- Email: [email protected]
Initial States and Time Evolution in Nonequilibrium Quantum Field Theory
Nuclear Physics A 785 (2007) 218c–221c
Initial States and Time Evolution in Nonequilibrium Quantum Field Theory ´ R. F. Alvarez-Estrada
a ∗†
a
Departamento de Fisica Teorica I, Facultad de Ciencias Fisicas Universidad Complutense, 28040, Madrid, Spain We consider a statistical system, described by a relativistic neutral scalar massive quantum field with t-dependent mass parameter. By assumption, the system is represented at time t0 by a nonequilibrium initial gaussian density operator ρin,g with spatial inhomogeneities ( potentially interesting for the cosmological inflaton and, qualitatively, for relativistic plasmas). In order to describe the out-of-equilibrium time evolution for t > t0 , as determined by ρin,g , one requires the corresponding nonequilibrium free correlators, as a necessary step before introducing interactions. We report, in outline, the space-time structures of those free nonequilibrium correlators. 1. INTRODUCTION Let us consider a large (statistical) quantum system, the dynamics of which is accounted for by Relativistic Quantum Field Theory (RQFT). By assumption, the system is not at thermal equilibrium at the initial time t = t0 . Two central questions, to be addressed here, will be some specific characterization of its initial non-equilibrium state ρin at t = t0 , and the subsequent evolution for t > t0 , as determined by ρin . In order to clarify the issue, let us consider the inflationary evolution of the Early Universe, modelled by means of the inflaton field, regarded as a relativistic neutral scalar one. For suitable early stages in that evolution, it appears to be adequate to treat the inflaton field (with its excitations) not only as a classical one but as a quantized field [1,2]. Then, one could characterize, at an early t0 , some initial non-equilibrium ρin of that inflaton quantum field, and study how the details of ρin are erased by inflation during the subsequent evolution for t > t0 [2]. Other examples could be borrowed from nonequilibrium fully relativistic plasmas in either QED or QCD ( with additional difficulties, due to half-integral spin and gauge degrees of freedom). For accounts of nonequilibrium statistical RQFT, see [3–5]. Section 2 will summarize the essentials of the model in RQFT (namely, a relativistic neutral scalar massive quantum field with t-dependent mass parameter), present one possible class of initial nonequilibrium gaussian density operators ρin = ρin,g and discuss ∗
e-mail: [email protected] The author acknowledges the financial support of CICYT (Project FPA2004-02602), Spain. He is grateful to Mr. D. Fernandez-Fraile and to Drs. A. Gomez Nicola and A. Lopez Maroto for discussions and information. †
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.11.062
R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
219c
a few aspects of the subsequent time evolution. Section 3 will report the results of our computations for the free non-equilibrium correlators for t > t0 , determined by ρin,g . 2. GAUSSIAN INITIAL STATE AND TIME EVOLUTION FOR t > t0 In (1 + d)-dimensional Minkowski space, let x = (t, x), where t denotes ordinary (real) time and x = (x1 , .., xd ) are the spatial coordinates. d = 1, 2, 3 is the spatial dimension. We shall consider a large statistical system, described by an (unrenormalized) relativistic hermitean scalar massive quantum field operator φ(t, x). This field has (unrenormalized) real mass parameter m = m(t) ( t-dependent, in general) and its dynamics is described by RQFT in (1 + d)-dimensional space. The (unrenormalized) quantum hamiltonian operator is H = dd xh(Π; φ). The operator h(Π; φ) is the standard (unrenormalized) quantum hamiltonian density: h(Π; φ) = N[2−1 [Π2 + di=1 (∂φ/∂xi )2 + m2 φ2 ] + V (φ)] = h0 (Π; φ) + V (φ), where Π = ∂φ/∂t, N denotes normal product of operators, V = V (φ) is the selfcoupling interaction hamiltonian and h0 (Π; φ) is the free hamiltonian density. The free hamiltonian is H0 = dd xh0 (Π; φ). Regarding the connection with quantum single inflaton models [1,2], we remind that φ is the associated quantum field in the standard conformal formalism and that t is interpreted as the conformal time. Then, in the conformal formalism, the potential V is a4 Vc (a−1 φ), where a = a(t) is the universal scale factor. Let us assume that Vc (φ) = V0 + 2−1 m20 φ2 , V0 and m20 being real constants. Then, the actual m(t)2 equals −a−1 d2 a/dt2 + m20 a2 . Such a model can be related to standard formulations for inflation by transforming to the comoving formalism, neglecting spatial variations, assuming the validity of Friedmann’s equation and imposing the slow-roll approximation [1]. Let the system be represented, at t0 , by some statistical mixture of states, given by the initial quantum density operator ρin . Gaussian (g) structures for ρin are rather popular: see [6], [7], [8] and [5], among others. Generically speaking, gaussian ρin ’s exclude, in the initial state, interactions of the same kind as those acting for t > t0 . Here, we shall also treat gaussian initial nonequilibrium mixed states. Let | φi >, i = 1, 2, be eigenstates of the quantized field operator φ(t0 , x), at time t0 , with real field eigenvalues φi (t0 , x): φ(t0 , x) | φi >= φi (t0 , x) | φi >. We shall denote φi (t0 , x) also as φi (x). Thus, we assume that ρin = ρin,g , with matrix elements: 1 d d < φ2 | ρin,g | φ1 >= exp[− d x d x [(φ2 (x)φ2 (x ) + φ1 (x)φ1 (x ))Kd (x, x ) 2 (1) −(φ1 (x)φ2 (x ) + φ2 (x)φ1 (x ))Ko (x, x )]] Kd (x, x ) and Ko (x, x ) are suitable real kernels (symmetric under x → x and x → x). We shall decompose the real symmetric kernel Kd (x, x ) − Ko (x, x )) into its eigenfunctions (fγ,g (x)) as: Kd (x, x ) − Ko (x, x )) =
fγ,0,g γ
2
fγ,g (x)fγ,g (x )∗
with real eigenvalues fγ,0,g (> 0, by assumption). The functions fγ,g are orthonormalized and constitute a complete set:
dd xfγ,g (x)[fγ ,g (x)]∗ = δγ,γ ,
γ
fγ,g (x)[fγ,g (x )]∗ = δ (d) (x − x )
(2)
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R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
γ denotes suitable integrations (continuous spectrum) and summations (discrete spectrum) and δ (d) is the d-dimensinal Dirac’s delta function. We also introduce:
fγ,g,2(k) =
dd xfγ,g (x) exp(ikx)
(3)
The time evolution of the system for t > t0 , as determined by the initial condition ρin,g , is given by the quantum density operator ρg = ρg (t, t0 ). The latter fulfills both the equation i∂ρg /∂t = Hρg − ρg H and ρg (t0 , t0 ) = ρin,g . Then, one has the formal solution: ρg (t, t0 ) = U(t, to )ρin,g U(t, to )+ . where U(t, t0 ) is the full evolution operator. The superscript (+) denotes the adjoint. Four two-point correlators are considered in Equilibrium Thermal Field Theory in the real-time formalism [9]. The four free ( V = 0) two-point correlators enable to formulate perturbation theory for systems at equilibrium [9]. The same holds for nonequilibrium systems [3]. One of those free two-point nonequilibrium correlators is: (0)
−Δ+,+,g (t2 , x2 ; t1 , x1 ) = T r[U0 (t2 , t0 )+ φ(t0 , x2 )U0 (t2 , t0 )U0 (t1 , t0 )+ φ+ (t0 , x1 ) ×U0 (t1 , t0 )ρin,g ] ≡< φ+ (t2 , x2 )φ+ (t1 , x1 ) >g
(4)
T r denotes the usual trace operation. φ(t0 , xj ), j = 1, 2, is the quantum field operator at the initial time t0 . U0 (t, t ) is the free evolution operator, determined by H0 . Similar expressions, with the adequate changes, define the other three free two(0) (0) point nonequilibrium correlators, namely, Δ−,−,g (t2 , x2 ; t1 , x1 ), Δ+,−,g (t2 , x2 ; t1 , x1 ) and (0) Δ−,+,g (t2 , x2 ; t1 , x1 ). For V = 0, similar equations yield the four full two-point nonequilibrium correlators, in terms of the full evolution operator U(t, t ). (0)
3. THE FREE NONEQUILIBRIUM CORRELATORS Δα,α ,g We give below the space-time structures of the four free nonequilibrium correlators (0) Δα,α ,g (α, α = ±): (0)
(0)
Δ(0) α,α,g (x; x ) = Δ0,g (x; x ) + iαΔ1,g (x; x ) (0) Δα,−α,g (x; x )
=
(0) −Δ0,g (x; x )
+
(5)
(0) iαΔ2,g (x; x )
(6)
The contributions in the right-hand-sides of those equations are ( k = |k|): (0)
(0)
(0)
Δ0,g (x; x ) = Δ0,1,g (x; x ) + Δ0,2,g (x; x )
(7) d
1 d k d k f1,k (t) × f1,k (t ) dd x dd x exp(ik(x − x)) 2 (2π)d (2π)d (8) × exp(ik (x − x ))(Kd (x , x ) + Ko (x , x )) d 1 d k (0) exp(−ikx)fγ,g,2 (k) × [f2,k (t) + f1,k (t)f3,k )] Δ0,2,g (x; x ) = − (2π)d γ 4fγ,0,g (0)
Δ0,1,g (x; x ) = −
×
d
dd k exp(−ik x )f−γ,g,2 (k )[f2,k (t ) + f1,k (t )f3,k ] (9) (2π)d f2,k (t)f1,k (t ) + f2,k (t )f1,k (t) dd k (0) + f Δ1,g (x; x ) = exp [ik(x − x )] − (t, t ) (10) k (2π)d 4 dd k f2,k (t)f1,k (t ) − f2,k (t )f1,k (t) (0) exp [ik(x − x )] × (11) Δ2,g (x; x ) = (2π)d 4
R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
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Let the functions fk,± (t), associated to m = m(t), be defined as the solutions of the following homogeneous second-order differential equation, in t0 ≤ t ≤ T : [
d2 + k2 + m(t)2 ]fk,± (t) = 0 dt2
(12)
with the following boundary conditions: fk,+ (t0 ) = 0, fk,+ (T ) = 1 and fk,− (t0 ) = 1, fk,− (T ) = 0. We have: fk,+ (t)[Jf−1 (dfk,− /dt)(T )] + fk,+ (t) (dfk,− /dt)(T ) − (dfk,+ /dt)(t0 ) dfk,+ fk,− (t) f2,k (t) = −( )(t0 ) + fk,− (t) dt Jf θ(t − t)fk,− (t )fk,+ (t) + θ(t − t )fk,− (t)fk,+ (t ) fk (t, t ) = Jf dfk,−(t) dfk,+(t) − fk,− (t) Jf = fk,+ (t) dt dt f1,k (t) =
(13) (14)
(15)
and f3,k = 2(dfk,−/dt)(t0 ) ((dfk,± /dt)(τ ) ≡ dfk,± (t)/dt, at t = τ ). T -dependences cancel (0) (0) out in Δα,α ,g . For t-independent m, the Δα,α ,g ’s simplify, as fk,+ (t) = sin ω(k)(t − t0 )/ sin ω(k)(T − t0 ) and fk,− (t) = sin ω(k)(T − t)/ sin ω(k)(T − t0 ) (ω(k) = [k2 + m2 ]1/2 ). The above free nonequilibrium correlators are consistent with general requirements: compare with [3]. They can be employed in perturbation theory, for the case in which interactions are included (say, V (φ) = 0). ρin,g and the free nonequilibrium correlators for m = m(t) could be interesting in connection with the single quantum inflaton. For t-independent mass, they can also be employed, qualitatively, for fully relativistic plasmas (leaving aside half-integral spin, gauge degrees of freedom,...,). For other studies of nonequilibrium RQFT, see, for instance, [10] and, for much wider referencing, [5]. REFERENCES 1. A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, Cambridge, 2000. 2. D. Boyanovsky, F. J. Cao and H. J. de Vega, Nucl. Phys. B 632 (2002) 121. 3. K.C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. 4. N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145 (1987) 141. 5. J. Berges, Introduction to Nonequilibrium Quantum Field Theory, arXiv:hepph/0409233 v1 20 Sep. (2004). 6. R. Jackiw, Physica A 158 (1989) 269. 7. J. Baacke, K. Heitmann and C. Patzold, Phys. Rev. D 58 (1998) 125013. 8. P. Jizba and E.S. Tututi, Phys. Rev. D 60 (1999) 105013. 9. M. Le Bellac, Thermal Field Theory, Cambridge University Press, Cambridge, 1996. 10. G. Aarts, J. Smit, Phys. Rev. D 61 (2000) 025002.
Initial States and Time Evolution in Nonequilibrium Quantum Field Theory ´ R. F. Alvarez-Estrada
a ∗†
a
Departamento de Fisica Teorica I, Facultad de Ciencias Fisicas Universidad Complutense, 28040, Madrid, Spain We consider a statistical system, described by a relativistic neutral scalar massive quantum field with t-dependent mass parameter. By assumption, the system is represented at time t0 by a nonequilibrium initial gaussian density operator ρin,g with spatial inhomogeneities ( potentially interesting for the cosmological inflaton and, qualitatively, for relativistic plasmas). In order to describe the out-of-equilibrium time evolution for t > t0 , as determined by ρin,g , one requires the corresponding nonequilibrium free correlators, as a necessary step before introducing interactions. We report, in outline, the space-time structures of those free nonequilibrium correlators. 1. INTRODUCTION Let us consider a large (statistical) quantum system, the dynamics of which is accounted for by Relativistic Quantum Field Theory (RQFT). By assumption, the system is not at thermal equilibrium at the initial time t = t0 . Two central questions, to be addressed here, will be some specific characterization of its initial non-equilibrium state ρin at t = t0 , and the subsequent evolution for t > t0 , as determined by ρin . In order to clarify the issue, let us consider the inflationary evolution of the Early Universe, modelled by means of the inflaton field, regarded as a relativistic neutral scalar one. For suitable early stages in that evolution, it appears to be adequate to treat the inflaton field (with its excitations) not only as a classical one but as a quantized field [1,2]. Then, one could characterize, at an early t0 , some initial non-equilibrium ρin of that inflaton quantum field, and study how the details of ρin are erased by inflation during the subsequent evolution for t > t0 [2]. Other examples could be borrowed from nonequilibrium fully relativistic plasmas in either QED or QCD ( with additional difficulties, due to half-integral spin and gauge degrees of freedom). For accounts of nonequilibrium statistical RQFT, see [3–5]. Section 2 will summarize the essentials of the model in RQFT (namely, a relativistic neutral scalar massive quantum field with t-dependent mass parameter), present one possible class of initial nonequilibrium gaussian density operators ρin = ρin,g and discuss ∗
e-mail: [email protected] The author acknowledges the financial support of CICYT (Project FPA2004-02602), Spain. He is grateful to Mr. D. Fernandez-Fraile and to Drs. A. Gomez Nicola and A. Lopez Maroto for discussions and information. †
0375-9474/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.11.062
R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
219c
a few aspects of the subsequent time evolution. Section 3 will report the results of our computations for the free non-equilibrium correlators for t > t0 , determined by ρin,g . 2. GAUSSIAN INITIAL STATE AND TIME EVOLUTION FOR t > t0 In (1 + d)-dimensional Minkowski space, let x = (t, x), where t denotes ordinary (real) time and x = (x1 , .., xd ) are the spatial coordinates. d = 1, 2, 3 is the spatial dimension. We shall consider a large statistical system, described by an (unrenormalized) relativistic hermitean scalar massive quantum field operator φ(t, x). This field has (unrenormalized) real mass parameter m = m(t) ( t-dependent, in general) and its dynamics is described by RQFT in (1 + d)-dimensional space. The (unrenormalized) quantum hamiltonian operator is H = dd xh(Π; φ). The operator h(Π; φ) is the standard (unrenormalized) quantum hamiltonian density: h(Π; φ) = N[2−1 [Π2 + di=1 (∂φ/∂xi )2 + m2 φ2 ] + V (φ)] = h0 (Π; φ) + V (φ), where Π = ∂φ/∂t, N denotes normal product of operators, V = V (φ) is the selfcoupling interaction hamiltonian and h0 (Π; φ) is the free hamiltonian density. The free hamiltonian is H0 = dd xh0 (Π; φ). Regarding the connection with quantum single inflaton models [1,2], we remind that φ is the associated quantum field in the standard conformal formalism and that t is interpreted as the conformal time. Then, in the conformal formalism, the potential V is a4 Vc (a−1 φ), where a = a(t) is the universal scale factor. Let us assume that Vc (φ) = V0 + 2−1 m20 φ2 , V0 and m20 being real constants. Then, the actual m(t)2 equals −a−1 d2 a/dt2 + m20 a2 . Such a model can be related to standard formulations for inflation by transforming to the comoving formalism, neglecting spatial variations, assuming the validity of Friedmann’s equation and imposing the slow-roll approximation [1]. Let the system be represented, at t0 , by some statistical mixture of states, given by the initial quantum density operator ρin . Gaussian (g) structures for ρin are rather popular: see [6], [7], [8] and [5], among others. Generically speaking, gaussian ρin ’s exclude, in the initial state, interactions of the same kind as those acting for t > t0 . Here, we shall also treat gaussian initial nonequilibrium mixed states. Let | φi >, i = 1, 2, be eigenstates of the quantized field operator φ(t0 , x), at time t0 , with real field eigenvalues φi (t0 , x): φ(t0 , x) | φi >= φi (t0 , x) | φi >. We shall denote φi (t0 , x) also as φi (x). Thus, we assume that ρin = ρin,g , with matrix elements: 1 d d < φ2 | ρin,g | φ1 >= exp[− d x d x [(φ2 (x)φ2 (x ) + φ1 (x)φ1 (x ))Kd (x, x ) 2 (1) −(φ1 (x)φ2 (x ) + φ2 (x)φ1 (x ))Ko (x, x )]] Kd (x, x ) and Ko (x, x ) are suitable real kernels (symmetric under x → x and x → x). We shall decompose the real symmetric kernel Kd (x, x ) − Ko (x, x )) into its eigenfunctions (fγ,g (x)) as: Kd (x, x ) − Ko (x, x )) =
fγ,0,g γ
2
fγ,g (x)fγ,g (x )∗
with real eigenvalues fγ,0,g (> 0, by assumption). The functions fγ,g are orthonormalized and constitute a complete set:
dd xfγ,g (x)[fγ ,g (x)]∗ = δγ,γ ,
γ
fγ,g (x)[fγ,g (x )]∗ = δ (d) (x − x )
(2)
220c
R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
γ denotes suitable integrations (continuous spectrum) and summations (discrete spectrum) and δ (d) is the d-dimensinal Dirac’s delta function. We also introduce:
fγ,g,2(k) =
dd xfγ,g (x) exp(ikx)
(3)
The time evolution of the system for t > t0 , as determined by the initial condition ρin,g , is given by the quantum density operator ρg = ρg (t, t0 ). The latter fulfills both the equation i∂ρg /∂t = Hρg − ρg H and ρg (t0 , t0 ) = ρin,g . Then, one has the formal solution: ρg (t, t0 ) = U(t, to )ρin,g U(t, to )+ . where U(t, t0 ) is the full evolution operator. The superscript (+) denotes the adjoint. Four two-point correlators are considered in Equilibrium Thermal Field Theory in the real-time formalism [9]. The four free ( V = 0) two-point correlators enable to formulate perturbation theory for systems at equilibrium [9]. The same holds for nonequilibrium systems [3]. One of those free two-point nonequilibrium correlators is: (0)
−Δ+,+,g (t2 , x2 ; t1 , x1 ) = T r[U0 (t2 , t0 )+ φ(t0 , x2 )U0 (t2 , t0 )U0 (t1 , t0 )+ φ+ (t0 , x1 ) ×U0 (t1 , t0 )ρin,g ] ≡< φ+ (t2 , x2 )φ+ (t1 , x1 ) >g
(4)
T r denotes the usual trace operation. φ(t0 , xj ), j = 1, 2, is the quantum field operator at the initial time t0 . U0 (t, t ) is the free evolution operator, determined by H0 . Similar expressions, with the adequate changes, define the other three free two(0) (0) point nonequilibrium correlators, namely, Δ−,−,g (t2 , x2 ; t1 , x1 ), Δ+,−,g (t2 , x2 ; t1 , x1 ) and (0) Δ−,+,g (t2 , x2 ; t1 , x1 ). For V = 0, similar equations yield the four full two-point nonequilibrium correlators, in terms of the full evolution operator U(t, t ). (0)
3. THE FREE NONEQUILIBRIUM CORRELATORS Δα,α ,g We give below the space-time structures of the four free nonequilibrium correlators (0) Δα,α ,g (α, α = ±): (0)
(0)
Δ(0) α,α,g (x; x ) = Δ0,g (x; x ) + iαΔ1,g (x; x ) (0) Δα,−α,g (x; x )
=
(0) −Δ0,g (x; x )
+
(5)
(0) iαΔ2,g (x; x )
(6)
The contributions in the right-hand-sides of those equations are ( k = |k|): (0)
(0)
(0)
Δ0,g (x; x ) = Δ0,1,g (x; x ) + Δ0,2,g (x; x )
(7) d
1 d k d k f1,k (t) × f1,k (t ) dd x dd x exp(ik(x − x)) 2 (2π)d (2π)d (8) × exp(ik (x − x ))(Kd (x , x ) + Ko (x , x )) d 1 d k (0) exp(−ikx)fγ,g,2 (k) × [f2,k (t) + f1,k (t)f3,k )] Δ0,2,g (x; x ) = − (2π)d γ 4fγ,0,g (0)
Δ0,1,g (x; x ) = −
×
d
dd k exp(−ik x )f−γ,g,2 (k )[f2,k (t ) + f1,k (t )f3,k ] (9) (2π)d f2,k (t)f1,k (t ) + f2,k (t )f1,k (t) dd k (0) + f Δ1,g (x; x ) = exp [ik(x − x )] − (t, t ) (10) k (2π)d 4 dd k f2,k (t)f1,k (t ) − f2,k (t )f1,k (t) (0) exp [ik(x − x )] × (11) Δ2,g (x; x ) = (2π)d 4
R.F. Álvarez-Estrada / Nuclear Physics A 785 (2007) 218c–221c
221c
Let the functions fk,± (t), associated to m = m(t), be defined as the solutions of the following homogeneous second-order differential equation, in t0 ≤ t ≤ T : [
d2 + k2 + m(t)2 ]fk,± (t) = 0 dt2
(12)
with the following boundary conditions: fk,+ (t0 ) = 0, fk,+ (T ) = 1 and fk,− (t0 ) = 1, fk,− (T ) = 0. We have: fk,+ (t)[Jf−1 (dfk,− /dt)(T )] + fk,+ (t) (dfk,− /dt)(T ) − (dfk,+ /dt)(t0 ) dfk,+ fk,− (t) f2,k (t) = −( )(t0 ) + fk,− (t) dt Jf θ(t − t)fk,− (t )fk,+ (t) + θ(t − t )fk,− (t)fk,+ (t ) fk (t, t ) = Jf dfk,−(t) dfk,+(t) − fk,− (t) Jf = fk,+ (t) dt dt f1,k (t) =
(13) (14)
(15)
and f3,k = 2(dfk,−/dt)(t0 ) ((dfk,± /dt)(τ ) ≡ dfk,± (t)/dt, at t = τ ). T -dependences cancel (0) (0) out in Δα,α ,g . For t-independent m, the Δα,α ,g ’s simplify, as fk,+ (t) = sin ω(k)(t − t0 )/ sin ω(k)(T − t0 ) and fk,− (t) = sin ω(k)(T − t)/ sin ω(k)(T − t0 ) (ω(k) = [k2 + m2 ]1/2 ). The above free nonequilibrium correlators are consistent with general requirements: compare with [3]. They can be employed in perturbation theory, for the case in which interactions are included (say, V (φ) = 0). ρin,g and the free nonequilibrium correlators for m = m(t) could be interesting in connection with the single quantum inflaton. For t-independent mass, they can also be employed, qualitatively, for fully relativistic plasmas (leaving aside half-integral spin, gauge degrees of freedom,...,). For other studies of nonequilibrium RQFT, see, for instance, [10] and, for much wider referencing, [5]. REFERENCES 1. A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge University Press, Cambridge, 2000. 2. D. Boyanovsky, F. J. Cao and H. J. de Vega, Nucl. Phys. B 632 (2002) 121. 3. K.C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. 4. N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145 (1987) 141. 5. J. Berges, Introduction to Nonequilibrium Quantum Field Theory, arXiv:hepph/0409233 v1 20 Sep. (2004). 6. R. Jackiw, Physica A 158 (1989) 269. 7. J. Baacke, K. Heitmann and C. Patzold, Phys. Rev. D 58 (1998) 125013. 8. P. Jizba and E.S. Tututi, Phys. Rev. D 60 (1999) 105013. 9. M. Le Bellac, Thermal Field Theory, Cambridge University Press, Cambridge, 1996. 10. G. Aarts, J. Smit, Phys. Rev. D 61 (2000) 025002.