Effective Action Approach to Quantum and Thermal Effects: From One Particle to Bose-Einstein Condensates

Year: 2025

Authors: Salasnich., L.

Autors Affiliation: Univ Padua, Dipartimento Fis & Astron Galileo Galilei, via Marzolo 8, I-35131 Padua, Italy; Univ Padua, Padua QTech Ctr, via Marzolo 8, I-35131 Padua, Italy; Ist Nazl Fis Nucleare, Sez Padova, via Marzolo 8, I-35131 Padua, Italy; CNR, Ist Nazl Ott, via Carrara 2, I-50019 Sesto Fiorentino, Italy.

Abstract: We present a detailed derivation of the quantum and quantum-thermal effective action for non-relativistic systems, starting from the single-particle case and extending to the Gross-Pitaevskii (GP) field theory for weakly interacting bosons. In the single-particle framework, we introduce the one-particle-irreducible (1PI) effective action formalism, taking explicitly into account the choice of the initial quantum state, its saddle-point plus Gaussian-fluctuation approximation, and its finite-temperature extension via Matsubara summation, yielding a clear physical interpretation in terms of zero-point and thermal contributions to the Helmholtz free energy. The formalism is then applied to the GP action, producing the 1PI effective potential at zero and finite temperature, including beyond-mean-field Lee-Huang-Yang and thermal corrections. We discuss the gapless and gapped Bogoliubov spectra, their relevance to equilibrium and non-equilibrium regimes, and the role of regularization. Applications include the inclusion of an external potential within the local density approximation, the derivation of finite-temperature Josephson equations, and the extension to D-dimensional systems, with particular attention to the zero-dimensional limit. This unified approach provides a transparent connection between microscopic quantum fluctuations and effective macroscopic equations of motion for Bose-Einstein condensates.

Journal/Review: ATOMS

Volume: 13 (12)      Pages from: 95-1  to: 95-11

More Information: PRIN 2022 Project of MUR Quantum Atomic Mixtures: Droplets, Topological Structures, and Vortices (2023-2024); Dipartimenti di Eccellenza Project of MUR Frontiere Quantistiche; Iniziativa Specifica Quantum of INFN.
KeyWords: ultracold atoms; effective action; Bose-Einstein condensation; Josephson effect
DOI: 10.3390/atoms13120095