Abstract
In this study, we introduce and investigate a family of quantum mechanical models in 0 + 1 dimensions, known as generalized Born quantum oscillators. These models represent a one-parameter deformation of a specific system obtained by reducing the Nambu-Goto theory to 0 + 1 dimensions. Despite these systems showing significant similarities with \( \textrm{T}\overline{\textrm{T}} \)-type perturbations of two-dimensional relativistic models, our analysis reveals their potential as interesting regularizations of the Berry-Keating theory. We quantize these models using the Weyl quantization scheme up to very high orders in ħ. By examining a specific scaling limit, we observe an intriguing connection between the generalized Born quantum oscillators and the Riemann-Siegel θ function.
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Acknowledgments
We would like to express our gratitude to Gianni Coppa for fruitful discussions, during the initial stages of this work and in the period associated with his Master thesis project [11]. Additionally, we are grateful to him for sharing a draft of his article [12]. We also thank Michael Berry, Zoltan Bajnok and Alexander Zamolodchikov for useful discussions. This project was partially supported by the INFN project SFT, grant PHY-2210349, by the Simons collaboration on Confinement and QCD Strings, and by the FCT Project PTDC/MAT-PUR/30234/2017 “Irregular connections on algebraic curves and Quantum Field Theory”. S.N. wishes to thank the Department of Physics of the Università degli Studi di Torino for its kind hospitality.
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Giordano, F., Negro, S. & Tateo, R. The generalized Born oscillator and the Berry-Keating Hamiltonian. J. High Energ. Phys. 2023, 99 (2023). https://doi.org/10.1007/JHEP10(2023)099
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DOI: https://doi.org/10.1007/JHEP10(2023)099