalibus definiri potest. deinde descriptioni singulorum particulorum quanticorum sub gravitatione studebimus atque ultimo in systemate composito facili versabimur. ab praesumptioni- bus clare conceptis systematice procedenti post Newtonum copulationem systematium mechanicorum quanticorum ad gravitationem proprie dedicare poterimus. qui modus procedendi ab aliis, heuristicis, velut ad experimenta optica quantica describendum utuntur, differt. quod attinet ad singula particula, quantificationem canonicam facilem particulorum nullas vires experientum in spatiotemporibus curvatis cum expansionibus formali- bus equationis Kleini Gordonique minime copulatae, quae ex ratione quanticorum camporum motivari possunt, comparabimus. proprie ad dicendum post Newtonum expansionem generalem WKB-bilem equitationis Kleini Gordonique ad quamlibet or- dinem in inverso velocitatis lucis faciemus. quod praeterea attinet ad spatiotempora stationaria, operatores Hamiltoni de expansionibus equationis Kleini Gordonique aut cum quantificatione canonica dedicatos in ordine lineali inter se impetu particulorum consentire demonstrabimus. quod non obnoxium cuicumque expansioni in inverso velocitatis lucis est. quod attinet ad systemata coniuncta, operatorem Hamiltoni in prima ordine post Newtonum radicite dedicemus, qui dynamiken systematis ex duobus particulis electro- magnetice coniuncti describit, quod et in campo electromagnetico et in campo gravi- tale est. quae ratio in opere Sonnleitneri Barnettique posita est, quod gravitationem comprehendendo augebimus multo. ultima in parte absoluta huius operis duos exitus perspicuitatis de loci quantitate Newtoni Wignerique, quod attinet ad systemata Hamiltoni classica invarianta secun- dum Poincareum, demonstrabimus. alius est analogon classicum directum mechanici quantici theorematis Newtoni Wignerique; alio locus Newtoni Wignerique pro ‘me- dio impetus rotationis interni’ geometrice interpretatur, ut Flemingus proposit anno MMDCCXVIII ab urbe condita. proposita: systemata quantica in gravitatione, expansio post Newtonum, gravitatio post Newtonum, gravitatio parva 1translatus de Philippo Sandero vii Acknowledgements First and foremost, I wish to thank my supervisor Domenico Giulini. Apart from guiding me through the research that lead up to this thesis and always being available for smaller as well as bigger questions of mine, Nico has deeply influenced my whole way of thinking about physics and mathematics. I am very grateful to have had a supervisor with such a sharp and conceptually clear personal way of understanding and explaining physical ideas. Furthermore, I am grateful to Klemens Hammerer, Claus Kiefer, and Elmar Schrohe for their willingness to act as referees for my thesis and/or being part of the examination commission. I wish to thank Klemens as well for bringing to my attention the article [SB18], on which a significant part of my work in this thesis is built. For taming all the probable and improbable uncertainties of university bureaucracy at several stages, even at very short notice by my side, I want to thank Birgit Ohlendorf. My research was funded by the DFG through the Collaborative Research Centre 1227 DQ-mat, projects B08 and A05. I want to thank my good friends Anke, Flo, and Rökki for their friendship, and for keeping me sane in phases of high workload. Special thanks go to my friends and colleagues from our institute, who made the last few years into a very enjoyable time for me: Johannes, with