Scale and Conformal Invariance in Quantum Field Theory

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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. One of the most important discoveries in physics of the twentieth century, the renormalization group, is an unavoidable consequence of the framework 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 systematic study of long-distance physics (emergent phenomena) through universality classes.******The proposed research program deals with the implications of scale and conformal invariance in quantum field theory. Scale-invariant and conformal-invariant quantum field theories exhibit interesting renormalization group flows and have applications in several areas of research including string theory and second-order phase transitions in condensed matter physics. A better understanding of scale and 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. More specifically, the proposal investigates two important issues: the connections between scale invariance, conformal invariance and the c-theorem (a long-lasting conjecture which has not been proved in all generality as of now) in quantum field theory; and the existence of scale-invariant but not conformal-invariant non-relativistic field theories.******Over the past years, some remarkable progress has been made on the implications of scale invariance and conformal invariance for quantum field theory. The research program proposes to extend these results to the more technically-challenging realm of strongly-coupled quantum field theory as well as the non-relativistic limit relevant to condensed matter physics. For the non-relativistic case, charting the novel behaviours encountered in such theories could lead to interesting developments in condensed matter physics.