The fate of conformal symmetry in the non-linear Schrödinger theory

The free Schrödinger theory in d space dimensions is a non-relativistic conformal field theory. The interacting non-linear theory preserves this symmetry in specific numbers of dimensions at the classical (tree) level. This holds in particular for the |Φ|-theory in d = 2. We compute the full quantum corrections to the 4-point function to show that the symmetry is broken by an anomalous contribution proportional to the exact β-function. ar X iv :0 71 2. 36 86 v2 [ he pth ] 1 1 Ja n 20 08 1. Conformal symmetry of the free Schrödinger theory Conformal symmetry, in exact or broken form, plays an important role in field theory. As it governs the scale dependence of a theory, it implicitly determines the regime of applicability of the theory. String theory, as an important candidate for a unified theory of quantum gravity, has an exact conformal invariance when formulated as a 2-D field theory on the world sheet. On the other hand, effective field theories in four space-time dimensions, like QED or the standard model, possess an approximate scale invariance, broken explicitly by mass terms and/or by quantum effects. Indeed, very few theories are known to be exactly scale invariant, except in 2 dimensions [1]. In 4 dimensions N = 4 Yang-Mills theory is known to have vanishing β-function to all orders [2], but in general gauge theories which are classically scale invariant, such as QCD with massless quarks, have this symmetry broken at the quantum level. In this paper we consider conformal symmetry in the context of nonrelativistic field theory. Such symmetries were first identified in [3, 4]. The symmetries and their realization in classical and quantum field theory have been studied by several authors [5]-[9]. We consider in particular theories describing Bose gases in d space dimensions. Conformal symmetry is an exact symmetry of the free theory, but it can be implemented in certain interacting models as well. The quantum field theoretical aspects of Bose gases in general have been studied widely in the literature; reviews and references can be found e.g. in [10, 11]. In d space dimensions the free theory is defined by the action S0 = ∫ dt ∫ dx ( i 2 Ψ∗ ↔ ∂ t Ψ− 1 2m ∇Ψ∗ ·∇Ψ ) . Without loss of generality the theory can be simplified by rescaling the time variable τ = t m , which is equivalent to choosing units in which m = 1. The action then reads S0 = ∫ dτ ∫ dx ( i 2 Ψ∗ ↔ ∂ τ Ψ− 1 2 ∇Ψ∗ ·∇Ψ ) , (1) which is stationary when Ψ satisfies the linear Schrödinger equation for free particles with unit mass: i∂τΨ = − 1 2 ∆Ψ. (2)