Towards the genuinely ghost–free massive vector Horndeski theory

Maier, Markus (30. März 2020): Towards the genuinely ghost–free massive vector Horndeski theory. On the Lagrangian characterization of first–order gravity. Masterarbeit, Fakultät für Physik, Ludwig-Maximilians-Universität München.

Abstract

We put forward an extension to already existing Lagrangian constraint algorithms, which is readily applicable to (almost all) first–order classical field theories. Our algorithm is optimized to obtain the explicit constraints and thus count the number of propagating degrees of freedom in said theories. This is the main result of the thesis. We employ both the renowned Dirac–Bergmann procedure and our own formalism to obtain the constraint structure of the H(P)–formulation of non–linear electrodynamics and two–dimensional Palatini gravity. Both approaches yield the same results. We observe that our proposed method is an algebraically simpler and conceptually clearer way to calculate the number of physical modes. The relevance and usefulness of our novel Lagrangian iterative procedure are twofold. On the one hand, it simplifies the determination of the constraint structure of these theories. This is particularly pertinent for effective theories of multiple interacting fields of different spins, whose analysis is in general cumbersome and which are prone to the presence of additional unphysical modes — ghosts. On the other hand, it constitutes an essential first step towards establishing a Lagrangian building principle for genuinely ghost–free theories. Indeed, given a first–order Lagrangian, our method yields its associated constraint structure. It is then possible to reverse the logic and find out the conditions a Lagrangian must satisfy in order to possess a certain constraint structure. This natural follow–up is work in progress.

Abstract

Wir schlagen eine Erweiterung zu bereits existierenden Lagrange Zwangsbedingungsalgorithmen vor, welche ohne weiteres auf (fast alle) klassischen Feldtheorien erster Ordnung anwendbar ist. Unser Algorithmus ist optimiert, um die explizite Form der Zwangsbedingungen zu erhalten und ermöglicht damit das Zählen der propagierenden Freiheitsgrade in diesen Theorien. Das ist das Hauptergebnis dieser Arbeit. Wir wenden sowohl das renommierte Dirac–Bergmann Verfahren, als auch unseren eigenen Formalismus an, um die Zwangsbedingungsstruktur der nicht–linearen Elektrodynamik in ihrer H(P)–Formulierung und der zweidimensionalen Palatini Gravitation zu bestimmen. Beide Ans¨atze liefern das gleiche Ergebnis. Wir beobachten, dass unsere Methode der algebraisch einfachere und konzeptuell klarere Weg ist, um die Anzahl der physikalischen Moden zu berechnen. Relevanz und Nutzen unseres neuen iterativen Lagrange Verfahrens ist zweifältig. Einerseits vereinfacht es die Bestimmung der Zwangsbedingungsstruktur besagter Theorien. Das ist besonders relevant für effektive Feldtheorien mehrerer wechselwirkender Felder mit verschiedenen Spins. Deren Analyse ist im Allgemeinen umständlich und sie sind anfällig für das Auftauchen zusätzlicher, unphysikalischer Moden — Geister. Andererseits stellt es den ersten essenziellen Schritt in Richtung eines Konstruktionsprinzips für wirklich geistfreie Theorien dar. Unter Voraussetzung einer Lagrangedichte erster Ordnung kann mit unserer Methode die zugehörige Struktur der Zwangsbedingungen bestimmt werden. Es ist dann möglich die Logik umzukehren und Bedingungen zu finden, die eine Lagrangedichte erfüllen muss, um eine bestimmte Zwangsbedingungsstruktur aufzuweisen. Dies ist Teil einer Follow–up–Arbeit in Vorbereitung.

• Dokumententyp: LMU München: Studienabschlussarbeit
• Keywords: constraint analysis; Horndeski; gravity; Lagrangian; first-order classical field theory; degree of freedom; Dirac-Bergmann
• Fakultät: Physik
• Institut oder Departement: Universitäts-Sternwarte München
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik ; 500 Naturwissenschaften und Mathematik > 520 Astronomie ; 500 Naturwissenschaften und Mathematik > 530 Physik
• Sprache: Englisch

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