On Conformal Field Theory and Dark Matter

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The two pillars of modern theoretical physics are quantum mechanics and general relativity. Although they cannot be combined together into a consistent framework yet (a notable exception being string theory), the union of quantum mechanics and special relativity, the basis for general relativity, leads to a consistent framework. This framework, called quantum field theory, gives rise to the most precise theory of elementary particles found thus far. However, there are many open questions concerning the formal aspects of quantum field theory or its use in particle physics that are still unanswered. The research proposal focuses on two different aspects of quantum field theory: conformal field theory and dark matter. 1) One of the most important discoveries in physics of the twentieth century, the renormalization group, is an unavoidable consequence of quantum field theory. The renormalization group, which is an iterative process where high-momentum degrees of freedom are integrated out, generates a flow in the space of all theories which determines the changes a physical system undergoes as viewed at different distance scales. It is of paramount importance in modern physics, especially in quantum field theory and condensed matter physics, since it allows the study of long-distance physics (emergent phenomena) through universality classes. Conformal field theories are theories that live at fixed points of the renormalization group flow. Their symmetry group is extended and the extra symmetries give a better handle on the typical observables present in conformal field theory. By themselves, consistency conditions originating from conformal invariance could be enough to solve the theories, and the first part of the research proposal aims to tackle this problem. Conformal field theories have applications in several areas of research including string theory and second-order phase transitions in condensed matter physics. A better understanding of conformal invariance could have far-reaching consequences, leading to a complete classification of all quantum field theories in terms of relevant deformations of conformal field theories. 2) Although the standard model of particle physics, based on quantum field theory, successfully explains a host of observations, unsolved questions remain. Dark matter, introduced to account for the missing mass inferred from astrophysical observations (for example from orbital speed measurements of visible stars in galaxies), is one such unexplained conundrum. The second part of the research proposal focuses on the issue of dark matter by investigating astrophysical constraints on dark matter models of weakly-interacting and very light hypothetical particles. These particles are well-motivated theoretically and lead to viable candidates for dark matter. Understanding the puzzle associated with dark matter could lead to a better understanding of our Universe and answer some of our most profound questions about it.