At the heart of quantum mechanics, the physical theory of elementary particles, lies a wave function. This wave function relates to the configuration of many particles in space. In this project, we consider so-called multi-time wave functions which instead relate to configurations of particles in space *and* time, instead of purely spatial ones. This is required to make the wave function compatible with the theory of relativity, our best theory of space and time.
The key idea is that multi-time wave functions allow us to formulate a new kind of equation which expresses direct interactions with time delay at the quantum level. That means, the action of one particle on another happens only after a delay which depends on their distance. In this way, an alternative mechanism of interaction besides fields can be achieved. The hope is to avoid a fundamental problem with fields, "ultraviolet-divergences". These result from the fact that the field of a particle is infinitely strong at its own location - which is where the field needs to be evaluated, and this infinity is non-sensical. In our approach particles do not act upon themselves, only on each other, and the problem does not occur. The ultimate significance of our project thus is to contribute to a better understanding of the fundamental interactions between subatomic particles.
Our objective has been to investigate the mathematical properties of the new equation, an integral equation for a multi-time wave function. Firstly, it has to be demonstrated that the equation makes sense mathematically (i.e. has solutions). Secondly, a link between the abstract concept of a multi-time wave function and actual experiments has to be established. Usually, the modulus squared of the wave function provides the probability to detect particles at locations in space at a common time (the same for all particles). Our goal has been to extend this rule (the "Born rule") to multi-time wave functions, such that particle detections at different times can be described. The third goal has been to study how the new equation relates to a known classical (i.e. non-quantum) theory of direct interactions with time delay, Wheeler-Feynman electrodynamics, in a suitable limit.
Our conclusions are the following: First, we have shown that the multi-time integral equations are indeed mathematically well-defined for a variety of cases. Furthermore, we identified the data that classify the solutions. Multi-time integral equations, therefore, yield a new, rigorous mechanism how fundamental interactions could work for finite times, not only scattering situations.
Second, we demonstrated for wide classes of quantum field theories that multi-time wave functions yield the probability density to detect configurations of particles not only at equal times but whenever the configuration is spacelike (the most general relativistic notion for "elsewhere in space"). Our integral equation, however, does not fall into these classes, and it will require more work to establish a similar statement for it.
Third, we have shown that common techniques for the classical limit can be extended to the multi-time case, at least in non-interacting (but entangled) situations. The case of delayed interactions, however, has turned out difficult and will be left to future work. On the other hand, we have obtained two major unexpected results: (a) we have shown that the technique of "interior boundary conditions", a novel approach of describing particle creation and annihilation, can be extended to the multi-time case and (b) we have constructed equations of a relativistic (multi-time) photon-electron system with contact interactions.
