Revisiting Schrödinger CFTs: Factorization, Massless Particles, and a Path to the Bootstrap

TL;DR

The paper develops a comprehensive framework for non-relativistic Schrödinger CFTs by introducing the harmonic trap geometry and a thermofield double-based state–operator correspondence that extends to all operators, including massless and normal-ordered composites. It provides a non-perturbative route to unitarity bounds, classifies the HT spectrum into massive and massless sectors, and clarifies the roles of genuine versus non-genuine operators, with massless states described by an emergent 1d CFT and potential implications for bootstrap approaches. A central theme is the interplay between factorization, non-renormalization theorems, and the massless sector: at fixed points without nontrivial M=0 states, a canonical normal-ordered composite exists and renormalization preserves OPEs between daggered and undaggered sectors; presence of massless states breaks factorization and induces novel 1d-CFT-like dynamics. The work also outlines how genuine massless theories could be constructed or coupled to massive NR CFTs, suggesting rich future directions for NR bootstrap, holography, and higher-conservation-law structures in Schrödinger-invariant systems.

Abstract

We revisit Schrödinger CFTs from a modern point of view. We introduce the ''harmonic trap geometry,'' analogous to the cylinder picture in relativistic CFTs, and demonstrate a state-operator correspondence that applies to all operators, including descendant, massless, and ''normal-ordered operators.'' A thermofield double construction plays an extremely important role. We systematically classify all physical spectra in the harmonic trap and their unitarity bounds, extending earlier results to include both massless and massive states of all spins, providing a new analytic treatment of unitarity bounds, and establishing foundations for a bootstrap. In our reformulation, previously known perturbative non-renormalization theorems follow immediately from non-perturbative factorization at fixed points and along RG flows. Massless states are described by an effective 1d CFT, as predicted by DLCQ, and violate the non-renormalization theorems. We include a self-consistent review of Schrödinger CFTs in our framework, making the paper accessible to anyone with a field theory background.

Figures (7)

• Figure 1: Left, the scaling dimensions of $N$-fermion states with orbital angular momentum $\ell$ at one-loop order in the $\bar{\epsilon}$ and $\epsilon$ expansions. Table reproduced from Nishida:2007pj; we have provided an independent check of the composite entries up to $N=3$. Right, the energy spectrum of the operators in the harmonic trap, with $\omega = 1$ and $\epsilon = \bar{\epsilon} = 1$. We see the Schrödinger CFT matches the harmonic trap with great success at leading order.
• Figure 2: Left, a commuting diagram explaining the relation between different quantization schemes in relativistic CFT. Right, a commuting diagram explaining the analogous quantization schemes in Schrödinger CFT. The main difference between the two is that Schrödinger-Weyl transformations effectively only act on time, and deform space in a completely determined way from the transform in time.
• Figure 3: Left, periodic Euclidean time on $S^1$ can be mapped to the Euclidean cylinder $\mathop{\mathrm{\mathbb{R}}}\nolimits \times S^0$ (center) by a Weyl transform -- splitting it into two branches. Time translations along the Euclidean cylinder coordinate are orientation reversed relative to the $-1$ branch. Right, the Euclidean "plane" is shown with its radial quantization surfaces, which slice the line along two disconnected points and opposite orientations.
• Figure 4: Left: flat spacetime $\mathop{\mathrm{\mathbb{R}}}\nolimits^{d+1}$ in coordinates $(t, \vec{x})$. A non-relativistic Weyl transformation brings flat space to a patch of the harmonic trap geometry $M_{\mathop{\mathrm{\mathrm{HT}}}\nolimits}$. In this picture, $(\tau,y^i)\in (-\tfrac{\pi}{2\omega},\tfrac{\pi}{2\omega}) \times \mathop{\mathrm{\mathbb{R}}}\nolimits^{d}$ describes a "Poincaré patch" on one branch of the harmonic trap geometry, with future and past infinities of the patch given by the boundaries $\tau = \pm \pi/2\omega$. $M_{\mathop{\mathrm{\mathrm{HT}}}\nolimits}$ is a Schrödinger version of the Lorentzian cylinder.
• Figure 5: Left, the dilatation vector field $\tfrac{\omega}{2}\mathscr{D}_E$ in the plane. The fixed point(s) of the vector field is at the origin (red), and infinity (not shown). Right, the NS vector field $\mathscr{P}_{E,0}+\omega^2 \mathscr{C}_{E,0}$. The fixed points (red) are now at $(t_E,\vec{x}) = (\pm 1,\vec{0})$. Plots are in units $\omega = 1$.