## Periodic Reporting for period 2 - Multi-time Integral Eqs. (Interacting relativistic quantum dynamics via multi-time integral equations)

Reporting period: 2018-06-01 to 2019-05-31

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.

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.

First, the basic physical and mathematical concept of our multi-time integral equation was made precise and connected with known theories in limiting cases. Second, the mathematical theory of the equation was investigated in detail. It was rigorously shown that the equation for spinless particles has solutions. Moreover, we showed which data uniquely classify the solutions. Third, both the equation and the results were generalized to curved space-times with a Big Bang singularity. Fourth, we extended the results to the most relevant case of fermions (e.g. electrons). Fifth, we generalized the Born rule to multi-time wave functions and particle detection at different times. This generalized rule was then derived rigorously as a theorem, given certain assumptions about the nature of interactions. Sixth, we started analyzing the classical limit of our integral equation. While we have obtained first results for the non-interacting (but entangled) multi-time case, the research for the interacting case turned out difficult and has not been concluded yet. To compensate for this, we started work in two new directions: (a) Seventh, we showed that the technique of “interior-boundary conditions” for describing particle creation and annihilation can be given a multi-time formulation. (b) Eighth, we constructed a multi-time model for an electron-photon system with contact interactions. In addition, we wrote an overview paper about multi-time wave functions which is accessible to non-specialists. Furthermore, we contrasted the concept of multi-time wave functions with wave functions on spacetimes with multiple timelike dimensions.

Concerning exploitation and dissemination activities, our results have been published as research papers (9 in total). These papers are available as Green Open Access via the arXiv.org repository. In addition, numerous talks in specialized workshops, conferences and seminars were given. A workshop,a course with a matching exercise class and a spring school about topics closely related with the project were organized. Moreover, the main ideas, as well as generally accessible introductions to the research topic were presented in several outreach events such as public talks, science fairs, information- and open-house days.

Concerning exploitation and dissemination activities, our results have been published as research papers (9 in total). These papers are available as Green Open Access via the arXiv.org repository. In addition, numerous talks in specialized workshops, conferences and seminars were given. A workshop,a course with a matching exercise class and a spring school about topics closely related with the project were organized. Moreover, the main ideas, as well as generally accessible introductions to the research topic were presented in several outreach events such as public talks, science fairs, information- and open-house days.

All of the above work is progress beyond the state of the art in relativistic quantum physics. To the best of our knowledge, the class of equations we consider is also new for the mathematicians specializing in integral equations. In our view the most crucial innovation is the way we found to express interactions with time delay at the quantum level using multi-time wave functions. With the conventional single-time wave function this would not have been possible.

Potential impacts: Our project deals with fundamental research in mathematics and physics. The ultimate goal is to contribute to a rigorous version of quantum field theory, our best theory describing fundamental interactions between elementary particles. In particular, such a rigorous formulation must be free of the ultraviolet-divergence problem, and it has to work for finite times, not only for idealized "scattering situations" where one waits infinitely long to see what happens. In our work, we have provided simple models that are free of these problems.

Overall, our results may help overcome present-day obstacles to a full understanding of the fundamental physical interactions. The wider impact on society is, as with all foundational research, indirect and hard to predict. But we note that technological innovations often depend on a clear understanding of the physical theory. On the other hand, such an understanding by itself often attracts curiosity from other fields of science and the public.

Potential impacts: Our project deals with fundamental research in mathematics and physics. The ultimate goal is to contribute to a rigorous version of quantum field theory, our best theory describing fundamental interactions between elementary particles. In particular, such a rigorous formulation must be free of the ultraviolet-divergence problem, and it has to work for finite times, not only for idealized "scattering situations" where one waits infinitely long to see what happens. In our work, we have provided simple models that are free of these problems.

Overall, our results may help overcome present-day obstacles to a full understanding of the fundamental physical interactions. The wider impact on society is, as with all foundational research, indirect and hard to predict. But we note that technological innovations often depend on a clear understanding of the physical theory. On the other hand, such an understanding by itself often attracts curiosity from other fields of science and the public.