Multi-time Wave Functions Research Articles

Particle Trajectories For Compton Scattering in One Space Dimension

Using a relativistic extension of Bohmian Me-chanics known as Multi-Time Wave Function formula-tion, we examine a two-body, one-dimensional sys-tem consisting of one photon and one electron that interact only upon contact. We investigate the effects that various parameters in this theory including mo-mentum of the incoming photon and mass of the electron have on the dynamics of the two interact-ing bodies with the goal of understanding conser-vation of momentum and energy in the system. We show that the core principles of Compton scattering remain when we use this alternative formulation of quantum mechanics. Although a complete relativ-istic theory of Bohmian mechanics has yet to be de-veloped, our work aims to make the ideas in this the-ory more accessible to a wider audience.

Existence of relativistic dynamics for two directly interacting Dirac particles in 1 + 3 dimensions

Here we prove the existence and uniqueness of solutions of a class of integral equations describing two Dirac particles in 1+3 dimensions with direct interactions. This class of integral equations arises naturally as a relativistic generalization of the integral version of the two-particle Schrödinger equation. Crucial use of a multi-time wave function [Formula: see text] with [Formula: see text] is made. A central feature is the time delay of the interaction. Our main result is an existence and uniqueness theorem for a Minkowski half-space, meaning that the Minkowski spacetime is cut off before [Formula: see text]. We furthermore show that the solutions are determined by Cauchy data at the initial time; however, no Cauchy problem is admissible at other times. A second result is to extend the first one to particular FLRW spacetimes with a Big Bang singularity, using the conformal invariance of the Dirac equation in the massless case. This shows that the cutoff at [Formula: see text] can arise naturally and be fully compatible with relativity. We thus obtain a class of interacting, manifestly covariant and rigorous models in [Formula: see text] dimensions.

Consistency Proof for Multi-time Schr\xf6dinger Equations with Particle Creation and Ultraviolet Cut-Off

For multi-time wave functions, which naturally arise as the relativistic particle-position representation of the quantum state vector, the analog of the Schrödinger equation consists of several equations, one for each time variable. This leads to the question of how to prove the consistency of such a system of PDEs. The question becomes more difficult for theories with particle creation, as then different sectors of the wave function have different numbers of time variables. Petrat and Tumulka (2014) gave an example of such a model and a non-rigorous argument for its consistency. We give here a rigorous version of the argument after introducing an ultraviolet cut-off into the creation and annihilation terms of the multi-time evolution equations. These equations form an infinite system of coupled PDEs; they are based on the Dirac equation but are not fully relativistic (in part because of the cut-off). We prove the existence and uniqueness of a smooth solution to this system for every initial wave function from a certain class that corresponds to a dense subspace in the appropriate Hilbert space.

A Lorentz-covariant interacting electron–photon system in one space dimension

A Lorenz-covariant system of wave equations is formulated for a quantum-mechanical two-body system in one space dimension, comprised of one electron and one photon. Manifest Lorentz covariance is achieved using Dirac's formalism of multi-time wave functions, i.e., wave functions $\Psi(\mathbf{x}_{{ph}},\mathbf{x}_{{el}})$ where $\mathbf{x}_{{el}},\mathbf{x}_{{ph}}$ are the generic spacetime events of the electron and photon, respectively. Their interaction is implemented via a Lorentz-invariant no-crossing-of-paths boundary condition at the coincidence submanifold $\{\mathbf{x}_{{el}}=\mathbf{x}_{{ph}}\}$, compatible with particle current conservation. The corresponding initial-boundary-value problem is proved to be well-posed. Electron and photon trajectories are shown to exist globally in a Hypersurface Bohm--Dirac theory, for typical particle initial conditions. Also presented are the results of some numerical experiments which illustrate Compton scattering as well as a new phenomenon: photon capture and release by the electron.

Born’s rule for arbitrary Cauchy surfaces

Suppose that particle detectors are placed along a Cauchy surface $\Sigma$ 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 $\Sigma$ has density $|\psi_\Sigma|^2$, where $\psi_\Sigma$ is a suitable wave function on $\Sigma$, and the operation $|\cdot|^2$ 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 $\Sigma$ 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 $|\psi_\Sigma|^2$. For this result, we need two hypotheses on the time evolution: that there is no interaction faster than light, and that there is no propagation faster than light. The wave function $\psi_\Sigma$ can be obtained from the Tomonaga--Schwinger equation, or from a multi-time wave function by inserting configurations on $\Sigma$. Thus, our result establishes in particular how multi-time wave functions are related to detection probabilities.

Multi-time formulation of particle creation and annihilation via interior-boundary conditions

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.

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

Dirac, Fock, and Podolsky devised a relativistic model in 1932 in which a fixed number of N Dirac electrons interact through a second-quantized electromagnetic field. It is formulated with the help of a multitime wave function ψ(t1,x1,…,tN,xN) that generalizes the Schrödinger multiparticle wave function to allow for a manifestly relativistic formulation of wave mechanics. The dynamics is given in terms of N evolution equations that have to be solved simultaneously. Integrability imposes a rather strict constraint on the possible forms of interaction between the N particles and makes the rigorous construction of interacting dynamics a long-standing problem, also present in the modern formulation of quantum field theory. For a simplified version of the multitime model, in our case describing N Dirac electrons that interact through a relativistic scalar field, we prove well-posedness of the corresponding multitime initial value problem and discuss the mechanism and type of interaction between the charges. For the sake of mathematical rigor, we are forced to employ an ultraviolet cutoff in the scalar field. Although this again breaks the desired relativistic invariance, this violation occurs only on the arbitrarily small but finite length-scale of this cutoff. In view of recent progress in this field, the main mathematical challenges faced in this work are, on the one hand, the unboundedness from below of the free Dirac Hamiltonians and the unbounded, time-dependent interaction terms, and on the other hand, the necessity of pointwise control of the multitime wave function.

Direct interaction along light cones at the quantum level

Here, we point out that interactions with time delay can be described at the quantum level using a multi-time wave function , i.e. a wave function depending on one spacetime variable per particle. In particular, such a wave function makes it possible to implement direct interaction along light cones (not mediated by fields), as in the Wheeler–Feynman formulation of electrodynamics. Our results are as follows. (1) We derive a covariant two-particle integral equation and discuss it in detail. (2) It is shown how this integral equation (or equivalently, a system of two integro-differential equations) can be understood as defining the time evolution of ψ in a consistent way. (3) We demonstrate that the equation has strong analogies with Wheeler–Feynman electrodynamics and therefore suggests a possible new quantization of that theory. (4) We propose two natural ways how the two-particle equation can be extended to N particles. It is shown that exactly one of them leads to the usual Schrödinger equation with Coulomb-type pair potentials if time delay effects are neglected.

Multi-Time Wave Functions Versus Multiple Timelike Dimensions

Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables (one per particle). In this paper we contrast them with solutions of wave equations on a space-time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as ${+}{+}{-}{-}$ or ${+}{+}{-}{-}{-}{-}{-}{-}$, instead of ${+}{-}{-}{-}$. Despite the superficial similarity, the two behave very differently: Whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant Schr\"odinger equations for multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states.

Multi-time wave functions

In non-relativistic quantum mechanics of N particles in three spatial dimensions, the wave function ψ(q1, …, qN, 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((t1, q1), …, (tN, qN)), 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.

Consistency of multi-time Dirac equations with general interaction potentials

In 1932, Dirac proposed a formulation in terms of multi-time wave functions as candidate for relativistic many-particle quantum mechanics. A well-known consistency condition that is necessary for existence of solutions strongly restricts the possible interaction types between the particles. It was conjectured by Petrat and Tumulka that interactions described by multiplication operators are generally excluded by this condition, and they gave a proof of this claim for potentials without spin-coupling. Under suitable assumptions on the differentiability of possible solutions, we show that there are potentials which are admissible, give an explicit example, however, show that none of them fulfills the physically desirable Poincaré invariance. We conclude that in this sense, Dirac’s multi-time formalism does not allow to model interaction by multiplication operators, and briefly point out several promising approaches to interacting models one can instead pursue.

Interacting relativistic quantum dynamics for multi-time wave functions

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.

A simple explicitly solvable interacting relativistic N-particle model

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.

On the question of current conservation for the two-body Dirac equations of constraint theory

The two-body Dirac (2BD) equations of constraint theory are of special interest not only in view of applications for phenomenological calculations of mesonic spectra but also because they avoid no-go theorems about relativistic interactions. Furthermore, they provide a quantum mechanical description in a manifestly Lorentz invariant way using the concept of a multi-time wave function. In this paper, we place them into the context of the multi-time formalism of Dirac, Tomonaga and Schwinger for the first time. A general physical and mathematical framework is outlined and the mechanism which permits relativistic interaction is identified. The main requirement derived from the general framework is the existence of conserved tensor currents with a positive component which can play the role of a probability density. We analyze this question for a general class of 2BD equations thoroughly and comprehensively. While the free Dirac current is not conserved, it is possible to find replacements. Improving on previous research, we achieve definite conclusions whether restrictions of the function space or of the interaction terms can guarantee the positive definiteness of the currents—and whether such restrictions are physically adequate. The consequences of the results are drawn, with respect to both applied and foundational perspectives.

A relativistically interacting exactly solvable multi-time model for two massless Dirac particles in 1 + 1 dimensions

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.

Multi-time wave functions for quantum field theory

Multi-time wave functions such as ϕ(t1,x1,…,tN,xN) have one time variable tj for each particle. This type of wave function arises as a relativistic generalization of the wave function ψ(t,x1,…,xN) of non-relativistic quantum mechanics. We show here how a quantum field theory can be formulated in terms of multi-time wave functions. We mainly consider a particular quantum field theory that features particle creation and annihilation. Starting from the particle–position representation of state vectors in Fock space, we introduce multi-time wave functions with a variable number of time variables, set up multi-time evolution equations, and show that they are consistent. Moreover, we discuss the relation of the multi-time wave function to two other representations, the Tomonaga–Schwinger representation and the Heisenberg picture in terms of operator-valued fields on space–time. In a certain sense and under natural assumptions, we find that all three representations are equivalent; yet, we point out that the multi-time formulation has several technical and conceptual advantages.

Multi-time Schrödinger equations cannot contain interaction potentials

Multi-time wave functions are wave functions that have a time variable for every particle, such as \documentclass[12pt]{minimal}\begin{document}$\phi (t_1,{\bm x}_1,\ldots ,t_N,{\bm x}_N)$\end{document}ϕ(t1,x1,...,tN,xN). They arise as a relativistic analog of the wave functions of quantum mechanics but can be applied also in quantum field theory. The evolution of a wave function with N time variables is governed by N Schrödinger equations, one for each time variable. These Schrödinger equations can be inconsistent with each other, i.e., they can fail to possess a joint solution for every initial condition; in fact, the N Hamiltonians need to satisfy a certain commutator condition in order to be consistent. While this condition is automatically satisfied for non-interacting particles, it is a challenge to set up consistent multi-time equations with interaction. We prove for a wide class of multi-time Schrödinger equations that the presence of interaction potentials (given by multiplication operators) leads to inconsistency. We conclude that interaction has to be implemented instead by creation and annihilation of particles, which, in fact, can be done consistently [S. Petrat and R. Tumulka, “Multi-time wave functions for quantum field theory,” Ann. Physics (to be published)]. We also prove the following result: When a cut-off length δ > 0 is introduced (in the sense that the multi-time wave function is defined only on a certain set of spacelike configurations, thereby breaking Lorentz invariance), then the multi-time Schrödinger equations with interaction potentials of range δ are consistent; however, in the desired limit δ → 0 of removing the cut-off, the resulting multi-time equations are interaction-free, which supports the conclusion expressed in the title.

Multi-time formulation of pair creation

In a recent work (Petrat and Tumulka 2013 arXiv:1309.0802), we have described a formulation of a model quantum field theory in terms of a multi-time wave function and proposed a suitable system of multi-time Schrödinger equations governing the evolution of that wave function. Here, we provide further evidence that multi-time wave functions provide a viable formulation of relevant quantum field theories by describing a multi-time formulation, analogous to the one in Petrat and Tumulka (2013 arXiv:1309.0802), of another model quantum field theory. This model involves three species of particles, say x-particles, anti-x-particles, and y-particles, and postulates that a y-particle can decay into a pair consisting of an x and an anti-x particle, and that an x–anti-x pair, when they meet, annihilate each other creating a y-particle. (Alternatively, the model can also be interpreted as representing beta decay.) The wave function is a multi-time version of a time-dependent state vector in Fock space (or rather, the appropriate product of Fock spaces) in the particle-position representation. We write down multi-time Schrödinger equations and verify that they are consistent, provided that an even number of the three particle species involved are fermionic.