Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: [Formula: see text] at an interior point [Formula: see text] and [Formula: see text] at a boundary point [Formula: see text], typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac’s concept of multi-time wave functions, i.e. wave functions [Formula: see text] depending on [Formula: see text] space-time coordinates [Formula: see text] for [Formula: see text] particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that, apart from a constant matrix (which is required to transform in a certain way), the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.
Interior-boundary conditions (IBCs) have been suggested as a possibility to circumvent the problem of ultraviolet divergences in quantum field theories. In the IBC approach, particle creation and annihilation is described with the help of linear conditions that relate the wave functions of two sectors of Fock space: [Formula: see text] at an interior point [Formula: see text] and [Formula: see text] at a boundary point [Formula: see text], typically a collision configuration. Here, we extend IBCs to the relativistic case. To do this, we make use of Dirac’s concept of multi-time wave functions, i.e. wave functions [Formula: see text] depending on [Formula: see text] space-time coordinates [Formula: see text] for [Formula: see text] particles. This provides the manifestly covariant particle-position representation that is required in the IBC approach. In order to obtain rigorous results, we construct a model for Dirac particles in 1+1 dimensions that can create or annihilate each other when they meet. Our main results are an existence and uniqueness theorem for that model, and the identification of a class of IBCs ensuring local probability conservation on all Cauchy surfaces. Furthermore, we explain how these IBCs relate to the usual formulation with creation and annihilation operators. The Lorentz invariance is discussed and it is found that, apart from a constant matrix (which is required to transform in a certain way), the model is manifestly Lorentz invariant. This makes it clear that the IBC approach can be made compatible with relativity.
Abstract Interior-boundary conditions (IBCs) are boundary conditions on wave functions for Schrödinger equations that allow that probability can flow into (and thus be lost at) a boundary of configuration space while getting added in another part of configuration space. IBCs are of particular interest because they allow for defining Hamiltonians involving particle creation and annihilation (as used in quantum field theories) without the need for renormalization or ultraviolet cut-off. For those Hamiltonians, the relevant boundary has codimension 3. In this paper, we develop (what we conjecture is) the general form of IBCs for the Laplacian operator (or Schrödinger operators), but we focus on the simpler case of boundaries with codimension 1.
The question how to Lorentz transform an N-particle wave function naturally leads to the concept of a so-called multi-time wave function, i.e., a map from (space-time)N to a spin space. This concept was originally proposed by Dirac as the basis of relativistic quantum mechanics. In such a view, interaction potentials are mathematically inconsistent. This fact motivates the search for new mechanisms for relativistic interactions. In this paper, we explore the idea that relativistic interaction can be described by boundary conditions on the set of coincidence points of two particles in space-time. This extends ideas from zero-range physics to a relativistic setting. We illustrate the idea at the simplest model which still possesses essential physical properties like Lorentz invariance and a positive definite density: two-time equations for massless Dirac particles in 1 + 1 dimensions. In order to deal with a spatio-temporally non-trivial domain, a necessity in the multi-time picture, we develop a new method to prove existence and uniqueness of classical solutions: a generalized version of the method of characteristics. Both mathematical and physical considerations are combined to precisely formulate and answer the questions of probability conservation, Lorentz invariance, interaction, and antisymmetry.
Carrying over the idea of a wave function to relativistic quantum mechanics naturally leads to a multi-time wave function of the form ψ(t 1 , x 1 , ..., t N , x N ).It depends on one time coordinate t k and one space coordinate x k for each particle k = 1, ..., N .As already proposed by Dirac in 1932, the dynamics of a multi-time wave function can be described by a system of N equations of motion, which determine the evolution along the N time coordinates.The existence of a common solution to such a multi-time system is associated with an integrability condition, the so-called consistency condition, that substantially restricts the possibilities to introduce an interaction between the particles.The aim of this doctoral thesis is to treat several physically relevant possibilities for interacting dynamics in a mathematically rigorous fashion.After a physical introduction to multi-time wave functions and their meaning as probability amplitudes similarly as in non-relativistic quantum mechanics, the mathematical solution theory of systems of equations for a multi-time wave function is discussed.The consistency conditions in Dirac equations with added interaction potentials are the topic of the first main part of the thesis.A generalization of the no-go result by Petrat and Tumulka (2014) about non-existence of solutions in multi-time systems with interaction potentials is proven.It entails that no Poincaré invariant interacting potentials are compatible with the consistency condition.This possibility for the formulation of relativistic interacting quantum dynamics is hence ruled out.Dirac's suggestion from 1932 to circumvent the problem with interaction potentials is to employ second quantized potentials, which is taken on in the subsequent chapter.Together with Fock and Podolsky, he devised a model of quantum electrodynamics that describes the interaction of N Dirac fermions through an electromagnetic field.In the second main part of this thesis, that model is formulated mathematically, using a scalar field for simplicity and a cut-off to bypass the well-known ultraviolet divergences.First, the single-time equations are treated by proving essential self-adjointness of the Hamiltonians and also invariance of certain regular functions under the respective time-evolutions.Then, a consistent multitime evolution is constructed by glueing together the single-time evolutions.Besides the results on existence and uniqueness of solutions, it is also demonstrated that the model is indeed interacting.In the subsequent part, a well-defined interaction is constructed in another way, namely by particle creation and annihilation via interior-boundary conditions (IBCs).This proposed solution to the ultraviolet problem of quantum field theory is treated in a relativistic setting for the first time.In a simplified model in 1 + 1 dimensions where Dirac fermions create and annihilate each other when they meet, the class of IBCs that lead to probability conservation is identified.The theorem on existence and uniqueness in this model is the first rigorous result in a multi-time system with a variable number of particles.As final results, general theorems on linear systems of arbitrary order in the time derivatives are presented.Those show how the solution theory of multi-time equations changes if time derivatives of higher order are present.v ZUSAMMENFASSUNG Die Übertragung des Konzepts einer Wellenfunktion auf die relativistische Quantenmechanik führt auf natürliche Weise zu einer Vielzeiten-Wellenfunktion, welche die Form ψ(t 1 , x 1 , ..., t N , x N ) besitzt.Sie hängt von einer Zeitkoordinate t k und einer Ortskoordinate x k für jedes Teilchen k = 1, ..., N ab.Wie schon von Dirac 1932 vorgeschlagen, kann die Dynamik einer Vielzeiten-Wellenfunktion durch ein System aus N Bewegungsgleichungen beschrieben werden, welche die Entwicklung entlang der N Zeitkoordinaten bestimmen.Die Existenz einer gemeinsamen Lösung zu solch einem Vielzeiten-System ist hierbei mit einer Integrabilitätsbedingung verbunden, der so genannten Konsistenzbedingung, die die Möglichkeiten, eine Wechselwirkung zwischen den Teilchen einzuführen, erheblich einschränkt.Das Ziel dieser Doktorarbeit ist es, verschiedene physikalisch relevante Möglichkeiten für wechselwirkende Dynamiken auf mathematisch rigorose Weise zu behandeln.Nach einer physikalischen Einleitung über Vielzeiten-Wellenfunktionen und deren Bedeutung als Wahrscheinlichkeitsamplitude, ähnlich wie in der nicht-relativistischen Quantenmechanik, wird die mathematische Lösungstheorie von Gleichungssystemen für eine Vielzeiten-Wellenfunktion besprochen.Die Konsistenzbedingung in Dirac-Gleichungen mit addierten Wechselwirkungspotentialen ist Gegenstand des ersten Hauptteils der Arbeit.Es wird eine Verallgemeinerung des Unmöglichkeits-Resultats von Petrat und Tumulka (2014) über die Nichtexistenz von Lösungen in Vielzeiten-Systemen mit Wechselwirkungspotentialen bewiesen.Es beinhaltet, dass keine Poincaré-invarianten wechselwirkenden Potentiale mit der Konsistenzbedingung konform gehen.Diese Möglichkeit für die Formulierung von relativistischer, wechselwirkender Quantendynamik ist damit ausgeschlossen.Diracs Vorschlag aus dem Jahr 1932, das Problem mit Wechselwirkungspotentialen zu umgehen, ist durch Benutzung zweitquantisierter Potentiale, was im folgenden Kapitel angegangen wird.Zusammen mit Fock und Podolsky ersann er ein Modell der Quantenelektrodynamik, das die Wechselwirkung von N Dirac-Fermionen durch ein elektromagnetisches Feld beschreibt.Im zweiten Hauptteil der Arbeit wird das Modell mathematisch formuliert, wobei der Einfachheit halber ein skalares Feld und zur Verhinderung der wohlbekannten Ultraviolett-Divergenzen ein Cut-Off benutzt wird.Zuerst werden die Ein-Zeit-Gleichungen behandelt, indem die wesentliche Selbstadjungiertheit der Hamiltonians und auch die Invarianz gewisser regulärer Funktionen unter den entsprechenden Zeitentwicklungen gezeigt wird.Sodann wird eine konsistente Vielzeiten-Entwicklung konstruiert, indem man die Ein-Zeit-Entwicklungen zusammenfügt.Neben den Resultaten über Existenz und Eindeutigkeit von Lösungen wird auch aufgezeigt, dass das Modell tatsächlich wechselwirkend ist.Im darauf folgenden Abschnitt wird eine wohldefinierte Wechselwirkung mittels einer anderen Methode konstruiert, und zwar durch Teilchenerzeugung und -vernichtung via interiorboundary conditions (IBCs).Diese vorgeschlagene Lösung des Ultraviolettproblems in der Quantenfeldtheorie wird zum ersten Mal in einem relativistischen Rahmen behandelt.In einem vereinfachten Modell in 1 + 1 Dimensionen, in welchem Dirac-Fermionen sich bei Kontakt gegenseitig erzeugen und vernichten können, werden jene IBCs klassifiziert, die vii zu Wahrscheinlichkeitserhaltung führen.Das Existenz-und Eindeutigkeitstheorem in diesem Modell ist das erste rigorose Resultat in einem Vielzeiten-System mit einer variablen Teilchenzahl.Als letzte Resultate werden allgemeine Theoreme über lineare Systeme beliebiger Ordnung in den Zeitableitungen präsentiert.Diese zeigen auf, wie sich die Lösungstheorie von Vielzeiten-Gleichungen verändert, wenn höhere Zeitableitungen vorkommen.viii
The present work deals with the idea of a multi-time wave function, i.e. a wave function with N space-time arguments for N particles. Firstly, a careful derivation of the necessity of multi-time wave functions in relativistic quantum mechanics is given and a general formalism is developed. Secondly, the physical meaning of multi-time wave functions is discussed in connection with realistic relativistic quantum theories, in particular the "Hypersurface Bohm-Dirac" model. Thirdly, a first interacting model for multi-time wave functions of two Dirac particles in 1+1 space-time dimensions is constructed. Interaction is achieved by means of boundary conditions on configuration space-time, a mechanism closely related to zero-range physics. This is remarkable, as a restrictive consistency condition rules out various types of interaction and consequently no rigorous interacting model was known before. Fourthly, the model is extended to more general types of interaction and to the N-particle case. Higher dimensions are also discussed. Finally, the "Two-Body Dirac equations" of constraint theory are placed within the context of the multi-time formalism. In particular, the question of probability conservation is critically discussed, leading to further implications both for fundamental and applied questions.
In non-relativistic quantum mechanics of N particles in three spatial dimensions, the wave function ψ ( q 1 , ..., q N , t ) is a function of 3 N position coordinates and one time coordinate. It is an obvious idea that in a relativistic setting, such functions should be replaced by ϕ (( t 1 , q 1 ), ..., ( t N , q N )), a function of N space-time points called a multi-time wave function because it involves N time variables. Its evolution is determined by N Schrödinger equations, one for each time variable; to ensure that simultaneous solutions to these N equations exist, the N Hamiltonians need to satisfy a consistency condition. This condition is automatically satisfied for non-interacting particles, but it is not obvious how to set up consistent multi-time equations with interaction. For example, interaction potentials (such as the Coulomb potential) make the equations inconsistent, except in very special cases. However, there have been recent successes in setting up consistent multi-time equations involving interaction, in two ways: either involving zero-range ( δ potential) interaction or involving particle creation and annihilation. The latter equations provide a multi-time formulation of a quantum field theory. The wave function in these equations is a multi-time Fock function, i.e., a family of functions consisting of, for every n = 0, 1, 2, ..., an n -particle wave function with n time variables. These wave functions are related to the Tomonaga–Schwinger approach and to quantum field operators, but, as we point out, they have several advantages.