Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

CRC Press eBooks

Path Integral for Bosons

Luca Salasnich

Published Oct 6, 2025• 0 citations
Publisher

Abstract

In this chapter we consider the quantum statistical mechanics of identical nonrelativistic bosons by introducing the formalism of functional integration for the bosonic field. We then derive the equation of state of non-interacting and weakly-interacting massive bosons taking into account beyond-mean-field regularized Gaussian fluctuations. We also analyze various properties of the mean-field time-dependent Gross-Pitaevskii equation. In particular, we show that this equation is equivalent to the zero-temperature equations of superfluid hydrodynamics proposed in 1949 by Landau. Finally, we discuss topological and solitonic solutions of the Gross-Pitaevskii equation.

DOI
10.1201/9781003588337-3

Related papers

Path Integral for Fermions

Luca Salasnich

In this chapter we consider the quantum statistical mechanics of identical nonrelativistic fermions by introducing the formalism of functional integration for the fermionic field. We then derive the equation of state of non-interacting and weakly-interacting repulsive fermions. We also discuss the pairing of electrons within the mean-field Bardeen-Cooper-Schrieffer (BCS) theory of low-temperature superconductivity and the Ginzburg-Landau theory, which is quite successful near the critical temperature. Finally, we analyze the BCS-BEC crossover that was experimentally achieved with ultracold fermionic atoms twenty years ago and also Josephson junctions, which are the building blocks of superconducting quantum computers.

CRC Press eBooks•Oct 6, 2025•0 citations

Phase transitions in quantum systems

Uwe Claus Täuber

This chapter addresses phase transitions and dynamic scaling occurring in systems comprised of interacting indistinguishable quantum particles, for which entanglement correlations are crucial. It first describes how the dynamics (in real time) and thermodynamics (in imaginary time) of quantum many-particle Hamiltonians can be mapped onto field theories based on coherent-state path integrals. While bosons are described by complex-valued fields, fermions are represented by anticommuting Grassmann variables. Since quantum-mechanical systems are of inherently dynamical nature, the corresponding field theory action entails d + 1 dimensions, with time playing a special role. For Hamiltonians that incorporate only two-particle interactions, we can make contact with the previously studied Langevin equations, yet with effectively multiplicative rather than additive noise. As an illustration, this formalism is applied to deduce fundamental properties of weakly interacting boson superfluids. Whereas Landau–Ginzburg theory already provides a basic hydrodynamic description, the Gaussian approximation allows the computation of density correlations, the Bose condensate fraction, and the normal- and superfluid densities from the particle current correlations. We next establish that quantum fluctuations are typically irrelevant for thermodynamic critical phenomena, provided that T c > 0, and readily extend finite-size scaling theory to the imaginary time axis to arrive at general scaling forms for the free energy. Intriguing novel phenomena emerge in the realm of genuine quantum phase transitions at zero temperature, governed by other control parameters such as particle density, interaction or disorder strengths.

Cambridge University Press eBooks•Mar 6, 2014•0 citations

Hydrodynamics of Bose and Fermi superfluids at zero temperature: the superfluid nonlinear Schrödinger equation

Luca Salasnich

We discuss the zero-temperature hydrodynamics equations of bosonic and fermionic superfluids and their connection with generalized Gross-Pitaevskii and Ginzburg-Landau equations through a single superfluid nonlinear Schrödinger equation.

arXiv:0804.1277•Laser Physics•Apr 1, 2009•37 citations

Nonuniform Bose Gases at Zero Temperature

Лев П. Питаевский, S. Stringari

Abstract This chapter develops the theory of weakly interacting nonuniform Bose gases at zero temperature. Interaction effects are accounted for by a single coupling constant fixed by the value of the s-wave scattering length. The nonlinear Gross–Pitaevskii equation is derived in both the stationary and time-dependent cases. The irrotational nature of the flow and the crucial role played by the chemical potential are highlighted. The classical nature of the equation is emphasized and the analogy with the Maxwell equations of classical electrodynamics is pointed out. The stationary solutions of the Gross–Pitaevskii equation, corresponding to quantized vortex lines, vortex rings, and solitons, are discussed. The time-dependent solutions corresponding to small-amplitude oscillations are derived and the corresponding dispersion relation is shown to coincide with the predictions of Bogoliubov theory for uniform media. The Thomas–Fermi limit is also discussed.

•Jan 1, 2016•1 citation

Quantum turbulence in the many-body regime

Sayak Bhattacharjee, Mahendra K. Verma, Alexander V. Balatsky +1 more

We discuss phenomenology associated with turbulent hydrodynamics in quantum fluids from a condensed-matter perspective. We begin with weakly-interacting superfluids, often modeled by a mean-field theory governed by the Gross-Pitaevskii equation. Considering the effect of quantum fluctuations beyond the mean-field approximation, we propose a study of many-body quantum effects in turbulent hydrodynamics, especially near zero temperature. We motivate examples of quantum many-body systems where such effects may be uncovered. These include bosons confined in a periodic potential in low spatial dimensions (one and two), and the associated quantum critical point of the superfluid-insulator transition, realized in present-day ultracold-atom and quantum computing platforms. We conclude by listing a set of (open) questions that may be answered using modern quantum many-body techniques. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.

cond-mat.quant-gasCross-listedarXiv:2606.23822•Jun 24, 2026•0 citations

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

Luca Salasnich

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.

arXiv:2511.18020•Atoms•Dec 1, 2025•1 citation
In this chapter we consider the quantum statistical mechanics of identical nonrelativistic bosons by introducing the formalism of functional integration for the bosonic field. We then derive the equation of state of non-interacting and weakly-interacting massive bosons taking into account beyond-mean-field regularized Gaussian fluctuations. We also analyze various properties of the mean-field time-dependent Gross-Pitaevskii equation. In particular, we show that this equation is equivalent to the zero-temperature equations of superfluid hydrodynamics proposed in 1949 by Landau. Finally, we discuss topological and solitonic solutions of the Gross-Pitaevskii equation.
In this chapter we consider the quantum statistical mechanics of identical nonrelativistic fermions by introducing the formalism of functional integration for the fermionic field. We then derive the equation of state of non-interacting and weakly-interacting repulsive fermions. We also discuss the pairing of electrons within the mean-field Bardeen-Cooper-Schrieffer (BCS) theory of low-temperature superconductivity and the Ginzburg-Landau theory, which is quite successful near the critical temperature. Finally, we analyze the BCS-BEC crossover that was experimentally achieved with ultracold fermionic atoms twenty years ago and also Josephson junctions, which are the building blocks of superconducting quantum computers.
This chapter addresses phase transitions and dynamic scaling occurring in systems comprised of interacting indistinguishable quantum particles, for which entanglement correlations are crucial. It first describes how the dynamics (in real time) and thermodynamics (in imaginary time) of quantum many-particle Hamiltonians can be mapped onto field theories based on coherent-state path integrals. While bosons are described by complex-valued fields, fermions are represented by anticommuting Grassmann variables. Since quantum-mechanical systems are of inherently dynamical nature, the corresponding field theory action entails d + 1 dimensions, with time playing a special role. For Hamiltonians that incorporate only two-particle interactions, we can make contact with the previously studied Langevin equations, yet with effectively multiplicative rather than additive noise. As an illustration, this formalism is applied to deduce fundamental properties of weakly interacting boson superfluids. Whereas Landau–Ginzburg theory already provides a basic hydrodynamic description, the Gaussian approximation allows the computation of density correlations, the Bose condensate fraction, and the normal- and superfluid densities from the particle current correlations. We next establish that quantum fluctuations are typically irrelevant for thermodynamic critical phenomena, provided that T c > 0, and readily extend finite-size scaling theory to the imaginary time axis to arrive at general scaling forms for the free energy. Intriguing novel phenomena emerge in the realm of genuine quantum phase transitions at zero temperature, governed by other control parameters such as particle density, interaction or disorder strengths.
Abstract This chapter develops the theory of weakly interacting nonuniform Bose gases at zero temperature. Interaction effects are accounted for by a single coupling constant fixed by the value of the s-wave scattering length. The nonlinear Gross–Pitaevskii equation is derived in both the stationary and time-dependent cases. The irrotational nature of the flow and the crucial role played by the chemical potential are highlighted. The classical nature of the equation is emphasized and the analogy with the Maxwell equations of classical electrodynamics is pointed out. The stationary solutions of the Gross–Pitaevskii equation, corresponding to quantized vortex lines, vortex rings, and solitons, are discussed. The time-dependent solutions corresponding to small-amplitude oscillations are derived and the corresponding dispersion relation is shown to coincide with the predictions of Bogoliubov theory for uniform media. The Thomas–Fermi limit is also discussed.
We discuss phenomenology associated with turbulent hydrodynamics in quantum fluids from a condensed-matter perspective. We begin with weakly-interacting superfluids, often modeled by a mean-field theory governed by the Gross-Pitaevskii equation. Considering the effect of quantum fluctuations beyond the mean-field approximation, we propose a study of many-body quantum effects in turbulent hydrodynamics, especially near zero temperature. We motivate examples of quantum many-body systems where such effects may be uncovered. These include bosons confined in a periodic potential in low spatial dimensions (one and two), and the associated quantum critical point of the superfluid-insulator transition, realized in present-day ultracold-atom and quantum computing platforms. We conclude by listing a set of (open) questions that may be answered using modern quantum many-body techniques. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.
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.