We discuss the equilibrium states at finite temperature \(\beta \) with a nonzero chemical potential \(\mu \). We first discuss the free theory (\(\lambda =0\)); afterward, we study the corresponding states for the interacting theory. We are interested in finding states which can be interpreted as exhibiting Bose–Einstein condensation. The traditional way of defining BEC with particle numbers and occupation numbers cannot be applied in relativistic quantum systems. Instead, we look at states with nonvanishing one-point functions, thus showing spontaneous breakdown of the internal U(1) symmetry of the theory. This symmetry is generated by a conserved current \(J_{\mu }\). The charge density \(J_0\),
$$\begin{aligned} J_0(x)\doteq -i:{\dot{\varphi }}^*\varphi -\varphi ^*{\dot{\varphi }}:_H, \end{aligned}$$
replaces the particle density of the nonrelativistic theory. The mean of the charge density then distinguishes between different phases.
In the free theory, at fixed inverse temperature \(\beta \), there is a critical value for the mean charge density. Below this value, the pure phases correspond to unique gauge invariant states, with a chemical potential \(\mu \) depending on the charge density and with \(\mu ^2<m^2\). If the charge density is above this threshold, the chemical potential has to satisfy \(\mu ^2=m^2\); the states corresponding to pure phases have nonvanishing one-point functions which are related by the gauge symmetry. (See [10] for the concept of chemical potential in an algebraic formulation.)
Due to the nonvanishing one-point function, the two-point function is not decaying at large separations. Similar nonvanishing long- distance correlations are the basis of the criterion for BEC in the nonrelativistic theory. There one says that the ground state has a macroscopic occupation if the one-particle density matrix smeared in both entries over a spatial box of dimension L grows at least as particle number N in the limit where \(L\rightarrow \infty \) keeping \(N/L^3\) finite, see [52] for a more extensive discussion.
Condensate in the Free Theory
Let us start discussing the condensate for a free massive complex scalar quantum field theory propagating in a Minkowski spacetime. Let us denote by \(\varphi \) the associated field configuration. Its equilibrium states with inverse temperature \(\beta >0\) and chemical potential \(\mu \), \(|\mu |<m\) are the states which satisfy the KMS condition with respect to the time evolution
$$\begin{aligned} \tau _{t,\mu }(\varphi (x))=\varphi (x+te_0)e^{it\mu } \end{aligned}$$
(15)
where \(e_0\) denotes the unit vector in time direction. The theory possesses an internal U(1) symmetry which might be spontaneously broken in some of equilibrium states. Hence, an interesting observable to distinguish these states is the current density
$$\begin{aligned} J_0(f)\doteq \int J_0(x)f(x) \mathrm{d}^4x \doteq -i\int \left( :{\dot{\varphi }}^*\varphi -\varphi ^*{\dot{\varphi }}:_H \right) f \mathrm{d}^4x \end{aligned}$$
(16)
where this is seen as an element of \({\mathcal {A}}\). We observe that, in view of the symmetry of the Hadamard coefficients U, \(V_i\) of \(V=V_n\sigma ^n\) in (2), and \(W_i\) of \(W=W_n\sigma ^n = (\omega _0-H-i\Delta /2)\), \(\omega _0\) being the two-point function of the vacuum state, we have that \(:{\dot{\varphi }}^*\varphi -\varphi ^*{\dot{\varphi }}:_H = :{\dot{\varphi }}^*\varphi -\varphi ^*{\dot{\varphi }}:_{H+W}.\) Hence, \(J_0(f)\) can be seen as the current normal-ordered with respect to vacuum state. The possible pure phases are thus characterized by the following proposition:
Proposition 3.1
For inverse temperature \(\beta >0\) and chemical potential \(|\mu |<m\), there exists an unique KMS state with respect to \(\tau _{t,\mu }\) whose n-point functions are tempered distributions. This state, denoted by \(\omega _{\beta ,\mu }\) is quasi-free and its two-point functions are
$$\begin{aligned} \omega _{\beta ,\mu }(\varphi ^*(x)\varphi (y))\doteq \frac{1}{(2\pi )^{3}}\int {\mathrm{d}^4p}\,\delta (p^2+m^2)\epsilon (p_0)e^{ip(x-y)}\frac{1}{1-e^{-\beta (p_0-\mu )}} \end{aligned}$$
(17)
and \(\omega _{\beta ,\mu }(\varphi (x)\varphi (y)))=\omega _{\beta ,\mu }(\varphi ^*(x)\varphi ^*(y)))=0\). The charge density in this state is
$$\begin{aligned} \omega _{\beta ,\mu }(J_0(x))&=\int {d}^4p\,2|p_0|\delta (p^2+m^2)\left( \frac{1}{e^{\beta (|p_0|-\mu )}-1}-\frac{1}{e^{\beta (|p_0|+\mu )}-1}\right) . \end{aligned}$$
It holds that
$$\begin{aligned} |\omega _{\beta ,\mu }(J_0(x)) | \le \rho _{cr}(\beta )\doteq \omega _{\beta ,m}(J_0(x)) \end{aligned}$$
(18)
where \(\rho _{cr}(\beta )\) is the critical charge density. For \(\beta >0\) and \(\mu = \pm m\), there exist various KMS states with respect to \(\tau _{t,\mu }\). Let us denote by \(\Omega _{\beta ,\pm m}\) the set of quasi-free KMS states. The pure phases are the extremal points in \(\Omega _{\beta ,\pm m}\), and these states are
$$\begin{aligned} \omega _{\beta ,c}^\pm = \omega _{\beta ,\pm m}\circ \gamma _c^{\pm } \end{aligned}$$
where \(\gamma _c^{\pm }\) is an automorphism which is generated by
$$\begin{aligned} \gamma ^{\pm }_c(\varphi (x))=\varphi (x)+e^{\pm ix^0m}c({\mathbf {x}}) \end{aligned}$$
(19)
where c is a harmonic function of the spatial variables \({\mathbf {x}}\), \(\Delta c=0\), with the spatial Laplacian \(\Delta \). In this case,
$$\begin{aligned} \omega _{\beta ,c}^{\pm }(J_0(x))&=\int {d}^4p\,2|p_0|\delta (p^2+m^2)\left( \frac{1}{e^{\beta (|p_0|\mp m)}-1}-\frac{1}{e^{\beta (|p_0|\pm m)}-1}\right) \pm 2m|c|^2. \end{aligned}$$
Furthermore,
$$\begin{aligned} |\omega _{\beta ,c}^{\pm }(J_0(x))| \ge \rho _{cr}(\beta ). \end{aligned}$$
Proof
First of all, we observe that the KMS states corresponding to pure phases are quasi-free states with at most a nontrivial one-point function. A proof of this fact can be found in [61]. Furthermore, the truncated two-point function is constrained by the KMS condition to be equal to (17). The one-point function \(\omega (\varphi )\) is constrained by the equation of motion and by request of invariance under the action of \(\tau _{t,\mu }\). In particular, invariance under the action \(\tau _{t,\mu }\) implies that the function
$$\begin{aligned} t\mapsto \omega (\varphi (x))e^{-i\mu t} \end{aligned}$$
is constant in time. This function needs to be a solution of \(-\Delta + m^2 -\mu ^2\); however, for \(|\mu |<m\) these solutions cannot be tempered distributions. The inequalities involving the critical charge density \(\rho _{cr}(\beta )\) are an immediate consequence of the form of the expectation value of \(J_0\) in the analyzed states.
\(\square \)
The state \(\omega _{\beta ,\mu }\) respects the U(1)-symmetry of the theory. For chemical potentials \(\mu =\pm m\), there exist many equilibrium states at fixed temperature and the U(1) symmetry is spontaneously broken in the states \(\omega _{\beta ,c}^{\pm }\) for \(c\ne 0\). At zero temperature, all the states with chemical potential \(|\mu |<m\) coincide with the vacuum. Hence, the vacuum expectation value of the charge density \(J_0(x)\) vanishes in that limit. Thus, any nonvanishing charge density in the limit \(T=\beta ^{-1}\rightarrow 0\) of vanishing temperature requires a condensate.
At finite \(\beta \) and \(\mu =m\), we have \(\omega _{\beta ,c}^{+}(J_0(x)) = \rho _{cr}(\beta ) + 2m|c|^2\). A nonvanishing condensate can occur only if \(|\omega _{\beta ,c}^{+}(J_0)|>\rho _{cr}(\beta )\). Since \(\rho _{cr}(\beta )>0\) is monotonically decreasing in \(\beta \), diverges for \(\beta \rightarrow 0\) and tends to 0 for large \(\beta \), at a fixed charge density \(\omega _{\beta ,c}^{\pm }(J_0)\), there is a critical temperature \(T_{cr}>0\) such that only for \(T<T_{cr}\) a condensate can be formed.
In “Appendix B,” we shall compute the nonrelativistic limit of the states analyzed in this section showing that the charged scalar field tends to the nonrelativistic scalar field and the states tend to the known equilibrium states of a nonrelativistic system of spinless noninteracting bosons in the thermodynamic limit. Furthermore, the charge density converges to the particle density. Finally, we see that the nontrivial one-point function shows up in the long-distance behavior of the 2-point function which coincides with the one-particle-reduced density matrix in the thermodynamic limit.
Massive Complex Scalar Field with \(\varphi ^4\) Interaction over the Condensate
In this section, we start discussing the perturbative construction of the \(\varphi ^4\) interacting theory over a suitable classical solution of the equation of motion which represents the condensate in the Minkowski spacetime. The Lagrangian of the theory we are considering is thus
$$\begin{aligned} {\mathcal {L}} = -\frac{1}{2} \partial {\overline{\varphi }}\partial {\varphi } - \frac{1}{2}m^2 |\varphi |^2 - \frac{\lambda }{4} |\varphi |^4 \end{aligned}$$
where \(\varphi \) is a complex scalar field. Following a similar procedure presented in section III of [1], we expand \({\mathcal {L}}\) around a real classical solution \(\phi \) which represents the condensate. Hence,
$$\begin{aligned} \varphi = e^{-i\mu x^0}(\phi + \psi ) \end{aligned}$$
(20)
where \(\mu \) is again the chemical potential, \(x^0\) is a fixed Minkowski time and \(\psi \) is a complex scalar field which describes the perturbations. Its real and imaginary parts are denoted by \(\psi _1\) and \(\psi _2\), and thus,
$$\begin{aligned} \psi = \psi _1+i\psi _2. \end{aligned}$$
The Lagrangian density can now be written as a sum of contributions homogenous in the number of fields \(\psi \) as follows:
$$\begin{aligned} {\mathcal {L}} = {\mathcal {L}}_0+{\mathcal {L}}_2+{\mathcal {L}}_3+{\mathcal {L}}_4 \end{aligned}$$
where
$$\begin{aligned} {\mathcal {L}}_0&= \frac{1}{2} \left| (\partial _0-i\mu )\phi \right| ^2 - \frac{1}{2} |\nabla \phi |^2 - \frac{\lambda }{4}|\phi |^4 - \frac{1}{2}m^2|\phi |^2 \\ {\mathcal {L}}_2&= \frac{1}{2} \left| (\partial _0-i\mu )\psi \right| ^2 - \frac{1}{2} |\nabla \psi |^2 -\lambda \phi ^2|\psi _1|^2 -\frac{1}{2}(\lambda \phi ^2+m^2)|\psi |^2 \\ {\mathcal {L}}_3&= -\lambda \phi \psi _1 |\psi |^2 \\ {\mathcal {L}}_4&= -\frac{\lambda }{4}|\psi |^4. \end{aligned}$$
The term \({\mathcal {L}}_1\) vanishes because \(\phi \) is chosen to be a stationary point for the classical action \(\int {\mathcal {L}}_0 \mathrm{d}^4x\). In the following, we shall choose a nonvanishing \(\phi \) to describe the condensate, we discuss the quantization of the linearized theory (\({\mathcal {L}}_2\)), and finally, we use perturbation theory over the linearized theory to take into account \({\mathcal {L}}_3+{\mathcal {L}}_4\).
Contrary to the case of the free theory, in the interacting theory the chemical potential is not restricted to the interval \([-m,m]\). A chemical potential outside of this interval induces a spontaneous breakdown of symmetry showing up in a nonvanishing one-point-function and, as a consequence, in long-range behavior of the two-point-function, similar to the nonrelativistic case. In contrast to the free case, states with different condensates are not in mutual thermal equilibrium, since their chemical potentials differ.
The Condensate in the Vacuum Theory
We look for the case of a translation invariant background \(\phi \). Then, the kinetic term in \({\mathcal {L}}_0\) has no effect and \(\phi \) is a stationary point for
$$\begin{aligned} I=\int U(|\phi ^2|) \mathrm{d}^4x \end{aligned}$$
where
$$\begin{aligned} U(|\phi ^2|) = -\frac{\lambda }{4}|\phi |^4-\frac{1}{2}(m^2-\mu ^2)|\phi |^2; \end{aligned}$$
hence, it holds
$$\begin{aligned} |\phi |^2 = \frac{\mu ^2-m^2}{\lambda } \end{aligned}$$
(21)
and only one real, positive and translational invariant background solution \(\phi \) is thus available for \(\mu ^2>m^2\). We notice that for fixed \(\mu ^2>m^2\) the background value of the field \(\phi \) is of order \(1/\sqrt{\lambda }\). In this case, we observe that \({\mathcal {L}}_2\) does not depend on \(\lambda \), \({\mathcal {L}}_3\) is of order \(\sqrt{\lambda }\) while \({\mathcal {L}}_4\) is of order \(\lambda \). In the next, we shall construct the interacting field theory with perturbation methods considering \({\mathcal {L}}^I=({\mathcal {L}}_3+{\mathcal {L}}_4)\) the interaction Lagrangian. Hence, the solution we shall obtain will be a formal power series in \(\sqrt{\lambda }\).
In the next, we shall discuss the construction of the quantum theory over the background discussed so far.
We argue that there exists a limit in which all the correlation functions are dominated by the classical background \(\phi \). Actually, in the limit \(\lambda \rightarrow 0\) keeping \(|\mu |^2-m^2\) finite, the classical background \(\phi \) diverges as \(\lambda ^{-1/2}\), furthermore, the linearized theory is not affected by changes of \(\lambda \) while the S-matrix constructed with the interacting Lagrangian tends to 1.
Hence, we expect that, under this limit, the one-point function rescaled by \(\sqrt{\lambda }\) tends to the background value \({\tilde{\phi }} e^{-i m x^0}\) where \({\tilde{\phi }} = \sqrt{\lambda } \phi \), and similarly, the rescaled charge density \(\lambda J_0\) tends to the charge density of the background \(2\mu |{\tilde{\phi }}|^2\). Both these quantities do not depend on \(\lambda \). We finally observe that the rescaled background \({\tilde{\phi }}\) is a solution of the equation of motion descending from the rescaled classical Lagrangian density
$$\begin{aligned} \tilde{{\mathcal {L}}}_0 = \lambda {\mathcal {L}}_0 = \frac{1}{2} |(\partial _0-i\mu ){\tilde{\phi }}|^2 - \frac{1}{2} |\nabla {\tilde{\phi }}|^2 - \frac{1}{4}|{\tilde{\phi }}|^4 - \frac{1}{2}m^2|{\tilde{\phi }}|^2 \end{aligned}$$
which is also independent on \(\lambda \).
This is in analogy to what happens in the nonrelativistic case; actually there under the Gross–Pitaevskii limit, the density of the ground state tends to the density of a suitable classical solution of the Gross–Pitaevskii equation [49, 50], see in particular Theorem 1.1 and Theorem 1.2 in [50].
We thus argue that the equation of motion corresponding to the rescaled zeroth-order Lagrangian \(\tilde{{\mathcal {L}}}_0\) can be interpreted as an analogous of the Gross–Pitaevskii equation in the relativistic setting, and thus, the limit \(\lambda \rightarrow 0\) taken with m and \(\mu \) fixed can be understood as the analogous of the Gross–Pitaevskii limit discussed in introduction.
Linearized Theory
The first step to construct the quantization of \(\varphi \) is the analysis of the linearized equations of motion for the fluctuations \((\psi _1,\psi _2)\) around \(\phi \). They have the form
$$\begin{aligned} \begin{aligned} (\square -M_1^2)\psi _1-2\mu {\dot{\psi }}_2&=0 \\ (\square -M_2^2)\psi _2+2\mu {\dot{\psi }}_1&=0 \end{aligned} \end{aligned}$$
(22)
where
$$\begin{aligned} M_1^2=(m^2-\mu ^2) + 3\lambda \phi ^2 \quad \text {and}\quad M_2^2= (m^2-\mu ^2)+ \lambda \phi ^2. \end{aligned}$$
(23)
Notice that if (21) holds, \(M_1^2=2(\mu ^2-m^2) \) and \(M_2^2=0\). Hence, we assume \(M_1>M_2\ge 0\). Let us introduce
$$\begin{aligned} M^2 =\frac{M_1^2+M_2^2}{2}, \quad \delta M^2 =\frac{M_1^2-M_2^2}{2}. \end{aligned}$$
We observe that Eq. (22) for \({\tilde{\psi }}=(\psi _1,\psi _2)\) can be written in a compact form \(D {{\tilde{\psi }}} = 0\) where D is given in terms of the standard Pauli matrices
$$\begin{aligned} {{\varvec{\sigma }}}_1=\begin{pmatrix} 0&{}\quad 1\\ 1&{}\quad 0 \end{pmatrix}, \quad {{\varvec{\sigma }}}_2=\begin{pmatrix} 0&{}\quad -i\\ i&{}\quad 0 \end{pmatrix} ,\quad {{\varvec{\sigma }}}_3=\begin{pmatrix} 1&{}\quad 0\\ 0&{}\quad -1 \end{pmatrix}. \end{aligned}$$
as
$$\begin{aligned} \begin{aligned} D&= (\square -M^2) {\mathbb {I}} - \delta M^2 {{\varvec{\sigma }}}_3 - i 2\mu \partial _0 {{\varvec{\sigma }}}_2 \\ {\overline{D}}&= (\square -M^2) {\mathbb {I}} + \delta M^2 {{\varvec{\sigma }}}_3 + i 2\mu \partial _0 {{\varvec{\sigma }}}_2. \end{aligned} \end{aligned}$$
(24)
Notice that
$$\begin{aligned} D{\overline{D}}= \left( (-\square +M^2)^2-(\delta M^2)^2 + 4\mu ^2 \partial _0^2\right) {\mathbb {I}}. \end{aligned}$$
The retarded and advanced propagators of the theory can be obtained as
$$\begin{aligned} \Delta _R \doteq {\overline{D}} (D{\overline{D}})_R,\quad \Delta _A \doteq {\overline{D}} (D{\overline{D}})_A, \end{aligned}$$
where \((D{\overline{D}})_R\) and \((D{\overline{D}})_A\) are the retarded and advanced fundamental solutions of \(D{\overline{D}}\). Let us thus study
$$\begin{aligned} \widehat{D{\overline{D}}}&= \left( (p^2+M^2)^2 - (\delta M^2)^2 -4\mu ^2p_0^2 \right) {\mathbb {I}} \\&= \left( p_0^4 -2p_0^2 ({\mathbf {p}}^2+M^2 +2\mu ^2) + ({\mathbf {p}}^2+M^2)^2 - (\delta M^2)^2\right) {\mathbb {I}} \end{aligned}$$
Hence, the four solutions of \(p_0^4 -2p_0^2 ({\mathbf {p}}^2+M^2 +2\mu ^2) + ({\mathbf {p}}^2+M^2)^2 - (\delta M^2)^2=0\) are \(\pm \omega _\pm \) where
$$\begin{aligned} \omega _\pm ^2&= w^2 +2\mu ^2 \pm \sqrt{(w^2 +2\mu ^2)^2 - w^4 +(\delta M^2)^2} \nonumber \\&= w^2 +2\mu ^2 \pm \sqrt{4\mu ^4 + 4\mu ^2w^2 +(\delta M^2)^2} \nonumber \\&= w^2 +2\mu ^2 \pm \sqrt{(w^2 +2\mu ^2)^2 - w_1^2w_2^2} \end{aligned}$$
(25)
where now \(w^2 \doteq {\mathbf {p}}^2+M^2\) and \(w_i^2\doteq {\mathbf {p}}^2+M^2_i\).
We notice that if \(M_2=0\) we have that \(w_2=0\) for \(|{\mathbf {p}}|=0\), and thus,
$$\begin{aligned} \lim _{|{\mathbf {p}}| \rightarrow 0 }\omega _-^2 = 0; \end{aligned}$$
hence, a massless mode is present in this system as expected by the Goldstone theorem. However, if the linearized theory is not in a ground state, it could happen that the normal-ordered interaction Lagrangian with respect to the state, as in (14), contains quadratic terms that could contribute to the masses of the fluctuations. If we use the formula
$$\begin{aligned} \prod _i \frac{1}{x-x_i} = \sum _i \frac{1}{x-x_i} \prod _{j\ne i} \frac{1}{x_i-x_j} \end{aligned}$$
valid for pairwise different \(x_1,\ldots , x_n\), a couple of times, we get
$$\begin{aligned} {{\widehat{\Delta }}}_R&= \frac{\widehat{{\overline{D}}}}{(\omega _+^2-\omega _-^2)} \left( \frac{1}{(p_0+i\epsilon )^2-\omega _+^2} -\frac{1}{(p_0+i\epsilon )^2-\omega _-^2}\right) \end{aligned}$$
where recalling (24)
$$\begin{aligned} \widehat{{\overline{D}}} = -(p^2+M^2) {\mathbb {I}} + \delta M^2 {{\varvec{\sigma }}}_3 + 2\mu p_0 {{\varvec{\sigma }}}_2. \end{aligned}$$
(26)
We can construct \(\Delta _A\) just changing \(i\epsilon \rightarrow - i\epsilon \), while the Feynman propagator \(\Delta _F\) is obtained substituting \((p_0+i\epsilon )^2\) with \(p_0^2+i\epsilon \) and multiplying by i. Finally, the commutator function is
$$\begin{aligned} {{\widehat{\Delta }}} = \frac{2\pi i \widehat{{\overline{D}}}}{\omega _+^2-\omega _-^2} \epsilon {(p_0)}\left( \delta (p_0^2-\omega _+^2)-\delta (p_0^2-\omega _-^2) \right) \!. \end{aligned}$$
With \(\Delta \) at disposal, the quantum product can be given as in Sect. 2.1; in this way we obtain the \(*\)-algebra of field observables \({\mathcal {A}}_0\). The analog of the Hadamard singularity H (2) for this theory can be given. The form of some of the corresponding Hadamard coefficients is discussed in “Appendix C.” The extended \(*\)-algebra of field observables \({\mathcal {A}}\) containing Wick polynomials normal-ordered with respect to H is obtained as in 2.1.
KMS States for the Linearized Theory
In view of the decomposition of the field \(\varphi \) given in (20), the action of \(\tau _t\) on \(\psi \) as time translation is equivalent to the action of \(\tau _{t,\mu }\) on \(\varphi \) as given in (15). Hence, having the causal propagator of the linearized theory at disposal, we can construct the two-point function of the quasi-free \(\beta \)-KMS state with respect to time translation \(\tau _t\) of the \(\psi \) fields as
$$\begin{aligned} {\widehat{\omega }}_{\beta ,\psi } = \frac{i{{\widehat{\Delta }}}}{1-e^{-\beta p_0}}. \end{aligned}$$
Introducing
$$\begin{aligned} {\mathcal {S}} \doteq \begin{pmatrix} \psi _1(x)\psi _1(y) &{}\quad \psi _1(x)\psi _2(y)\\ \psi _2(x)\psi _1(y)&{}\quad \psi _2(x)\psi _2(y) \end{pmatrix} \end{aligned}$$
we have that the two-point function of the quasi-free \(\beta \)-KMS state \(\omega _{\beta ,\psi }\) is in position space
$$\begin{aligned} \omega _{\beta ,\psi }({\mathcal {S}})= & {} \frac{1}{(2\pi )^{3}} \int {d}^4p\; e^{ip(x-y)} \frac{ \epsilon {(p_0)}}{\omega _+^2-\omega _-^2} \nonumber \\&\left( \delta (p_0^2-\omega _+^2)-\delta (p_0^2-\omega _-^2) \right) \frac{(-\widehat{{\overline{D}}})}{1-e^{-\beta p_0}}. \end{aligned}$$
(27)
Recalling the form of \(\omega _\pm \) in (25), we notice that if \(M_1>M_2>0\)
$$\begin{aligned}&\omega _\pm ^2=w^2 +2\mu ^2 \pm \sqrt{(w^2 +2\mu ^2)^2 - w_1^2w_2^2}>0, \\&\omega _+^2-\omega _-^2 =2\sqrt{4\mu ^4 + 4\mu ^2w^2 +(\delta M^2)^2} >0, \end{aligned}$$
this means that no infrared divergences are present in \(\omega _{\beta ,\psi }\) if \(M_2>0\). The two-point function of the ground state of the \(\psi _i\) theory (keeping the condensate \(\phi \ne 0\)) can be obtained taking the limit \(\beta \rightarrow \infty \) of (27). Hence, to study expectation values in the state \(\omega _{\beta ,\psi }\) of observables normal-ordered with respect to the vacuum \(\omega _{\infty ,\psi }\) we consider \(W=\omega _{\beta ,\psi }-\omega _{\infty ,\psi }\) and we obtain
$$\begin{aligned} W({\mathcal {S}})&= \frac{1}{(2\pi )^{3}} \int {d}^4p\; e^{ip(x-y)} \frac{1}{\omega _+^2-\omega _-^2} \left( \delta (p_0^2-\omega _+^2)-\delta (p_0^2-\omega _-^2) \right) \frac{(-\widehat{{\overline{D}}})}{e^{\beta |p_0|}-1}. \end{aligned}$$
(28)
We observe that in the coinciding point limit, the off-diagonal expectation values are vanishing
$$\begin{aligned} W(\psi _1(x)\psi _2(x))=0, \quad W(\psi _2(x)\psi _1(x))=0, \end{aligned}$$
and introducing \(2\delta \omega ^2\doteq \omega _+^2-\omega _-^2 = 2\sqrt{4\mu ^4 + 4\mu ^2 w^2 +(\delta M^2)^2}\) we have that \(W(\psi _i^2)\); the coinciding point limits of the diagonal elements of \(W({\mathcal {S}})\) are
$$\begin{aligned} W(\psi _1^2)&= \frac{1}{(2\pi )^{3}} \int {d}^3{\mathbf {p}}\; \left( \frac{\delta \omega ^2+2\mu ^2+\delta M^2 }{\delta \omega ^2} \frac{1}{2\omega _+} \frac{1}{e^{\beta \omega _+}-1} \right. \nonumber \\&\quad \left. + \frac{ \delta \omega ^2 -2\mu ^2-\delta M^2 }{\delta \omega ^2}\frac{1}{2\omega _-}\frac{1}{e^{\beta \omega _-}-1} \right) \end{aligned}$$
(29)
$$\begin{aligned} W(\psi _2^2)&= \frac{1}{(2\pi )^{3}} \int {d}^3{\mathbf {p}}\; \left( \frac{\delta \omega ^2+2\mu ^2-\delta M^2 }{\delta \omega ^2} \frac{1}{2\omega _+} \frac{1}{e^{\beta \omega _+}-1} \right. \nonumber \\&\quad \left. + \frac{ \delta \omega ^2 -2\mu ^2+\delta M^2 }{\delta \omega ^2}\frac{1}{2\omega _-}\frac{1}{e^{\beta \omega _-}-1} \right) \end{aligned}$$
(30)
Notice that the integrand in both \(W(\psi _1^2)\) and \(W(\psi _2^2)\) is positive.
To analyze some properties of the condensate in the linearized theory, we compare the expectation values of the current density in the state \(\omega _{\beta ,\psi }\) with (16), namely the current density of the free theory analyzed in Sect. 3.1. To this end, we recall the decomposition (20) and we get that
$$\begin{aligned} J_0=-i (:\dot{{\overline{\varphi }}}{\varphi }-{{\overline{\varphi }}}{\dot{\varphi }}:_H) = {\tilde{j}} -i \phi (\dot{{\overline{\psi }}}-{\dot{\psi }})+2\mu :|\phi +\psi |^2 :_H \end{aligned}$$
(31)
where now
$$\begin{aligned} {\tilde{j}}=-i \left( :\dot{{\overline{\psi }}}\psi -{\overline{\psi }}{{\dot{\psi }}} :_H\right) = 2\left( :{{\dot{\psi }}}_1\psi _2-\psi _1{{\dot{\psi }}}_2:_H\right) \end{aligned}$$
and H is the distinguished Hadamard function constructed in (57). We furthermore observe that, up to some choice of the renormalization freedom, \(:{\dot{\psi }}_1\psi _2-\psi _1{\dot{\psi }}_2:_H=:{\dot{\psi }}_1\psi _2-\psi _1{\dot{\psi }}_2:_{\omega _{\infty ,\psi }}\) and \(:|\psi |^2:_{H}=:|\psi |^2:_{\omega _{\infty ,\psi }}=:\psi _1^2:_{\omega _{\infty ,\psi }}+:\psi _2^2:_{\omega _{\infty ,\psi }}\). Hence,
$$\begin{aligned} \omega _{\beta ,\psi }(:|\psi |^2:_H)&=\frac{1}{(2\pi )^{3}} \int {d}^3{\mathbf {p}} \nonumber \\&\quad \left( \left( 1+ \frac{2\mu ^2 }{\delta \omega ^2}\right) \frac{1}{\omega _+} \frac{1}{e^{\beta \omega _+}-1}+ \left( 1-\frac{2\mu ^2}{\delta \omega ^2}\right) \frac{1}{\omega _-}\frac{1}{e^{\beta \omega _-}-1} \right) \end{aligned}$$
(32)
and
$$\begin{aligned} \omega _{\beta ,\psi }({\tilde{j}})&= \frac{4\mu }{(2\pi )^{3}} \int {d}^3{\mathbf {p}}\; \frac{1}{\delta \omega ^2} \left( \frac{\omega _-}{e^{\beta \omega _-}-1} - \frac{\omega _+}{e^{\beta \omega _+}-1} \right) . \end{aligned}$$
(33)
Hence,
$$\begin{aligned} \omega _{\beta ,\psi }(J_0) = \omega _{\beta ,\psi }({\tilde{j}}) +2\mu \; \omega _{\beta ,\psi }(:|\psi |^2:_H) +2\mu |\phi |^2. \end{aligned}$$
Notice that \(\omega _+^2>\omega _-^2\) and that \(\delta \omega ^2 \ge 2 \mu ^2\); hence, the integrand in \(\omega _{\beta ,\psi }(:|\psi |^2:_H) \) given in (32) is always positive and monotonically decreasing in \(\beta \). Similarly, for positive \(\mu \), the integrand in \(\omega _{\beta ,\psi }({\tilde{j}})\) given in (33) is also always positive and monotonically decreasing in \(\beta \). Finally, both expressions (32) and (33) are diverging for \(\beta \rightarrow 0\) and vanishes for \(\beta \rightarrow \infty \). Hence, similar to the discussion given in Sect. 3.1 we have that \(2\mu |\phi ^2|\) plays the role of the condensate charge density.
Consider now the case where \(M_i\) are given in (23) with \(\phi \) chosen to satisfy (21). In this case, \(\lambda |\phi |^2 = \mu ^2-m^2\), and thus, the linearized theory does not depend on \(\lambda \) while the background field scales as \(\lambda ^{-1}\). The charge density is thus dominated by the charge density of the background \(2\mu |\phi |^2\) thus confirming that the limit \(\lambda \rightarrow 0\) taken with fixed \(\mu \) and m is the relativistic analogous of the Gross–Pitaevskii limit discussed in introduction and at the end of Sect. 3.2.1.
Following closely the discussion given at the end of Sect. 3.1, we also see that in this case the critical charge density equals \(\rho _{cr}(\beta )\) given implicitly in (18). Finally, in the limit \(\lambda \rightarrow 0\) taken keeping the ratio \((\mu ^2-m^2)/\lambda \) finite we have that the states \(\omega _{\beta ,\psi }\) of the linearized theory discussed so far tend to \(\omega ^\pm _{\beta ,c}\) with \(c=\phi \).
Thermal Masses
Having analyzed the equilibrium state of the free theory on field observables \({\mathcal {A}}\), the next step in the construction of an equilibrium state for the interacting theory will be an application of the analysis given in [30] and summarized in Sect. 2.2, namely to use (10) starting with a quasi-free state whose two-point function is given in (27). However, we expect that the limit \(h\rightarrow 1\) cannot be directly taken because, as discussed above, if (21) holds, the mass \(M_2\) given in (23) vanishes; hence, for vanishing spatial momentum, \(\omega _-^2\) is also vanishing. This implies that various propagators of the linearized theory diverge for \(p\rightarrow 0\). Hence, in agreement with Goldstone theorem a massless mode is present in this case. This implies a slow decay in the connected n-point functions constructed with \(\omega _{\beta , \mu }\) given in (27).
In order to cure this problem, we use a different splitting of the Lagrangian into the free and interacting part. Actually, we add a virtual mass \(m_v^2\) to the linearized fields and we remove them in the interaction Lagrangian. More precisely, the Lagrangian of the free theory is now
$$\begin{aligned} {\mathcal {L}}_2' = {\mathcal {L}}_2 - \frac{m_v^2}{2}|\psi |^2 \end{aligned}$$
(34)
while the modified interaction Lagrangian is
$$\begin{aligned} {{\mathcal {L}}'}^I = {\mathcal {L}}^I + \frac{m_v^2}{2}|\psi |^2 = {\mathcal {L}}_3+ {\mathcal {L}}_4+ \frac{m_v^2}{2}|\psi |^2. \end{aligned}$$
The elements of the interacting algebra are now given in terms of two parameters \(\lambda \) and \(m_v\). More precisely, keeping \(\mu \) fixed, as in (21), they are formal power series in \(\sqrt{\lambda }\) with coefficients depending on \( m_v^2\), which can be understood as a partial resummation of the original perturbative expansion. The advantage of this new expansion is in the fact that the coefficients remain finite in the adiabatic limit, when they are evaluated in the state representing the condensate at finite temperature. We furthermore observe that the principle of perturbative agreement discussed below implies that the final theory does not depend on this extra parameter \(m_v\).
Let \(H_{\psi ,\beta }\) be the symmetrized two-point function of the \(\beta \)-KMS given in (27), we observe that if \(m_v\) is chosen to be sufficiently small, the interaction Lagrangian normal-ordered with respect to \(H_{\psi ,\beta }\) is again convex. To see this in detail, let T be a time-ordering operator such that \(TF = :F:_H\) where H is the distinguished Hadamard function constructed in (57). We have up to a choice of renormalization freedom [the lengthscale \(\xi \) in (57) chosen in such a way that \(:|\psi |^4:_H=:|\psi |^4:_{H_{\infty ,\psi }}\) where \(H_{\infty ,\psi }\) is the symmetrized two-point function of the vacuum obtained taking the limit \(\beta \rightarrow \infty \) in (27)]
$$\begin{aligned} T\left( \frac{1}{4}|\psi |^4\right)= & {} \frac{1}{4}:|\psi |^4:_{H_\beta }+ \frac{1}{2} (3 m_{\beta ,1}^2 + m_{\beta ,2}^2) :|\psi _1|^2:_{H_\beta } \nonumber \\&+ \frac{1}{2} (3 m_{\beta ,2}^2 + m_{\beta ,1}^2) :|\psi _2|^2:_{H_\beta } +C \end{aligned}$$
(35)
where C is a constant which can be discarded and the two thermal masses \(m_{\beta ,i}\) have been computed above in (29) and (30)
$$\begin{aligned} m_{\beta ,1}^2 \doteq W(\psi _1^2), \quad m_{\beta ,2}^2 \doteq W(\psi _2^2). \end{aligned}$$
Hence \(T(\frac{1}{4}|\psi |^4 -\frac{m_v^2}{2}|\psi |^2)\) remains convex, provided \(m_v^2< \lambda (3 m_{\beta ,1}^2 + m_{\beta ,2}^2) \) and \(m_v^2 < \lambda (3 m_{\beta ,2}^2 + m_{\beta ,1}^2)\). For this reason, it is expected that the stability properties of the theory are not altered adding the virtual masses \(m_v\) in the free theory.
Condensate and Perturbative Agreement
We need to check that the Wick monomials in the interaction Lagrangian originally constructed over the linearized theory \({\mathcal {L}}_2\) are not corrected because of the new splitting. In other words, we prove that the principle of perturbative agreement holds also when a condensate is present. Let us recall the form of the equation of motion for \({{\tilde{\psi }}} = (\psi _1,\psi _2)\) given in (24)
$$\begin{aligned} D{{\tilde{\psi }}} = 0, \quad D = (\square -M^2) {\mathbb {I}} - \delta M^2 {{\varvec{\sigma }}}_3 - i 2\mu \partial _0 {{\varvec{\sigma }}}_2 \end{aligned}$$
and consider the preferred Hadamard function \(H_{M^2,\delta M^2,\mu }\), with a lengthscale \(\xi \), associated with this operator constructed in “Appendix C.”
We prove now that the time ordering operator \(T_{M,\delta M,\mu }(F) = :F:_{H_{M^2,\delta M^2,\mu }}\) satisfies the principle of perturbative agreement. To this end, consider the \(2\times 2\) matrix \(\Psi =\{\psi _i\psi _j\}_{i,j\in \{1,2\}}\), following the discussion presented in “Appendix A” we want to prove that
$$\begin{aligned} \Delta \Psi = \gamma T_{0,0,\mu }\Psi - T_{M,\delta M,\mu } \Psi \end{aligned}$$
vanishes, where \(\gamma \) is the map which intertwines \(T_{0,0,\mu }(\psi _i(x)\psi _j(y))\) to \(T_{M,\delta M,\mu }(\psi _i(x)\psi _j(y))\). Formally, indicating with the subscript c the quantities referred to the condensate \((M,\delta M,\mu )\) and with the subscript 0 those referred to the vacuum (0, 0, m) we have to compute
$$\begin{aligned} \Delta \Psi = \lim _{y\rightarrow x}\left( (H_F^0(x,y))_{\text {ren}} - H_F^c(x,y)\right) \end{aligned}$$
where \(H_F^{c/0}\) are the time-ordered/Feynman propagator associated with the Hadamard functions \(H^{c/0}\). By power counting, we notice that all the contributions larger than order two in \((D^c-D)\) are removed from \(\Delta \Psi \) by renormalization. In order to check if there is a finite reminder after this renormalization, we analyze the form of the Hadamard singularity \(H_{M^2,\delta M^2,\mu }\) given in “Appendix C”; we remove the contributions of order lower than the third in \(x_i\) from \(H_{M^2+x_1,\delta M^2+x_2,\mu +x_3}\) before computing the coinciding point limit. Let us recall the form of some Hadamard coefficient given in “Appendix C.” From Eqs. (59) and (60), we have
$$\begin{aligned} U =\cos (\mu x^0){\mathbb {I}} - i{{\varvec{\sigma }}}_2 \sin (\mu x^0) \end{aligned}$$
and
$$\begin{aligned} V_0 = -\frac{1}{2} U \left( (\mu ^2+M^2) {\mathbb {I}} + \delta M^2 \left( \frac{\sin (2\mu x^0)}{2\mu x^0}{{\varvec{\sigma }}}_3 +\frac{\cos (2\mu x^0)-1}{2\mu x^0} {{\varvec{\sigma }}}_1\right) \right) . \end{aligned}$$
Hence, in \(H_{M^2+x_1,\delta M^2+x_2,\mu +x_3}\), \(U/\sigma \) does not depend on \(x_1\) and \(x_2\) and the contributions in \(x_3\) larger than the second order vanish in the coinciding point limits. Similarly, the contributions larger than second order vanish also in \(V\log (\frac{\sigma }{\xi ^2})\). As for the mass perturbations, we thus have that the Wick monomials \(\Psi ^n\) computed with respect to the Hadamard parametrix do not change under the action of the map which intertwines the time ordering constructed with two different sets of parameters \(M,\delta M, \mu \).
Construction of the Condensate, Cluster Estimates
To construct the state at finite temperature over the condensate, we follow the construction given in [30] and summarized in Sect. 3.2. In particular, at fixed spatial cutoff h, the equilibrium state at inverse temperature \(\beta \) can be constructed as in (11). Using the spatial translation invariance of the interacting hamiltonian and denoting by \(\tau _{t,{\mathbf {x}}}\) the \(*\)-automorphisms realizing a spacetime translation of step \((t,{\mathbf {x}})\), we have, for any element A of \({\mathcal {A}}_I({\Sigma _\epsilon })\),
$$\begin{aligned} \omega ^{\beta ,V}_h(A) \doteq&\sum _{n}\int _{0\le u_1\le \cdots u_n\le \beta } d u_1\ldots d u_n\int _{{{\mathbb {R}}}^{3n}}\mathrm{d}^3{\mathbf {x}}_1\ldots \mathrm{d}^3{\mathbf {x}}_n h({\mathbf {x}}_1)\ldots h({\mathbf {x}}_n)\nonumber \\&\omega ^\beta _T\left( A;\tau _{iu_{1},{\mathbf {x}}_{1}}({\mathcal {H}}_I(0));\ldots ;\tau _{iu_n;{\mathbf {x}}_{n}}({\mathcal {H}}_I(0))\right) \end{aligned}$$
(36)
where \(\omega ^\beta _T\) denotes the truncated n-point function of the state \(\omega _{\beta ,\psi }\). Hence, in order to discuss the limit \(h\rightarrow 1\) we need to control the decay for large spatial directions of the truncated n-point functions. We have actually the following theorem
Theorem 3.1
(Cluster expansions). Consider \(A_i\in {\mathcal {A}}({\mathcal {O}})\) where \({\mathcal {O}}\subset B_R\) the open ball of radius R centered at the origin of the Minkowski spacetime and
$$\begin{aligned} F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n)\doteq \omega _T(A_0 ; \tau _{iu_1,{\mathbf {x}}_1}(A_1);\ldots ;\tau _{iu_n,{\mathbf {x}}_1}(A_n)). \end{aligned}$$
There exists a constant C such that
$$\begin{aligned} |F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n)| \le C e^{-\frac{m}{\sqrt{n}}r},\quad r=\sqrt{\sum _i |{\mathbf {x}}_i|^2} \end{aligned}$$
for \(r>4cR\), uniformly in u for \(0< u_1< \cdots< u_n <\beta \) with \(\beta -u_n \ge \frac{\beta }{n+1}\).
Proof
Thanks to the decay property for large spatial separations of the locally smeared two-point functions given in Proposition D.2, the proof of this theorem can be done in a similar way as the proof of Theorem 3 of [30]. We recall here the main steps of that proof, and we adapt them to the case studied here.
The truncated n-point functions can be written as a sum over all possible connected graphs joining n points. We shall denote the set of connected graphs, without tadpoles, with \(n+1\) vertices \(V=\{0,\ldots , n\}\) as \({\mathcal {G}}_{n+1}^c\). Furthermore, for any \(G\in {\mathcal {G}}_{n+1}\), E(G) denotes the set of edges of G. For any \(l\in E(G)\), s(l) and r(l) denote the source and the range of l. A graph G is considered to be in \({\mathcal {G}}_{n+1}\) only if, for every l, \(s(l)<r(l)\). Finally, \(l_{ij}(G)\) is the number of lines connecting \(i,j\in V\). With these definitions,
$$\begin{aligned} F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n)\doteq \sum _{G\in {\mathcal {G}}^c_{n+1}} \frac{1}{{\text {sym}(G)}} F_G(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n) \end{aligned}$$
where \(\text {sym}(G)=\prod _{i<j} l_{ij}(G)!\) is a numerical factor and
$$\begin{aligned}&F_G(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n) \\&\quad \doteq \left( \prod _{0\le i<j\le n} \Gamma ^{ij} \right) \left. (A_0 \otimes \tau _{iu_1,{\mathbf {x}}_1}(A_1)\otimes \cdots \tau _{iu_n,{\mathbf {x}}_1}(A_n))\right| _{(\psi ^0,\ldots ,\psi ^n) = 0}. \end{aligned}$$
Furthermore,
$$\begin{aligned} \Gamma ^{ij} = \int \mathrm{d}^4x \mathrm{d}^4y \;{\mathcal {K}}(x-y) \frac{\delta }{\delta \psi ^i(x)} \otimes \frac{\delta }{\delta \psi ^j(y)} \end{aligned}$$
with the integral kernel \({\mathcal {K}}(x-y) = \omega _{\beta ,\psi }(\psi (x)\psi (y))\), given in terms of the thermal two-point function of the background theory (27). Furthermore, \(\psi ^j=\psi ^j_1+i\psi ^j_2\) is the field configuration in the jth factor of the tensor product and the functional derivative \(\frac{\delta }{\delta \psi ^j}\) acts on the jth factor of the tensor product. We have that
$$\begin{aligned} F_G(U,{\mathbf {X}}) \doteq \int dP \left( \prod _{l\in E(G)} e^{p_l^0(u_{s(l)}-u_{r(l)})} e^{i {\mathbf {p}}_l ({\mathbf {x}}_{s(l)}-{\mathbf {x}}_{s(l)})} \hat{{\mathcal {K}}}(p_l) \right) {{\hat{\Psi }}}(-P,P) \end{aligned}$$
where \(U=(u_0,\ldots , u_n)\), \({\mathbf {X}}=({\mathbf {x}}_0, \ldots ,{\mathbf {x}}_n)\) with \(u_0=0\) and \({\mathbf {x}}_0=0\), while \(P=(p_1,\ldots ,p_{|E(G)|})\) and
$$\begin{aligned} \Psi (Z,Y) = \left. \left( \prod _{l\in E(G)} \frac{\delta }{\delta \psi ^{s(l)}(z_l)}\otimes \frac{\delta }{\delta \psi ^{r(l)}(y_l)} \right) (A_0 \otimes A_1\otimes \cdots A_n)\right| _{(\psi ^0,\ldots ,\psi ^n) = 0}. \end{aligned}$$
We observe that
$$\begin{aligned} \hat{{\mathcal {K}}}(p) = \left( \lambda _+(p) + \lambda _-(p)\right) (-\widehat{{\overline{D}}}) \end{aligned}$$
where \(\lambda _+\) and \(\lambda _-\) are the positive and negative frequency part
$$\begin{aligned} \lambda _+(p)&= \frac{1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0-\omega _+)}{2\omega _+}- \frac{\delta (p_0-\omega _-)}{2\omega _-}\right) \frac{1}{1-e^{-\beta p_0}}\\ \lambda _-(p)&= -\frac{1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0+\omega _+)}{2\omega _+}- \frac{\delta (p_0+\omega _-)}{2\omega _-}\right) \frac{1}{1-e^{-\beta p_0}}. \end{aligned}$$
Hence, separating the positive and negative contributions in \(F_G\) we get
$$\begin{aligned} F_G(U,{\mathbf {X}})&= \sum _{P_2(E(G))}\int d{\mathbf {P}} \\&\quad \left( \prod _{l_+\in E_+(G)} e^{p_{l_+}^0(u_{s(l_+)}-u_{r(l_+)})} e^{i {\mathbf {p}}_{l_+} ({\mathbf {x}}_{s(l_+)}-{\mathbf {x}}_{r(l_+)})} \lambda _+(p_{l_+})(-\hat{{\overline{D}}}(p_{l_+}))\right) \\&\quad \cdot \left( \prod _{l_-\in E_-(G)} e^{p_{l_-}^0(u_{s(l_-)}-u_{r(l_-)})} e^{i {\mathbf {p}}_{l_-} ({\mathbf {x}}_{s(l_-)}-{\mathbf {x}}_{s(l_-)})} \lambda _-(p_{l_-})(-\hat{{\overline{D}}}(p_{l_-}))\right) \\&\quad {\hat{\Psi }}(-P,P) \end{aligned}$$
where the sum is taken over all possible partitions of E(G) in up to two sets \(\{E_+(G),E_-(G)\}\in P_2(E(G))\). We proceed now splitting again these contributions over the two possible frequencies \(\omega _{\pm }\). Hence, denoting by
$$\begin{aligned} \lambda _{++}(p)&\doteq \frac{1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0-\omega _+)}{2\omega _+}\right) \frac{1}{1-e^{-\beta p_0}}\\ \lambda _{+-}(p)&\doteq \frac{-1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0-\omega _-)}{2\omega _-}\right) \frac{1}{1-e^{-\beta p_0}}\\ \lambda _{-+}(p)&\doteq -\frac{1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0+\omega _+)}{2\omega _+}\right) \frac{1}{1-e^{-\beta p_0}}\\ \lambda _{--}(p)&\doteq \frac{1}{\omega _+^2-\omega _-^2} \left( \frac{\delta (p_0+\omega _-)}{2\omega _-}\right) \frac{1}{1-e^{-\beta p_0}} \end{aligned}$$
we have
$$\begin{aligned} F_G(U,{\mathbf {X}})&= \sum _{P_2(E(G))} \sum _{P_2(E_+(G))}\sum _{P_2(E_-(G))} \\&\quad \int d{P} \left( {\mathcal {Q}}_{++} \cdot {\mathcal {Q}}_{+-} \cdot {\mathcal {Q}}_{-+}\cdot {\mathcal {Q}}_{--}\right) {\hat{\Psi }}(-P,P) \end{aligned}$$
where
$$\begin{aligned}&{\mathcal {Q}}_{\sigma \sigma '} \doteq \left( \prod _{l\in E_{\sigma \sigma '}(G)} e^{p_{l}^0(u_{s(l)}-u_{r(l)})} e^{i {\mathbf {p}}_{l} ({\mathbf {x}}_{s(l)}-{\mathbf {x}}_{r(l)})} \lambda _{\sigma \sigma '}(p_{l})(-\hat{{\overline{D}}}(p_{l}))\right) \\&\quad \sigma ,\sigma '\in \{+,-\}. \end{aligned}$$
The function \({\hat{\Psi }}\) is an entirely analytic function which grows at most polynomially in every direction. We might thus integrate over all possible \(p_0\) to get
$$\begin{aligned} F_G(U,{\mathbf {X}}) \doteq \sum _{P_2(E(G))} \sum _{P_2(E_+(G))}\sum _{P_2(E_-(G))}\int d{\mathbf {P}} \left( \tilde{{\mathcal {Q}}}_{++} \cdot \tilde{{\mathcal {Q}}}_{+-} \cdot \tilde{{\mathcal {Q}}}_{-+}\cdot \tilde{{\mathcal {Q}}}_{--}\right) \Phi ({\mathbf {P}}) \end{aligned}$$
where now
$$\begin{aligned} \Phi ({\mathbf {P}}) \doteq \left. {\hat{\Psi }}(-P,P)\right| _{p_0^{l_{\sigma \sigma '}} = \sigma \omega _{\sigma '}({\mathbf {p}}_l)} \end{aligned}$$
and
$$\begin{aligned}&\tilde{{\mathcal {Q}}}_{\sigma \sigma '} \\&\quad = (\sigma 1) (\sigma ' 1)\left( \prod _{l\in E_{\sigma \sigma '}(G)} \frac{e^{ \sigma \omega _{\sigma '}({\mathbf {p}}_{l})(u_{s(l)}-u_{r(l)})} e^{i {\mathbf {p}}_{l} ({\mathbf {x}}_{s(l)}-{\mathbf {x}}_{r(l)})} }{(\omega _+^2-\omega _-^2)2\omega _{\sigma '}} \frac{(-\hat{{\overline{D}}}(\sigma \omega _{\sigma '},{\mathbf {p}}_{l}))}{1-e^{-\sigma \beta \omega _{\sigma '}}} \right) \\&\quad = (\sigma ' 1) \left( \prod _{l\in E_{\sigma \sigma '}(G)} \frac{e^{-\left( \frac{(1-\sigma )}{2}\beta + \sigma (u_{r(l)}-u_{s(l)})\right) \omega _{\sigma '}} e^{i {\mathbf {p}}_{l} ({\mathbf {x}}_{s(l)}-{\mathbf {x}}_{r(l)})} }{(\omega _+^2-\omega _-^2)2\omega _{\sigma '}} \frac{(-\hat{{\overline{D}}}(\sigma \omega _{\sigma '},{\mathbf {p}}_{l}))}{1-e^{-\beta \omega _{\sigma '}}} \right) , \\&\qquad \sigma ,\sigma '\in \{+,-\}. \end{aligned}$$
Since, by hypothesis,
$$\begin{aligned} u_{i+1}>u_i,\quad \beta -u_{n} \ge \frac{\beta }{n+1} \end{aligned}$$
and \(r(l)>s(l)\), we have that
$$\begin{aligned} e^{-\left( \beta - (u_{r(l)}-u_{s(l)})\right) \omega _{\sigma '}} \le e^{-\frac{n}{n+1}\beta \omega _{\sigma '}}. \end{aligned}$$
Hence,
$$\begin{aligned} {\tilde{\Phi }}({\mathbf {P}})\doteq \tilde{{\mathcal {Q}}}_{-+}\tilde{{\mathcal {Q}}}_{--}\Phi ({\mathbf {P}}) \end{aligned}$$
is rapidly decreasing, in every direction, because \(F_G\) is a microcausal functional and \(\Phi ({\mathbf {P}})\) is the restriction on a particular subdomain of \({\hat{\Psi }}(-P,P)\) which is an entire analytic function which grows at most polynomially. Hence, the negative frequencies are exponentially suppressed, and if directions containing only positive frequencies are considered, they are also rapidly decreasing by Proposition D.1. The integral over \({\mathbf {P}}\) can now be taken and we may apply Proposition D.2 to estimate the decay of the result of that integral. We obtain
$$\begin{aligned} |F_G(U,{\mathbf {X}})| \le c' \prod _{l\in E(G)} e^{- M_- \sqrt{|{\mathbf {x}}_{r(l)}-{\mathbf {x}}_{s(l)}|^2}}\le c' e^{- \frac{M_-}{\sqrt{n}} \sqrt{\sum _{i=1}^n |{\mathbf {x}}_{i}|^2}} \end{aligned}$$
where the constant \(c'\) does not depend on \(u_i\). In the last inequality, we used the fact that G is a connected graph, and thus, every \(x_{i}\) can be reached from the origin \(({\mathbf {x}}_0=0)\). Hence,
$$\begin{aligned} \sum _{l\in E(G)} \sqrt{{|{\mathbf {x}}_{r(l)}-{\mathbf {x}}_{s(l)}|^2}} \ge \text {max}_i{\sqrt{|{\mathbf {x}}_{i}|^2}} \ge \sqrt{\frac{1}{n}\sum _{i=1}^n |{\mathbf {x}}_{i}|^2} \end{aligned}$$
thus concluding the proof. \(\square \)
Theorem 3.2
Let \(A \in {\mathcal {A}}_I({\mathcal {O}})\) where \({\mathcal {O}}\subset \Sigma _\epsilon \), the adiabatic limit
$$\begin{aligned} \begin{gathered} \omega ^{\beta ,V}(A)= \lim _{h\rightarrow 1}\sum _{n}\int _{0\le u_1\le \cdots u_n\le \beta } \mathrm{d} u_1\ldots \mathrm{d} u_n\int _{{{\mathbb {R}}}^{3n}}\mathrm{d}^3{\mathbf {x}}_1\ldots \mathrm{d}^3{\mathbf {x}}_n h({\mathbf {x}}_1)\ldots h({\mathbf {x}}_n) \\ \omega _T^\beta \left( A;\tau _{iu_{1},{\mathbf {x}}_{1}}(K);\ldots ; \tau _{iu_n;{\mathbf {x}}_{n}}(K)\right) , \end{gathered} \end{aligned}$$
where \(K\doteq \lim _{h\rightarrow 1} {\mathcal {H}}_I(0)\), exists in the sense of perturbation theory and defines an equilibrium state for the interacting theory.
Proof
Since \({\mathcal {O}}\) is of compact support, it exists and \(R>0\) such that the open ball \(B_R\) centered in the origin of Minkowski spacetime contains \({\mathcal {O}}\), namely \({\mathcal {O}}\subset B_R\). Furthermore, thanks to the temporal cutoff \(\chi \), and in view of the causal properties of the Bogoliubov map, \(K = \lim _{h\rightarrow 1} {\mathcal {H}}_I\) is supported in \(B_R\) for a sufficiently large R.
Consider the nth order contribution in the sum defining \(\omega ^{\beta ,V}_h\) given in (36)
$$\begin{aligned} \begin{aligned} \Omega _{n,h}(A)&\doteq \int _{0\le u_1\le \cdots u_n\le \beta } d u_1\ldots d u_n \\&\quad \int _{{{\mathbb {R}}}^{3n}}\mathrm{d}^3{\mathbf {x}}_1\ldots \mathrm{d}^3{\mathbf {x}}_n h({\mathbf {x}}_1)\ldots h({\mathbf {x}}_n) F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n) \end{aligned} \end{aligned}$$
(37)
where
$$\begin{aligned} F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n)\doteq \omega ^\beta _T\left( A;\tau _{iu_{1},{\mathbf {x}}_{1}}(K);\ldots ;\tau _{iu_n;{\mathbf {x}}_{n}}(K))\right) , \quad A\in {\mathcal {A}}_I({\mathcal {O}}). \end{aligned}$$
To apply the results of Theorem 3.1, we observe that if R is sufficiently large \({\mathcal {H}}_I(0) \in {\mathcal {A}}_I(B_R)\), furthermore, the form of the integration domain of the u variables as given in (37) is such that
$$\begin{aligned} 0 \le u_1 \le \cdots \le u_n \le \beta . \end{aligned}$$
(38)
Using the KMS condition, we might restrict attention to the case where \(\beta - {u_{n}} \ge \frac{\beta }{n+1}\). In fact, if this is not the case, there must exist an m for which \(u_{m}-u_{m-1} \ge \frac{\beta }{n+1}\). Actually, for \(A_i\in {\mathcal {A}}_I({\mathcal {O}})\) by the KMS condition we have that
$$\begin{aligned}&\omega _T^\beta (\tau _{iu_0}(A_0) ; \tau _{iu_1,{{\mathbf {x}}}_1}(A_1); \ldots ; \tau _{iu_n,{{\mathbf {x}}}_n}(A_n))\\&\quad =\omega _T^{\beta }(\tau _{iu_{m},{\mathbf {x}}_{m} } (A_{m}) ; \ldots ; \tau _{iu_n,{{\mathbf {x}}}_n}(A_n) \\&\qquad \otimes \tau _{i\beta +iu_0}(A_0) ; \ldots ; \tau _{i\beta +iu_{m-1},{{\mathbf {x}}}_{m-1}}(A_{m-1}) ); \end{aligned}$$
hence, we might now consider
$$\begin{aligned}&F'(v_1,{\mathbf {y}}_1;\ldots ;v_n,{\mathbf {y}}_n) \\&\quad \doteq \omega ^\beta _T(K ; \tau _{iv_1,{\mathbf {y}}_1}(K);\ldots ; \tau _{iv_{n-m},{\mathbf {y}}_{n-m}}(K); \tau _{iv_{n-m+1},{\mathbf {y}}_{n-m+1}}(A_0); \\&\qquad \tau _{iv_{n-m+2},{\mathbf {y}}_{n-m+2}}(K);\ldots \tau _{iv_n,{\mathbf {y}}_v}(K)) \end{aligned}$$
in place of F. In fact, the previous equality obtained with the KMS condition together with translation invariance of the state implies that
$$\begin{aligned} F(u_1,{\mathbf {x}}_1;\ldots ;u_n,{\mathbf {x}}_n) = F'(v_1,{\mathbf {y}}_1;\ldots ;v_n,{\mathbf {y}}_n) \end{aligned}$$
if
$$\begin{aligned} ({\mathbf {y}}_1,\ldots ,{\mathbf {y}}_n) = ({\mathbf {x}}_{m+1}-{\mathbf {x}}_m,\ldots , {\mathbf {x}}_{n}-{\mathbf {x}}_m, -{\mathbf {x}}_m,{\mathbf {x}}_1-{\mathbf {x}}_m,\ldots ,{\mathbf {x}}_{m-1}-{\mathbf {x}}_m) \end{aligned}$$
and
$$\begin{aligned} (v_1,\ldots ,v_n) = (u_{m+1}-u_m,\ldots , u_{n}-u_m,\beta -u_m,\beta +u_1-u_m,\ldots ,). \end{aligned}$$
The arguments of the function \(F'\) have the desired property, actually \(\beta - v_n= u_m - u_{m-1} \ge \beta /(n+1)\). We might thus use \(F'\) in place of F, because the integration over the u variables is over a compact set and because the points where \(u_i=u_j\) for some \(i\ne j\) form a zero measure set. Hence, Theorem 3.1 implies that the integral over \({\mathbf {x}}_i\) can be taken for all i to conclude the proof. \(\square \)