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

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.

1. Introduction

Quantum and thermal fluctuations play a crucial role in determining the properties of many-body systems, particularly in the realm of ultracold atomic gases and Bose–Einstein condensates (BECs). Since the first theoretical descriptions of spontaneous symmetry breaking and effective potentials by Goldstone and Weinberg [1], and the seminal works by Jona-Lasinio on effective action methods [2,3], the field-theoretic approach has become a foundational tool to bridge microscopic quantum dynamics and emergent macroscopic phenomena. Coleman and Weinberg’s pioneering analysis of radiative corrections in effective potentials [4] further enriched the understanding of fluctuation effects beyond classical approximations [5,6,7]. For non-relativistic and non-interacting bosons in the presence of an external potential, an effective action approach was developed by Toms and Kirsten [8,9]. For interacting bosons the Gross–Pitaevskii (GP) equation [10,11] provides a mean-field description of the condensate wavefunction. However, capturing corrections beyond mean-field theory, such as the Lee–Huang–Yang quantum fluctuations [12,13,14] and thermal effects [15,16,17], requires a more comprehensive framework. The quantum effective action formalism, especially within the one-particle-irreducible (1PI) scheme [5,6,7], offers a systematic pathway to incorporate these fluctuations by deriving effective potentials and excitation spectra that reflect both quantum and thermal contributions. Recent experimental advances in ultracold gases have highlighted the importance of precise theoretical tools for describing weakly interacting bosons under various conditions, including finite temperature, external trapping potentials, and low-dimensional regimes [15,18,19]. Moreover, the interplay between gapless and gapped Bogoliubov modes has implications for equilibrium properties and non-equilibrium dynamics, such as Josephson oscillations [20,21], which require refined effective descriptions [22].

Building on these developments, this work presents a detailed derivation of the quantum and quantum–thermal effective action for non-relativistic systems. Starting from the fundamental single-particle case, we clarify the construction of the 1PI effective action and its approximation at the Gaussian (one-loop) level. This formalism is then extended to the GP field theory for weakly interacting bosons, capturing beyond-mean-field Lee–Huang–Yang corrections and finite-temperature generalizations. We also discuss the inclusion of external potentials within local density approximations and the dimensional crossover to analytically tractable limits, including the zero-dimensional case. By providing a transparent and unified framework that connects microscopic quantum fluctuations to macroscopic condensate dynamics, our approach complements and extends previous theoretical efforts [17,22] and lays the groundwork for future studies involving dissipative effects, stronger interactions, and non-equilibrium phenomena.

2. Single-Particle Non-Relativistic Quantum Mechanics

2.1. Quantum Effective Action

Let us consider a particle of mass m and coordinate $q\left(t\right)$ described by the action functional

$$S\left[q\left(t\right)\right]={\int }_{t_0}^{t_1}\left({\displaystyle \frac{m}{2}}\dot{q}{\left(t\right)}^2-V\left(q\left(t\right)\right)\right)dt$$

(1)

It is well known that the classical trajectory $q_c\left(t\right)$ is the one that extremizes this action, namely

$${\displaystyle \frac{\delta S\left[q_c\right]}{\delta q\left(t\right)}}=0$$

(2)

We now introduce the following decomposition

$$q\left(t\right)=\overline{q}\left(t\right)+\eta \left(t\right)$$

(3)

where

$$\overline{q}\left(t\right)=⟨q\left(t\right)⟩$$

(4)

is the average of $q\left(t\right)$ with respect some pure or mixed quantum state. For instance, in the case of a pure quantum state $|\chi ⟩$, such that $\chi (q,t)=⟨q,t|\chi ⟩$ is its wavefunction (normalized to one) in the coordinate representation at time t, we have

$$⟨q\left(t\right)⟩={\int }_{-\infty }^{+\infty }dqq{\left|\chi (q,t)\right|}^2={\int }_{-\infty }^{+\infty }dqq|{\int }_{-\infty }^{+\infty }d{q}_0{\int }_{q\left(t_0\right)=q_0}^{q\left(t\right)=q}\mathcal{D}\left[q\left(t^{\prime }\right)\right]e^{{\displaystyle \frac{i}{\hslash }}S\left[q\left(t^{\prime }\right)\right]}\chi (q_0,t_0){|}^2$$

(5)

by using the Feynman path integral representation of the quantum propagator

$$K(q,t|q_0,t_0)=⟨q,t|q_0,t_0⟩={\int }_{q\left(t_0\right)=q_0}^{q\left(t\right)=q}\mathcal{D}\left[q\left(t^{\prime }\right)\right]e^{{\displaystyle \frac{i}{\hslash }}S\left[q\left(t^{\prime }\right)\right]}$$

(6)

that gives the conditional probability amplitude of finding the particle in the position q at time t having it in the position $q_0$ at time $t_0$, and consequently

$$\chi (q,t)={\int }_{-\infty }^{+\infty }d{q}_0{\int }_{q\left(t_0\right)=q_0}^{q\left(t\right)=q}\mathcal{D}\left[q\left(t^{\prime }\right)\right]e^{{\displaystyle \frac{i}{\hslash }}S\left[q\left(t^{\prime }\right)\right]}\chi (q_0,t_0)$$

(7)

From Equation (5) one explicitly shows that $⟨q\left(t\right)⟩$ crucially depends on the choice of the wavefunction of the pure quantum state at the initial time $t_0$. Notice that $\overline{q}\left(t\right)$ is sometimes called the background dynamical variable and, in general, $\overline{q}\left(t\right)\ne q_c\left(t\right)$. Thus, usually

$${\displaystyle \frac{\delta S\left[\overline{q}\right]}{\delta q\left(t\right)}}\ne 0$$

(8)

We observe that, as a consequence of Equations (3) and (4), it follows

$$⟨\eta \left(t\right)⟩=0$$

(9)

The main issue of this paper is to find the functional of $\overline{q}\left(t\right)$ whose Euler-Lagrange equation gives the exact equation of motion of $\overline{q}\left(t\right)$. This functional is usually called the quantum (one-particle irreducible, 1PI) effective action $\Gamma \left[\overline{q}\right]$. It is important to stress that, in general, $\Gamma \left[\overline{q}\right]$ cannot explicitly be calculated without doing some approximation or some perturbative expansion. Usually $\Gamma \left[\overline{q}\right]$ is obtained by introducing a source term $J\left(t\right)$ and performing a Legendre transformation [4,5]. Here, we will derive the 1PI quantum effective action of the system without the use of source terms or Legendre transformations. In our approach, $\Gamma \left[\overline{q}\right]$ is simply given by

$$e^{{\displaystyle \frac{i}{\hslash }}\Gamma \left[\overline{q}\right]}=\int \mathcal{D}\left[\eta \right]e^{{\displaystyle \frac{i}{\hslash }}S[\overline{q}+\eta ]}$$

(10)

provided that Equation (9) holds. In the absence of the condition (9), the action of Equation (10) is called the background effective action.

The stationary phase approximation (saddle-point plus Gaussian fluctuations) of this functional integral around $q=\overline{q}$ gives

$$e^{{\displaystyle \frac{i}{\hslash }}\Gamma \left[\overline{q}\right]}\simeq F\left[\overline{q}\right]e^{{\displaystyle \frac{i}{\hslash }}S\left[\overline{q}\right]}$$

(11)

where

$$F\left[\overline{q}\right]=\det {\left[{\displaystyle \frac{{\delta }^{2}S\left[\overline{q}\right]}{\delta q\left(t\right)\delta q\left(t^{\prime }\right)}}\right]}^{-1/2}=\det {\left[\left(-m{\displaystyle \frac{d^2}{d{t}^2}}-V^{″}\left(\overline{q}\right)\right)\delta (t-t^{\prime })\right]}^{-1/2}=\det {\left[\widehat{G}\right]}^{-1/2}$$

(12)

is the contribution due to Gaussian fluctuations. Quite remarkably, although expanding $S[\overline{q}+\eta ]$ produces a linear term in $\eta$ whenever $\overline{q}\left(t\right)$ is not equal to the classical trajectory $q_c\left(t\right)$, this contribution does not appear in the effective action $\Gamma \left[\overline{q}\right]$. The reason is that quantum corrections from higher-order fluctuation terms (tadpoles) exactly compensate the bare linear term, leaving $\Gamma \left[\overline{q}\right]$ free of such contributions by construction. One can prove [6,7] that this compensation is ensured by the condition (9), or equivalently by the definition of the background field (4). This guarantees that the linear term in the fluctuation expansion cancels between the bare action and the quantum tadpole contributions.

To conclude the discussion, we notice that $F\left[\overline{q}\right]$ can be exponentiated:

$$F\left[\overline{q}\right]=e^{ln\left(F\left[\overline{q}\right]\right)}$$

(13)

This means that we can write

$$\Gamma \left[\overline{q}\right]\simeq S\left[\overline{q}\right]+{\displaystyle \frac{\hslash }{i}}ln\left(F\left[\overline{q}\right]\right)$$

(14)

Thus, at the Gaussian level, the quantum effective action $\Gamma \left[\overline{q}\right]$ is the classical action $S\left[\overline{q}\right]$ plus the one-loop quantum correction $(\hslash /i)ln\left(F\left[\overline{q}\right]\right)$.

The exact analytical calculation of $F\left[\overline{q}\right]$ can be performed only in the very simple case where $\overline{q}$ in $V\left(\overline{q}\right)$ is time independent. If $\overline{q}$ is time dependent, the standard trick (lowest-order derivative expansion, aka adiabatic approximation, aka local field approximation) is to calculate $F\left[\overline{q}\right]$ assuming that $\overline{q}\left(t\right)$ is time independent and restoring the time dependence only at the end of the calculation. Under this assumption we have

$$\begin{array}{ccc}\hfill ln\left(F\left[\overline{q}\right]\right)& =& -{\displaystyle \frac{1}{2}}ln\left(\det \left[\widehat{G}\right]\right)=-{\displaystyle \frac{1}{2}}\mathrm{Tr}[ln\left(\widehat{G}\right)]\hfill \\ & \simeq & -{\displaystyle \frac{1}{2}}{\int }_{-\infty }^{+\infty }dt{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d\omega }{2\pi }}ln\left({\omega }^2-{\displaystyle \frac{V^{″}\left(\overline{q}\left(t\right)\right)}{m}}\right)\hfill \end{array}$$

(15)

In this way we identify the effective (one-loop) potential of the system

$$V_{\mathrm{eff}}\left(\overline{q}\right)=V\left(\overline{q}\right)+{\displaystyle \frac{\hslash }{2i}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{d\omega }{2\pi }}ln\left({\omega }^2-{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}\right)$$

(16)

After integration, and discarding spurious divergent terms, we obtain

$$V_{\mathrm{eff}}\left(\overline{q}\right)=V\left(\overline{q}\right)+{\displaystyle \frac{\hslash }{2}}\sqrt{{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}}$$

(17)

and the corresponding one-loop 1PI action functional

$${\Gamma }^{(1-\mathrm{loop})}\left[\overline{q}\right]=\int \left({\displaystyle \frac{m}{2}}{\dot{\overline{q}}}^2-V_{\mathrm{eff}}\left(\overline{q}\right)\right)dt$$

(18)

The quantum effective action method is rather complicated but the final result is quite simple: the quantum effective action ${\Gamma }^{(1-\mathrm{loop})}\left[\overline{q}\right]$ it is nothing else than the one derived from the stationary phase approximation (saddle point plus Gaussian fluctuations) of the path integral. The only caveat is that the saddle point classical (mean field) solution $q_c\left(t\right)$, which extremizes the classical action $S\left[q\right]$, must be re-interpreted as the quantum average $\overline{q}\left(t\right)$ of the dynamical variable $q\left(t\right)$ of the problem. Moreover, the Gaussian beyond-mean-field correction can be interpreted as the zero-point energy of the harmonic oscillator of quantum fluctuations with effective frequency

$${\omega }_{\mathrm{eff}}\left(\overline{q}\right)=\sqrt{{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}}$$

(19)

In conclusion, we observe that it is possible to extend this effective action approach to the case of a mass m (see Equation (1)) that depends on the dynamical variable $q\left(t\right)$: $M\left(q\right(t\left)\right)$. For details see Refs. [5,23,24].

2.2. Quantum-Thermal Effective Action

In the spirit of the adiabatic approximation, Equation (15) can be generalized to the case of finite temperature T as follows

$$\begin{array}{ccc}\hfill \omega & \to & {\omega }_n={\displaystyle \frac{2\pi }{\hslash \beta }}n\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill \int {\displaystyle \frac{d\omega }{2\pi }}& \to & {\displaystyle \frac{1}{\hslash \beta }}\sum _{n=-\infty }^{+\infty }\hfill \end{array}$$

(21)

where ${\omega }_n$ are the bosonic Matsubara frequencies and $\beta =1/\left(k_{B}T\right)$ with $k_B$ the Boltzmann constant. Thus, the quantum-thermal effective potential reads

$$V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{q}\right)=V\left(\overline{q}\right)+{\displaystyle \frac{1}{2i\beta }}\sum _{n=-\infty }^{+\infty }ln\left({\displaystyle \frac{4{\pi }^2{n}^2}{{\hslash }^2{\beta }^2}}-{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}\right)$$

(22)

namely

$$V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{q}\right)=V\left(\overline{q}\right)+{\displaystyle \frac{\hslash }{2}}\sqrt{{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}}+k_{B}Tln\left(1-exp\left(-{\displaystyle \frac{\hslash }{k_{B}T}}\sqrt{{\displaystyle \frac{V^{″}\left(\overline{q}\right)}{m}}}\right)\right)$$

(23)

This final result has a clear physical interpretation: the Gaussian beyond-mean-field quantum-thermal correction is the Helmholtz free energy of the harmonic oscillator, with effective frequency given Equation (19), of quantum-thermal fluctuations around the mean-field (classical) result. Notice that for $T\to 0$ one has $V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{q}\right)\to V_{\mathrm{eff}}\left(\overline{q}\right)$. The key point is that to obtain an action functional containing both time t and temperature T it is useful to have a two times action functional and then performing a Wick rotation with respect to one of the two times. Equation (15) contains implicitly two times: explicitly t and implicitly $t^{\prime }$ because $\omega$ is the Fourier dual of $t^{\prime }$.

3. Gross-Pitaevskii Quantum Field Theory

Let us now face the problem of the non-relativistic quantum field theory for the Schrödinger field $\psi (\mathbf{r},t)$ with action functional

$$S\left[\psi \right]=\int dt\int d^3\mathbf{r}{\psi }^{\ast }\left(i\hslash {\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+\mu -{\displaystyle \frac{g}{2}}{\left|\psi \right|}^2\right)\psi$$

(24)

where g is the strength of the contact interaction of the identical bosonic particles of mass m with chemical potential $\mu$. The action functional (24) is called Gross-Pitaevskii (GP) action. Also in this case we are looking for the quantum effective action of the average $\overline{\psi }(\mathbf{r},t)$ of the Schrödinger field $\psi (\mathbf{r},t)$. To maintain a connection with the discussion of the previous section, we denote by $\mathcal{X}\left[\psi \right(\mathbf{r}),t]=⟨\psi \left(\mathbf{r}\right),t|\mathcal{X}⟩$ the normalized wavefunctional of the field $\psi \left(\mathbf{r}\right)$ at time t, where $\left|\psi \right(\mathbf{r}),t⟩$ is the coherent state, i.e., the eigenstate, of the quantum field operator $\widehat{\psi }(\mathbf{r},t)$ with eigenvalue $\psi (\mathbf{r},t)$, while $|\mathcal{X}⟩$ is a many-body quantum state at the initial time. The expectation value

$$\overline{\psi }(\mathbf{r},t)=⟨\psi (\mathbf{r},t)⟩$$

(25)

of the field $\psi (\mathbf{r},t)$ is computed as the average of the configuration field $\psi \left(\mathbf{r}\right)$ with respect to the probability density ${\left|\mathcal{X}[\psi \left(\mathbf{r}\right),t]\right|}^2$, namely

$$\begin{array}{ccc}\hfill ⟨\psi (\mathbf{r},t)⟩& =& \int \mathcal{D}\left[\psi \left(\mathbf{r}\right)\right]\psi \left(\mathbf{r}\right)|\mathcal{X}[\psi \left(\mathbf{r}\right),t]{|}^2\hfill \\ & =& \int \mathcal{D}\left[\psi \left(\mathbf{r}\right)\right]\psi \left(\mathbf{r}\right)|\int \mathcal{D}\left[{\psi }_0\left(\mathbf{r}\right)\right]{\int }_{\psi (\mathbf{r},t_0)={\psi }_0\left(\mathbf{r}\right)}^{\psi (\mathbf{r},t)=\psi \left(\mathbf{r}\right)}\mathcal{D}\left[\psi (\mathbf{r},t^{\prime })\right]\hfill \\ & & e^{{\displaystyle \frac{i}{\hslash }}S\left[\psi [\mathbf{r},t^{\prime }]\right]}\mathcal{X}[{\psi }_0\left(\mathbf{r}\right),t_0]{|}^2\hfill \end{array}$$

(26)

which is a generalization of Equation (5).

Without repeating the procedure developed in the previous section, taking into account well established results for zero-temperature Gaussian fluctuations [14], we find that the quantum effective action is given by

$$\Gamma \left[\overline{\psi }\right]=\int dt\int d^3\mathbf{r}\left\{{\overline{\psi }}^{\ast }\left(i\hslash {\displaystyle \frac{\partial }{\partial t}}+{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\right)\overline{\psi }-V_{\mathrm{eff}}\left(\overline{\psi }\right)\right\}$$

(27)

where

$$V_{\mathrm{eff}}\left(\overline{\psi }\right)=-\mu {\left|\overline{\psi }\right|}^2+{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^4+{\displaystyle \frac{1}{2}}\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{E}_k(\overline{\psi },\mu )$$

(28)

In this formula appears the zero-point energy of Gaussian quantum fluctuations, characterized by the gapped Bogoliubov spectrum

$$E_k(\overline{\psi },\mu )=\sqrt{{\left({\displaystyle \frac{{\hslash }^2{k}^2}{2m}}-\mu +2g{\left|\overline{\psi }\right|}^2\right)}^2-g^2{\left|\overline{\psi }\right|}^2}$$

(29)

which reduces to the gapless Bogoliubov spectrum

$$E_k\left(\overline{\psi }\right)=\sqrt{{\displaystyle \frac{{\hslash }^2{k}^2}{2m}}\left({\displaystyle \frac{{\hslash }^2{k}^2}{2m}}+2g{\left|\overline{\psi }\right|}^2\right)}$$

(30)

only under the very strong assumption

$$\mu =g|\overline{\psi }{|}^2$$

(31)

that is justified only at equilibrium. By the way, the Goldstone theorem (gapless spectrum) works only at equilibrium and at $T=0$. The Gaussian beyond-mean-field correction can be interpreted as the zero-point energy of a set of harmonic oscillators with effective frequencies $E_k/\hslash$. It is important to stress that $V_{\mathrm{eff}}\left(\overline{\psi }\right)$ is divergent but one can extract a meaningful finite contribution performing an appropriate regularization, for details see Ref. [14]. In three spatial dimensions one obtains the so-called Lee–Huang–Yang quantum correction.

We can easily extend this result at finite temperature. On the basis of the physical interpretation of quantum fluctuations as a gas of Bogoliubov excitations, by using Equation (5) of Ref. [17] we immediately obtain

$$\begin{array}{ccc}\hfill V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)& =& -\mu {\left|\overline{\psi }\right|}^2+{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^4\hfill \\ & +& \int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left[{\displaystyle \frac{1}{2}}{E}_k(\overline{\psi },\mu )+k_{B}Tln\left(1-exp\left(-{\displaystyle \frac{E_k(\overline{\psi },\mu )}{k_{B}T}}\right)\right)\right]\hfill \end{array}$$

(32)

This is the 1PI effective potential of the GP action at finite temperature. Indeed, $V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)$ is nothing else than the grand canonical potential $\Omega (\mu ,n_0,T)$ of Ref. [17] with the identification $n_0={\left|\overline{\psi }\right|}^2$.

The equation of motion of $\overline{\psi }(\mathbf{r},t)$ is given by

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\overline{\psi }=-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2\overline{\psi }+{\displaystyle \frac{\partial V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)}{\partial {\overline{\psi }}^{\ast }}}=0$$

(33)

where $\mu$ is fixed by imposing that

$$N_0=\int d^3\mathbf{r}{\left|\overline{\psi }(\mathbf{r},t)\right|}^2$$

(34)

Our Equation (33) reduces to the stationary generalized Gross-Pitaevskii equation discussed in Refs. [18,19] setting ${\displaystyle \frac{\partial \overline{\psi }}{\partial t}}=0$ and adopting the spectrum (29) instead of (30) in $V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)$.

The standard approach is to adopt the gapless Bogoliubov spectrum in Equation (33) under the assumption of working near thermal equilibrium. Introducing the local number density

$$n_0(\mathbf{r},t)={\left|\overline{\psi }(\mathbf{r},t)\right|}^2$$

(35)

Equation (33) can be then written as

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\overline{\psi }=\left[-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2-\mu +g{n}_0+{\mu }_{\mathrm{LHY}}\left(n_0\right)+{\mu }_{\mathrm{th}}(n_0,T)\right]\overline{\psi }$$

(36)

where

$${\mu }_{\mathrm{LHY}}\left(n_0\right)=g{n}_0{\displaystyle \frac{32}{3\sqrt{\pi }}}{\left(n_0{a}_s^3\right)}^{1/2}$$

(37)

is the renormalized zero-temperature Lee–Huang–Yang beyond-mean-field correction, i.e., a sort of additional bulk chemical potential with $a_s$ the s-wave scattering length such that $g=4\pi {\hslash }^2{a}_s/m$, while

$${\mu }_{\mathrm{th}}(n_0,T)=2g{n}_{\mathrm{th}}(n_0,T)$$

(38)

is the quantum-thermal correction, where

$$n_{\mathrm{th}}(n_0,T)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}\left({\displaystyle \frac{{\hslash }^2{k}^2}{2m}}+g{n}_0\right){\displaystyle \frac{1}{E_k\left(n_0\right)}}{\displaystyle \frac{1}{e^{E_k\left(n_0\right)/\left(k_{B}T\right)}-1}}$$

(39)

plays the role of the density of a thermal bath, such that

$$n_{\mathrm{th}}(n_0,T)\to \zeta (3/2){\left({\displaystyle \frac{m{k}_{B}T}{2\pi {\hslash }^2}}\right)}^{3/2}\mathrm{for}{n}_0\mathrm{very}\mathrm{small}$$

(40)

with $\zeta (3/2)\simeq 2.612$ and

$$n_{\mathrm{th}}(n_0,T)\to {\displaystyle \frac{\zeta \left(3\right)}{{\pi }^2}}{\left({\displaystyle \frac{k_{B}T}{\hslash }}\right)}^3{\left({\displaystyle \frac{m}{g{n}_0}}\right)}^{3/2}\mathrm{for}{n}_0\mathrm{very}\mathrm{large}$$

(41)

with $\zeta \left(3\right)\simeq 1.202$. We stress that Equation (36) can also be seen as the time-dependent extension of the stationary Gross-Pitaevskii equation for the Bose–Einstein condensate which appears in the Zaremba-Nikuni-Griffin (ZNG) formalism [15]. By adopting the ZNG formalism we recover our thermal density $n_{\mathrm{th}}$ from a Boltzmann equation of non-condensed bosons. At equilibrium and zero temperature the Lee–Huang–Yang term of Equation (36) is consistent with the modified Gross–Pitaevskii equation of Ref. [13].

3.1. Including an External Potential

In the spirit of the local density approximation (LDA) we can also consider the inclusion of an external potential $U\left(\mathbf{r}\right)$ in Equation (36), namely

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\overline{\psi }=\left[-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+U\left(\mathbf{r}\right)-\mu +g{n}_0+{\mu }_{\mathrm{LHY}}\left(n_0\right)+{\mu }_{\mathrm{th}}(n_0,T)\right]\overline{\psi }$$

(42)

Remember that here, and in Equation (36), the chemical potential $\mu$ can be formally removed with an appropriate redefinition of the phase of time-dependent field $\overline{\psi }(\mathbf{r},t)$.

A drawback of Equation (42) is that, for $g\to 0$ the thermal density $n_{\mathrm{th}}$ becomes uniform despite the presence of $U\left(\mathbf{r}\right)$. To cure this drawback it is sufficient to make the substitution $g{n}_0\to g{n}_0+U\left(\mathbf{r}\right)$ in Equation (39). However, just to simplify a bit the problem one usually considers the following Hartree approximation of the Bogoliubov spectrum in the single-particle phase space [16]

$$E_k^{\left(\mathrm{HF}\right)}(\mathbf{r},t)={\displaystyle \frac{{\hslash }^2{k}^2}{2m}}+U\left(\mathbf{r}\right)+g{n}_0(\mathbf{r},t)$$

(43)

that is reliable for large k. In this way we have

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\overline{\psi }=\left[-{\displaystyle \frac{{\hslash }^2}{2m}}{\nabla }^2+U\left(\mathbf{r}\right)-\mu +g{n}_0+{\mu }_{\mathrm{LHY}}\left(n_0\right)+2g{\tilde{n}}_{\mathrm{th}}(n_0,T)\right]\overline{\psi }$$

(44)

with

$${\tilde{n}}_{\mathrm{th}}(n_0,T)=\int {\displaystyle \frac{d^3\mathbf{k}}{{\left(2\pi \right)}^3}}{\displaystyle \frac{1}{e^{(\frac{{\hslash }^2{k}^2}{2m}+U\left(\mathbf{r}\right)+g{n}_0(\mathbf{r},t))/\left(k_{B}T\right)}-1}}$$

(45)

At this point an important remark is needed. Depending on the adopted formalism slightly different versions of Equations (44) and (45) are derived. For instance, in the ZNG approach $n_0$ is substituted by the total number density n in the LHY chemical potential and also in the thermal density.

3.2. Josephson Equations at Finite Temperature

Under the assumption of an external potential $U\left(\mathbf{r}\right)$ which separates our three-dimensional ($D=3$) system in two weakly linked regions and that $\overline{\psi }(\mathbf{r},t)$ of Equation (42) truly describes Bose-condensed particles at finite temperature T, a straightforward generalization of the Josephson equations [20,21] at finite temperature reads

$$\begin{array}{ccc}\hfill \hslash \dot{z}& =& -2J\sqrt{1-z^2}sin\left(\varphi \right)\hfill \end{array}$$

(46)

$$\begin{array}{ccc}\hfill \hslash \dot{\varphi }& =& 2\left(f_T\left[{\displaystyle \frac{{\overline{n}}_0}{2}}(1+z)\right]-f_T\left[{\displaystyle \frac{{\overline{n}}_0}{2}}(1-z)\right]\right)+2J{\displaystyle \frac{z}{\sqrt{1-z^2}}}cos\left(\varphi \right)\hfill \end{array}$$

(47)

where $z\left(t\right)$ is the population imbalance of the Bose–Einstein condensate, $\varphi \left(t\right)$ is the relative phase of the Bose condensate, J is the tunneling energy of condensed bosons, ${\overline{n}}_0$ is the space-time independent average number density of condensed bosons in the two regions, and

$$f_T\left[x\right]=gx+{\mu }_{\mathrm{LHY}}\left(x\right)+{\mu }_{\mathrm{th}}(x,T)$$

(48)

Working with a fixed total average number density $\overline{n}$ of bosons, the average number density ${\overline{n}}_0$ of condensed bosons can be extracted from the finite-temperature Bogoliubov formula

$${\overline{n}}_0=\overline{n}-{\displaystyle \frac{8}{3\sqrt{\pi }}}{\left(a_s\overline{n}\right)}^{3/2}-\overline{n}I[a_s{\overline{n}}^{1/3},{\displaystyle \frac{m{k}_{B}T}{{\hslash }^2{\overline{n}}^{2/3}}}]$$

(49)

with $I[x,y]$ given by

$$I[x,y]={\displaystyle \frac{1}{24\sqrt{\pi }}}{\displaystyle \frac{y^2}{x^{1/2}}}$$

(50)

A detailed derivation of Equation (49) with Equation (50) is discussed in Ref. [17]. The main idea is to write the thermodynamic grand potential $\Omega$ of the system at equilibrium as a function of both the chemical potential $\mu$ and the condensate density ${\overline{n}}_0={\left|\overline{\psi }\right|}^2$ by using the gapped Bogoliubov spectrum (29). The number density is then obtained as $\overline{n}=-{\displaystyle \frac{\partial \Omega }{\partial \mu }}{\displaystyle \frac{1}{L^3}}$. Finally, setting $\mu =g{\overline{n}}_0$ one finds $\overline{n}$ as a function of ${\overline{n}}_0$ and T.

3.3. 1PI GP Effective Potential in D Dimensions

The generalization of Equation (32) to the case of D-dimensional bosonic system is immediate:

$$\begin{array}{c}\hfill V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)=-\mu {\left|\overline{\psi }\right|}^2+{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^4+\int {\displaystyle \frac{d^D\mathbf{k}}{{\left(2\pi \right)}^D}}\left[{\displaystyle \frac{1}{2}}{E}_k(\overline{\psi },\mu )+k_{B}Tln\left(1-exp\left(-{\displaystyle \frac{E_k(\overline{\psi },\mu )}{k_{B}T}}\right)\right)\right]\end{array}$$

(51)

As previously discussed, for $D\ne 0$, we can safely use the gapless spectrum $E_k(\overline{\psi },\mu =g|\overline{\psi }{|}^2)$ instead of the gapped spectrum $E_k(\overline{\psi },\mu )$ in this effective potential. However, this substitution cannot be performed for $D=0$, where only the mode $\mathbf{k}=\mathbf{0}$ survives, because it will imply $E_k=E_0=0$, see Equation (30).

In the zero-dimensional case ($D=0$) one must use the gapped spectrum of Equation (29) with $k=0$, namely

$$E_0(\overline{\psi },\mu )=\sqrt{{\left(\mu -2g|\overline{\psi }{|}^2\right)}^2-g^2{\left|\overline{\psi }\right|}^2}$$

(52)

Thus, the 1PI GP effective potential for $D=0$ reads

$$V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)=-\mu {\left|\overline{\psi }\right|}^2+{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^4+{\displaystyle \frac{1}{2}}{E}_0(\overline{\psi },\mu )+k_{B}Tln\left(1-exp\left(-{\displaystyle \frac{E_0(\overline{\psi },\mu )}{k_{B}T}}\right)\right)$$

(53)

Actually, a better treatment at $T=0$ is obtained considering the gapless Bogoliubov spectrum with a finite $\mathbf{k}$ and dimensional regularization around $D=0$ [25]. At finite temperature one instead uses the gapped $E_0(\overline{\psi },\mu )$. At the end, we obtain

$$V_{\mathrm{eff}}^{\left(T\right)}\left(\overline{\psi }\right)=-\mu {\left|\overline{\psi }\right|}^2+{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^4-{\displaystyle \frac{g}{2}}{\left|\overline{\psi }\right|}^2+k_{B}Tln\left(1-exp\left(-{\displaystyle \frac{E_0(\overline{\psi },\mu )}{k_{B}T}}\right)\right)$$

(54)

Quite remarkably, the term $(g/2)\left(\right|\overline{\psi }{|}^4-|\overline{\psi }{|}^2)=(g/2)|\overline{\psi }{|}^2\left(\right|\overline{\psi }{|}^2-1)$ gives for $T=0$ the exact internal energy $(g/2)N(N-1)$ at equilibrium with $N=|\overline{\psi }{|}^2$.

4. Conclusions

We have reviewed and extended the quantum effective action formalism for non-relativistic systems, with emphasis on its Gaussian (one-loop) implementation and finite-temperature generalization. Starting from the single-particle case, we showed how the 1PI effective action can be systematically derived and how the saddle-point plus Gaussian-fluctuation approximation leads to simple yet physically transparent expressions for the effective potential. The finite-temperature extension, obtained through Matsubara summation, yields quantum–thermal corrections with a clear interpretation as the Helmholtz free energy of harmonic modes with effective Bogoliubov frequencies. Applied to the Gross–Pitaevskii field theory, this approach naturally incorporates beyond-mean-field effects, including the Lee–Huang–Yang correction and its finite-temperature counterpart, and accommodates both gapped and gapless excitation spectra. The formalism is flexible enough to handle external potentials within the local density or Hartree approximations, to describe Josephson dynamics at finite temperature, and to generalize to arbitrary spatial dimensions, including the analytically interesting zero-dimensional limit. Overall, the quantum–thermal effective action framework offers a unified and physically intuitive route from microscopic fluctuations to macroscopic dynamical equations for Bose–Einstein condensates. It connects field-theoretic principles with experimentally relevant phenomena, providing a solid base for further developments, such as the inclusion of dissipative effects, stronger interaction regimes, or non-equilibrium dynamics. Regarding the connection with experiments, it is important to stress that, in the context of superconducting Josephson junctions, the zero-point correction of Equation (17) explains the shift of the quantized energy levels of qubits [26]. Instead, in the context of ultracold bosonic atoms, the beyond-mean-field Gaussian quantum corrections described here have been found in several experiments (see, for instance, Refs. [27,28,29]). In this review we have not considered open quantum systems and the impact of external environments. This is a hot topic of research where, however, a clear connection between the formalism of master equations [30,31,32] and the quantum effective action approach is still lacking.