Alberto Nicolis

The short-distance singular structure of the two-point function of a free scalar field in curved spacetime has a universal behavior that characterizes well-behaved states (called Hadamard states). This includes a non-analytic term proportional to the Ricci scalar curvature known as the Hadamard tail. This is usually derived by solving a differential equation for the Green's function of a Klein-Gordon field in curved spacetime. We present an alternative derivation which leverages the equivalence principles and makes use of perturbative field theory methods. This allows for the computation of the short-distance singular behavior of correlators of QFTs in curved space, including for interacting field theories, where the traditional Green's function strategy cannot be easily generalized. As an example, we apply these ideas to the two-point function of two scalar primary operators of an arbitrary Conformal Field Theory placed in an arbitrary curved background.

We investigate the analytic structure of scattering amplitudes in theories in which Lorentz invariance is spontaneously broken. We do so by computing and studying the S-matrix for a simple example: a superfluid described by a complex scalar with quartic interactions. The computation is confined to tree-level, for there are no absolutely stable single-particle states, though the lifetime can be made long by lowering the chemical potential. For the 2 → 2 amplitude in center-of-mass configurations, not only is crossing symmetry violated, there appears a tree level branch cut for unphysical kinematics. Its appearance is a consequence of non-analyticity in the dispersion relation. The branch point defines a new scale in the problem, which scales inversely with the chemical potential. In this example, even derivatives of the forward amplitude are positive while odd derivatives are negative. This pattern can be understood in a general way in the limit of a small chemical potential, or weak Lorentz breaking.

We consider the superfluid phase of a specific renormalizable relativistic quantum field theory. We prove that, within the regime of validity of perturbation theory and of the superfluid effective theory, there are consistent and regular vortex solutions where the superfluid’s velocity field as traditionally defined smoothly interpolates between zero and arbitrarily large superluminal values. We show that this solution is free of instabilities and of superluminal excitations. We show that, in contrast, a generic vortex solution for an ordinary fluid does develop an instability if the velocity field becomes superluminal. All this questions the characterization of a superfluid velocity field as the actual velocity of “something”.

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ⁴ theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ⁶ conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ⁴ superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

At momenta much higher than the Hubble scale, the cosmological expansion can be thought of as a weak gravitational field. We consider QFT in a particularly convenient set of coordinates that makes this manifest, so that, for those high momenta, the effects of the cosmological expansion can be dealt with using the standard tools of perturbation theory in Minkowski space. In this way, we re-derive standard results of QFT in a cosmological background, such as mode-stretching and gravitational particle production. We discuss the implications of our results for the trans-Planckian problem.

At momenta much higher than the Hubble scale, the cosmological expansion can be thought of as a weak gravitational field. We consider QFT in a particularly convenient set of coordinates that makes this manifest, so that, for those high momenta, the effects of the cosmological expansion can be dealt with using the standard tools of perturbation theory in Minkwoski space. In this way, we re-derive standard results of QFT in a cosmological background, such as mode-stretching and gravitational particle production. We discuss the implications of our results for the trans-Planckian problem.