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 mass-less 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.

Suppose that particle detectors are placed along a Cauchy surface Σ in Minkowski space-time, and consider a quantum theory with fixed or variable number of particles (i.e., using Fock space or a subspace thereof). It is straightforward to guess what Born's rule should look like for this setting: The probability distribution of the detected configuration on Σ has density |ψ Σ | 2 , where ψ Σ is a suitable wave function on Σ, and the operation | · | 2 is suitably interpreted. We call this statement the "curved Born rule." Since in any one Lorentz frame, the appropriate measurement postulates referring to constant-t hyperplanes should determine the probabilities of the outcomes of any conceivable experiment, they should also imply the curved Born rule. This is what we are concerned with here: deriving Born's rule for Σ from Born's rule in one Lorentz frame (along with a collapse rule). We describe two ways of defining an idealized detection process, and prove for one of them that the probability distribution coincides with |ψ Σ | 2 . For this result, we need two hypotheses on the time evolution: that there is no interaction faster than light, and that there is no creation of particles from the Fock vacuum. The wave function ψ Σ can be obtained from the Tomonaga-Schwinger equation, or from a multi-time wave function by inserting configurations on Σ. Thus, our result establishes in particular how multi-time wave functions are related to detection probabilities.

In this paper, we generalize a previous relativistic 1 + 1-dimensional model for two mass-less Dirac particles with relativistic 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 δ-potential at equal times. Finally, we answer the question 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 3N 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 spacetime 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.

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: ψ (n) (p) at an interior point p and ψ (n+m) (q) at a boundary point q, 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 ψ(x 1 , ..., x N ) depending on N space-time coordinates x i for N 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.