Spectral Properties of the Symmetry Generators of Conformal Quantum Mechanics: A Path-Integral Approach

Abstract

A path-integral approach is used to study the spectral properties of the generators of the SO(2,1) symmetry of conformal quantum mechanics (CQM). In particular, we consider the CQM version that corresponds to the weak-coupling regime of the inverse square potential. We develop a general framework to characterize a generic symmetry generator $G$ (linear combinations of the Hamiltonian $H$, special conformal operator $K$, and dilation operator $D$), from which the path-integral propagators follow, leading to a complete spectral decomposition. This is done for the three classes of operators: elliptic, parabolic, and hyperbolic. We also highlight novel results for the hyperbolic operators, with a continuous spectrum, and their quantum-mechanical interpretation. The spectral technique developed for the eigensystem of continuous-spectrum operators can be generalized to other operator problems.

Updated with published version

References in corpus (26)

• Anomaly in conformal quantum mechanics: From molecular physics to black holes
• The Multi-faceted Inverted Harmonic Oscillator: Chaos and Complexity
• Physics of the Inverted Harmonic Oscillator: From the lowest Landau level to event horizons
• Near-horizon aspects of acceleration radiation by free fall of an atom into a black hole
• Horizon induces instability locally and creates quantum thermality
• Near horizon local instability and quantum thermality
• Anomalous Commutator Algebra for Conformal Quantum Mechanics
• Renormalization in conformal quantum mechanics
• Acceleration radiation of an atom freely falling into a Kerr black hole and near-horizon conformal quantum mechanics
• SO(2,1) conformal anomaly: Beyond contact interactions
• Semiclassical Methods in Curved Spacetime and Black Hole Thermodynamics
• Chaos and Complexity for Inverted Harmonic Oscillators
• Quantum optics meets black hole thermodynamics via conformal quantum mechanics: II. Thermodynamics of acceleration radiation
• Schwarzian mechanics via nonlinear realizations
• Conformal mapping of Unruh temperature
• Conformal quantum mechanics of causal diamonds
• Cooper Triples in Attractive Three-Component Fermions: Implication for Hadron-Quark Crossover
• Non-commutative $Ad{S}_2/CF{T}_1$ duality: the case of massless scalar fields
• Generating multi-partite entanglement from the quantum vacuum with a finite-lifetime mirror
• On the Inequivalence of Renormalization and Self-Adjoint Extensions for Quantum Singular Interactions
• Thermal effect in a causal diamond: open quantum systems approach
• Duality in the dynamics of Unruh-DeWitt detectors in conformally related spacetimes
• Virial Expansion for a Three-Component Fermi Gas in One Dimension: The Quantum Anomaly Correspondence
• Note on correlation functions in conformal quantum mechanics
• Vacuum thermal effects in flat space-time from conformal quantum mechanics
• Conformal quantum mechanics as a Floquet-Dirac system

Cited by in corpus (8)

• Entanglement Degradation in Causal Diamonds
• Conformal quantum mechanics of causal diamonds: Time evolution, thermality, and instability via path integral functionals
• Quantum aspects of spacetime: A quantum optics view of acceleration radiation and black holes
• Horizon brightened acceleration radiation entropy in causal diamond geometry: A near-horizon perspective
• Superpositions of thermalisations in relativistic quantum field theory
• $n$-point functions in Conformal Quantum Mechanics: A Momentum Space Odyssey
• Thermal nature of the causal diamond horizon: A hidden property of the inertial propagator
• Generalized Gross-Pitaevskii Equation for 2D Bosons with Attractive Interactions