In this paper, we report on recent progress about a rigorous and manifestly covariant interacting model for two Dirac particles in 1+1 dimensions [9, 10]. It is formulated using the multi-time formalism of Dirac, Tomonaga and Schwinger. The mechanism of interaction is a relativistic generalization of contact interactions, and it is achieved going beyond the usual functional-analytic Hamiltonian method.
In this paper, we report on recent progress about a rigorous and manifestly covariant interacting model for two Dirac particles in 1+1 dimensions [9, 10]. It is formulated using the multi-time formalism of Dirac, Tomonaga and Schwinger. The mechanism of interaction is a relativistic generalization of contact interactions, and it is achieved going beyond the usual functional-analytic Hamiltonian method.
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 this paper, we generalize a previous relativistic -dimensional model for two mass-less Dirac particles with contact interactions to the N -particle case. Our model is based on the notion of a multi-time wave function which, according to Dirac, is the central object in a relativistic multi-particle quantum theory in the Schrödinger picture. Consequently, we achieve a manifestly Lorentz invariant formulation on configuration space–time. Our model is constructed to be compatible with antisymmetry and probability conservation in a relativistic sense. On the mathematical side, we further develop the method of multi-time characteristics and show that uniqueness of solutions follows from probability conservation. We prove that the model is interacting and outline how one can understand the interaction as effectively given by a -potential at equal times. Finally, we answer the question of whether Lorentz invariant and probability-conserving dynamics can also be obtained when the particles are confined in a region with a non-zero minimal space-like distance, a question relevant for an extension to higher dimensions.
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.
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.
On the relativistic quantum mechanics of a photon between two electrons in 1+1 dimensions
Abstract A Lorentz-covariant system of wave equations is formulated for a quantum-mechanical three-body system in one space dimension, comprised of one photon and two identical massive spin one-half Dirac particles, which can be thought of as two electrons (or alternatively, two positrons). Manifest covariance is achieved using Dirac’s formalism of multi-time wave functions, i.e., wave functions Ψ(xph,xe1,xe2) Ψ ( x ph , x e 1 , x e 2 ) where xph,xe1,xe2 x ph , x e 1 , x e 2 are generic spacetime events of the photon and two electrons, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifolds {xph=xe1} { x ph = x e 1 } and {xph=xe2} { x ph = x e 2 } compatible with conservation of probability current. The corresponding initial-boundary value problem is shown to be well-posed, and it is shown that the unique solution can be represented by a convergent infinite sum of Feynman-like diagrams, each one corresponding to the photon bouncing between the two electrons a fixed number of times.
A relativistic configuration space (many-time) presentation of quantum electrodynamics is developed, which, being equivalent to the Tomonaga-Schwinger formalism, may be expected to have the advantage of its direct applicability to bound-state problems of multi-electron systems. The main purpose of this paper is therefore to get an evidently relativistically invariant description of the interactions between individual electrons, in such a way, however, as to avoid those divergence difficulties which may be accounted for by the use of electron plane waves for building up the actual state of the system. The order of the presentation is as follows: a general relativistically invariant proof of equivalence of the Tomonaga-Schwinger formalism and the many-time formalism of Dirac, Fock, and Podolski given in Sec. I will provide a necessary starting point for further generalization in the form of Eq. (53). This equation, when eventually solve in Sec. II, gives rise to a formal extension of the multi-electron wave function concept into regions for which the latter remained hitherto undefined. In Sec. III a method of elimination of virtual processes is outlined and invariant expressions for the multi-electron interactions are derived. All explicit derivations in this paper are carried out to terms of the order of e2. An example, a relativistic two-electron wave equation, is given. This equation accounts for the Coulomb and Breit interactions.