Periodic Reporting for period 1 - QFT.zip (Compressing many-body quantum states in continuous space-time with tensor networks)
Reporting period: 2023-01-01 to 2025-06-30
The most fundamental description we currently have of the world is quantum field theory. It is well tested in various limits, and makes us confident it is correct in a wider range of settings. However, solving the equations of quantum field theory is brutally difficult, and can be done only approximately, thus limiting our ability to test and predict.
Currently, the most efficient method to extract predictions is undoubtedly the Monte Carlo method. It first works by discretizing the continuum of space-time in which the quantum field theory ""lives"" into a finite lattice. The Monte Carlo method has impressive successes already, but is extremely expensive in numerical resources, and limited in the physical regimes it can access. A newer promising method is tensor network states, which are in principle not limited in the same ways. Currently the method also relies on a discretization. But very recently, it has been shown that one could use the technique directly in the continuum, without first having to introduce errors by discretizing space-time.
This method -using tensor networks in the continuum- is still in its infancy and can only deal with extremely simple models, very unlike those appearing in the real world. The main objective of the project is to push the method to make it applicable to a broader class of theories, that share more features with the ones of the real world. Another objective is to develop the method itself so that it can answer more complicated questions about the theories at hand. In the long run, the hope is that progress done during this project will enhance our quantitative and qualitative understanding of the most fundamental description we have of the world.
Currently, the most efficient method to extract predictions is undoubtedly the Monte Carlo method. It first works by discretizing the continuum of space-time in which the quantum field theory ""lives"" into a finite lattice. The Monte Carlo method has impressive successes already, but is extremely expensive in numerical resources, and limited in the physical regimes it can access. A newer promising method is tensor network states, which are in principle not limited in the same ways. Currently the method also relies on a discretization. But very recently, it has been shown that one could use the technique directly in the continuum, without first having to introduce errors by discretizing space-time.
This method -using tensor networks in the continuum- is still in its infancy and can only deal with extremely simple models, very unlike those appearing in the real world. The main objective of the project is to push the method to make it applicable to a broader class of theories, that share more features with the ones of the real world. Another objective is to develop the method itself so that it can answer more complicated questions about the theories at hand. In the long run, the hope is that progress done during this project will enhance our quantitative and qualitative understanding of the most fundamental description we have of the world.
After two years, most of our effort has been dedicated to building a coherent group of researchers, and train them in the intricacies of tensor networks and their continuum version. Indeed, it is a fairly new subject that only a handful of people know in the world.
Now with a strong and balanced team we have started working on most parts of the project, and in particular the extension of the method to new models of quantum field theories, so far exclusively in low dimension.
More precisely, we have worked on extending the continuous tensor network technology to two new models 1) one with symmetries, the so called complex scalar field 2) one with interactions similar to that of electromagnetism, the Schwinger model. This work is still in progress, but in particular for the first model, we expect concrete results next year.
Our effort has also revolved around the calculation of new quantities in quantum field theory, that are not trivial to access with other techniques. We have written a preprint on the inclusion of so called ""defect operators"", which are an important new class of observables, and are currently working on estimating spectral data, an example of which is the mass of fundamental particles.
Finally, we had to take into account new developments in the field during the project, the most important being the hybridization of tensor networks with semi-definite programming. We strongly believe that most parts of the project could benefit from this new technique. To grow our own expertise on it, we applied it to a simple example outside of quantum field theory. We obtained unexpectedly interesting results that are written in a preprint. We hope that it will soon be formally accepted in a journal.
Now with a strong and balanced team we have started working on most parts of the project, and in particular the extension of the method to new models of quantum field theories, so far exclusively in low dimension.
More precisely, we have worked on extending the continuous tensor network technology to two new models 1) one with symmetries, the so called complex scalar field 2) one with interactions similar to that of electromagnetism, the Schwinger model. This work is still in progress, but in particular for the first model, we expect concrete results next year.
Our effort has also revolved around the calculation of new quantities in quantum field theory, that are not trivial to access with other techniques. We have written a preprint on the inclusion of so called ""defect operators"", which are an important new class of observables, and are currently working on estimating spectral data, an example of which is the mass of fundamental particles.
Finally, we had to take into account new developments in the field during the project, the most important being the hybridization of tensor networks with semi-definite programming. We strongly believe that most parts of the project could benefit from this new technique. To grow our own expertise on it, we applied it to a simple example outside of quantum field theory. We obtained unexpectedly interesting results that are written in a preprint. We hope that it will soon be formally accepted in a journal.
The two preprints we have so far put online, even if preliminary, certainly push beyond the state of the art. However, they admittedly remain fairly theoretical, and quite far from the main achievements we expect to obtain during the project. Most results we have so far obtained remain unpublished, in particular regarding the estimation of spectral data and the study of new models. These have required a lot of effort (and still will require more), but should push more decisively beyond the frontier of what is currently known.