Anomaly in conformal quantum mechanics: From molecular physics to black holes

Abstract

A number of physical systems exhibit a particular form of asymptotic conformal invariance: within a particular domain of distances, they are characterized by a long-range conformal interaction (inverse square potential), the apparent absence of dimensional scales, and an SO(2,1) symmetry algebra. Examples from molecular physics to black holes are provided and discussed within a unified treatment. When such systems are physically realized in the appropriate strong-coupling regime, the occurrence of quantum symmetry breaking is possible. This anomaly is revealed by the failure of the symmetry generators to close the algebra in a manner shown to be independent of the renormalization procedure.

A relation between massive scalar field inAdSd+1and diffusion in channels

“…The spectrum of the Hamiltonian (9) was found by de Alfaro, Fubini and Furlan [34]. It is worth to mention that the conformal quantum mechanics appears in different contexts, from black-holes to atomic physics [35,36,37] and has been proposed as the CF T 1 dual to AdS 2 [38,39]. We will show that this Hamiltonian appears in diffusion phenomena too.…”

“…Now, the equation (7) is the static Schrödinger equation for the conformal quantum mechanics [34], which appears in different contexts, from black-holes to atomic physics [35,36,37]. Notice that, if a system is related with the conformal quantum mechanics, then it is also related with a massive scalar field in AdS d+1 .…”

Effective field theory of black hole echoes

“…Because this is an inverse-square potential the system's near-horizon behaviour naturally becomes scale-invariant and universal. As has been remarked elsewhere for the Schwarzschild black hole [25], this inverse-square potential turns out always to be attractive because r 2 s ≥ 4a 2 . Because this leading inverse-square interaction is attractive the solutions to the Schrödinger equation are particularly sensitive to the boundary conditions chosen as r → r + , as we now describe.…”

On the landscape of scale invariance in quantum mechanics

“…This feature is a signature of residual DSIThus, * danny.brattan@gmail.com † somrie@campus.technion.ac.il ‡ eric@physics.technion.ac.il a quantum phase transition occurs at λ c between a continuous scale invariant (CSI) phase and a discrete scale invariant phase (DSI). This transition has been associated with Berezinskii-Kosterlitz-Thouless (BKT) transitions [13, 18-23] and has found applications in the Efimov effect [24-26], graphene [15], QED3 [27] and other phenomena [18,[28][29][30][31][32][33][34][35].A useful tool in the characterisation of this phenomenon is the renormalisation group (RG) [36]. For the case ofĤ S,D,L , it consists of introducing an initial short distance scale L and defining model dependent parameters such as λ, and the boundary conditions, according to physical information.…”

“…Thus, a quantum phase transition occurs at λ c between a continuous scale invariant (CSI) phase and a discrete scale invariant phase (DSI). This transition has been associated with Berezinskii-Kosterlitz-Thouless (BKT) transitions [13,[18][19][20][21][22][23] and has found applications in the Efimov effect [24][25][26], graphene [15], QED3 [27] and other phenomena [18,[28][29][30][31][32][33][34][35]. A useful tool in the characterisation of this phenomenon is the renormalisation group (RG) [36].…”