Seminario di fisica teorica

In this talk I will review my recent work on anisotropic (Lifshitz) scale invariance in quantum theories. Broadly this talk will cover two related but distinct topics: (1) generalisations of the infamous problem of the inverse square potential in quantum mechanics and (2) scale invariant, charged, non-relativistic scalar field theories in (2+1)-dimensions. It is well known that the spectrum of the inverse square potential, important to a range of fields from ultracold gases to black hole physics, is pathological. Less well known is that this is also true for the Dirac Hamiltonian with an external electric potential, as may be relevant for example to graphene sheets with a charged impurity. These Hamiltonians share the property of scale invariance and, as it turns out, this invariance must be broken for the Hamiltonian to be well-defined. Hence these systems have a quantum anomaly and many of their properties are determined in terms of this anomaly. In work with collaborators I demonstrated that a scale anomaly is also present for a large class of Hamiltonians having a more generic scale invariance. I will discuss both the energy spectrum and renormalisation group methods for analysing these systems. I will also touch on results for their supersymmetrisations as these may be important for polynomial supersymmetry. Separately, I will discuss scale anomalies for non-relativistic, charged scalar field theories in (2+1)-dimensions. The prototypical example of such a theory is phi^4 which is the unique, interacting, classically scale invariant field theory of a charged scalar with Schrodinger invariance. Unsurprisingly, it has a scale anomaly and one finds a two-particle bound state which breaks the scale invariance. I will discuss the generalisation of this simple model to anisotropic scaling symmetry, where there is much more freedom to add interactions. In contrast to phi^4, I will argue that there is an exactly marginal deformation of these anisotropic theories. I will also show that other interactions lead to a scale anomaly accompanied by a multi-particle bound state.