Depending on conditions like temperature or pressure, a given substance can appear in different phases, such as solid, liquid or gaseous. This project is devoted to the investigation of phases of matter under extreme conditions, addressing two main questions.
What happens to matter when compressed to extremely high density? The answer to this question lies in the phase diagram of the theory of the strong nuclear force, Quantum ChromoDynamics (QCD), yet in a region accessible neither to current experiments, nor to theoretical computations. The goal of this project is to obtain, for the first time, model-independent constraints on the phase diagram using a combination of indirect input from numerical simulations and general rigorous arguments. I will study properties of dense matter in theories similar to QCD which are amenable to computer simulations due to their ``positivity''. With the help of already existing numerical data, I will extract information about how the dense medium affects the nuclear force, in particular its ability to ``confine'' the fundamental constituents of matter, the quarks. Moreover, I will make model predictions that will facilitate the comparison to future simulations, thus preparing ground for further improvement of the expected results.
Why are some systems in equilibrium homogeneous, while others develop order? This second question is more general and although seemingly unrelated, it is in fact deeply connected to the first one. According to a little-known conjecture made a decade ago, positivity of a theory implies that its equilibrium state must be homogeneous. No counterexample has been found so far, but a definitive proof is still missing. I will investigate this intriguing link between positivity and homogeneity and strive to either prove or disprove the hypothesis. In any case, the work will provide further rigorous constraints on the structure of matter at high density. In a complementary study, I will develop a model-independent, effective description of systems with spontaneous order, valid at low energy and temperature. While this research is directly motivated by the desire to understand the behavior of dense nuclear matter, the results will be of much wider relevance, with applications for instance to condensed-matter or atomic physics.
To summarize, in this project I put forward a novel approach to matter under extreme conditions, based on exploiting available numerical data for QCD-like theories, and on the systematic use of symmetry. Whereas it does not pretend to answer all questions at once, it does intend to show an alternative way to tackle some longstanding problems.
