On quantization in background scalar fields
OOn quantization in background scalar fields
E. T. Akhmedov ∗ , E. N. Lanina † , and D. A. Trunin ‡ Institutskii per, 9, Moscow Institute of Physics and Technology, 141700, Dolgoprudny, Russia B. Cheremushkinskaya, 25, Institute for Theoretical and Experimental Physics, 117218, Moscow, Russia
February 25, 2020
Abstract
We consider (0+1) and (1+1) dimensional Yukawa theory in various scalar field backgrounds, whichare solving classical equations of motion: ¨ φ cl = 0 or (cid:3) φ cl = 0, correspondingly. The (0+1)–dimensionaltheory we solve exactly. In (1+1)–dimensions we consider background fields of the form φ cl = E t and φ cl = E x , which are inspired by the constant electric field. Here E is a constant. We study thebackreaction problem by various methods, including the dynamics of a coherent state. We also calculateloop corrections to the correlation functions in the theory using the Schwinger–Keldysh diagrammatictechnique. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ h e p - t h ] F e b ontents B.1 Path integral calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49B.2 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
C Definite-frequency and definite-momentum operators 51D Derivation of the coherent state 53 Introduction
The main goal of quantum field theory is to find the response of the fields to external perturbations,i.e. to find correlation functions or, more generically, correlations between an external influence on thesystem and its backreaction on it. In classical field theory correlation functions are solutions of equationsof motion. In quantum field theory one also should take into account quantum fluctuations, i.e. calculateloop corrections to the tree-level correlation functions. Usually one treats quantum fluctuations usingFeynman diagrammatic technique. It implicitly assumes that external perturbations do not change theinitial state of the theory, i.e. the system remains stationary.However, strong background fields usually take the state of the quantum field theory out of equilibrium;in this situation standard (stationary or Feynman) technique incorrectly describes the dynamics of thefields. For instance, stationary approximation is violated in an expanding space–time (see, e.g., [1–5]),in strong electric fields [6, 7], during the gravitational collapse [8] and in a number of other non-trivialphysical situations [9–12]. In such situations loop corrections to the tree-level correlation functions growwith time. This indicates the breakdown of the perturbation theory. Namely, every power of the smallcoupling constant is accompanied by a large (growing with evolution time) factor. This raises the questionof the loop resummation.Such a resummation was performed only in a limited number of cases [2–7]. Moreover, even in thesecases one can catch only the leading qualitative effects in the limit of long evolution period and smallcoupling constant. In this respect it would be nice to find a simple but nontrivial example of a non-equilibrium field theory, in which calculations and dynamics itself are more transparent than in complexgravitational and electromagnetic analogs.As an example of such a non-equilibrium situation we propose to consider the Yukawa theory ofinteracting fermions and massless bosons in ( D + 1)-dimensional Minkowski spacetime: S = (cid:90) d D +1 x (cid:20)
12 ( ∂ µ φ ) + i ¯ ψ /∂ψ − λφ ¯ ψψ (cid:21) . (1.1)We start with D = 0 ,
1. Usually one quantizes this theory on the trivial background φ cl = 0, ψ cl = 0and uses the standard equilibrium approach to find scattering amplitudes [13]. This approach is notapplicable in the presence of a strong background scalar field φ cl , at least if there is a pumping ofenergy into the system, which may generate an increase of the higher level populations and anomalousquantum averages. To study such an out of equilibrium situation, we quantize the fields on a non-zeroclassical background and then calculate correlation functions using non-equilibrium Schwinger–Keldyshdiagrammatic technique [14–20].Namely, in this paper we rely on the following program. First, we assume that there is a strong scalarfield, i.e. a classical solution φ cl ( x ) (cid:29) D +1)-dimensional x and ψ cl = 0. For instance,we separately study linearly growing in two dimensions background fields of the form φ cl = mλ + Et and φ cl = mλ + Ex inspired by the strong background electric field in QED [6, 7]. Whereas the separatepaper [12] considers the case of the strong scalar wave background of the form φ cl = λ Φ (cid:16) t − x √ (cid:17) . Second,we split each field into the sum of the “classical background” and “quantum fluctuations”: φ = φ cl + φ q , ψ = ψ q , quantize the “quantum” part and find tree-level correlation functions. We use the exact fermionmodes instead of plane waves; thereby we explicitly find the response of the fermion field (at least at thetree–level in such backgrounds). Then we find at tree–level the response of the scalar field itself on thebackground.Finally, we calculate loop corrections to the correlation functions using non-equilibrium Schwinger–Keldysh diagrammatic technique. In particular, we are interested in the loop corrections to the Keldyshpropagators for scalar and fermion fields, because these propagators reflect the change of the state of thetheory. Namely, at the loop level they show the time dependence of the corresponding level populationsand anomalous quantum averages. The usual equilibrium technique is not applicable if these quantities3re non-zero. For instance, this is the case of strong electric [6, 7] and gravitational [2, 8] fields, whereloop corrections to the Keldysh propagator grow with time.Feynman technique takes into account only contributions of the zero point fluctuations into correlationfunctions. To take into account the change of the initial state of the theory (change in the anomalousaverages) and of the excitation of higher than zero point levels (for the exact modes in background fields)one has to apply the Schwinger–Keldysh technique.However, in this paper we show that strong scalar fields under consideration do not share the propertiesof the background electric and gravitational fields: even in the limit of indefinitely long evolution periodloop corrections to the level population and anomalous quantum average remain finite in the first looplevel. Which means that while in the strong electric and gravitational fields to understand the dynamicsone has to resum the leading contributions from all loops (see e.g. [2] for a review), in the backgroundscalar fields under consideration one does not need to do that.Let us also emphasize the other two apparent important differences between strong scalar field andstrong electric and gravitational fields. The equations of motion for a point like relativistic particle inthe φ cl = mλ + Et , φ cl = mλ + Ex or φ cl = λ Φ (cid:16) t − x √ (cid:17) backgrounds does not have Euclidean world–lineinstanton solutions and the effective actions in the scalar background fields are real [12]. Therefore, thereis no particle tunneling in the strong scalar fields under consideration. This distinguishes strong scalarfield from the strong electric [21–23] or gravitational [24] ones. However, the situation with the particlecreation in the scalar field background φ cl = λ Φ (cid:16) t − x √ (cid:17) is not that trivial as is shown in [12] on thetree–level. This can signal that in the latter background field loop corrections also may grow with time,but that is a subject for a separate paper and is not considered here.The paper is organized as follows. In section 2 we discuss the one-dimensional problem. This is thesimplest case to our knowledge, because in (0 + 1) dimensions the scalar current λ (cid:104) ¯ ψψ (cid:105) can be calculatedexactly. Moreover, the theory can be solved exactly. Using operator formalism we show that first loopcorrections to the scalar two-point functions are fully determined by corrections to one-point functions.Then we reproduce this result in Schwinger–Keldysh diagrammatic technique and extend it to all ordersof perturbation theory.In sections 3 and 4 we consider the case of linearly growing in time, φ cl = mλ + Et , or in space, φ cl = mλ + Ex , scalar field in (1 + 1) dimension. We discuss the subtleties of choosing the correct fermionmodes and quantize the fermion field. Using these modes we calculate the tree-level scalar current andfirst loop corrections to the scalar and fermion propagators. We find that in both cases these correctionsremain finite in the limit of infinitely long evolution periods.In section 5 we consider another approach to the scalar field background: we examine the timeevolution of the “coherent state” corresponding to the initial value of the field φ cl ( x ) = mλ + Ex : (cid:68) φ cl (cid:12)(cid:12)(cid:12) ˆ φ ( t = 0 , x ) (cid:12)(cid:12)(cid:12) φ cl (cid:69) = φ cl ( x ) . Such an approach corresponds to a different setup for the background field, which at first sight seems tobe the same. On one side, if we consider the background field φ cl = mλ + Ex for all times and find theexact fermion modes in it, this should correspond to the situation that the background field is maintainedsomehow for all times in its fixed form under consideration. Or this approach is applicable when thebackreaction on the background is very week. On the other hand, if we consider a background field set upby the initial coherent state | φ cl (cid:105) , which is then released to evolve freely, such an approach can be usedfor the case when the backreaction is strong.To the best of our knowledge, the last approach has not yet been considered for other non-equilibriumsystems. However, we find that the behavior of the scalar field in different setups are qualitatively thesame, which seems to be a peculiarity of the scalar background fields under consideration.Finally, we discuss the results and conclude in section 6. In addition, we discuss the asymptoticexpansion for the parabolic cylinder functions in appendix A, review textbook derivation of the Feynman4ffective action and renormalizations for the scalar field in appendix B, discuss some subtleties in themode decomposition in appendix C and derive the coherent state in appendix D. To start with we consider the most simple situation — the (0 + 1)–dimensional quantum field theory ofinteracting fermions and real scalar field. In considering this simplest (0 + 1)–dimensional situation wewill show many technical details for pedagogical reasons to introduce the non–stationary technique andset up the notations.There are two options to describe fermions in one dimension. First one is determined by the followingaction: S = (cid:90) dt (cid:20)
12 ˙ φ + i ¯ ψ ˙ ψ − λφ ¯ ψψ (cid:21) , (2.1)where we denoted the conjugated fermion as ¯ ψ = ψ † . The fermions become Grassmanian upon quantiza-tion. Another option is the theory with two-component spinors: S = (cid:90) dt (cid:20)
12 ˙ φ + i ¯ ψγ ˙ ψ − λφ ¯ ψψ (cid:21) , (2.2)where γ = (cid:18) − (cid:19) , ¯ ψ = ψ † γ .It can be shown that the situation in the latter theory is just a bit more complicated than in the formerone. Essentially the dynamics is the same. The main complication of (2.2) in comparison with the theory(2.1) is that in (2.2) upon quantization we have four fermion Fock space states, | , (cid:105) , | , (cid:105) , | , (cid:105) and | , (cid:105) , rather than two, | (cid:105) and | (cid:105) as it is the case for (2.1). In what follows we consider only the theory(2.1). We address the theory under consideration as if it is the simplest one dimensional quantum fieldtheory. Namely instead of calculating quantum mechanical transition amplitudes we calculate correlationfunctions. Our main goal is to find the backreaction on a strong scalar field, to be described below, inthese very simple settings.The equations of motion for the action (2.1) are as follows: (cid:40) ¨ φ = − λ ¯ ψψ,i ˙ ψ = λφψ. (2.3)These equations have the following classical solution: φ cl ( t ) = mλ + αλ t, ψ cl = ¯ ψ cl = 0 , (2.4)which we will consider as a background.Then we consider mode decomposition for quantum parts of these fields over the classical back-ground (2.4): ˆ ψ ( t ) = ˆ ap ( t ) , ˆ¯ ψ ( t ) = ˆ a † p ∗ ( t ) , (2.5)ˆ φ ( t ) = ˆ αf ( t ) + ˆ α † f ∗ ( t ) , where operators ˆ a and ˆ α obey the standard (anti)commutation relations: { ˆ a, ˆ a † } = 1 , [ ˆ α, ˆ α † ] = 1 . (2.6)The equations for the modes on this background are as follows:5 ¨ f = 0 , (cid:0) i ddt − m − αt (cid:1) p = 0 . (2.7)Thus, we have the first order differential equation for the fermion modes, hence, their form is p ( t ) = e − i t (cid:82) ( m + αt (cid:48) ) dt (cid:48) . (2.8)As a result, the tree–level expectation value of the equal-time product of two fermion operators does notdepend on time: (cid:104) | ¯ ψψ | (cid:105) = 0 and (cid:104) | ¯ ψψ | (cid:105) = 1 , (2.9)where ˆ a | (cid:105) = ˆ a † | (cid:105) = 0. To find (cid:104) ¯ ψψ (cid:105) exactly, note that the full Hamiltonian of the theory is as follows: H full = λφ ¯ ψψ + π , (2.10)where π is the momentum conjugate to the scalar field, [ φ, π ] = i , { ψ, ¯ ψ } = 1. Using such a Hamiltonianone can find that:[ ¯ ψψ, H full ] = 0 , hence , (cid:104) | ¯ ψψ | (cid:105) exact ( t ) = 0 and (cid:104) | ¯ ψψ | (cid:105) exact ( t ) = 1 . (2.11)Thus, we have two options for the backreaction problem:¨ (cid:104) φ (cid:105) ≡ − λ (cid:104) | ¯ ψψ | (cid:105) = 0 , and¨ (cid:104) φ (cid:105) ≡ − λ (cid:104) | ¯ ψψ | (cid:105) = − λ, (2.12)i.e. either the background force is zero or non–zero, but constant.It should be stressed at this point that the result under consideration does not depend whether wequantize in the background scalar field (2.4) or we put the background field to zero. However, to completethe solution of the problem, we also have to calculate the scalar and fermion two–point functions, whenthe points do not coincide.To do that let us point out one important issue. Consider one–dimensional scalar with a non–zeromass: S = 12 (cid:90) dt (cid:104) ˙ φ − ω φ (cid:105) . (2.13)The standard mode in this case is f ( t ) = √ ω e − iωt .Consider the two–point Wightman functions in this theory in the limit ω → φ ( t ) = 1 √ ω ( αe − iωt + α † e iωt ) ω → −−−→ α + α † √ ω + i (cid:114) ω α † − α ) t, and (cid:104) φ ( t ) φ ( t (cid:48) ) (cid:105) = e − iω ( t − t (cid:48) ) ω ω → −−−→ ω − i t − t (cid:48) ) . (2.14)Note that if we just omit the term ω in the propagator it can be used as the tree–level Wightman scalarfunction in the theory (2.1). In fact, the latter one does solve the appropriate differential equation: (cid:18) d dt + ω (cid:19) G ( t − t (cid:48) ) = 0 , and can be used as a basis for the construction of other propagators. Such as e.g. Feynman, retarded andKeldysh two–point functions. 6n the other hand, consider the direct quantization of the scalar part of the theory (2.1). Then themode is f ( t ) = − it √ and the expansion of the field operator is: φ ( t ) = 1 √ (cid:104) ( α + α † ) + i ( α † − α ) t (cid:105) . (2.15)It is easy to check that such φ satisfies the equation of motion and [ φ, π ] = i .Now we can calculate the tree–level boson Wightman propagator: (cid:104) φ ( t ) φ ( t (cid:48) ) (cid:105) = 12 (cid:104) | (cid:104) ( α + α † ) + i ( α † − α ) t (cid:105) (cid:104) ( α + α † ) + i ( α † − α ) t (cid:48) (cid:105) | (cid:105) = 1 − i ( t − t (cid:48) ) + tt (cid:48) . (2.16)This provides another option for the two–point function in the theory. The two choices of the Wightmanpropagators in the theory correspond to two different choices of states. While the second choice corre-sponds to a ground state in the Fock space, the first one is a sort of a coherent state. A somewhat similarsituation appears for the massless scalar field in two–dimensional flat space or in de Sitter space [25].Please also note that while the first choice of the propagator respects the time translational invariance,but does not respect so called positivity, (cid:104) φ ( t ) (cid:105) > (cid:104) φ ( t ) (cid:105) is just vanishing, whilein two–dimensions similar Wightman function can become negative), the second choice does respectpositivity, but violates the time translational invariance.What remains to be done now is to calculate the exact two–point Wightman function for the scalarsand fermions. In the next two subsections we will do that in two different, but related, ways. But beforedoing this let us explain the resulting solution of the problem in simple terms. Consider a solution of thesecond equation in (2.12): ¨ (cid:104) φ (cid:105) = − λ. (2.17)It is given by (cid:104) φ (cid:105) = − λ t + c t + c , (2.18)where c , are integration constants. Hence, the field operator ˆ φ ( t ) can be written in the following form:ˆ φ ( t ) = mλ + αλ t + 1 √ (cid:104)(cid:16) ˆ α + ˆ α † (cid:17) + i (cid:16) ˆ α † − ˆ α (cid:17) t (cid:105) − λ t + c t + c . (2.19)Then, the boson propagator has the following form:∆ (cid:104) φ ( t ) φ ( t ) (cid:105) = (cid:104) φ ( t ) (cid:105)(cid:104) φ ( t ) (cid:105) = λ t t − λ c ( t t + t t ) − λ c ( t + t )+ c c ( t + t )+ c t t + c . (2.20)This expression coincides with the exact result shown e.g. in eq. (2.34) if we set c = λt , (2.21) c = − λ t , (2.22)That is true because the exact expression follows from the “tadpole” diagram, which corresponds to thesolution of the equation (2.17). 7 .1 Two-point functions and perturbative corrections Let us make the field φ dynamical and calculate corrections to the tree–level propagators. The potentialoperator in the interaction picture is as follows: V ( t ) = U † ( t, t ) (cid:0) λφ ( t ) ¯ ψψ (cid:1) U ( t, t ) = λφ ( t ) ¯ ψψ = λ (cid:16) ˆ αf ( t ) + ˆ α † f ∗ ( t ) (cid:17) ˆ a † ˆ a, (2.23)where t is the time after which the self-interaction λφ ¯ ψψ is adiabatically turned on. We recall that ¯ ψψ does not depend on time and f ( t ) = − it √ . Evolution operator in the interaction picture is as follows: U ( t b , t a ) = T exp (cid:20) − i (cid:90) t b t a dηV ( η ) (cid:21) = 1 − i (cid:90) t b t a dηV ( η ) + ( − i ) (cid:90) t b t a dηV ( η ) (cid:90) ηt a dξV ( ξ ) + · · · ≡≡ U ( t b , t a ) + U ( t b , t a ) + · · · (2.24)One can explicitly calculate the first and second order corrections to the evolution operator: U ( t b , t a ) = − iλ √ a † ˆ a (cid:20) ( t a − t b ) (cid:18) − i t a + t b ) (cid:19) ˆ α + h.c. (cid:21) ,U ( t b , t a ) = − λ a † ˆ a (cid:34)
124 ( t a − t b ) (cid:0)
12 + 3 t a + t b (3 t b + 4 i ) + t a (6 t b − i ) (cid:1) ˆ α † ˆ α −−
18 ( t a − t b ) (2 i + t a + t b ) ˆ α ˆ α + h.c. (cid:35) , (2.25)where we have used the identity ˆ a † ˆ a ˆ a † ˆ a = ˆ a † ˆ a . Now let us calculate the Wightman function of two bosonfields in the vacuum state of the scalar field, ˆ α | (cid:105) = 0: D exact ( t , t ) = (cid:104) φ ( t ) φ ( t ) (cid:105) = (cid:68) U † ( t , t ) φ ( t ) U ( t , t ) φ ( t ) U ( t , t ) (cid:69) == (cid:10) [1 + U ( t , t ) + U ( t , t ) + . . . ] φ [1 + U ( t , t ) + U ( t , t ) + . . . ] φ ×× [1 + U ( t , t ) + U ( t , t ) + . . . ] (cid:11) = D ( t , t ) + ∆ D ( t , t ) + . . . , (2.26)where we denote φ ( t a ) ≡ φ a for short.Note that if we average over the vacuum for fermions, a | (cid:105) = 0, all contributions except the bare bosonpropagator vanish because they always contain the combination ψ | (cid:105) = 0. So in this case the tree-levelexpression for the boson propagator is exact: D exact ( t , t ) = D ( t , t ) . (2.27)Now consider the averaging over the state ˆ a † | (cid:105) = 0 for fermions, which gives a less trivial result. Usingthe decomposition of the evolution operator, one finds that the correction to the tree–level propagatorgrows with time:∆ D ( t , t ) = λ t − t )( t − t ) (cid:8) ( t + t − i )( t + t + 2 i ) f ( t ) f ∗ ( t )++ ( t + t − i )( t + t − i ) f ( t ) f ( t ) + h.c (cid:9) = λ t − t ) ( t − t ) . (2.28)To calculate (cid:104) φ ( t ) φ ( t ) (cid:105) we should simply change t ↔ t . For the future reference we show hereexpressions for the Keldysh and retarded/advanced (R/A) propagators [16–20]: D K ( t , t ) = 12 (cid:104){ φ ( t ) , φ ( t ) }(cid:105) ,D R/A ( t , t ) = ± θ ( ± t ∓ t ) (cid:104) [ φ ( t ) , φ ( t )] (cid:105) . (2.29)8ote that D A ( t , t ) = D R ( t , t ) . (2.30)This means that advanced and retarded propagators behave similarly and we need to calculate only theretarded one. Thus, it follows that D K = 12 [ f ( t ) f ∗ ( t ) + f ∗ ( t ) f ( t )] = 1 + t t , ∆ D K = λ t − t ) ( t − t ) ,D R = θ ( t − t ) [ f ( t ) f ∗ ( t ) − f ∗ ( t ) f ( t )] = iθ ( t − t )( t − t ) , ∆ D R = 0 . (2.31)Here subscript 0 denotes tree–level propagators, while ∆ D — perturbative corrections which we calculatehere. To understand the obtained result let us calculate the expectation value of the single operator: (cid:104) φ (cid:105) = (cid:68) U † ( t , t ) φ U ( t , t ) (cid:69) . (2.32)Up to the first order in λ the correction looks as follows:∆ (cid:104) φ (cid:105) = − iλ t (cid:90) t dt ( (cid:104) φ φ (cid:105) − (cid:104) φ φ (cid:105) ) = λ t (cid:90) t dt ( t − t ) = − λ t − t ) . (2.33)Hence, we see that ∆ D is completely determined by the correction to the one-point correlation function:∆ D ( t , t ) = ∆ D K ( t , t ) = ∆ (cid:104) φ (cid:105) ∆ (cid:104) φ (cid:105) = λ t − t ) ( t − t ) . (2.34)In the following subsection we will see that this contribution corresponds to the so-called “tadpole”diagrams. And, thus, although we have obtained the result under consideration only at the order λ inthe expansion of (2.26) it is actually the exact expression.Apart from other things the observations that we have made in this section indicate that the growthof the two–point function with times t , has no connection to the change of the state in the theoryunlike the case of non–stationary situations in higher dimensional quantum field theories. Namely, thetime evolution in the theory does not lead to a generation of the anomalous quantum averages and levelpopulations neither for fermions nor for the boson. In other words, the initial state does not changedespite the non-stationarity of the theory. In this subsection we recalculate the results of the previous subsection with the use of the diagrammatictechnique. We use Schwinger-Keldysh diagrammatic technique [14–20]. This technique uses the followingfermionic propagators: iG −− ( x , x ) ≡ (cid:104) T ψ ( x ) ¯ ψ ( x ) (cid:105) = θ ( t − t ) iG + − ( x , x ) + θ ( t − t ) iG − + ( x , x ) ,iG ++ ( x , x ) ≡ (cid:104) ˜ T ψ ( x ) ¯ ψ ( x ) (cid:105) = θ ( t − t ) iG − + ( x , x ) + θ ( t − t ) iG + − ( x , x ) ,iG + − ( x , x ) ≡ (cid:104) ψ ( x ) ¯ ψ ( x ) (cid:105) ,iG − + ( x , x ) ≡ −(cid:104) ¯ ψ ( x ) ψ ( x ) (cid:105) , (2.35)where (cid:104) · · · (cid:105) denotes averaging over an appropriate initial state, T stands for the time ordering and ˜ T —for the anti-time ordering. Corresponding bosonic correlation functions are as follows:9 D −− ( x , x ) ≡ (cid:104) T φ ( x ) φ ( x ) (cid:105) = θ ( t − t ) iD + − ( x , x ) + θ ( t − t ) iD − + ( x , x ) ,iD ++ ( x , x ) ≡ (cid:104) ˜ T φ ( x ) φ ( x ) (cid:105) = θ ( t − t ) iD − + ( x , x ) + θ ( t − t ) iD + − ( x , x ) ,iD + − ( x , x ) ≡ (cid:104) φ ( x ) φ ( x ) (cid:105) ,iD − + ( x , x ) ≡ (cid:104) φ ( x ) φ ( x ) (cid:105) . (2.36)In what follows we will include the imaginary unit into the definition of the correlation functions (2.35)and (2.36) for short.One can also define these correlation functions using the Keldysh time contour, which starts at themoment t , goes to t → + ∞ and then returns back to the starting point [15]. The contour appearsdue to the simultaneous presence of time ordered U and anti–time ordered U † in (2.26). Ordering alongthis contour corresponds to the time-ordering on the “forward” part and to the anti-time-ordering on the“backward” part. Hence, one can assign “ ∓ ” signs to the fields sitting on the forward and backward partsof the contour, correspondingly, and define correlation functions G ±± ≡ (cid:104) ψ ± ¯ ψ ± (cid:105) , D ±± ≡ (cid:104) φ ± φ ± (cid:105) . Thisdefinition is equivalent to the definition (2.35) and (2.36). More details can be found in [16–18].Also note that functions G ±± and D ±± are not independent due to the relations: G ++ + G −− = G + − + G − + , D ++ + D −− = D + − + D − + . (2.37)It is convenient to do the Keldysh rotation from the forward and backward (“ ± ”) components of the fieldsto the so called classical and quantum components [15–17]: (cid:18) φ cl φ q (cid:19) = ˆ R (cid:18) φ + φ − (cid:19) , (cid:18) ψ cl ψ q (cid:19) = ˆ R (cid:18) ψ + ψ − (cid:19) , (cid:18) ¯ ψ cl ¯ ψ q (cid:19) = ˆ R (cid:18) ¯ ψ + ¯ ψ − (cid:19) , ˆ R = (cid:18)
12 12 − (cid:19) , (2.38)and introduce the Keldysh and retarded/andvanced propagators: G K ≡ (cid:104) ψ cl ¯ ψ cl (cid:105) = 12 (cid:0) G ++ + G −− (cid:1) , D K ≡ (cid:104) φ cl φ cl (cid:105) = 12 (cid:0) D ++ + D −− (cid:1) ,G R ≡ (cid:104) ψ cl ¯ ψ q (cid:105) = G −− − G − + , D R ≡ (cid:104) φ cl φ q (cid:105) = D −− − D − + ,G A ≡ (cid:104) ψ q ¯ ψ cl (cid:105) = G −− − G + − , D A ≡ (cid:104) φ q φ cl (cid:105) = D −− − D + − . (2.39)This definition is equivalent to the one used in (2.29).Note that in (0 + 1)–dimensions diagrammatic technique works only for correlation functions averagedover the vacuum or thermal (stationary) state, because diagrammatics is based on Wick’s theorem [11,26],which is applicable only in stationary situations in one dimension. However, in our case this restrictiondoes not bother us, because the state of the fields does not change in time. In higher dimensional quantumfield theory this restriction disappears because of the infinite space volume which kills unsuitable operatoraverages [20, 27].Let us calculate the first loop correction to the boson two-point correlation function using Schwinger-Keldysh diagrammatic technique. First, we consider the averaging over the state | (cid:105) ψ | (cid:105) φ . In this casetree–level propagators have the following form (it is easy to restore the remaining four correlators usingdefinitions (2.35) and (2.36)): In general, matrices which rotate fields φ , ψ , and ¯ ψ are independent, but here we choose them to be equal to each other. G + − ( t , t ) = exp − i t (cid:90) t ( m + αt (cid:48) ) dt (cid:48) ,G − +0 ( t , t ) = 0 ,D + − ( t , t ) = f ( t ) f ∗ ( t ) = (1 − it )(1 + it )2 ,D − +0 ( t , t ) = (cid:0) D + − ( t , t ) (cid:1) ∗ = (1 + it )(1 − it )2 . (2.40)The one-loop corrections to the scalar field propagators (Fig. 1) is vanishing:∆ D + − ( t , t ) = − λ (cid:90) dt dt (cid:88) σ , = { + , −} D + σ ( t , t ) G σ σ ( t , t ) G σ σ ( t , t ) D σ − ( t , t ) sgn( σ σ ) = 0 , (2.41)because G − + = 0 and θ θ = 0, where for short we denote θ ≡ θ ( t − t ). Thus, ∆ D K ( t , t ) =∆ D R/A ( t , t ) = 0. Due to the same reason the so-called “bubble” diagram (Fig. 2) is also equal tozero . Finally, the tadpole diagrams (Fig. 3) are zero because they contain the free fermion propagatorsin coincident points: (cid:104) | ψ ¯ ψ | (cid:105) = 0. Thus, one-loop corrections to the boson propagator is zero for thecase of averaging over the state | (cid:105) ψ | (cid:105) φ . This is exactly what we have seen in the previous subsection(see eq. (2.27)).Now let us take the average over the state | (cid:105) ψ | (cid:105) φ . In this case tree–level boson propagators do notchange, whereas tree–level fermion propagators acquire the following form: G + − ( t , t ) = 0 ,G − +0 ( t , t ) = − exp − i t (cid:90) t ( m + αt (cid:48) ) dt (cid:48) . (2.42)The diagrams (Fig. 1) and (Fig. 2) in this case are zero again for the same reasons. Hence, we recalculate In the Schwinger–Keldysh diagrammatic technique vacuum bubbles always cancel out. (cid:10) φ +1 (cid:11) = − iλ (cid:90) dt (cid:88) σ = { + , −} D + σ ( t , t ) G σσaa ( t , t )sgn( − σ ) == − iλ + ∞ (cid:90) t dt D R ( t , t ) = λ (cid:90) t t dt ( t − t ) = − λ t − t ) , ∆ (cid:10) φ − (cid:11) = − iλ (cid:90) dt (cid:88) σ = { + , −} D − σ ( t , t ) G σσaa ( t , t )sgn( − σ ) == − iλ + ∞ (cid:90) t dt D R ( t , t ) = λ (cid:90) t t dt ( t − t ) = − λ t − t ) = ∆ (cid:10) φ +1 (cid:11) . (2.43)Hence, the correction to the boson correlation function looks as follows:∆ D + − ( t , t ) = ∆ D K ( t , t ) = ∆ (cid:10) φ +1 (cid:11) ∆ (cid:10) φ − (cid:11) = λ t − t ) ( t − t ) , (2.44)which coincides with the result (2.34) from the previous subsection.Note that if we choose the bare scalar Wightman propagator as follows: (cid:104) φ φ (cid:105) = − i t − t ) , (2.45)which, as we have discuss around eq. (2.14), respects the time translational invariance, we will get thesame answer for the tadpole diagram:∆ (cid:10) φ − (cid:11) = ∆ (cid:10) φ +1 (cid:11) = − iλ + ∞ (cid:90) t dt D R ( t , t ) = λ t (cid:90) t dt ( t − t ) = − λ t − t ) , (2.46)because retarded propagators do not depend on the state. Thus, the diagrammatic technique gives thecorrect combinatoric factors and reproduces the result of the direct calculation performed above in thesubsection 2.1. As we have already pointed out in the subsection 2.1, the tree-level expression for the boson propagatoris exact if we average over the fermion vacuum ˆ a | (cid:105) ψ = 0: D exact ( t , t ) = D ( t , t ). So in this subsectionwe consider averaging over the state ˆ a † | (cid:105) ψ = 0. We will see that in this case the situation is nearly thesame.Let us classify what sort of diagrams can provide contributions to the exact boson propagator (cid:104) φ φ (cid:105) .First, note that corrections to fermion propagators vanish due to the fact that they come from theinteraction vertex V , which contains fields in coincident points, and (cid:10) ¯ ψψ (cid:11) exact = (cid:10) ¯ ψψ (cid:11) , as we havealready shown above . Many-loop diagrams containing (Fig. 1) and (Fig. 2) and even such diagramswith corrected vertexes, for example, (Fig. 4) vanish for the same reasons as have been discussed in theprevious subsection.Consider loops connected with more than one boson propagator, for example, (Fig. 5). To provethat this diagram also vanishes, we consider such diagrams as depicted on the Figs. 6 and 7. These twodiagrams are described by the following expression: This observation means that we know the exact value of the fermionic two–point functions in the theory under consid-eration. G σ σ σ σ ( t , t , t , t ) = λ (cid:90) dt dt dt dt (cid:88) σ , , , = { + , −} G σ σ ( t , t ) G σ σ ( t , t ) G σ σ ( t , t ) ×× G σ σ ( t , t ) G σ σ ( t , t ) G σ σ ( t , t ) D σ σ ( t , t ) D σ σ ( t , t )sgn( σ σ )sgn( σ σ ) . (2.47)Note that G + − = 0, so only expressions of the following form: G σ + ( t , t ) G ++ ( t , t ) G + σ ( t , t ) G σ + ( t , t ) G ++ ( t , t ) G + σ ( t , t ) D ++ ( t , t ) D ++ ( t , t ) ,G σ + ( t , t ) G ++ ( t , t ) G + σ ( t , t ) G σ − ( t , t ) G −− ( t , t ) G − σ ( t , t ) D + − ( t , t ) D + − ( t , t ) ,G σ − ( t , t ) G −− ( t , t ) G − σ ( t , t ) G σ + ( t , t ) G ++ ( t , t ) G + σ ( t , t ) D − + ( t , t ) D − + ( t , t ) ,G σ − ( t , t ) G −− ( t , t ) G − σ ( t , t ) G σ − ( t , t ) G −− ( t , t ) G − σ ( t , t ) D −− ( t , t ) D −− ( t , t ) (2.48)may give non-zero contributions. But due to the presence of theta-functions they are proportional to( t − t ) and, as we remember, such diagrams come from the interaction vertex V which contains fieldsin the coincident points, where ¯ ψ ( t ) ψ ( t ) does not depend on time t due to the form of modes (2.8).Hence, for example, in the diagram of the Fig. 5 we have, as a part, the four point correlation function G σ σ σ σ ( t, t, t, t ), in which we can set all its arguments equal to t . Hence, that enforces t = t andcontributions from the Fig. 6 and Fig. 7 vanish in the case under consideration. The contributions ofhigher-loop diagrams are also zero for the same reason.As a result, only the remaining tadpole diagrams (Fig. 3) can give the non–vanishing contribution.So the exact propagators are as follows: 13 + − exact ( t , t ) = D + − ( t , t ) + ∆ (cid:104) φ (cid:105) ∆ (cid:104) φ (cid:105) = (1 − it )(1 + it )2 + λ t − t ) ( t − t ) ,D Kexact ( t , t ) = D K ( t , t ) + ∆ (cid:104) φ (cid:105) ∆ (cid:104) φ (cid:105) = 1 + t t λ t − t ) ( t − t ) ,D R/Aexact ( t , t ) = D R/A ( t , t ) = ± iθ ( ± t ∓ t )( t − t ) . (2.49)This generalizes the result of the subsection 2.1 to the arbitrary order in λ and, as we have explainedabove, comes from the solution of the eq.(2.17) with the tadpole appearing due to non–zero right handside (cid:104) | ¯ ψψ | (cid:105) . In this section we consider the Yukawa model of interacting fermions and real scalar field in (1 + 1)-dimensional Minkowski space-time with (+ , − ) signature of the metric: S = (cid:90) d x (cid:20) ∂ µ φ∂ µ φ + i ¯ ψ /∂ψ − λφ ¯ ψψ (cid:21) , (3.1)where we denote /∂ ≡ γ µ ∂ µ , ¯ ψ = ψ † γ and assume that the coupling parameter is λ >
0. In this sectionwe use the Dirac-Pauli representation for the Clifford algebra: γ ≡ (cid:18) − (cid:19) , γ ≡ (cid:18) − (cid:19) . (3.2)The equations of motion for the action (3.1) are as follows: (cid:40) ∂ φ + λ ¯ ψψ = 0 , (cid:0) i /∂ − λφ (cid:1) ψ = 0 . (3.3)Their classical solutions can be taken as ψ cl = 0 , φ cl = F ( t − x ) + ˜ F ( t + x ), where F and ˜ F are arbitrarysmooth functions. In what follows we consider such classical solutions as external backgrounds and splitthe classical and quantum parts of the fields: φ = φ cl + φ q , ψ = ψ cl + ψ q . Our goal is to calculatecorrelation functions.Concretely, in this section we consider the background field which linearly grows with time: φ cl = Et ,where E is some real positive constant . Specifically, in the limit E → E (cid:54) = 0 the Hamiltonian depends on time, i.e. the situation is not stationary.Hence, one may expect the particle creation that is similar to the one in strong electric [14, 21–23] orgravitational fields [24].However, let us emphasize the difference between e.g. the pair creation in the electric field background(the well-known Schwinger effect [14]) and processes in the scalar field background. On the one hand, attree–level the particle creation in the electric field can be attributed to the quantum tunneling through theclassically forbidden region. The rate of such a process is described by an imaginary part of the effectiveaction; moreover, the expression for the rate is not an analytic function of the background field [23]. Onthe other hand, as we will see below the imaginary part of the Feynman effective action on the scalarfield background is zero (see Appendix B). Hence, non-zero quantum expectation values indicate rather Of course, one can obtain classical solutions with other values and signs of this constant via time shifts: t → t + δt = ⇒ φ cl = Eδt + Et , or reversals: t → − t = ⇒ φ cl = mλ − Et . E.g. one can give a mass m ψ to the fermion field by the timeshift δt = m ψ λE . However, these transformations do not bring anything substantially new into our discussion. So, we considerpositive E and zero mass without loss of generality. φ cl = Et is rather unrealistic, since an indefinitelygrowing field requires infinite amount of energy. However, it allows one to grasp the main propertiesof the model. It would be more appropriate to consider the pulse background φ cl = ET tanh tT , whichbecomes constant at the past and future infinities and reproduces the linear growth for | t | (cid:28) T . Such aconfiguration does not solve the equations of motion without a source in (3.3). Another possibility is toconsider a strong wave, i.e. F ( t − x ) which has compact support. The latter classical background wasconsidered in [12]. To set up the notations let us start with the consideration of the free massive fermion field without abackground scalar field. This field can be decomposed into the modes as follows: ψ ( t, x ) = (cid:90) dp π (cid:104) a p ψ (+) p ( t, x ) + b † p ψ ( − ) p ( t, x ) (cid:105) . (3.4)The functions ψ (+) p ( t, x ) ≡ u p e − ipx and ψ ( − ) p ( t, x ) ≡ v p e ipx , which are positive and negative frequencymodes, solve the free equations of motion: ( i /∂ − m ) ψ = 0 , (3.5)and creation and annihilation operators obey the standard anticommutation relations: (cid:110) a p , a † q (cid:111) = (cid:110) b p , b † q (cid:111) = 2 πδ ( p − q ) . (3.6)This fixes the equal-time anticommutation relations for ψ and ψ † : (cid:110) ψ a ( t, x ) , ψ † b ( t, y ) (cid:111) = δ ( x − y ) δ ab , (3.7)where we restored the spinor indices a, b = 1 ,
2. The form of u p and v p spinors is as follows: u p = (cid:18) u p, u p, (cid:19) = sgn( p ) (cid:112) ω ( ω − m ) (cid:18) pω − m (cid:19) , v p = (cid:18) v p, v p, (cid:19) = sgn( p ) (cid:112) ω ( ω − m ) (cid:18) ω − mp (cid:19) , (3.8)where ω = (cid:112) p + m and we have used the Dirac representation for gamma-matrices (3.2). For furtherpurposes (see footnote 5) we introduced the phase factor sgn( p ) which does not affect the conditions (3.5)and (3.7). In what follows we omit the index p of u p , v p and ψ p where it can be easily restored.The fermion field in the time-dependent background can be decomposed in the way similar to (3.4),except that functions ψ ( ± ) solve the equations of motion (3.3) with φ = φ cl instead of the free equa-tions (3.5): [ iγ µ ∂ µ − M ( t )] ψ = 0 , (3.9)where we define for short: M ( t ) = αt, α = λE. (3.10)Because of the spatial homogeneity it is convenient to represent the modes in the following form: ψ ( t, x ) = ψ p ( t ) e ipx . (3.11)Substituting this factorized solution into (3.9), one obtains the equation for the time dependent part ofthe modes: 15 iγ ∂ t − γ p − M ( t ) (cid:3) ψ p ( t ) = 0 . (3.12)One can decouple this system applying the operator (cid:2) − iγ ∂ t − γ p − M ( t ) (cid:3) to its left hand side andkeeping in mind that the eigenvalues of γ are ±
1. Hence, the equation reduces to: (cid:20) ∂ t + (cid:16) ω (1 , p (cid:17) ( t ) (cid:21) ψ , ( t ) = 0 , where (cid:16) ω (1 , p (cid:17) ( t ) ≡ p + α t ± iα. (3.13)Note that this resembles the equation for the massive charged scalar field on the constant electric fieldbackground [1, 6, 7]. Its exact solution is the sum of linearly independent parabolic cylinder functions D ν ( z ): ψ [ z ( t )] = A D ν [ z ( t )] + B D − ν − [ iz ( t )] ,ψ [ z ( t )] = A D ν − [ z ( t )] + B D − ν [ iz ( t )] , (3.14)where A , , B , are complex constants which we fix below, and we define for convenience: z ≡ i √ α M ( t ) , ν ≡ − ip α . (3.15)It is not possible to define usual in– and out– modes as well as positive and negative frequency solutionsin our case due to the fact that the external field is never switched off. Indeed, parabolic cylinder functionhas the following asymptotic behavior [28, 29]: D ν ( z ) = z ν e − z N (cid:88) n =0 (cid:0) − ν (cid:1) n (cid:0) − ν (cid:1) n n ! (cid:16) − z (cid:17) n + O (cid:12)(cid:12) z (cid:12)(cid:12) − N − , ( γ ) = 1 , ( γ ) n (cid:54) =0 = γ ( γ + 1) · · · ( γ + n − , (3.16)for | z | (cid:29) | ν | and | Arg( z ) | < π . In our case Arg( z ) = ± π and the condition | z | (cid:29) | ν | is satisfied forsufficiently large times | t | (cid:29) p α / . So, in the leading order as t → + ∞ one obtains: ψ , ( z ( t )) ∼ A , ( p ) exp (cid:18) − i αt − ip α log t (cid:19) + B , ( p ) exp (cid:18) i αt + ip α log t (cid:19) , (3.17)where A , ( p ) and B , ( p ) are some constants that do not depend on time (but depend on the momentum).Thus, the modes ψ , ( t, x ) cannot be reduced to the sum of positive and negative frequency plane waves,and the interpretation in terms of particles is meaningless. Please keep in mind that in non-stationarysituations it is more appropriate to calculate correlation functions rather than amplitudes, at least becausethere are no asymptotic particle states [30–32].However, let us check the other limit — the ultraviolet region, where | p | (cid:29) √ α for a fixed t . In sucha limit we expect that the modes in the strong scalar background and in the free theory have similarbehavior. In fact, in this case the parabolic cylinder function has the following asymptotic expansion (seeAppendix A for details): D ν [ z ( t )] (cid:39) e πp α √ (cid:32) M (cid:112) M + p + 1 (cid:33) e ip α − ip α log ( √ M p M ) α − iM √ M p α (cid:20) O (cid:18) αM + p (cid:19)(cid:21) . (3.18)Hence, for times | t | (cid:28) | p | α the exact modes behave as follows:16 , ( t, x ) ∼ A (cid:48) , ( p ) e − i | p | t + ipx + B (cid:48) , ( p ) e i | p | t + ipx , (3.19)which means that for fixed time and large momenta one obtains the standard flat space plane waves. Nowit is clear that functions D ν [ z ( t )] and D ν − [ z ( t )] correspond to “positive frequency” modes, i.e. the exactharmonics should be as follows: ψ (+) ( t ) ≡ (cid:32) ψ (+)1 ( t ) ψ (+)2 ( t ) (cid:33) = A (+) (cid:32) D ν [ z ( t )] ( i∂ t − M ( t )) p D ν [ z ( t )] (cid:33) , (3.20)where we used the system (3.12) to relate the first and second components of the spinor. One can simplifythis expression using the following relations for parabolic cylinder functions [28, 29]: ∂ z D ν ( z ) + 12 zD ν ( z ) − νD ν − ( z ) = 0 ,∂ z D ν ( z ) − zD ν ( z ) + D ν +1 ( z ) = 0 , (3.21)and represent the “positive frequency” modes in the form: ψ (+) p ( t, x ) = A (+) (cid:32) D ν [ z ( t )] i √ p √ α D ν − [ z ( t )] (cid:33) e ipx . (3.22)They behave as ψ ∼ e − i | p | t + ipx for sufficiently large momenta. We choose to consider such modes out ofall options present in eq. (3.14) because they have proper UV behavior, i.e. tend to the free fermion fieldmodes in the limit p → ∞ . Propagators expanded in such modes possess the proper Hadamard behaviour.Which means that they lead to the same UV renormalization as in the absence of the background field.On general grounds we think that this is the appropriate physical picture. We come back to the discussionof other options below at the end of this subsection.In the same way one obtains the “negative frequency” modes: ψ ( − ) p ( t, x ) = A ( − ) (cid:32) − i √ p √ α D ∗ ν − [ z ( t )] D ∗ ν [ z ( t )] (cid:33) e − ipx , (3.23)which behave as ψ ∼ e i | p | t − ipx for sufficiently large momenta.Let us fix the coefficients A (+) and A ( − ) using the equal-time anticommutation relations (3.7): (cid:110) ψ a ( t, x ) , ψ † b ( t, y ) (cid:111) == (cid:90) (cid:90) dp π dq π (cid:104)(cid:8) a p , a + q (cid:9) ψ (+) a,p ( t ) ψ (+) b,q ( t ) ∗ e i ( px − qy ) + (cid:8) b + p , b q (cid:9) ψ ( − ) a,p ( t ) ψ ( − ) b,q ( t ) ∗ e − i ( px − qy ) (cid:105) == (cid:90) dp π (cid:104) ψ (+) a,p ( t ) ψ (+) b,p ( t ) ∗ + ψ ( − ) a, − p ( t ) ψ ( − ) b, − p ( t ) ∗ (cid:105) e ip ( x − y ) = δ ( x − y ) δ ab , (3.24)where we use the canonical anticommutation relations (3.6). This condition is satisfied if ψ (+) a,p ( t ) ψ (+) b,p ( t ) ∗ + ψ ( − ) a, − p ( t ) ψ ( − ) b, − p ( t ) ∗ = δ ab ⇐⇒ | A (+) | | D ν ( z ) | + | A ( − ) | p α | D ν − ( z ) | = 1 , | A ( − ) | | D ν ( z ) | + | A (+) | p α | D ν − ( z ) | = 1 , (cid:0) | A (+) | − | A ( − ) | (cid:1) − i √ p √ α D ν ( z ) D ∗ ν − ( z ) = 0 , (3.25)for arbitrary times z ( t ). Note that this condition is time-independent due to the equations of motion andthe relation ψ ( − ) a, − p ( t ) = − γ ab (cid:16) ψ (+) b,p ( t ) (cid:17) ∗ which follows from the symmetry of the system (3.3):17 t (cid:16) ψ (+) a,p ( t ) ψ (+) b,p ( t ) ∗ + ψ ( − ) a, − p ( t ) ψ ( − ) b, − p ( t ) ∗ (cid:17) = 0 . (3.26)First, (3.25) implies that | A (+) | = | A ( − ) | = | A | . Second, it allows one to find the constant | A | bysetting the argument of parabolic cylinder functions equal to any convenient value, e.g. to zero: | A | π (cid:12)(cid:12)(cid:12) Γ (cid:16) + ip α (cid:17)(cid:12)(cid:12)(cid:12) + p α π (cid:12)(cid:12)(cid:12) Γ (cid:16) ip α (cid:17)(cid:12)(cid:12)(cid:12) = 1 . Using the properties of the Gamma function: | Γ( iy ) | = πy sinh( πy ) , (cid:12)(cid:12)(cid:12)(cid:12) Γ (cid:18)
12 + iy (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = π cosh( πy ) , we find that | A | = e − πp α . (3.27)Let us sum up the main results of this subsection, i.e. write down the asymptotic expressions for themodes.For t > α | t | (cid:28) | p | , | p | (cid:29) √ α one obtains up to a O (cid:16) M p (cid:17) that the modes behave as: ψ (+) ( t, x ) (cid:39) √ | M | | p | sgn( p ) (cid:16) − | M | | p | (cid:17) e − i | p | t + ipx + ip α − ip α log p α + i ˜ ϕ , (3.28)where ˜ ϕ is an arbitrary constant phase independent of t and p . Up to an irrelevant phase this asymptoticbehaviour coincides with the free modes (3.8).At the same time, for t > α | t | (cid:29) | p | , | t | (cid:29) √ α one obtains up to a O (cid:16) p M (cid:17) that the modes behaveas ψ (+) ( t, x ) (cid:39) (cid:18) p | M | (cid:19) (cid:0) αt (cid:1) ip α e − iαt + ipx + ip α log p α + i ˜ ϕ . (3.29)The “negative frequency” modes are obtained from the “positive frequency” ones by the charge conjuga-tion operation: ψ ( − ) p ( t, x ) = γ ψ (+) ∗ p ( t, x ) , (3.30)where γ = γ γ . Also one can check that the modes obey the following relation: ψ (+) p ( − t, x ) = sgn p γ ψ (+) ∗ p ( t, x ) . (3.31)Finally let us point out the following important issue. In this subsection we have found a complete basisof modes solving the classical equations of motion. But there is an ambiguity in the choice of such a basis.Depending on this choice, there are different “ground” Fock space states in the theory. In fact, insteadof (3.22) and (3.23) one could consider canonically transformed basis of modes: (cid:101) ψ (+) p ( t, x ) = (cid:90) dq π (cid:20) a pq ψ (+) q ( t, x ) + b pq ψ ( − ) q ( t, x ) (cid:21) , (cid:101) ψ ( − ) p ( t, x ) = (cid:90) dq π (cid:20) c pq ψ (+) q ( t, x ) + d pq ψ ( − ) q ( t, x ) (cid:21) . (3.32) For this reason we have introduced the phase factor sgn( p ) in eq. (3.8).
18o respect the canonical anti–commutation relations for the fermionic fields and for the correspondingcreation and annihilation operators the Bogoliubov coefficients, a pq , b pq , c pq and d pq , should satisfy certainrelations which are listed in [12].On physical grounds one also should demand that a pq ≈ d pq ≈ δ ( p − q ) , b pq ≈ c pq ≈ , (3.33)as p is taken to infinity. That is necessary for the propagators to have the proper Hadamard behaviour.Thus, there is no unique way to choose the basis of modes and all possibilities in (3.32) are in principleallowed and may lead to different physical situations. This fact is apparent when there is no preferablebasis of special functions found in XIX century and listed in the standard text books.For a given choice of modes one can define a new Fock space “ground” state:ˆ (cid:101) a p | a, b, c, d (cid:105) = ˆ (cid:101) b p | a, b, c, d (cid:105) = 0 , (3.34)where ˆ (cid:101) a p and ˆ (cid:101) b p are canonically transformed annihilation operators. For this new state certain physicalquantities will be different from those for the original state [12]. However, we will argue, as it was also donein [12], that the scalar current, (cid:104) ¯ ψψ (cid:105) , at leading approximation for large and slowly changing backgroundscalar field does not depend on the choice of the initial state. In the previous subsection we derived the exact modes for the fermion field, which in a sense describesthe fermion response to the strong scalar field background. In this subsection we find the response of thescalar field itself due to the presence of the non–trivial fermion zero–point fluctuations in the scalar fieldbackground under consideration.Quantizing the Hamiltonian of the theory (3.1):ˆ H = (cid:90) dx (cid:20) (cid:16) ∂ t ˆ φ (cid:17) + (cid:16) ∂ x ˆ φ (cid:17) − i ˆ¯ ψγ ∂ x ˆ ψ + λ ˆ φ ˆ¯ ψ ˆ ψ (cid:21) , (3.35)and using Hamilton’s equations:˙ˆ φ ( x ) = i (cid:104) ˆ H, ˆ φ ( x ) (cid:105) , ˙ˆ ψ ( x ) = i (cid:104) ˆ H, ˆ ψ ( x ) (cid:105) , (3.36)one obtains the following operator equation for the scalar field: ∂ ˆ φ + λ ˆ¯ ψ ˆ ψ = 0 , (3.37)which reproduces one of the classical equations of motion (3.3). Hence, one needs to calculate the scalarcurrent j cl ( t ) ≡ (cid:104) ˆ¯ ψ ˆ ψ (cid:105) to find the response of the classical field φ cl = (cid:104) ˆ φ (cid:105) . This current has the followingform: (cid:104) ¯ ψψ (cid:105) ( t ) = (cid:90) (cid:90) dp π dq π (cid:104) (cid:104) b p b † q (cid:105) (cid:16) ψ ( − )1 ,p ( t ) ψ ( − )1 ,q ( t ) ∗ − ψ ( − )2 ,p ( t ) ψ ( − )2 ,q ( t ) ∗ (cid:17) e i ( p − q ) x (cid:105) == (cid:90) dp π (cid:18)(cid:12)(cid:12)(cid:12) ψ ( − )1 ,p ( t ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12) ψ ( − )2 ,p ( t ) (cid:12)(cid:12)(cid:12) (cid:19) = (cid:90) dp π (cid:18) − e − πp α | D ν [ z ( t )] | (cid:19) , (3.38)where we have used the notations of Sec. 3.1 for short, and in the last line also we have used one of therelations (3.25). Note that in principle the equation under consideration provides an implicit expressionfor the current. However, this form of the current is hard to interpret in physical terms. To obtainphysically tractable equations we will consider only the leading contribution in the limit t → ∞ for small α . 19efore evaluating the integral (3.38), consider the case of a free fermion field with a mass m . Usingthe free modes (3.8) one obtains the following free current: (cid:104) ¯ ψψ (cid:105) free = − (cid:90) Λ − Λ dp π m (cid:112) m + p ≈ mπ log m , (3.39)where we have introduced the ultraviolet cutoff at the scale Λ. Note that the constant classical background φ cl = mλ , substituted into the system (3.3), reproduces this case. The analog of the mass parameter m inthe theory (3.1) is M ( t ) = λφ cl = λEt . Thus one expects the following behavior for the current (3.38): (cid:104) ¯ ψψ (cid:105) ( t ) (cid:39) λφ cl π log λφ cl . (3.40)Let us check this conjecture by calculating the integral (3.38) in such an approximation when φ cl is largeand slowly changing function. Note that M ( t ) = αt grows indefinitely with time, so it can overcome anarbitrarily large fixed scale Λ. Due to this fact we consider cases M <
Λ and
M >
Λ separately. In bothcases we assume M (cid:29) α to single out the leading contributions. The case M >
Λ is rather unphysicalas we have already mentioned. However, we consider it for integrity.In the case M (cid:28) Λ we divide the region of integration into two segments: [0 , Λ] = [0 , √ α ] + [ √ α, Λ],and estimate integrals over these segments using expansions (3.16) and (3.18) correspondingly: (cid:90) √ α dp (cid:18) − e − πp α | D ν [ z ( t )] | (cid:19) (cid:39) (cid:90) √ α dp (cid:20) − p M + O (cid:18) α M (cid:19)(cid:21) (cid:39)(cid:39) −√ α (cid:20) − αM + O (cid:18) α M (cid:19)(cid:21) ; (3.41) (cid:90) Λ √ α dp (cid:18) − e − πp α | D ν [ z ( t )] | (cid:19) (cid:39) (cid:90) Λ √ α dp (cid:34) − (cid:32) M (cid:112) M + p (cid:33) (cid:18) O (cid:18) αp (cid:19)(cid:19)(cid:35) (cid:39)(cid:39) M (cid:20) log M
2Λ + O (cid:18) M Λ , √ αM (cid:19)(cid:21) . (3.42)Hence, in the limit t → ∞ we obtain that: (cid:104) ¯ ψψ (cid:105) ( t ) (cid:39) αtπ log αt
2Λ + · · · , (3.43)where we denoted the subleading contribution as “ · · · ”. This expression coincides with (3.40) in theapproximation under consideration. It also reproduces the behaviour of the scalar current found in [12].In the case M (cid:29) Λ one can use the decomposition (3.16) in the entire domain [0 , Λ]: (cid:104) ¯ ψψ (cid:105) ( t ) ∼ (cid:90) Λ0 dp (cid:20) − p M + O (cid:18) α M (cid:19)(cid:21) (cid:39) − Λ + 16 Λ M + · · · , (3.44)i.e., in the leading order the current does not depend on time and linearly diverges as Λ → ∞ . We thinkthat this behavior has no physical sense, e.g., it does not allow us to treat UV divergences properly. Thismeans that an indefinitely growing scalar field is not self-consistent because it is not realistic, as we havementioned already.However, this problem can be avoided if one considers a pulse background φ cl = ET tanh tT instead ofthe φ cl = Et one. On the one hand, for times t (cid:28) T these backgrounds coincide, hence, the result (3.43)is valid. On the other hand, for times t (cid:29) T the pulse background reproduces the free Dirac field withconstant mass m = ± λET . Hence, if one chooses the UV cutoff Λ (cid:29) M ( T ), the condition λφ cl (cid:28) Λ isalways satisfied, and the equality (3.40) holds.Thus, the effective equation of motion for the boson field gets modified in the following way:20 (cid:104) φ (cid:105) + λ (cid:104) φ (cid:105) π log λ (cid:104) φ (cid:105) Λ ≈ . (3.45)This identity is valid for the fields from the interval √ λE (cid:28) λφ cl (cid:28) Λ and (cid:104) φ (cid:105) = φ cl + . . . . Note that φ cl = Et does not solve this equation, i.e. the classical field must restructure itself to satisfy the correctedequation. We discuss the origin of such a behavior in the concluding section and in the Appendix B.Also note that the true equation of motion cannot depend on the artificial UV cutoff Λ. This problemcan be solved by renormalization of the bare mass of scalar field. It turns out that quantum fluctuationsbreak the symmetry of the problem and bring to the scalar field constant non-zero value φ = (cid:104) φ (cid:105) GS (seeAppendix B). First, this means that the UV cutoff in the expression (3.45) is replaced by the vacuumvalue λ (cid:104) φ (cid:105) GS . Second, excitations of the scalar field near the new vacuum have the mass µ ∼ λ . Wereview the derivation of these statements in the Appendix B. The tree-level calculation of the subsection 3.2 indicates the decay of the strong scalar field φ = Et .Usually this means that loop corrections significantly perturb the ground state of the system. Whichmeans that the background field excites population of higher levels and anomalous averages [1–3, 6–8, 16].In this subsection we calculate loop corrections to the correlation functions and find that loop correctionsactually do not grow with time, unlike the case of strong electric and gravitational fields.Due to the non-stationarity of the theory in question we use the Schwinger-Keldysh diagrammatictechnique discussed in Sec. 2.2. Note that the definition (2.35) should be corrected to take into accountspinor indices of the fermions in two dimensions. For convenience we do the spatial Fourier transformation: G ±± ab ( x , x ) = (cid:90) dp π G ±± ab ( t , t ; p ) e ip ( x − x ) , (3.46)which gives the following expressions for the fermionic propagators: iG + − ab ( t , t ; p ) = ψ ap ψ c ∗ p (cid:0) γ (cid:1) cb = (cid:18) ψ p ψ ∗ p − ψ p ψ ∗ p ψ p ψ ∗ p − ψ p ψ ∗ p (cid:19) ,iG − + ab ( t , t ; p ) = − ˜ ψ ap ˜ ψ c ∗ p (cid:0) γ (cid:1) cb = (cid:32) − ˜ ψ p ˜ ψ ∗ p ˜ ψ p ˜ ψ ∗ p − ˜ ψ p ˜ ψ ∗ p ˜ ψ p ˜ ψ ∗ p (cid:33) = (cid:18) − ψ ∗ p ψ p − ψ ∗ p ψ p ψ ∗ p ψ p ψ ∗ p ψ p (cid:19) , (3.47)where a , b enumerate spinor indices and we denoted for short ψ (+) p,a ( t α ) = ψ apα , ψ ( − ) − p,a ( t α ) = ˜ ψ apα . Here wealso used the representation (3.2) for gamma-matrices, decomposition (3.4) and the relation ψ ( − ) − p ( t ) = − γ (cid:16) ψ (+) p ( t ) (cid:17) ∗ . Let us emphasize that we use the exact modes (3.22) and (3.23) rather than the planewaves (3.8).Corresponding bosonic propagators are as follows: iD + − ( t , t ; p ) = f p ( t ) f ∗ p ( t ) = 12 | p | e − i | p | ( t − t ) ,iD − + ( t , t ; p ) = f ∗ p ( t ) f p ( t ) = 12 | p | e i | p | ( t − t ) , (3.48)where functions f p ( t ) are nothing but the free modes of the scalar field: φ ( t, x ) = (cid:90) dp π (cid:104) α p f p ( t ) e ipx + α † p f p ( t ) ∗ e − ipx (cid:105) . (3.49)Operators α p and α † p satisfy the standard commutation relations: [ α p , α † q ] = 2 πδ ( p − q ).21sing mode decompositions for fermion and boson fields, one obtains that after the Keldysh rota-tion (2.38) the tree-level propagators have the following form: D K ( t , t ; p ) = 12 (cid:2) f p ( t ) f ∗ p ( t ) + f ∗ p ( t ) f p ( t ) (cid:3) ,D R/A ( t , t ; p ) = ± θ ( ± t ∓ t ) (cid:2) f p ( t ) f ∗ p ( t ) − f ∗ p ( t ) f p ( t ) (cid:3) , tr G Kab ( t , t ; p ) = 12 (cid:0) ψ p ψ ∗ p − ψ p ψ ∗ p + ψ ∗ p ψ p − ψ ∗ p ψ p (cid:1) , tr G R/Aab ( t , t ; p ) = ± θ ( ± t ∓ t ) (cid:0) ψ p ψ ∗ p − ψ p ψ ∗ p − ψ ∗ p ψ p + ψ ∗ p ψ p (cid:1) . (3.50)Apart from the other advantages (e.g. less bulky formulas), these notations allow one to study the behaviorof each p -mode separately. Namely, the retarded and advanced propagators carry information about thespectrum of quasi–particles, while the Keldysh propagators allow to specify the state of the theory. Infact, if one does the quantum average over an arbitrary state | χ (cid:105) which respects spatial translationalinvariance, the Keldysh propagators acquire the following form: D K ( t , t ; p ) = (cid:18) n p + 12 (cid:19) f p ( t ) f ∗ p ( t ) + κ p f p ( t ) f − p ( t ) + h.c., tr G Kab ( t , t ; p ) = (cid:18) − n (cid:48) p (cid:19) (cid:0) ψ p ψ ∗ p − ψ p ψ ∗ p (cid:1) − κ (cid:48) p (cid:0) ψ p ψ p + ψ p ψ p (cid:1) + ( c.c, p.c, h.c. ) , (3.51)where h.c. denotes Hermitian conjugation, p.c. denotes the change p → − p and c.c. denotes the change ψ (+) p → ψ ( − ) p . Also we introduced the notations as follows. First, the bosonic Keldysh propagatorincorporates the level population of bosons (cid:104) χ | α † p α p (cid:48) | χ (cid:105) ≡ πn p δ ( p − p (cid:48) ) and anomalous quantum av-erage (cid:104) χ | α p α − p (cid:48) | χ (cid:105) ≡ πκ p δ ( p − p (cid:48) ) and its complex conjugate. Second, the trace of the fermionicKeldysh propagator contains the level population of fermions (cid:104) χ | a † p a p (cid:48) | χ (cid:105) ≡ πn (cid:48) p δ ( p − p (cid:48) ), anti-fermions (cid:104) χ | b †− p b − p (cid:48) | χ (cid:105) ≡ π ˜ n (cid:48) p δ ( p − p (cid:48) ) and anomalous quantum average (cid:104) χ | a p b − p (cid:48) | χ (cid:105) ≡ πκ (cid:48) p δ ( p − p (cid:48) ) and itscomplex conjugate. Note that the tree–level retarded and advanced propagators are proportional to thecommutator [ φ, φ ] or anticommutator { ψ, ψ † } , correspondingly, which are c-numbers. I.e. the latterpropagators do not depend on the choice of the state | χ (cid:105) .Before turning on the interaction term, i.e. in the Gaussian theory, all these expectation values areexactly zero for the initial state ˆ a | (cid:105) = ˆ α | (cid:105) = 0 that we consider. However, they can grow in time in theinteracting case due to the non-stationarity of the background field. Namely, the secular growth of thelevel populations n p , n (cid:48) p or ˜ n (cid:48) p (if present) indicates the amplification of the higher levels (than zero pointfluctuations of the exact modes), whereas the growth of anomalous quantum averages (if present) meansthat the state of the theory at the start of the evolution is not the true vacuum state [2]. In the followingsections we estimate one–loop corrections to these averages (Fig. 8) and check their behavior at futureinfinity. In this subsubsection we calculate one-loop corrections to the boson two-point correlation functions(Fig. 8). For convenience we denote T = ( t + t ), τ = t − t , where t and t are the time argu-ments of the two-point functions. To simplify the expressions below we assume that the evolution of thesystem starts after the moment t = − T . Note that the full evolution time is T − t = 2 T . Then we takethe limit T → ∞ , fix τ (cid:28) T and single out the leading contributions in this limit. Such contributionsindicate the destiny of the state of the theory under consideration, because they tell about the timeevolution of n p ( T ) and κ p ( T ) introduced in the previous subsection. For short below we use the notation λEt a = αt a = M ( t a ) = M a .
22) b)Figure 8: One-loop corrections to the fermion (a) and boson (b) two-point functions. Solid lines correspondto the bare fermion propagators, dashed lines correspond to the bare boson propagatorsFirst, one can show that loop corrections to the retarded and advanced propagators never grow as T → ∞ and τ = const . In fact, due to the presence of the theta-function in these propagators one obtains thefollowing expression for the first loop correction to the retarded propagator: ∆ D R ( t , t ; p ) == − λ tr (cid:90) t t dt (cid:90) t t dt (cid:90) dq π D R ( t , t ; p ) G Rab (cid:18) t , t ; p + q (cid:19) G Kba (cid:18) t , t ; p − q (cid:19) D R ( t , t ; p ) . (3.52)Due to the limits of integration over t and t such an expression can grow only if τ → ∞ , but not when T → ∞ for fixed τ . The higher-order expressions posses similar behavior, because loop corrections do notchange the causal properties of the retarded and advanced propagators [2, 16–20].Now let us calculate the first loop correction to the Keldysh propagator:∆ D K ( t , t ; p ) = 12 (cid:2) ∆ D ++ ( t , t ; p ) + ∆ D −− ( t , t ; p ) (cid:3) == − λ (cid:90) dt dt (cid:90) dq π (cid:88) σ , , = { + , −} D σ σ ( p ) G σ σ (cid:18) p + q (cid:19) G σ σ (cid:18) p − q (cid:19) D σ σ ( p ) sgn( σ σ ) , (3.53)where we denote for short G ±± a a ( t , t ; p ) ≡ G ±± ( p ), D ±± ( t , t ; p ) ≡ D ±± ( p ) and assume the summationover the coincident spinor indices. Also we denote the one-loop corrections to the propagators D ++ and D −− as ∆ D ++ and ∆ D −− .Then we open the brackets in (3.53) and substitute the tree–level propagators (2.35), (2.36). As aresult, we obtain an expression of the form (3.51), in which leading contributions to the level populationand anomalous quantum average have the following form: n p ( T ) (cid:39) λ Re (cid:90) T − T dt (cid:90) t − T dt (cid:90) dq π e ip ( t − t ) p F ∗ ( t ) F ( t ) == λ p Re (cid:90) T dt (cid:90) t dt (cid:90) ∞ dqπ (cid:104) e ip ( t − t ) F ∗ ( t ) F ( t ) + sgn ( | p | − | q | ) e − ip ( t + t ) F ( t ) F ( t ) (cid:105) , (3.54)and κ p ( T ) (cid:39) − λ (cid:90) T − T dt (cid:90) t − T dt (cid:90) dq π e ip ( t + t ) p F ( t ) F ∗ ( t ) == − λ p (cid:90) T dt (cid:90) t dt (cid:90) ∞ dqπ [ F ( t ) F ∗ ( t ) cos ( p ( t + t )) + sgn ( | p | − | q | ) F ( t ) F ( t ) sin ( p ( t − t ))] . (3.55)23ere we neglect the subleading (in the limit T → ∞ , τ (cid:28) T ) contributions and introduce the function F ( t ) to simplify the expressions:tr (cid:20) G + − (cid:18) p + q (cid:19) G − +43 (cid:18) p − q (cid:19)(cid:21) = F ( t ) F ∗ ( t ) . (3.56)Using the expressions for the propagators (2.35) one obtains that: F ( t ) ≡ ψ (+) p + q , ( t ) ψ (+) p − q , ( t ) + ψ (+) p + q , ( t ) ψ (+) p − q , ( t ) , (3.57)In both identities (3.54) and (3.55) we have divided the area of the integration over t and t in a specificway and then used the property (3.31) of the modes. Also we assumed that p > F ( t ) under the change q → − q .It is instructive first to calculate integrals (3.54) and (3.55) in the theory without background field φ cl , i.e. when the fermion modes are just plane waves (3.8). Substituting these modes into the integrals,one finds: n p ( T ) (cid:39) λ (cid:90) Tt dt (cid:48) (cid:90) ∞−∞ dτ (cid:48) (cid:90) ∞−∞ dq π N e i ( ω p + q + ω p − q + | p | ) τ (cid:48) = (3.58)= λ (cid:90) Tt dt (cid:48) (cid:90) ∞−∞ dq π N δ (cid:16) ω p + q + ω p − q + | p | (cid:17) ∼ O ( λ T ) , and κ p ( T = + ∞ ) (cid:39) − λ (cid:90) ∞−∞ dt (cid:48) (cid:90) ∞ dτ (cid:48) (cid:90) ∞−∞ dq π N e i | p | t (cid:48) − i ( ω p + q + ω p − q ) τ (cid:48) = (3.59)= − λ (cid:90) ∞−∞ dq π N δ (2 | p | ) (cid:32) πδ (cid:16) ω p + q + ω p − q (cid:17) − P iω p + q + ω p − q (cid:33) ∼ O ( λ T ) . where N denotes the following expression: N = 116 ( p + q ) (cid:16) ω p + q − m (cid:17) + ( p − q ) (cid:16) ω p − q − m (cid:17) ω p + q ω p − q (cid:16) ω p + q − m (cid:17) (cid:16) ω p − q − m (cid:17) , (3.60)which depends on p and q and does not depend on t (cid:48) = t + t and τ (cid:48) = t − t . In the second integral weused the Sokhotski–Plemelj theorem and denoted the Cauchy principal value as P . Note that in κ p we putthe argument T = + ∞ and, hence, extended the limits of integrations over times to the infinity, becausewe would like to show that it is not divergent as T → + ∞ . Thus, one obtains either finite expression as T → + ∞ or an integration over delta-function whose argument is never zero. In other words, the one-loop correction to the free boson propagator does not grow with time T due to the energy conservationwhich is ensured by the delta-functions. This agrees with the fact that in stationary situations correlationfunctions depend only on the time difference t − t and do not depend on T = ( t + t ) / T → ∞ , τ (cid:28) T . Concretely, our goal here is to find if there arecontributions to n and κ which survive in the limit T → ∞ , λ → λ g ( T ) = const, where g ( T ) issome growing function of T (e.g. g ( T ) = T n for n ≥ g ( T ) = log T ).Using the expansions (3.28) and (3.29) one can estimate the function F ( t ): F ( t ) (cid:39) (cid:16)
1+ sgn( p − q )2 +
1+ sgn( q − p )2 2 pαt | q − p | + · · · (cid:17) e − i | p + q | + | p − q | t , if t < | p − q | α , (cid:16) √ + · · · (cid:17) e − iαt − i | p + q | t , if | p − q | α < t < | p + q | α , (cid:16) | p + q | αt + · · · (cid:17) e − iαt + i ( p q α log ( αt ) , if t > | p + q | α . (3.61) Note that the integral over dq in (3.59) converges. p < αT , grow faster, becausecorresponding low laying levels are easier to populate. Second, usually loop integrals receive leadingcontributions due to large virtual momenta, q > p — the main income into the lower p –levels comes fromthe higher q –levels. Finally, the intuition gained during the study of other background fields [2, 3, 6–11]tells us that the main contribution should come from the integrands of the form F ∗ ( t ) F ( t ) e ip ( t − t ) ,because in this case it is possible to single out the part of the integrand which does not depend on t (cid:48) = t + t . (Then the integral over dt (cid:48) may give the growing with T factor.) For all other combinationsof functions F ( t ) and e ipt this behavior is impossible , hence, their contributions are suppressed. Basedon this argumentation, consider the following integral ( p < αT ): I = (cid:90) ∞ dq (cid:90) T dt (cid:90) t dt F ∗ ( t ) F ( t ) e ip ( t − t ) (cid:39) (3.62) (cid:39) (cid:90) p dq (cid:34) (cid:90) p + q α dt (cid:90) t dt e ip ( t − t ) + (cid:90) T p + q α dt (cid:90) p + q α dt p αt e iαt − ipt ++ (cid:90) T p + q α dt (cid:90) t p + q α dt p α t t e iαt − iαt (cid:35) + (3.63)+ (cid:90) αT − pp dq (cid:34) (cid:90) q − p α dt (cid:90) t dt p α t t q e iq ( t − t ) + (cid:90) q + p αq − p α dt (cid:90) q − p α dt √ pαt q e iq ( t − t ) ++ 12 (cid:90) q + p αq − p α dt (cid:90) t q − p α dt e iq ( t − t ) + (cid:90) T q + p α dt (cid:90) q − p α dt q αt e iαt − iqt ++ (cid:90) T q + p α dt (cid:90) q + p αq − p α dt q √ αt e iαt − iqt + (cid:90) T q + p α dt (cid:90) t q + p α dt q α t t e iαt − iαt (cid:35) + (3.64)+ (cid:90) αT + p αT − p dq (cid:34) (cid:90) q − p α dt (cid:90) t dt p α t t q e iq ( t − t ) + (cid:90) T q − p α dt (cid:90) q − p α dt pαt q e iq ( t − t ) ++ 12 (cid:90) T q − p α dt (cid:90) t q − p α dt e iq ( t − t ) (cid:35) (3.65)+ (cid:90) ∞ αT + p dq (cid:90) T dt (cid:90) t dt p α t t q e iq ( t − t ) . (3.66)In this expression we threw away the subleading terms, i.e. held only leading absolute values and phasesof the integrands in the limit in question. However, even this rough estimate shows that there are only twoterms which can grow as T → ∞ (in the above formula these terms are enclosed in the boxes), whereasother contributions give constant or decaying with T corrections: I ≡ (cid:90) αT − pp dq (cid:90) q + p αq − p α dt (cid:90) t q − p α dt e iq ( t − t ) (cid:39) ip α log αTp + O (cid:16) pα (cid:17) , (3.67) I ≡ (cid:90) αT − pp dq (cid:90) T q + p α dt (cid:90) t q + p α dt q α t t e iαt − iαt (cid:39) i αT + ip αTp + O (cid:16) pα (cid:17) . (3.68) Except the combination F ( t ) F ∗ ( t ) e ip ( t − t ) , which is not presented in the integrals (3.54) and (3.55), and complexconjugated combinations. O (cid:0) pα (cid:1) denotes such a function g ( T ) that λg ( T ) = const as λ → T → ∞ . Now it is obvious thatsuch contributions cannot appear if the integrand contains F ( t ) F ( t ) instead of F ∗ ( t ) F ( t ), because inthis case oscillating terms do not cancel out: I ∼ (cid:90) αT − pp dq (cid:90) dt (cid:90) dt e iq ( t + t ) ∼ (cid:90) αT − pp dqq ∼ p , (3.69) I ∼ (cid:90) αT − pp dq (cid:90) T q + p α dt (cid:90) t q + p α dt q α t t e iαt + iαt ∼ (cid:90) αT − pp dq q e iαT α T − q e i ( q + p )24 α ( q + p ) ∼ p . (3.70)Also there is no any significant contribution if p > αT . In fact, in the latter case the line (3.64) is replacedby the line (3.65) which gives leading behavior similar to (3.67). However, this time it is bounded fromabove: I (cid:39) (cid:90) αT + pp dq (cid:90) T q − p α dt (cid:90) t q − p α dt e iq ( t − t ) + · · · (cid:39) (cid:18) ip α + iT (cid:19) log (cid:18) αTp (cid:19) − iT + · · · = O (cid:16) pα (cid:17) . (3.71)Thus, despite the fact that this integral grows at some time intervals, it is suppressed by big externalmomenta and does not diverge when T → ∞ .Now let us combine all the above observations to estimate the expressions (3.54) and (3.55). Keeping inmind the integrals (3.67) and (3.68), we consider small external momenta: p < αT , neglect the integrandsproportional to F ( t ) F ( t ) or F ∗ ( t ) F ∗ ( t ), and focus on the interval p < q < αT − p , q − p α < t < q + p α , q − p α < t < t . However, this time we calculate the integrals more accurately, i.e. we take into accountthe next-to-the-leading order terms in the phases of the exponents: n p ( T ) (cid:39) λ πp Re (cid:90) αT − pp dq (cid:90) q + p αq − p α dt (cid:90) t q − p α dt e i ( q + p )( t − t )+ ip ( t − t )+ iα ( t − t ) ++ λ πp Re (cid:90) αT − pp dq (cid:90) T q + p α dt (cid:90) t q + p α dt q α t t e iαt − iαt + ip ( t − t ) + λ πp O (cid:16) pα (cid:17) (cid:39)(cid:39) λ πp Re (cid:20) i αT + ip αTp + ipα log αTp + O (cid:16) pα (cid:17)(cid:21) ∼∼ λ O (cid:16) pα (cid:17) → , as λ → , T → ∞ , (3.72)and κ p ( T ) (cid:39) − λ πp (cid:90) αT − pp dq (cid:90) q + p αq − p α dt (cid:90) t q − p α dt e − i ( q + p )( t − t )+ iα ( t − t ) cos ( p ( t + t )) −− λ πp (cid:90) αT − pp dq (cid:90) T q + p α dt (cid:90) t q + p α dt q α t t e iαt − iαt cos ( p ( t + t )) − λ πp O (cid:16) pα (cid:17) (cid:39)(cid:39) − λ πp (cid:90) αT − pp dq i sin (cid:16) p α (cid:17) p cos (cid:0) pqα (cid:1) q + sin(2 pT )8 p q α T − sin (cid:0) pqα (cid:1) pq − λ πp O (cid:16) pα (cid:17) (cid:39)∼ λ p sin (cid:18) p α (cid:19) Ci (cid:18) p α (cid:19) + λ p sin(2 pT ) + λ O (cid:16) pα (cid:17) → , as λ → , T → ∞ , (3.73)where Ci( x ) is the cosine integral. In essence, integral (3.72) does not grow with T because it is real and theintegral (3.73) does not grow due to the oscillating term cos ( p ( t + t )). Thus, both level population andanomalous quantum average do not grow in the limit T → ∞ . They are generated, because the situation26s not stationary, but are suppressed by the small λ factor, which is not accompanied by a growing factor T n , n ≥
1. This situation is very different from the case of strong electric and gravitational fields [1–3,6–8].The technical reason for the absence of the secular growth in the background scalar field as opposedto its presence e.g. in constant electric field or de Sitter space can be explained as follows. In the constantelectric field (de Sitter space) all the quantities depend on the invariant/physical momenta p − eEt ( | (cid:126)p | e − t/H ). (Here p is the component of the momentum along the external electric field E and H is theHubble constant in the case of the de Sitter space.) As the result all physical quantities are invariant underthe simultaneous translations t → t − a and p → p − eEa ( | (cid:126)p | → | (cid:126)p | e − a/H ). Due to such symmetriesthe integrands of ( t + t ) / T .At the same time in the background scalar field under consideration there is no such a symmetry.Finally, note that Wightman functions D + − and D − + also do not receive growing corrections in thelimit λ → T → ∞ for the same reasons. As we have shown above, these correlation functions canreceive growing corrections only from the integrals of the form (3.67) and (3.68); however, both D + − and D − + contain only the real part of these integrals. This is consistent with our observations above, becauseimaginary part of such correlation functions is proportional to the retarded propagator, which does notgrow in the limit in question. In this subsubsection we calculate one-loop corrections to the fermion two-point functions (Fig. 8). Wealso work in the same limit for times T and τ as in the previous subsubsection and set t = − T .For convenience here we restore the mass of the boson field: S = (cid:90) d x (cid:20) ∂ µ φ∂ µ φ − µ φ + i ¯ ψ /∂ψ − λφ ¯ ψψ (cid:21) . (3.74)On one hand, it allows us to avoid uncontrollable infrared divergences in the loop integrals due to massless2D scalar field. On the other hand, it is a standard textbook exercise to show that the scalar fieldspontaneously acquires a mass µ ∼ λ (see Appendix B). We use this estimate to roughly check theself-consistency of the expressions below.Obviously, loop corrections to the fermion retarded and advanced propagators do not grow with time.In fact, these propagators have the same causal properties as boson retarded and advanced propagators,and hence the reasoning of the previous subsubsection also works for them.First loop correction to the fermionic Keldysh propagator is given by the following expression:∆ G Kab ( t , t ; p ) = 12 (cid:2) ∆ G ++ ab ( t , t ; p ) + ∆ G −− ab ( t , t ; p ) (cid:3) == − λ (cid:90) dt dt (cid:90) dq π (cid:88) σ , , = { + , −} G σ σ ( p ) G σ σ ( q ) D σ σ ( p − q ) G σ σ ( p ) sgn( σ σ ) . (3.75)Then we open the brackets, substitute the expressions (2.35) and (2.36), take the trace over the externalspinor indices and obtain the following leading contributions to the fermion level density and anomalousquantum average: 27 (cid:48) p ( T ) (cid:39) − λ Re (cid:90) T − T dt (cid:90) t − T dt (cid:90) dq π e i | p − q | ( t − t ) | p − q | (cid:0) ψ ∗ p, ψ ∗ q, + ψ ∗ p, ψ ∗ q, (cid:1) (cid:0) ψ q, ψ p, + ψ q, ψ p, (cid:1) , (cid:39) − λ Re (cid:90) T dt (cid:90) t dt (cid:90) dq π (cid:34) e i | p − q | ( t − t ) | p − q | H ∗ ( t ) H ( t ) + sgn q e i | p − q | ( t + t ) | p − q | H ∗ ( t ) H ∗ ( t ) (cid:35) , (3.76)and κ (cid:48) p ( T ) (cid:39) λ (cid:90) T − T dt (cid:90) t − T dt (cid:90) dq π e − i | p − q | ( t − t ) | p − q | (cid:0) ψ ∗ p, ψ q, − ψ ∗ p, ψ q, (cid:1) (cid:0) ψ ∗ q, ψ ∗ p, + ψ ∗ q, ψ ∗ p, (cid:1) (cid:39)(cid:39) λ (cid:90) T dt (cid:90) t dt (cid:90) dq π (cid:34) e i | p − q | ( t − t ) | p − q | (cid:16) ˜ H ( t ) H ∗ ( t ) − H ( t ) ˜ H ∗ ( t ) (cid:17) ++ sgn q e i | p − q | ( t + t ) | p − q | (cid:16) ˜ H ( t ) H ( t ) + H ( t ) ˜ H ( t ) (cid:17) (cid:35) , (3.77)where we introduced functions H ( t ) and ˜ H ( t ), which are defined as: H ( t ) ≡ ψ (+) p, ( t ) ψ (+) q, ( t ) + ψ (+) p, ( t ) ψ (+) q, ( t ) , ˜ H ( t ) ≡ ψ (+) ∗ p, ( t ) ψ (+) q, ( t ) − ψ (+) ∗ p, ( t ) ψ (+) q, ( t ) . (3.78)As in the previous subsubsection, we have divided the area of the integration over t and t in a specificway and then used the property (3.31) of the modes to obtain expressions (3.76) and (3.77). Also weassumed that p > φ cl = 0. As in the boson loop calculation (Sec. 3.3.1), it is straightforward to show that one-loopcorrections to the fermion quantum expectation values do not grow with T : n (cid:48) p ( T ) (cid:39) λ (cid:90) Tt dt (cid:48) (cid:90) ∞−∞ dτ (cid:48) (cid:90) dq π M e i ( ω p + ω q + | p − q | ) τ (cid:48) (cid:39)(cid:39) λ (cid:90) Tt dt (cid:48) (cid:90) dq M δ ( ω p + ω q + | p − q | ) ∼ O ( T ) , (3.79)and κ (cid:48) p ( T ) (cid:39) λ T (cid:90) t dt (cid:48) + ∞ (cid:90) dτ (cid:90) dq π N e iω p t (cid:48) e − i ( | p − q | + ω q ) τ (cid:48) == 2 λ (cid:90) dq N δ (2 ω p ) (cid:18) πδ ( | p − q | + ω q ) − P i | p − q | + ω q (cid:19) ∼ O ( T ) . (3.80)Here we have made the following substitutions: t (cid:48) = t + t , τ (cid:48) = t − t , and singled out the time-independent parts of the integrands:1 | p − q | (cid:0) ψ ∗ p, ψ ∗ q, + ψ ∗ p, ψ ∗ q, (cid:1) (cid:0) ψ q, ψ p, + ψ q, ψ p, (cid:1) = M e i ( ω p + ω q )( t − t ) , where M ≡ | p − q | ( p ( ω q − m ) + q ( ω p − m )) ω p ω q ( ω p − m )( ω q − m ) , | p − q | (cid:0) ψ ∗ p, ψ q, − ψ ∗ p, ψ q, (cid:1) (cid:0) ψ ∗ q, ψ ∗ p, + ψ ∗ q, ψ ∗ p, (cid:1) = N e − iω q ( t − t )+ iω p ( t + t ) , where N ≡ | p − q | ( pq − ( ω p − m )( ω q − m )) ( p ( ω q − m ) + q ( ω p − m ))4 ω p ω q ( ω p − m )( ω q − m ) . (3.81)28s in boson calculation (Sec. 3.3.1), integrals do not grow due to the delta-functions which ensure theenergy conservation law. As the result the two–point functions depend only on the time difference t − t ,as it should be in stationary situations.However, in the strong scalar field background there is no energy conservation. At the same time theintegrals (3.76) and (3.77) again cannot be taken exactly. Hence, we estimate them in the limit T → ∞ , τ (cid:28) T . Using expansion (3.28) and (3.29) one can find the behavior of H ( t ) and ˜ H ( t ): H ( t ) (cid:39) (cid:16) sgn q +12 + sgn q − α | t | ( | q |− p )2 | q | p (cid:17) e − i ( p + | q − p | ) t , if t < min( p, | q | ) , sgn q √ (cid:0) αt (cid:1) ip α e − iαt − i | q | t , if p < | q | and p < α | t | < | q | , √ (cid:0) αt (cid:1) iq α e − iαt − ipt , if | q | < p and | q | < α | t | < p, p + q α | t | (cid:0) αt (cid:1) i ( p q α e − iαt , if t > max( p, | q | ) , (3.82)˜ H ( t ) (cid:39) (cid:16) − sgn q + sgn q +12 α | t | ( | q | + p )2 | q | p (cid:17) e − i ( p −| q − p | ) t , if t < min( p, | q | ) , √ (cid:0) αt (cid:1) − ip α e iαt − i | q | t , if p < | q | and p < α | t | < | q | , √ (cid:0) αt (cid:1) iq α e − iαt + ipt , if | q | < p and | q | < α | t | < p, (cid:0) αt (cid:1) i ( q − p α , if t > max( p, | q | ) . (3.83)Here we showed only the leading terms in the exponents and their prefactors, as in the previous subsub-section.Note that integrals of H ( t ) H ( t ) and H ( t ) ˜ H ( t ) (and similar expressions) are suppressed in com-parison with the integral over H ∗ ( t ) H ( t ), because the former always contain oscillating factors of both t − t and t + t simultaneously. Hence, due to the same argumentation as in the previous subsubsection,if we would like to single out a growing contribution in the limit T → ∞ , it is sufficient to consider thefollowing integral (we assume p < αT ): I = (cid:90) ∞ dq (cid:90) T dt (cid:90) t dt (cid:32) H ∗ ( t ) H ( t ) e i | q − p | ( t − t ) | q − p | + ( q → − q ) (cid:33) = (3.84)= (cid:34) (cid:90) p − µ dq (cid:32) (cid:90) qα dt (cid:90) t dt + (cid:90) pαqα dt (cid:90) qα dt + (cid:90) pαqα dt (cid:90) t qα dt + (3.85)+ (cid:90) T pα dt (cid:90) qα dt + (cid:90) T pα dt (cid:90) pαqα dt + (cid:90) T pα dt (cid:90) t pα dt (cid:33) ++ (cid:90) p + µp − µ dq (cid:32) (cid:90) pα dt (cid:90) t dt + (cid:90) T pα dt (cid:90) pα dt + (cid:90) T pα dt (cid:90) t pα dt (cid:33) + (3.86)+ (cid:90) αTp + µ dq (cid:32) (cid:90) pα dt (cid:90) t dt + (cid:90) qαpα dt (cid:90) pα dt + (cid:90) qαpα dt (cid:90) t pα dt + (3.87)+ (cid:90) T qα dt (cid:90) pα dt + (cid:90) T qα dt (cid:90) qαpα dt + (cid:90) T qα dt (cid:90) t qα dt (cid:33) ++ (cid:90) ∞ αT dq (cid:32) (cid:90) pα dt (cid:90) t dt + (cid:90) T pα dt (cid:90) pα dt + (cid:90) T pα dt (cid:90) t pα dt (cid:33)(cid:35) × (3.88) × (cid:32) H ∗ ( t ) H ( t ) e i | q − p | ( t − t ) | q − p | + ( q → − q ) (cid:33) . µ (cid:54) = 0, i.e. excluded the integration interval q ∈ [ p − µ, p + µ ] to getrid of the logarithmic infrared divergencies from the virtual boson. Considering each term in the abovesum and using corresponding expansions from (3.82) one finds that the only terms which potentially cangrow with T come from the integrals in boxes: I (cid:39) i α log αTp + O (cid:18) α (cid:19) · log pµ + O (cid:18) α (cid:19) . (3.89)Here O (cid:0) α (cid:1) denotes such a function g ( T ) that λg ( T ) = const as λ → T → ∞ . Note that suchintegrals do not grow if p > αT (in this case they are bounded from above) or if the integrand containsother combinations of H ( t ), H ∗ ( t ), ˜ H ( t ) and ˜ H ∗ ( t ) (in this case time-oscillating functions reduce thegrowth rate at least by one power of T ). Therefore, we get only non-growing with T contributions bothin level population and anomalous quantum averages for fermions: n (cid:48) p ( T ) ∼ κ (cid:48) p ( T ) ∼ λ · log pµ · O (cid:18) α (cid:19) → , as λ → T → ∞ . (3.90)This limit holds even if we substitute the mass µ ∼ λ expected from the standard equilibrium analysis(Appendix B). Thus, for the fermions the situation is similar to the one for bosons. To make a thorough analysis in this subsubsection we calculate one-loop correction to the three-pointcorrelation function G ±±± ab ( x , x , x ), i.e. to the vertex (Fig. 9). Note that in non–stationary situationsin strong background fields vertexes potentially can also show a secular growth [4].To single out the growing contributions, if any, we consider the limit | t i − t j | (cid:28) T and ( t + t + t ) = T → ∞ . For convenience we work before the Keldysh rotation (2.35), (2.36) and do spatial Fouriertransformation. We set external momenta of the three-point correlation function | p | , | q | → | r | (cid:29) αT (see fig. 9). On general physical grounds one canexpect that the growing contribution, if any, comes from this region of physical parameters. A genericcontribution in this limit has the following form: 30 G ±±± ∼ (cid:90) Tt dt dt dt (cid:90) | r | >M dr | r | e ± i | r | ( t − t ) ± i | r − p − q | ( t − t ) ± i | r − p | ( t − t ) ± i | q | t ± iαt α ± iαt α ∼∼ (cid:90) Tt dt dt dt (cid:90) ∞ M drr e ± i | r | ( t − t ) ± i | r | ( t − t ) ± i | r | ( t − t ) ± i | q | t ± iαt ± iαt cos ( ±| p + q | ( t − t ) ± | p | ( t − t )) , (3.91)where we took into account different signs of the virtual momentum r . Let us estimate the expression (3.91)for different combinations of signs. For this purpose we need the following integral which is saturated inthe vicinity of zero: (cid:90) t e ix + iρx dx = 1 + i √ π + O (cid:18) t (cid:19) + O ( ρ ) , if ρ (cid:28) (cid:90) t e ix + iρx dx = iρ + O (cid:18) t (cid:19) + O (cid:18) ρ (cid:19) , if ρ (cid:29) . (3.92)First, consider the situation when the exponent in the second line of (3.91) vanishes, i.e. all termswhich are proportional to | r | cancel each other. In this case the integral (3.91) reduces to the followingexpression: ∆ G ±±± ∼ (cid:90) Tt dt e ± i ( | p + q |−| p |±| q | ) t (cid:90) Λ M drr (cid:46) ( T − t ) log Λ √ α . (3.93)Naively one can think that such a term gives growing with T contribution. However, such a term alwaysappears with the following products of theta-functions: θ θ θ or θ θ θ , which are identically zero.Hence, this growth does not occur in the vertex.Second, consider the case when the exponent (3.91) does not contain the term i | r | t , but contain terms ± i | r | t and ± i | r | t . Then:∆ G ±±± ∼ (cid:90) Tt dt e ± i ( | p + q |−| p |±| q | ) t (cid:90) Λ M drr (cid:46) T − t M (cid:46) O ( T ) . (3.94)Finally, consider a situation when the time t does not cancel out in the exponent (3.91). Integrating out t and t , one obtains the following expression:∆ G ±±± (cid:46) (cid:90) Tt dt (cid:90) Λ M drr e ± irt ∼ (cid:90) Tt dt αT t (cid:46) O ( T ) . (3.95)Our arguments here are generic and, hence, are applicable also to other vertex corrections and to othertypes of vertexes. Thus, we can conclude that one-loop corrections to the three-point correlation functionsalso do not grow in the limit T → ∞ . In this section we consider the same theory as above (3.1), but in a different background field. We usethe following representation for the Clifford algebra: γ = (cid:18) (cid:19) , γ = (cid:18) − i i (cid:19) , (4.1)and consider the background field which linearly grows with space coordinate: φ cl = mλ + Ex, ψ cl = 0 . (4.2)31ithout loss of generality, we restrict our attention to the case E >
0, since the case
E < x → − x . Specifically, in the limit E → m . However, note that in the background field this mass can be removed by the translation x → x − mλE . In this case the situation is obviously the same as in the time–dependent background above. To set up the notations consider again the free massive Dirac field. Unlike the case of the subsection 3.1,here we have to use the following decomposition for the field: ψ ( x, t ) = (cid:90) | ω | >m dω π (cid:104) a ω ψ ( x, ω ) e − iωt + b † ω (cid:101) ψ ( x, ω ) e iωt (cid:105) , (4.3)because in the background that we consider in this section there is time translational invariance ratherthan the spatial one.The functions ψ ( x, ω ) e − iωt and (cid:101) ψ ( x, ω ) e iωt solve the free equations of motion (3.5) and creation andannihilation operators a ω and b ω obey the standard anticommutation relations which are similar to (3.6).This fixes the equal-time anticommutation relations (3.7). The frequency in this expression runs in theinterval ω ∈ ( −∞ , − m ] ∪ [ m, ∞ ).We emphasize that the definite-frequency operators a ω , b ω and definite-momentum operators a p , b p do not coincide, in particular they act differently on the vacuum state. Namely, there are four non-zeroexpectation values of the product of two operators: (cid:104) | a ω a † ω (cid:48) | (cid:105) = (cid:104) | b ω b † ω (cid:48) | (cid:105) = θ ( ω ) × πδ ( ω − ω (cid:48) ) , (cid:104) | a † ω a ω (cid:48) | (cid:105) = (cid:104) | b † ω b ω (cid:48) | (cid:105) = θ ( − ω ) × πδ ( ω − ω (cid:48) ) , (4.4)where | (cid:105) is the standard vacuum state, which is annihilated by the operators a p and b p . This is due to thefact that the creation operators with definite positive frequency correspond to the creation operators withdefinite positive momentum, whereas the creation operators with definite negative frequency correspondto the charge-conjugated annihilation operators with definite negative momentum. For the details seeappendix C.The form of ψ ( x, ω ) and (cid:101) ψ ( x, ω ) spinors is as follows: ψ ( x, ω ) = 1 (cid:112) | ω | p (cid:18) ωm − ip (cid:19) e ipx , (cid:101) ψ ( x, ω ) = 1 (cid:112) | ω | p (cid:18) ω − m − ip (cid:19) e − ipx , (4.5)where p = √ ω − m and we have used the Dirac representation for gamma-matrices (4.1).Now, let us consider the Dirac field on the classical background φ cl = mλ + Ex . In this case we havethe analog of the decomposition (4.3), but with the modes that solve the following equation:( i /∂ − m − αx ) ψ ω ( x, t ) = 0 , (4.6)where we have defined for short α = λE .Because of the time translational invariance of the equations of motion one can do the time Fouriertransformation and obtain the equation for the spatial coordinate dependent part of the modes: (cid:2) iγ ( − iω ) + iγ ∂ x − m − αx (cid:3) ψ ( x, ω ) = 0 . (4.7)As in the time-dependent field case (Sec. 3.1), one can decouple this system applying the operator (cid:2) − γ ω − iγ ∂ x − m − αx (cid:3) to its left hand side. Then the system reduces to: (cid:40)(cid:2) ∂ x − ( m + αx ) + ω − α (cid:3) ψ ( x, ω ) = 0 , (cid:2) ∂ x − ( m + αx ) + ω + α (cid:3) ψ ( x, ω ) = 0 . (4.8) Note that in the subsection 3.1 we did the spatial Fourier transformation. D ν ( z ): ψ ( x, ω ) = C ( ω ) D ν − ( z ) + C ( ω ) D − ν ( iz ) ,ψ ( x, ω ) = B ( ω ) D ν ( z ) + B ( ω ) D − ν − ( iz ) , (4.9)where C , , B , are complex constants which we will fix below, and for convenience we define: ν ≡ ω α , z ( x ) ≡ (cid:114) α ( m + αx ) . (4.10)Note that these variables are real unlike the φ cl = Et case (3.15).In order to fix the integration constants C , , B , one should impose additional constraints on themodes (4.9). To do this, consider the limit | ω | (cid:29) √ α for a fixed x . We expect that the modes in the scalarbackground and without it have similar behavior in such a limit, because high energy modes should notbe sensitive to a smooth background field. In other words, the modes (4.9) must behave as plane waves(i.e. as e − iωt ± i | ω | x ) for ω → ∞ . We refer to functions with asymptotic behavior ∼ e − iωt + i | ω | x as “positivefrequency modes” and functions ∼ e iωt − i | ω | x as “negative frequency modes”. As above we choose suchmodes to have the proper Hadamard behaviour of the propagators. More generic choice of the modes isalso possible, as we have discussed at the end of the subsection 3.1.Note that one obtains the “negative frequency” modes from the “positive frequency” ones by thefollowing operation: (cid:101) ψ ( x, ω ) = iγ ψ ∗ ( x, ω ) or (cid:101) ψ ( x, ω ) = − ψ ∗ ( x, − ω ) . (4.11)Consider the anticommutation relation (3.7): (cid:110) ψ a ( t, x ) , ψ † b ( t, y ) (cid:111) = (cid:90) | ω | >m dω π (cid:104) ψ a ( x, ω ) ψ † b ( y, ω ) + (cid:101) ψ a ( x, − ω ) (cid:101) ψ † b ( y, − ω ) (cid:105) = δ ( x − y ) δ ab , (4.12)where a, b = 1 , ψ ( x, ω ) ∼ e i | ω | x in the limit ω → ∞ , we get the following asymptotic behavior at high frequencies: ψ a ( x, ω ) ψ † b ( y, ω ) = 12 e i | ω | ( x − y ) δ ab . (4.13)Hence, the asymptotic behavior of ψ ( x, ω ) for ω → ∞ is as follows: ψ ( x, ω ) = 1 √ e i | ω | x + iϕ ( ω ) , (4.14)where ϕ ( ω ) is some coordinate independent phase.Now, using the asymptotics of parabolic cylinder functions for large values of their parameter [28, 29,34], we choose the coefficients C , ( ω ) in (4.9) in order to get the exponent: ψ ( x, ω ) = √ e i | ω | x in the limit | ω | (cid:29) √ α , | m + αx | (cid:28) | ω | due to (4.14). Thus, we obtain the first component of the positive-frequencymode ψ ( x, ω ): ψ ( x, ω ) = 12 e iπω α − i | ω | mα (cid:40) e iπω α e − ω α + ω α log ω α D − ν ( iz ) − i | ω |√ α e ω α − ω α log ω α D ν − ( z ) (cid:41) . (4.15)We can get rid of the phase factor due to its arbitrariness:33 ( x, ω ) = 12 (cid:40) e iπω α e − ω α + ω α log ω α D − ν ( iz ) − i | ω |√ α e ω α − ω α log ω α D ν − ( z ) (cid:41) . (4.16)Then ψ ( x, ω ) can be found from the system of equations (4.7): ψ ( x, ω ) = 1 ω ( m + αx − ∂ x ) ψ ( x, ω ) = i (cid:40) e iπω α e − ω α + ω α log ω α | ω |√ α D − ν − ( iz ) − e ω α − ω α log ω α D ν ( z ) (cid:41) sgn( ω ) . (4.17)Here we have used the relations (3.21). The expressions for the negative frequency modes are obtainedusing the relation (4.11). According to the operator equations of motion (3.37), one needs to calculate the classical current j cl ( x ) ≡(cid:104) ˆ¯ ψ ˆ ψ (cid:105) to find the response of the classical field φ cl = (cid:104) ˆ φ (cid:105) . We expand the fermion field over the modes (4.3),substitute the expectation values (4.4) and use the symmetry (4.11) to find the expression for this current: (cid:104) | ¯ ψψ | (cid:105) = 2 Λ (cid:90) m dω π (cid:16) (cid:101) ψ (cid:101) ψ ∗ + (cid:101) ψ ∗ (cid:101) ψ (cid:17) , (4.18)where | (cid:105) is the state, which is annihilated by the definite-momentum annihilation operators (see AppendixC for the notations). Note that the integration is performed only over the positive frequencies.For the same reason as in the subsection 3.2 we expect the following dependence for the current (3.40)on the φ cl = mλ + Ex background: (cid:10) ¯ ψψ (cid:11) (cid:39) λφ cl π log λφ cl . (4.19)Note that in this case the analog of the mass parameter is M ( x ) = λφ cl = m + λEx. (4.20)Let us check this conjecture and calculate the integral (4.18). Note again that M ( x ) = m + αx indefinitely grows with x -coordinate, so it can overcome an arbitrarily large fixed scale Λ. However, thecase of M >
Λ is not realistic, because the infinitely growing field φ cl is not a physically meaningfulsituation, as we have already mentioned several times. Below we assume that M ( x ) (cid:29) α to single outthe leading contributions.In the case M <
Λ we divide the region of integration into two segments: [ m, Λ] = [ m, M ] + [ M, Λ].The asymptotics (3.16) for the parabolic cylinder functions is valid over the integration interval [ m, M ]: (cid:10) ¯ ψψ (cid:11) (cid:39) M (cid:90) m dω π M cosh (cid:20) M − ω α + ω α log ω M (cid:21) ∼ αM e M α . (4.21)In the interval [ M, Λ] we use the following asymptotic form of the function U ( A, z ) ≡ D − A − ( z ) , which works for A → −∞ , − √− A < | z | < √− A , − π < arg z < π , [28, 29, 34]: U (cid:18) − µ , µτ √ (cid:19) (cid:39) g ( µ )(1 − τ ) / (cid:32) cos κ ∞ (cid:88) s =0 ( − s ˜ A s ( τ ) µ s − sin κ ∞ (cid:88) s =0 ( − s ˜ A s +1 ( τ ) µ s +2 (cid:33) , (4.22)34here g ( µ ) (cid:39) h ( µ ) (cid:32) ∞ (cid:88) s =1 γ s (cid:0) µ (cid:1) s (cid:33) , h ( µ ) = 2 − µ − e − µ µ µ − ,κ = µ η − π , η = 12 arccos τ − τ (cid:112) − τ , ˜ A s ( τ ) = u s ( τ )(1 − τ ) s ,u s ( τ ) are polynomials of τ , γ s are numbers depending on s ; all that matters is that u ( τ ) = 1. In ourcase µ = ω α − , µτ = m + αx √ α . (4.23)Taking limits µ → + ∞ , τ →
0, we leave the first term from the asymptotic expansion (4.22): U (cid:18) − µ , µτ √ (cid:19) (cid:39) h ( µ )(1 − τ ) / cos (cid:18) µ τ − πµ π (cid:19) (4.24)where we used that g ( µ ) (cid:39) h ( µ ), κ (cid:39) π (cid:0) µ − (cid:1) − µ τ , η (cid:39) π − τ . Then we rotate the variable µ → iµ and obtain: U (cid:18) µ , iµτ √ (cid:19) (cid:39) e − iπ µ − iπ √ µh ( µ )(1 − τ ) / cos (cid:18) µ τ − πµ − π (cid:19) . (4.25)Using these formulas and multiplying by an x -independent phase, we find the asymptotic behavior of thecomponents of the Dirac field: ψ ( x, ω ) (cid:39) (cid:114) ω e i | ω | x ( ω − M ( x )) / , (cid:101) ψ ( x, ω ) (cid:39) (cid:114) ω e − i | ω | x ( ω − M ( x )) / ,ψ ( x, ω ) (cid:39) M ( x ) − i | ω |√ ω e i | ω | x ( ω − M ( x )) / , (cid:101) ψ ( x, ω ) (cid:39) − M ( x ) + i | ω |√ ω e − i | ω | x ( ω − M ( x )) / . (4.26)Then the integrand for the scalar current acquires the following form: (cid:101) ψ (cid:101) ψ ∗ + (cid:101) ψ ∗ (cid:101) ψ = − M ( x ) (cid:112) ω − M ( x ) + · · · , (4.27)where we denoted the subleading (in the limit in question) contribution by ellipsis.Finally, we obtain the following expression for the scalar current: (cid:104) ¯ ψψ (cid:105) (cid:39) (cid:104) ¯ ψψ (cid:105) (cid:39) − Λ (cid:90) M dω π M ( x ) (cid:112) ω − M ( x ) = − π M log Λ + √ Λ − M M (cid:39) M ( x ) π log M ( x )2Λ , (4.28)where we neglected the subleading contributions in the limit √ λE (cid:28) λφ cl ( x ) (cid:28) Λ. We also neglected (cid:104) ¯ ψψ (cid:105) supposing that Λ is very big: M α log Λ M (cid:29) e M α . The obtained result (4.28) coincides with theproposal (4.19).Thus, again we obtain a peculiar behaviour of the scalar current for the large and slowly changingbackground field, which agrees with the result of [12] and of the previous section. We explain such adependence of the scalar current on the background field in the Appendix B and in the Concludingsection. 35 .3 Loop corrections We do the time Fourier transformation of the two dimensional analog of (2.35): G ±± ab ( x , x ) = (cid:90) | ω | >m dω π G ±± ab ( x , x ; ω ) e − iω ( t − t ) , (4.29)where we denoted x = ( t, x ). Then: G + − ab ( x , x ; ω ) = 2Re ψ aω ψ c ∗ ω ( γ ) cb θ ( ω ) = 2Re (cid:18) ψ ω ψ ∗ ω ψ ω ψ ∗ ω ψ ω ψ ∗ ω ψ ω ψ ∗ ω (cid:19) θ ( ω ) ≡ G ( ω ) θ ( ω ) ,G − + ab ( x , x ; ω ) = − ψ aω ˜ ψ c ∗ ω ( γ ) cb θ ( − ω ) = − (cid:18) ψ ∗ ω ψ ω ψ ∗ ω ψ ω ψ ∗ ω ψ ω ψ ∗ ω ψ ω (cid:19) θ ( − ω ) = − G ( ω ) θ ( − ω ) , (4.30)where we use the notations ψ a ( ω, x α ) = ψ aωα , (cid:101) ψ a ( − ω, x α ) = ˜ ψ aωα . We also use the representation forgamma matrices (4.1), decomposition (4.3) and relation (4.11).At the same time, using the expansion for the boson field: φ ( t, x ) = + ∞ (cid:90) −∞ dω π (cid:104) α ω f ω ( x ) e − iωt + α † ω f ∗ ω ( x ) e iωt (cid:105) , (4.31)where f ω ( x ) = √ | ω | e i | ω | x , and doing the time Fourier transformation of the two dimensional analogof (2.36): D ±± ( x , x ) = + ∞ (cid:90) −∞ dω π D ±± ( x , x ; ω ) e − iω ( t − t ) , (4.32)we obtain (see appendix C for the details on definite-frequency operators α ω and α † ω ): D + − ( x , x ; ω ) = [ f ω ( x ) f ∗ ω ( x ) + h.c. ] θ ( ω ) = cos {| ω | ( x − x ) }| ω | θ ( ω ) ≡ D ( x − x ; | ω | ) θ ( ω ) ,D − + ( x , x ; ω ) = [ f ω ( x ) f ∗ ω ( x ) + h.c. ] θ ( − ω ) = cos {| ω | ( x − x ) }| ω | θ ( − ω ) = D ( x − x ; | ω | ) θ ( − ω ) . (4.33)It is convenient to do the Keldysh rotation (2.38) and keep in mind that if one does the quantum averageover an arbitrary state | χ (cid:105) , the Keldysh propagators acquire the following form: D K ( x , x ) = (cid:90) dω π (cid:90) dω (cid:48) π (cid:20) (cid:18) n ωω (cid:48) + 12 2 πδ ( ω − ω (cid:48) ) (cid:19) f ω ( x ) f ∗ ω (cid:48) ( x ) + κ ωω (cid:48) f ω ( x ) f ω (cid:48) ( x ) + h.c. (cid:21) e − iωt + iω (cid:48) t , tr G Kab ( x , x ) = (cid:90) dω π (cid:90) dω (cid:48) π (cid:20) (cid:18)
12 2 πδ ( ω − ω (cid:48) ) − n (cid:48) ωω (cid:48) (cid:19) (cid:0) ψ ω ψ ∗ ω (cid:48) + ψ ω ψ ∗ ω (cid:48) (cid:1) ++ κ (cid:48) ωω (cid:48) (cid:0) ψ ω ψ ω (cid:48) + ψ ω ψ ω (cid:48) (cid:1) + ( c.c, p.c, h.c. ) (cid:21) e − iωt + iω (cid:48) t . (4.34)Here we have introduced the following notations for the quantum averages. First, the bosonic Keldyshpropagator contains (cid:104) χ | α † ω α ω (cid:48) | χ (cid:105) ≡ n ωω (cid:48) and (cid:104) χ | α ω α † ω (cid:48) | χ (cid:105) ≡ ˜ n ωω (cid:48) , anomalous quantum averages (cid:104) χ | α ω α − ω (cid:48) | χ (cid:105) ≡ κ ωω (cid:48) and (cid:104) χ | α † ω α †− ω (cid:48) | χ (cid:105) ≡ ˜ κ ωω (cid:48) . Second, the trace of the fermionic Keldysh propagatorcontains (cid:104) χ | a † ω a ω (cid:48) | χ (cid:105) ≡ n (cid:48) ωω (cid:48) , (cid:104) χ | b ω b † ω (cid:48) | χ (cid:105) ≡ ˜ n (cid:48) ωω (cid:48) , anomalous quantum averages (cid:104) χ | a ω b − ω (cid:48) | χ (cid:105) ≡ κ (cid:48) ωω (cid:48) and (cid:104) χ | a † ω b †− ω (cid:48) | χ (cid:105) ≡ ˜ κ (cid:48) ωω (cid:48) . 36igure 10: One-loop correction to the boson two-point function with energy conservation laws Similarly to the time-dependent background field, one can show that loop corrections to the retarded andadvanced propagators do not grow, when | t − t | (cid:28) t + t = T → ∞ . Let us now calculate the one–loopcorrection to the Keldysh propagator:∆ D K ( x , x ) = 12 (cid:20) ∆ D ++ ( x , x ) + ∆ D −− ( x , x ) (cid:21) == − λ (cid:90) d x d x (cid:88) σ , , = { + , −} D σ σ ( x , x ) G σ σ ab ( x , x ) G σ σ ba ( x , x ) D σ σ ( x , x ) sgn( σ σ ) (cid:39)(cid:39) λ (cid:90) dx dx + ∞ (cid:90) dω π ω (cid:90) m dω (cid:48) π + (cid:90) −∞ dω π − m (cid:90) ω dω (cid:48) π D ( x − x ; | ω | ) J ( ω (cid:48) , ω (cid:48) − ω ) D ( x − x ; | ω | ) e − iω ( t − t ) (cid:39)(cid:39) λ (cid:90) dx dx + ∞ (cid:90) dω π ω (cid:90) m dω (cid:48) π + (cid:90) −∞ dω π − m (cid:90) ω dω (cid:48) π cos {| ω | ( x − x ) } · cos {| ω | ( x − x ) } ω J ( ω (cid:48) , ω (cid:48) − ω ) e − iω ( t − t ) , (4.35)where we have introduced the following notation: J ( ω , ω ) ≡ G ( ω ) G ( ω ) = (cid:2) ψ ∗ ω ( x ) ψ ∗ ω ( x ) + ψ ∗ ω ( x ) ψ ∗ ω ( x ) (cid:3) (cid:2) ψ ω ( x ) ψ ω ( x ) + ψ ω ( x ) ψ ω ( x ) (cid:3) ++ (cid:2) ψ ω ( x ) ψ ∗ ω ( x ) + ψ ∗ ω ( x ) ψ ω ( x ) (cid:3) (cid:2) ψ ω ( x ) ψ ∗ ω ( x ) + ψ ∗ ω ( x ) ψ ω ( x ) (cid:3) + h.c. (4.36)Thus, one can see that the loop corrections are finite in the limit ( T − t ) → ∞ because ∆ D K ( x , x )depends only on ( t − t ). 37igure 11: Correction to the four-point correlation function Again it can be similarly shown that the loop corrections to the retarded and advanced propagators donot grow with time. Let us then calculate the first loop correction to the Keldysh propagator:∆ G Kab ( x , x ) = 12 (cid:0) ∆ G ++ ab ( x , x ) + ∆ G −− ab ( x , x ) (cid:1) == − λ (cid:90) d x d x (cid:88) σ , , = { + , −} G σ σ ac ( x , x ) G σ σ cd ( x , x ) D σ σ ( x , x ) G σ σ db ( x , x ) sgn( σ σ ) (cid:39)(cid:39) λ (cid:90) dx dx + ∞ (cid:90) m dω π ω (cid:90) m dω (cid:48) π − − m (cid:90) −∞ dω π − m (cid:90) ω dω (cid:48) π G ( ω ) G ( ω (cid:48) ) D ( x − x ; | ω − ω (cid:48) | ) G ( ω ) e − iω ( t − t ) . (4.37)Again we see that the loop corrections are finite in the limit ( T − t ) → ∞ because ∆ G Kab ( x , x ) dependsonly on ( t − t ). In this subsubsection we will show that one-loop corrections to the vertexes (Fig. 9) do not grow withtime.Let us consider the four-point correlation function G σ σ σ σ , (Fig. 11). Note that it is a part of thethree-point correlation function G ±±± ab . To single out the growing contributions, if any, we consider thelimit | t i − t j | (cid:28) T and ( t + t + t + t ) = T → ∞ . A generic contribution in this limit has the following38orm:∆ G σ σ σ σ ∼ λ (cid:90) d x d x D + − (cid:88) σ , = { + , −} G σ σ G σ σ G σ σ G σ σ sgn( σ σ ) == λ (cid:90) dx dx (cid:90) dt dt ∞ (cid:90) m dω π + ∞ (cid:90) m dω π + ∞ (cid:90) m dω π + ∞ (cid:90) m dω π + ∞ (cid:90) dω π e − it ( ω + ω − ω ) e it ( ω + ω − ω ) ×× e − iω t e iω t e − iω t e iω t D + − ( ω ) (cid:88) σ , = { + , −} G σ σ ( ω ) G σ σ ( ω ) G σ σ ( ω ) G σ σ ( ω ) sgn( σ σ ) == λ (cid:90) dx dx ∞ (cid:90) m dω (cid:48) π + ∞ (cid:90) m dω (cid:48)(cid:48) π + ∞ (cid:90) dω π e iω ( t − t ) e iω (cid:48) ( t − t ) e iω (cid:48)(cid:48) ( t − t ) D + − ( ω ) ×× (cid:88) σ , = { + , −} G σ σ ( ω (cid:48) ) G σ σ ( ω (cid:48) − ω ) G σ σ ( ω (cid:48)(cid:48) − ω ) G σ σ ( ω (cid:48)(cid:48) ) sgn( σ σ ) . (4.38)Thus, we see that the last integral depends only on ( t i − t j ) and does not grow in the limit in question. In the previous section we have considered φ cl ( x ) = Ex + mλ and found the exact modes ψ ( t, x ) in sucha background. Such an approach means that the background field φ cl ( x ) is set by a brutal external forceto be the same for all times. Such an approach can work only when the backreaction on the backgroundis weak.The situation, which we consider in this section, corresponds to a different set up. Namely, at somepoint in time there was formed a state | φ cl (cid:105) which corresponds to the presence of the external field φ cl ( x )in the sense that we will see in a moment. And then this state is released to evolve freely. Our goal is tofind out how it will be changing in time.To start with, we define the coherent state | φ cl (cid:105) as follows: (cid:104) φ cl | ˆ φ ( y ) | φ cl (cid:105) = φ cl ( y ) . (5.39)In appendix D it is shown that one can represent the state as follows: | φ cl (cid:105) = e − i (cid:82) φ cl ˆ π φ dx | (cid:105) , where a p | (cid:105) = 0 . (5.40)In what follows we want to calculate the following expectation value: (cid:104) φ (cid:105) ( t, x ) = (cid:104) φ cl | ˆ U † ( t, t ) ˆ φ I ( t, x ) ˆ U ( t, t ) | φ cl (cid:105) , (5.41)where ˆ U ( t, t ) is the evolution operator and we use the interaction picture and the modes are ordinaryplane waves: φ I ( t, x ) = (cid:90) dp π (cid:112) | p | (cid:16) α p e ipx − i | p | t + α † p e − ipx + i | p | t (cid:17) ,ψ I ( t, x ) = (cid:90) dp π (cid:112) E p (cid:16) a p u p e ipx − iE p t + b † p v p e − ipx + iE p t (cid:17) , (5.42)unlike the case of the previous section. Here u p and v p are modes of the two-dimensional free massiveDirac field (3.8).One can find the equation for (cid:104) φ (cid:105) ( t, x ): 39 (cid:104) φ (cid:105) ( t, x ) = − λ (cid:104) φ cl | ˆ U † ( t, t ) ˆ¯ ψ I ( t, x ) ˆ ψ I ( t, x ) ˆ U ( t, t ) | φ cl (cid:105) . (5.43)Now let us transform the right hand side of this equation. We commute ˆ U with the exponent in thedefinition of the coherent state (5.40). Let us denote:ˆ X = − i (cid:90) d xλ ˆ φ I ˆ¯ ψ I ˆ ψ I , and ˆ Y = − i (cid:90) dyφ cl ˆ π φ , (5.44)and use that φ I ( x, t ) = e iH t φ I ( x ) e − iH t . (5.45)Then[ X, Y ] = − (cid:90) d xdyλφ cl ( y ) ¯ ψ I ( x, t ) ψ I ( x, t )[ φ I ( x, t ) , π φ ( y )] == − (cid:90) d xdyλφ cl ( y ) ¯ ψ I ( x, t ) ψ I ( x, t ) (cid:0) e iH t φ I ( x )[ e − iH t , π φ ( y )] + iδ ( x − y ) + [ e iH t , π φ ( y )] φ I ( x ) e − iH t (cid:1) . (5.46)To simplify the last expression we denote A = − iH t = − it (cid:90) dx (cid:18) π φ + 12 φ (cid:48) I (cid:19) , and B = − i (cid:90) φ cl π φ dx, (5.47)and check that the commutator of A and B is vanishing in our case:[ A, B ] = − t (cid:90) dx (cid:90) dy [ φ (cid:48) I ( y ) , π φ ( x )] φ cl ( x ) == − t (cid:90) dx (cid:90) dy φ (cid:48) I ( y ) φ cl ( x ) ∂ y [ φ I ( y ) , π φ ( x )] = it (cid:90) dx φ (cid:48)(cid:48) I ( x ) φ cl ( x ) = 0 , (5.48)because the background field that we consider here is the linear function of x : φ cl = mλ + Ex and φ I ( y )does vanish at infinity.Hence, due to (5.48) the first and the third terms in (5.46) are vanishing. Therefore[ X, Y ] = − i (cid:90) d xλφ cl ¯ ψ I ψ I , (5.49)and then ˆ U e − i (cid:82) φ cl ˆ π φ dx = e − i (cid:82) d xλφ cl ¯ ψ I ψ I e − i (cid:82) φ cl ˆ π φ dx ˆ U . (5.50)Thus, we obtain that: (cid:104) φ cl | ˆ U † ( t, t ) ˆ¯ ψ I ( t, x ) ˆ ψ I ( t, x ) ˆ U ( t, t ) | φ cl (cid:105) == (cid:104) | ˆ U † ( t, t ) e i (cid:82) d xλφ cl ¯ ψ I ψ I ˆ¯ ψ I ( t, x ) ˆ ψ I ( t, x ) e − i (cid:82) d xλφ cl ¯ ψ I ψ I ˆ U ( t, t ) | (cid:105) . (5.51)We will use these relations below.Meanwhile to find the relation between the problem of the previous section to the one here, note that: e i (cid:82) φ cl ˆ π φ dx O ( ˆ φ ) e − i (cid:82) φ cl ˆ π φ dx = O ( φ cl + ˆ φ ) , for any operator O ( ˆ φ ) in the theory. Using this relation on the right hand side of the eq. (5.43) one canassume that we obtain here the same scalar current as in the previous section. However, note that in theprevious section the average in the correlation function was done with respect to the ground Fock spacestate corresponding to the exact fermionic modes in the φ cl ( x ) background, while in (5.43) the expectationvalue is taken with respect to the ordinary Poincare invariant state for fermions.40 .1 Loop corrections (coherent state) In this subsection we calculate the right hand side of the equation (5.43). The tree–level result for thescalar current in the present case is obviously trivial (the same as in empty space). To restore the tree–level result of the previous section within the present settings one has to sum up infinite number of terms,as is explained in the footnote 11 in appendix B.In what follows we consider the corrections of the order λ to the propagators (Fig. 8) in the limit τ = t − t = const , T = 12 ( t + t ) → + ∞ , t → −∞ , (5.52)where t and t are arguments of the two-point functions, or, more specifically: | T | , | t | (cid:29) √ α , | T | (cid:29) | τ | . (5.53)Note that the variant with the averaging over the coherent state allows one not to specify the form of φ cl , therefore it allows to get more general result. It is also interesting to compare the answers in thesetwo problems (sections 4.3 and 5.1) and to find out if the expressions for the first loop corrections showa different behaviour.To do the calculation in question, we need to find φ ( x, t ) | φ cl (cid:105) = e iH t φ ( x ) e − iH t e − i (cid:82) φ cl ˆ π φ dx | (cid:105) , (5.54)where H = (cid:90) dx (cid:18) π φ + 12 φ (cid:48) (cid:19) . (5.55)Taking into account the Becker-Hausdorff formula and the result of (5.48), we obtain that: φ ( x, t ) | φ cl (cid:105) = e iH t φ ( x ) e − i (cid:82) φ cl ˆ π φ dx e − iH t | (cid:105) = e − iE t e iH t φ ( x ) | φ cl (cid:105) == e − iE t φ cl ( x ) e iH t e − i (cid:82) φ cl ˆ π φ dx | (cid:105) + e − iE t e iH t e − i (cid:82) φ cl ˆ π φ dx φ ( x ) | (cid:105) == φ cl ( x ) | φ cl (cid:105) + e − iE t e − i (cid:82) φ cl ˆ π φ dx e iH t φ ( x ) | (cid:105) = φ cl ( x ) | φ cl (cid:105) + e − i (cid:82) φ cl ˆ π φ dx φ ( x, t ) | (cid:105) . (5.56)Thus, if we consider quantum averages over the state | φ cl (cid:105) , then according to (5.56) instead of (3.48) wehave that: D + − ( x , x ) = φ cl ( x ) φ cl ( x ) + (cid:104) | φ ( x ) φ ( x ) | (cid:105) = φ cl ( x ) φ cl ( x ) + D ( x − x ) ,D − + ( x , x ) = φ cl ( x ) φ cl ( x ) + D ( x − x ) ,D −− ( x , x ) = φ cl ( x ) φ cl ( x ) + θ ( t − t ) D ( x − x ) + θ ( t − t ) D ( x − x ) ,D ++ ( x , x ) = φ cl ( x ) φ cl ( x ) + θ ( t − t ) D ( x − x ) + θ ( t − t ) D ( x − x ) , (5.57)where we denoted x = ( t, x ) and D ( x − x ) = (cid:90) dp π | p | e − i | p | ( t − t )+ ip ( x − x ) (5.58)is just the empty space scalar propagator. The fermion propagators in the situation under considerationare the same as in the theory without background field.41 .2 One-loop corrections to the fermion propagators We continue with the loop corrections to the boson and fermion correlation functions. For the retardedand advanced propagators we have the usual story as was described in the previous sections. Hence,below we concentrate on the calculations of the loop corrections to the Keldysh propagators.We start with the calculation of the first loop correction to the fermion Keldysh propagator (4.37).Due to the fact that in the one-loop correction to the fermion propagator we have only one tree-levelbosonic Green function and in the free case fermion Green functions does not receive growing with timecorrections (see subsubsection 3.3.2), we can use the tree-level bosonic propagator in the following form: D + − ( x , x ) = D − + ( x , x ) = D ++ ( x , x ) = D −− ( x , x ) = φ cl ( x ) φ cl ( x ) ≡ D ( x , x ) , (5.59)instead of (5.57), and define the following expressions: H ( p, q, p (cid:48) ) ≡ (cid:0) ψ ∗ p, ψ ∗ q, + ψ ∗ p, ψ ∗ q, (cid:1) (cid:0) ψ q, ψ p (cid:48) , + ψ q, ψ p (cid:48) , (cid:1) (cid:0) ψ p, ψ ∗ p (cid:48) , − ψ p, ψ ∗ p (cid:48) , (cid:1) ,K ( p, q, p (cid:48) ) ≡ (cid:0) ψ p, ψ p (cid:48) , + ψ p, ψ p (cid:48) , (cid:1) (cid:0) ψ ∗ p (cid:48) , ψ q, − ψ ∗ p (cid:48) , ψ q, (cid:1) (cid:0) ψ ∗ q, ψ ∗ p, + ψ ∗ q, ψ ∗ p, (cid:1) . (5.60)Then, taking into account the Fourier representation of φ cl ( x ):˜ φ cl ( p ) = (cid:90) dx φ cl ( x ) e − ipx = 2 π (cid:0) αδ ( p ) + iβδ (cid:48) ( p ) (cid:1) , (5.61)we obtain that the one loop correction to the fermion Keldysh propagator is as follows:∆ G Kab ( x , x ) = − λ (cid:90) d x d x D ( x , x ) (cid:88) σ , , = { + , −} G σ σ ac ( x , x ) G σ σ cd ( x , x ) G σ σ db ( x , x )sgn( σ σ ) (cid:39)(cid:39) − λ (cid:90) Tt dt (cid:90) Tt dt (cid:90) dp π dq π dp (cid:48) π ˜ φ cl ( p − q ) ˜ φ cl ( q − p (cid:48) ) e ipx − ip (cid:48) x (cid:2) H ( p, q, p (cid:48) ) + h.c. (cid:3) ++ λ (cid:90) Tt dt (cid:90) t t dt (cid:90) dp π dq π dp (cid:48) π ˜ φ cl ( p − q ) ˜ φ cl ( q − p (cid:48) ) e ipx − ip (cid:48) x (cid:2) K ( p, q, p (cid:48) ) + K ( p (cid:48) , q, p ) + h.c (cid:3) == − λ (cid:90) dp π e ip ( x − x ) (cid:90) Tt dt (cid:90) Tt dt (cid:20) H ( p, p, p ) (cid:26) α + αβ ( x + x ) − αβ pω p ( t + t − t − t )++ β (cid:18) i pω p ( t − t ) − ix + 2 m − p ω p p (cid:19)(cid:18) i pω p ( t − t ) + ix + 2 m − p ω p p (cid:19)(cid:27) + h.c. (cid:21) ++ λ (cid:90) dp π e ip ( x − x ) (cid:90) Tt dt (cid:90) t t dt (cid:20) K ( p, p, p ) (cid:26) α + αβ ( x + x ) − αβ pω p ( t − t )++ β (cid:18) i pω p t + t − t − ix + 3 m − p ω p p (cid:19)(cid:18) i pω p t + t − t ix + 3 m − p ω p p (cid:19)(cid:27) + h.c. (cid:21) == O ( T ) , (5.62)where we have used that (cid:0) ψ ∗ p, ψ ∗ p, + ψ ∗ p, ψ ∗ p, (cid:1) (cid:0) ψ p, ψ p, + ψ p, ψ p, (cid:1) = p ω p e iω p ( t − t ) , (cid:0) ψ ∗ p, ψ p, − ψ ∗ p, ψ p, (cid:1) (cid:0) ψ ∗ p, ψ ∗ p, + ψ ∗ p, ψ ∗ p, (cid:1) = pmω p e − iω p ( t − t ) e iω p ( t + t ) . (5.63)and the following expressions for the derivatives of the fermion field components: ∂ p ψ p,α = − m ( ω p − m )2 ω p p ψ p,α − it α pω p ψ p,α , ∂ p ψ p,α = mp ω p ( ω p − m ) ψ p,α − it α pω p ψ p,α . (5.64)42hus, as follows follows from (5.62) one-loop correction to the fermion propagator does not grow as T → ∞ . The one-loop correction to the free bosonic Keldysh propagator does not grow with time (see subsubsec-tion 3.3.1). To show that in the present case let us denote: F ( p, q ) ≡ (cid:0) ψ ∗ q, ψ ∗ p, + ψ ∗ p, ψ ∗ q, (cid:1) (cid:0) ψ q, ψ p, + ψ p, ψ q, (cid:1) = p ω p e iω p ( t − t ) . (5.65)Therefore, keeping in mind eq. (5.61), we obtain:∆ D K ( x , x ) (cid:39) − λ (cid:90) Tt dt (cid:90) Tt dt (cid:90) dp π dq π (cid:20)(cid:18) φ cl ( x ) ˜ φ cl ( q − p ) φ cl ( x ) ˜ φ cl ( p − q ) + φ cl ( x ) ˜ φ cl ( q − p ) ×× e − i | p − q | ( t − t ) | p − q | e − i ( p − q ) x + φ cl ( x ) ˜ φ cl ( p − q ) e − i | p − q | ( t − t ) | p − q | e i ( p − q ) x (cid:19) F ( p, q ) + h.c. (cid:21) ++ λ (cid:90) Tt dt (cid:90) t t dt (cid:90) dp π dq π (cid:20)(cid:18) φ cl ( x ) ˜ φ cl ( q − p ) φ cl ( x ) ˜ φ cl ( p − q ) + φ cl ( x ) ˜ φ cl ( q − p ) ×× e − i | p − q | ( t − t ) (cid:112) | p − q | + µ e − i ( p − q ) x + φ cl ( x ) ˜ φ cl ( p − q ) e − i | p − q | ( t − t ) (cid:112) | p − q | + µ e i ( p − q ) x + φ cl ( x ) ˜ φ cl ( q − p ) ×× e − i | p − q | ( t − t ) (cid:112) | p − q | + µ e − i ( p − q ) x + φ cl ( x ) ˜ φ cl ( p − q ) e − i | p − q | ( t − t ) (cid:112) | p − q | + µ e i ( p − q ) x (cid:19) F ∗ ( p, q ) + h.c. (cid:21) (cid:39)(cid:39) − λ µ ( T − t ) (cid:90) + ∞−∞ dτ (cid:48) (cid:90) dp π (cid:20) α ( φ cl ( x ) + φ cl ( x )) + 2 β ( φ cl ( x ) x − φ cl ( x ) x ) (cid:21) p ω p e − iω p τ (cid:48) == O ( T ) , (5.66)where we have restored the spontaneously acquired mass of the boson field µ ∼ λ (see appendix B) toeliminate the singularity | p − q | in the denominator. Note that the growing factor ( T − t ) is multiplied by δ ( ω p ) which is never zero. This situation is similar to the free cases from subsections 3.3.1 and 3.3.2. We consider one of the simplest examples of nontrivial quantum field theory out of equilibrium — theYukawa model in strong scalar field backgrounds in (0 + 1) and (1 + 1) dimensions. Our main interest isin the response of the dynamical scalar and fermion fields to such a background. To find this response,we calculate the tree–level scalar current (cid:104) ¯ ψψ (cid:105) (i.e. the fermion propagator at coincident points) andloop corrections to both fermion and boson correlation functions. To take into account possible non-equilibrium effects we use Schwniger–Keldysh diagrammatic technique instead of the Feynman one. Inthis section we summarize our results and explain their physical meaning. In (0 + 1) dimensions the dynamics of fermion and scalar fields is nearly trivial. First of all, dueto the properties of one–dimensional fermions the scalar current can be exactly calculated from the verybeginning. Then the corrections to the two-point correlation functions of the scalar field basically reduceto the disconnected corrections to the one-point functions — so-called “tadpoles”. We show this fact bothin operator formalism and diagrammatic approach. Moreover, it is not difficult to generalize this resultto arbitrary orders of the perturbation theory and arbitrary n -point functions, because “tadpoles” do notreceive any loop corrections in one dimension. This result means that no external scalar perturbation canchange the initial state of the theory. 43 . The dynamics in (1 + 1) dimensions is more interesting. First, in the case of indefinitely growingscalar field, in particular, φ cl = mλ + Et and φ cl = mλ + Ex , one should accurately choose the exactmodes. Namely, one should demand a correct UV behavior of the modes, because at the space-timeinfinities they do not tend to the plane waves. Such a correct UV behaviour is necessary to have the sameUV renormalization in the background field as is in its absence, which is meaningful on general physicalgrounds.Second, in the leading order (when φ (cid:48) cl is small, while φ cl itself is large) the scalar current on thebackgrounds φ cl = mλ + Et and φ cl = mλ + Ex coincides with the current in the theory of free fermionswith the time-dependent mass m ( t ) = λφ cl ( t ): (cid:104) ¯ ψψ (cid:105) (cid:39) λφ cl π log λφ cl Λ , (6.1)where Λ is the UV cut-off.The last equation is an analog of the causal equations that have been derived in, e.g., [45] for thescalar and electromagnetic fields. In the expression under consideration we have explicitly calculated theright hand side (the scalar current) for the given background fields in the tree–level approximation and forlarge and slowly changing backgrounds. This result indicates that the leading expressions in the strongscalar fields are insensitive to the choice of the initial state. Note that subleading corrections to the scalarcurrent do depend on the choice of the initial state [12].Third, neither level population nor anomalous quantum average of the scalar and fermion fields dogrow with time. Hence, in the limit of small coupling constant time–dependent corrections to the tree-level correlation functions (including scalar current) are negligible despite the strength of the background.This type of behavior does not resemble the one in strong electric [6, 7] or gravitational [2, 8] fields, inwhich loop corrections to these quantities do grow with time .This is a very strange phenomenon for the case of time dependent scalar background. In fact, it seemsthat the secular growth under discussion is forbidden due to a specific behaviour of the exact modesin the background. In particular, it means that even if one starts with any non-stationary state (e.g.non-plankian initial distribution) there will not be any substantial change of the level population andof the anomalous averages, if the mass of a particle changes in time, M ( t ) = λ φ ( t ). This we find asquite a non–trivial observation, which should be compared to the time-dependent gauge and gravitationalbackgrounds.The technical reason why there is no secular growth in the background scalar field as opposed to itspresence e.g. in constant electric field or de Sitter space can be explained as follows. In the constantelectric field (de Sitter space) all the quantities depend on the invariant/physical momenta p − eEt ( | (cid:126)p | e − t/H ). (Here p is the component of the momentum along the external electric field E and H is theHubble constant in the case of the de Sitter space.) As the result all physical quantities are invariantunder the simultaneous translations t → t − a and p → p − eEa ( | (cid:126)p | → | (cid:126)p | e − a/H ). Furthermore, inthe field theory without background field, but with an initial non–stationary (non–plankian) distributionthere is the time translational invariance at the tree–level. That is the reason why there is secular growthin the loops in all the listed in this paragraph situations. Meanwhile in the background fields φ cl = Et there is no time translational invariance.At the same time in the background φ cl = mλ + Ex the simple explanation for the absence of thesecular growth is not yet clear to us. What remains to be checked now if there is a secular growth for anyother states of the type (3.34) for the spatial coordinate dependent background.We should probably stress here that for finite coupling constants the corrections to the quantumaverages, (cid:104) a + a (cid:105) and (cid:104) aa (cid:105) , are non-zero, i.e. the theory (3.1) is indeed non-stationary. Also let us emphasizethat the scalar currents calculated in different ground states (e.g. currents (3.43) and (4.28)) coincide Note that by calculating corrections to the Keldysh propagator (to the level population and anomalous quantum averages)we examine if there are contributions to the scalar current which grow with time, but do not check if there are correctionswhich are large, when φ cl is large, but does not change the state of the system. (cid:68) φ cl (cid:12)(cid:12)(cid:12) ˆ φ ( t = 0 , x ) (cid:12)(cid:12)(cid:12) φ cl (cid:69) = φ cl ( x ) , i.e. a self-guided dynamics of a freely evolving in time initially set up “coherent state”. Namely, wewere attempting to calculate (cid:68) φ cl (cid:12)(cid:12)(cid:12) ˆ φ ( t, x ) (cid:12)(cid:12)(cid:12) φ cl (cid:69) for arbitrary t in the full theory. We have found that thebehavior of the correlation functions in this case is qualitatively the same as the one previously found forthe strong fixed scalar backgrounds. As soon as the dynamics in the strong scalar field (when φ (cid:48) cl is small, while φ cl itself is large) isweakly sensitive to the choice of the ground state, we can estimate its effective action in the equilibriumapproach. It is a standard textbook exercise to show that in this approach the effective action for thescalar field (the action one obtains after the integration over the fermion degrees of freedom) in the leadingorder looks as follows: S eff = (cid:90) d x (cid:20)
12 ( ∂ µ φ ) − V eff [ φ ] (cid:21) , where V eff [ φ ] (cid:39) ( λφ ) π log φ (cid:104) φ (cid:105) GS − ( λφ ) π (6.2)and (cid:104) φ (cid:105) GS is the minimum of the renormalized effective potential V eff [ φ ]. Note that scalar field acquiresnon-zero mass µ = λ √ π at the bottom of the potential due to the quantum fluctuations. Also we remindthat the derivation of (6.2) assumes that the scalar field is non-dynamical, large and slowly changing, (cid:12)(cid:12) /∂φ (cid:12)(cid:12) (cid:28) λφ . We review the derivation of this expression in appendix B.The equation of motion that follows from the action (6.2): ∂ φ cl + λ φ cl π log φ cl (cid:104) φ (cid:105) GS = 0 , (6.3)obviously reproduces the results of sections 3.2 and 4.2 with classical backgrounds φ cl = Et and φ cl = mλ + Ex . In fact, it generalizes these results to arbitrary large, but slowly changing scalar field backgrounds.So it is not surprising that the calculations in the strong scalar wave background [12] result in the sameanswer for the scalar current. However, we emphasize again that this result is correct only in the leadingorder, whereas the subleading corrections can be different for different choices of initial states. The “universality” of the leading order approximation to the effective action can be interpreted asfollows. First, note that fermion modes with high enough momenta behave as plane waves. The criticalscale is roughly p ∼ λφ . Such a behavior is necessary for the proper treatment of UV divergences, as wehave already mentioned above in this section.Second, the main contribution to the scalar current and effective potential comes from exactly suchhigh-momenta modes (e.g. see equations (3.41) and (3.42)). This is due to the spatiotemporal oscillationsof nearly-zero momenta modes, which are significantly faster than oscillations of higher-momenta modes.In fact, compare asymptotic behaviors (3.29), ψ ( t ) ∼ e iαt , and (3.28), ψ ( t ) ∼ e i | p | t .Third, when calculating the contribution from such modes, the variations of the background scalarfield can be neglected. Roughly speaking, plane waves with large momenta ( p (cid:38) λφ ) dominate in regionswith small spatial and temporal size. Hence, at each moment the background plays the role of a fixedmass of the fermion field. Therefore, one can just substitute m → λφ cl ( t, x ) (cid:39) const in the expressionsfor the free case (3.40).In summary, one expects that the effective action coincides for arbitrary strong scalar fields becausesuch fields are not sensitive, at the leading order, to the properties of the low lying initial state. In thenext orders this sensitivity does manifest itself [12].45 . Thus, the calculation with the use of the Feynman approach shows that zero point fluctuations ofthe fermion field polarize the vacuum and deform the classical scalar field background . However, weremind that this calculation is valid only if (cid:12)(cid:12) /∂φ (cid:12)(cid:12) (cid:28) λφ and λ → λ → t → ∞ for φ cl = Et or λ → x → ∞ for φ cl = mλ + Ex . Obviously, they also hold near theminimum of the effective potential. Therefore, in this limit the scalar field just classically rolls down tothe minimum of such a potential.It would be interesting to calculate loop corrections to the quantum averages on top of such a rollingclassical solution. In principle such corrections can change the situation under consideration [1–3, 6–11].If one considers a coherent state decay, most likely this would not happen: of course, strong initialperturbation can induce complex dynamics for a while, but one expects that eventually the field fallson the classical trajectory described by the equation (6.3) for large and slowly changing values of φ cl .However, if one pumps energy into the system, i.e., maintains a strong field with substantial derivatives,loop corrections potentially can grow. In this case the choice of the initial state is important, the leadingapproximation (6.2) is not valid anymore, and the dynamics of the field is less predictable. This case willbe studied elsewhere. We would like to thank Artem Alexandrov, Francesco Bascone, Kirill Gubarev, Olexander Diatlyk, An-drew Semenov, Ugo Moschella, Fedor Popov and Daniil Sherstnev for useful comments and discussions.We would like to thank Hermann Nicolai and Stefan Theisen for the hospitality at the Albert EinsteinInstitute, Golm, where the work on this project was partly done. AET would like to thank Ugo Moschellafor the hospitality during his stay at the INFN, Sez di Milano and the University of Insubria at Como,where the work on this project was partly done. AET also gratefully acknowledges support from theSimons Center for Geometry and Physics, Stony Brook University at which some of the research for thispaper was performed.The work of DAT and ENL was supported by the Russian Ministry of Science and Education, projectnumber 3.9911.2017/BasePart. The work of AET was supported by the Russian Ministry of Science andEducation, project number 3.9904.2017/BasePart. The work of ETA and DAT was supported by thegrant from the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”. Thework of ETA was partially supported by the RFBR grant 19-02-00815.
A Asymptotic behavior of parabolic cylinder function for large order
The asymptotic behavior of the parabolic cylinder functions has been widely studied in the literature(e.g., see [28, 29, 34, 40]). But the only asymptotic expansion for an arbitrary complex | ν | (cid:29) D ν ( z ) = 1 √ (cid:20) ν log( − ν ) − ν − √− νz (cid:21) (cid:34) O (cid:32) (cid:112) | ν | (cid:33)(cid:35) , (A.1)where | arg( − ν ) | ≤ π and | z | is bounded. The error of this expansion is too large for our purposes: e.g.when one integrates D − ip α ( z ) over dp , due to terms of the order O (cid:18) √ | ν | (cid:19) ∼ O (cid:16) p (cid:17) the integral candiverge. Thus, we have to obtain a more accurate asymptotic expansion.Following [40], we start with the integral representation of the parabolic cylinder function: Note that the Feymnan diagrammatic technique takes care of only zero point fluctuations. To see excitations of higherlevels one has to apply the Schwinger–Keldysh technique. That is the reason why we calculate loops in the latter technique. D ν ( z ) = Γ(1 + ν )2 πi e − z z (cid:90) C exp (cid:20) z (cid:18) v − v (cid:19) − (1 + ν ) log( zv ) (cid:21) dv, (A.2)where the integration contour C is depicted on the Fig. 12. One can check that this expression indeedsolves the differential equation for parabolic cylinder function. For the case φ cl = Et we have ν = − ip α , z = e iπ/ (cid:114) α M ( t ) , M ( t ) = αt, α = λE. (A.3)Using the saddle-point approximation in (A.2), one obtains its decomposition as follows: D ν ( z ) (cid:39) Γ(1 + ν ) i √ π ze − z (cid:88) j =0 , exp( iα j + f ( v j )) | f (cid:48)(cid:48) ( v j ) | ∞ (cid:88) l =2 (2 l − ilα j ) v lj | f (cid:48)(cid:48) ( v j ) | l (cid:88) λ n l (cid:89) n =3 [(1 + ν ) /n ] λ n λ n ! , (A.4)where f ( v ) = z ( v − v ) − (1 + ν ) log( zv ) ,α j = 12 π −
12 arg( f (cid:48)(cid:48) ( v j )) , and we denoted the critical points of the function f ( v ) as v , = − ± (cid:16) − ν ) z (cid:17) . The innermostsum in eq. (A.4) is taken over all distinct partitions of 2 l given by non-negative integer solutions λ n suchthat (cid:80) ln =3 nλ n = 2 l . Let us estimate this sum. The l -th term in it contains the l -th power of the followingexpression: 1 v , f (cid:48)(cid:48) ( v , ) = 12(1 + ν ) (cid:34) ∓ (cid:18) − ν ) z (cid:19) − (cid:35) , and not greater than the (cid:98) l (cid:99) -th power of [(1 + ν ) /n ]. In the case | ν | (cid:29) v , f (cid:48)(cid:48) ( v , ) ∼ ν ) = O (cid:0) ν (cid:1) for both signs “ ∓ ”. This means that the innermost sum is47 (cid:18) ν [ v , f (cid:48)(cid:48) ( v , )] (cid:19) = O (cid:0) ν (cid:1) , so we neglect it in the integrals over dp . Substituting the values of the saddlepoints into the decomposition (A.4) we obtain: D ν ( z ) = 1 √ (cid:20) − ν (cid:18) ν + 12 (cid:19) log ν − (cid:18) ν (cid:19) log z −
14 log (cid:18) − ν ) z (cid:19)(cid:21) ×× (cid:88) ± exp (cid:34)(cid:18) − − ν (cid:19) log (cid:32) ± (cid:18) − ν ) z (cid:19) (cid:33) ± z (cid:18) − ν ) z (cid:19) (cid:35) (cid:20) O (cid:18) ν (cid:19)(cid:21) . (A.5)Then we note that in the notations of (A.3) and limit | ν | (cid:29) p (cid:29) α ) or | z | (cid:29) M (cid:29) α ), wehave that (cid:18) − ν ) z (cid:19) = (cid:112) M + p M (cid:34) iαM + p + O (cid:18) αM + p (cid:19) (cid:35) . Hence, denoting V = (cid:112) M + p for short, we obtain: D − ip α (cid:18) i √ α M (cid:19) (cid:39) e πp α √ (cid:18) MV + 1 (cid:19) e ip α − ip α log ( V + M )22 α − iMV α (cid:104) O (cid:16) αV (cid:17)(cid:105) . (A.6)Then for the squared module of the parabolic cylinder function we get: (cid:12)(cid:12)(cid:12)(cid:12) D − ip α (cid:18) i √ α M (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) e πp α (cid:32) M (cid:112) M + p + 1 (cid:33) (cid:20) O (cid:18) αM + p (cid:19)(cid:21) . (A.7)Here we neglected the second term in the sum because it contains the factor of e − πp α . Note that we havechosen the sheet on the complex plane in which − e − iπ . One can check that (A.6) coincides with (A.1)up to O (cid:16) p (cid:17) . But the new equation also contains the next term of the asymptotic expansion.We emphasize that eqs. (A.6) and (A.7) work for arbitrary values M (cid:29) α . However, they simplifyin extremal cases. For instance, D − ip α (cid:18) i √ α M (cid:19) (cid:39) √ e πp α + ip α − ip α log p α e − i | p | Mα − iM | p | α + M | p | (cid:20) O (cid:18) M + αp (cid:19)(cid:21) , (A.8)if M (cid:28) p , and D − ip α (cid:18) i √ α M (cid:19) (cid:39) (cid:16) − p M (cid:17) e πp α − iM α − ip α log M α (cid:104) O (cid:16) p + αM (cid:17)(cid:105) , M > , p | M | e πp α + iM α + ip α log M α − ip α log p α (cid:104) O (cid:16) p + αM (cid:17)(cid:105) , M < M (cid:29) p .In the opposite case | ν | (cid:28) ν ∼ | ν | (cid:28) (cid:28) | z | (i.e., p (cid:28) α (cid:28) M ) we can usethe following decomposition, which can be obtained from another integral representation for paraboliccylinder function [28]: D ν ( z ) = z ν e − z N (cid:88) n =0 (cid:0) − ν (cid:1) n (cid:0) − ν (cid:1) n n ! (cid:16) − z (cid:17) n + O (cid:16)(cid:12)(cid:12) z (cid:12)(cid:12) − N − (cid:17) , ( γ ) = 1 , ( γ ) n (cid:54) =0 = γ ( γ + 1) · · · ( γ + n − . (A.10)48ence, we find that: D − ip α (cid:18) i √ α M (cid:19) (cid:39) e πp α − iM α − ip α log M α (cid:18) − p M (cid:19) (cid:20) O (cid:18) α M (cid:19)(cid:21) , (A.11)and for the squared module: (cid:12)(cid:12)(cid:12)(cid:12) D − ip α (cid:18) i √ α M (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) (cid:39) e πp α (cid:20) − p M + O (cid:18) α M (cid:19)(cid:21) . (A.12)Note that expressions (A.6) and (A.11) approximately coincide if | ν | (cid:28) | z | , | ν | (cid:29)
1, as it should be.
B Effective action
B.1 Path integral calculation
In sections 3 and 4 we have shown that the leading behavior of the fermion current does not dependon the ground state of the theory (see also [12]). Moreover, in the limit of small coupling constantsloop corrections to the scalar and fermion propagators do not grow. Therefore, if φ is large and slowlychanging function we can estimate the effective action using standard equilibrium technique, assumingthat the field φ is not dynamical. In this appendix we review the textbook calculation of the Feynmaneffective action [41–44] for the theory (3.1).To find the effective action for scalars, we integrate out the fermionic degrees of freedom in thefunctional integral: e iS eff [ φ ] = (cid:82) D ¯ ψ D ψe i (cid:82) d x ( ( ∂ µ φ ) + ¯ ψ ( i/∂ − λφ ) ψ ) (cid:82) D ¯ ψ D ψe i (cid:82) d x ¯ ψi/∂ψ = exp (cid:20) i (cid:90) d x
12 ( ∂ µ φ ) + tr log i /∂ − λφi /∂ (cid:21) , (B.1)which we normalize to the partition function of a free massless fermion for the correct definition of theoperator determinant.As we have just mentioned, in this section we consider the situation, when the scalar field is non-dynamical. At the same time in (B.1) we calculate the time–ordered Feynman effective action ratherthan Schwinger–Keldysh one. Note that this approximation in general is not valid if one takes intoaccount the quantum fluctuations of the scalar field. In this calculation it is implicitly assumed that thestate of the theory does not change in time. However, we have seen in the sections 3.3 and 4.3 that bothof these approximations are good enough if we work in the limit of large and slowly changing backgroundscalar field.Let us evaluate the determinant in (B.1). For simplicity we consider scalar fields smaller than the UVcut-off: λφ (cid:28) Λ (these fields still can be strong: φ (cid:29) x and p [44]. Using the reflection symmetry, i.e multiplyingthe expression by 1 = ( γ ) , and anti-commuting γ and γ µ , one obtains:tr log i /∂ − λφi /∂ = 12 tr log ( i /∂ − λφ )( − i /∂ − λφ )( i /∂ )( − i /∂ ) = 12 tr log ∂ + ( λφ ) − iλ /∂φ∂ (cid:39) tr log (cid:16) λφ ) ∂ (cid:17) , (B.2)where we took the trace over the spinor indices and neglected the derivatives ∂ t φ (cid:28) λφ and ∂ x φ (cid:28) λφ .E.g. for φ cl = Et we have exactly such situation when t (cid:29) √ λE and for φ cl = mλ + Ex when x (cid:29) | √ λE − m | λE .To evaluate the tr log we do Wick rotation into the Euclidean space [13]. One can actually do such atransformation, which is not valid in non–stationary situation, in the approximation that we are adoptinghere. Then we expand the logarithm: 49r log (cid:18) λφ ) ∂ (cid:19) = (cid:90) d x (cid:90) i d p (2 π ) log (cid:18) λφ ) p (cid:19) (cid:39) i (cid:90) d x π (cid:20) ( λφ ) log Λ ( λφ ) + ( λφ ) (cid:21) . (B.3)For the last equality we neglected the terms of the order ( λφ ) Λ and smaller. Thus, in the leading order forlarge φ and small derivatives of φ the effective action has the following form : S eff (cid:39) (cid:90) d x (cid:20) ∂ µ φ∂ µ φ − V eff [ φ ] (cid:21) , where V eff [ φ ] (cid:39) ( λφ ) π log λφ Λ − ( λφ ) π . (B.4)The partition function Z = (cid:82) D φ e iS eff [ φ ] is predominantly gained on the functions which solve the classicalequation of motion. Hence: ∂ (cid:104) φ (cid:105) + λ (cid:104) ¯ ψψ (cid:105) ≈ ∂ (cid:104) φ (cid:105) + λ (cid:104) φ (cid:105) π log λ (cid:104) φ (cid:105) Λ = 0 . (B.5)This expression is consistent with the values of the scalar currents (3.43) and (4.28) for φ cl = Et and φ cl = mλ + Ex , respectively, which were obtained in the main body of the text. However, (B.5) worksfor strong, but slowly changing classical backgrounds (see also [12]). Note that subleading corrections tothe scalar current (and, hence, to the effective action) do depend on the state with respect to which theaveraging is done in the correlation functions [12]. The corrections should be calculated with the use ofthe Schwinger–Keldysh technique.Now the classical fields φ cl = Et and φ cl = mλ + Ex do not solve the corrected equation of motion (B.5),although they do solve the free equation (3.3). This basically means that such classical fields have todecay due to quantum fluctuations of the fermions. This resembles the decay of strong constant electricfield [6, 7]. However, in contrast to the strong electric field in this case loop corrections to boson andfermion level populations do not grow, as we have shown in the main body of the text. B.2 Renormalization
One can see that expressions (B.4) and (B.5) explicitly depend on the UV cut-off, i.e. they are seeminglynot invariant with respect to renormalization group. Of course, this dependence has no physical sense,because observables must be renormalization group invariant. To resolve the issue we restore the mass ofthe scalar field and take into account UV counterterms (we recall that Yukawa theory in two dimensionsis renormalizable, since coupling constant λ has positive mass dimension): S eff = (cid:90) d x (cid:20)
12 ( ∂ µ φ ) − µ φ − V eff [ φ ] + 12 A ( ∂ µ φ ) − Bφ (cid:21) . (B.6)Usually, one defines the renormalized mass as the value of the inverse propagator at zero momentum: µ = ∂ V∂φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B.7) Let us recall that the calculation of the effective action corresponds to the summation of the Feynman diagrams (e.g.see [13, 41]). Indeed, consider the soft bosonic corrections to the free fermion propagator: G ( p ) = i/p + i(cid:15) + ( − iλφ ) (cid:18) i/p + i(cid:15) (cid:19) + 2!2! ( − iλφ ) (cid:18) i/p + i(cid:15) (cid:19) + · · · = i/p + i(cid:15) − λφ/p + i(cid:15) = i/p − λφ + i(cid:15) . Such corrections take into account the interaction between the fermion field and fixed scalar field background, so it is notsurprising that we have obtained the inverse operator of the second equation in the system (3.3) in almost constant φ cl background. The fermion current corresponds to the exact propagator with the coincident initial and end points, i.e. to thesum of the closed fermionic loops with an even number of external legs (diagrams with an odd number of legs are zero dueto Furry’s theorem [13]). Hence, the summation of such diagrams should reproduce the result (B.3) in the limit that weconsider in this section. V includes both the effective potential, mass term and counterterms. However, in the present casethis definition is meaningless: the second derivative of V at the origin does not exist due to the logarithmicsingularity. Due to this reason we define the mass at an arbitrary but non-zero value M R : µ = ∂ V∂φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M R . (B.8)This implies the following expression for the counterterm B : B = − λ π log λM R Λ − λ π . (B.9)and for the renormalized potential: V = 12 µ φ + ( λφ ) π log λφM R − λφ ) π . (B.10)It is easy to check that this expression is invariant under the change of renormalization scale. Also onecan note that the effective potential has the minimum, which is not φ = 0. This situation is obviouslysimilar to the well-known Coleman–Weinberg potential [13, 41].Finally, we set µ = 0, replace an arbitrary parameter M R by the ground state expectation value ofthe scalar field which minimizes the renormalized potential (we emphasize that this value differs from theaverage over the original state): M R = 1 e λ (cid:104) φ (cid:105) GS , (B.11)where e is the Euler’s constant, and obtain the following renormalization group invariant expression forthe effective potential: V eff = ( λφ ) π log φ (cid:104) φ (cid:105) GS − ( λφ ) π . (B.12)The expansion of this potential near the minimum φ = (cid:104) φ (cid:105) GS + ˜ φ has the following form: V eff (cid:39) − λ (cid:104) φ (cid:105) GS π + λ π ˜ φ + · · · , (B.13)i.e. the field spontaneously acquires the mass µ = λ π .Note that eqs. (B.5) and (B.12) were obtained in the approximation λφ (cid:28) Λ which is obviously notsatisfied near the minimum of the potential. However, higher loops corrections do not change the form ofthe potential near φ = 0. Therefore, loop corrections cannot shift the minimum of the effective potentialto zero, although they can affect its absolute value [41]. I.e. the expression (B.12) provides a goodqualitative description of the situation. C Definite-frequency and definite-momentum operators
In this appendix we find the relation between the definite-frequency and definite-momentum creation andannihilation operators. In order to do this let us consider the standard mode decomposition for the freemassive fermion field: 51 ( t, x ) = (cid:90) (cid:90) ∞−∞ dωdp π δ ( ω − p − m ) (cid:104) a p ψ p e − iωt + ipx + b † p ˜ ψ p e iωt − ipx (cid:105) == (cid:90) ∞−∞ dp π (cid:104) a p ψ p,ω p e − iω p t + ipx + b † p ˜ ψ p,ω p e iω p t − ipx (cid:105) == (cid:90) ∞ dp π (cid:104) a p ψ p,ω p e − iω p t + ipx + b † p ˜ ψ p,ω p e iω p t − ipx + a − p ψ − p,ω p e − iω − p t − ipx + b †− p ˜ ψ − p,ω p e iω − p t + ipx (cid:105) == (cid:90) ∞ m dω π (cid:104) a p ω ψ p ω ,ω e − iωt + ip ω x + b † p ω ˜ ψ p ω ,ω e iωt − ip ω x + a − p ω ψ − p ω ,ω e − iωt − ip ω x + b †− p ω ˜ ψ − p ω ,ω e iωt + ip ω x (cid:105) == (cid:90) | ω | >m dω π (cid:104) (cid:16) θ ( ω ) a p ω ψ p ω ,ω + θ ( − ω ) b †− p ω ˜ ψ − p ω , − ω (cid:17) e − iωt + ip ω x ++ (cid:16) θ ( ω ) b † p ω ˜ ψ p ω ,ω + θ ( − ω ) a − p ω ψ − p ω , − ω (cid:17) e iωt − ip ω x (cid:105) == (cid:90) | ω | >m dω π (cid:2) a ω χ ω e − iωt + ip ω x + b † ω ˜ χ ω e iωt − ip ω x (cid:3) . (C.1) Here ω p = (cid:112) p + m and p ω = √ ω − m , both functions are even and always positive; ψ p,ω and ˜ ψ p,ω correspond to fermion and antifermion spinors. The creation and annihilation operators with definitefrequency ( a ω and b ω ) are expressed via the corresponding operators with definite momentum as follows: a ω = θ ( ω ) a p ω + θ ( − ω ) b †− p ω , b ω = θ ( ω ) b p ω + θ ( − ω ) a †− p ω . (C.2)In other words, for positive frequencies new operators coincide with definite-momentum operators, butfor negative frequencies fermion creation operator and antifermion annihilation operators switch places.The definite-frequency and definite-momentum spinors are connected by similar expressions: χ ω = θ ( ω ) ψ p ω ,ω + θ ( − ω ) ˜ ψ − p ω , − ω = ψ p ω ,ω , ˜ χ ω = θ ( ω ) ˜ ψ p ω ,ω + θ ( − ω ) ψ − p ω , − ω = ˜ ψ p ω ,ω , (C.3)where we have used that ˜ ψ − p ω , − ω = ψ p ω ,ω . Note that in sections 3 and 4 we use different representationsfor the Clifford algebra. So the modes (4.5) and (3.8) are related by the expression (C.3) plus the unitarytransformation, which maps gamma-matrices (3.2) into (4.1).We emphasize that operators (C.2) obey the standard anticommutation relations: (cid:110) a ω , a † ω (cid:48) (cid:111) = (cid:110) b ω , b † ω (cid:48) (cid:111) = 2 πδ ( ω − ω (cid:48) ) , (C.4)if these relations hold for the operators a p and b p . However, the expectation value for the standard vacuumstate, which is annihilated by a p and b p , is not trivial: (cid:104) | a ω a † ω (cid:48) | (cid:105) = (cid:104) | b ω b † ω (cid:48) | (cid:105) = θ ( ω ) × πδ ( ω − ω (cid:48) ) , (cid:104) | a † ω a ω (cid:48) | (cid:105) = (cid:104) | b † ω b ω (cid:48) | (cid:105) = θ ( − ω ) × πδ ( ω − ω (cid:48) ) . (C.5)The operators a ω and b ω in the decomposition (4.3) are related to the operators a p and b p via the relation(C.3).Similarly one can show that the definite-frequency and definite-momentum boson creation and anni-hilation operators are related as follows: α ω = θ ( ω ) α p ω + θ ( − ω ) α †− p ω , (C.6)which implies that: (cid:104) | α ω α † ω (cid:48) | (cid:105) = θ ( ω ) × πδ ( ω − ω (cid:48) ) , (cid:104) | α † ω α ω (cid:48) | (cid:105) = θ ( − ω ) × πδ ( ω − ω (cid:48) ) . (C.7)52 Derivation of the coherent state
In this appendix we show that the coherent state that we use in the main body of the text has thefollowing form: | φ cl (cid:105) = e − i (cid:82) φ cl ˆ π φ dx | (cid:105) , where a p | (cid:105) = 0 . (D.1)Let us apply the operator ˆ φ ( y ) to the state | φ cl (cid:105) :ˆ φ ( y ) | φ cl (cid:105) = ˆ φ ( y ) e − i (cid:82) φ cl ˆ π φ dx | (cid:105) == ∞ (cid:88) n =0 ( − i ) n n ! (cid:90) dx . . . dx n φ cl ( x ) . . . φ cl ( x n ) ˆ φ ( y )ˆ π φ ( x ) . . . ˆ π φ ( x n ) | (cid:105) . (D.2)Commuting ˆ φ ( y ) with ˆ π φ ( x i ): [ ˆ φ ( y ) , ˆ π φ ( x i )] = iδ ( x i − y ) , (D.3)we get that:ˆ φ ( y ) | φ cl (cid:105) = (cid:40) ∞ (cid:88) n =0 ( − i ) n n ! iφ cl ( y ) n (cid:18)(cid:90) φ cl ( x )ˆ π φ ( x ) dx (cid:19) n − + e − i (cid:82) φ cl ˆ π φ dx ˆ φ ( y ) (cid:41) | (cid:105) == (cid:40) φ cl ( y ) ∞ (cid:88) n =1 ( − i ) n − ( n − (cid:18)(cid:90) φ cl ( x )ˆ π φ ( x ) dx (cid:19) n − + e − i (cid:82) φ cl ˆ π φ dx ˆ φ ( y ) (cid:41) | (cid:105) == φ cl | φ cl (cid:105) + e − i (cid:82) φ cl ˆ π φ dx ˆ φ ( y ) | (cid:105) . (D.4)From the last expression it is straightforward to show that eq. (5.39) is true.Now let us find the normalization factor. Define | φ cl (cid:105) = C ( φ cl ) e − i (cid:82) φ cl ˆ π φ dx | (cid:105) . (D.5)Then (cid:104) φ cl | φ cl (cid:105) = | C ( φ cl ) | (cid:104) | e i (cid:82) φ cl ˆ π φ dx e − i (cid:82) φ cl ˆ π φ dx | (cid:105) = | C ( φ cl ) | = 1 . (D.6)Hence, C ( φ cl ) = 1, because (cid:104) | (cid:105) = 1. Thus, we confirm the expression (5.40) for the coherent state. References [1] D. Krotov and A. M. Polyakov, “Infrared Sensitivity of Unstable Vacua,” Nucl. Phys. B , 410(2011) [arXiv:1012.2107 [hep-th]].[2] E. T. Akhmedov, “Lecture notes on interacting quantum fields in de Sitter space,” Int. J. Mod. Phys.D , 1430001 (2014) [arXiv:1309.2557 [hep-th]].[3] E. T. Akhmedov and F. Bascone, “Quantum heating as an alternative of reheating,” Phys. Rev. D , no. 4, 045013 (2018) [arXiv:1710.06118 [hep-th]].[4] E. T. Akhmedov, U. Moschella and F. K. Popov, “Characters of different secular effects in variouspatches of de Sitter space,” Phys. Rev. D , no. 8, 086009 (2019) [arXiv:1901.07293 [hep-th]].[5] E. T. Akhmedov, U. Moschella, K. E. Pavlenko and F. K. Popov, “Infrared dynamics of massivescalars from the complementary series in de Sitter space,” Phys. Rev. D , no. 2, 025002 (2017)[arXiv:1701.07226 [hep-th]]. 536] E. T. Akhmedov, N. Astrakhantsev and F. K. Popov, “Secularly growing loop corrections in strongelectric fields,” JHEP , 071 (2014) [arXiv:1405.5285 [hep-th]].[7] E. T. Akhmedov and F. K. Popov, “A few more comments on secularly growing loop corrections instrong electric fields,” JHEP , 085 (2015) [arXiv:1412.1554 [hep-th]].[8] E. T. Akhmedov, H. Godazgar and F. K. Popov, “Hawking radiation and secularly growing loopcorrections,” Phys. Rev. D , no. 2, 024029 (2016) [arXiv:1508.07500 [hep-th]].[9] E. T. Akhmedov and S. O. Alexeev, “Dynamical Casimir effect and loop corrections,” Phys. Rev. D , no. 6, 065001 (2017) [arXiv:1707.02242 [hep-th]].[10] L. Astrakhantsev and O. Diatlyk, “Massive quantum scalar field theory in the presence of movingmirrors,” International Journal of Modern Physics A 1 Vol. 33 (2018) [arXiv:1805.00549 [hep-th]].[11] D. A. Trunin, “Comments on the adiabatic theorem,” Int. J. Mod. Phys. A , no. 24, 1850140(2018) [arXiv:1805.04856 [hep-th]].[12] E. T. Akhmedov, O. Diatlyk and A. G. Semenov, “Out of equilibrium two-dimensional Yukawatheory in a strong scalar wave background,” arXiv:1909.12805 [hep-th].[13] M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field theory”[14] J. S. Schwinger, “On gauge invariance and vacuum polarization,” Phys. Rev. , 664 (1951).[15] L. V. Keldysh, “Diagram technique for nonequilibrium processes,” Zh. Eksp. Teor. Fiz. , 1515(1964) [Sov. Phys. JETP , 1018 (1965)].[16] A. Kamenev, “Many-body theory of non-equilibrium systems,” Cambridge, UK: Univ. Pr. (2011)[arXiv:cond-mat/041229].[17] J. Berges, “Introduction to nonequilibrium quantum field theory,” AIP Conf. Proc. , no. 1, 3(2004) [hep-ph/0409233].[18] J. Rammer, “Quantum field theory of non-equilibrium states,” Cambridge, UK: Univ. Pr. (2007).[19] E. A. Calzetta and B. L. B. Hu, “Nonequilibrium Quantum Field Theory,” Cambridge, UK: Univ.Pr. (2008).[20] L. D. Landau and E. M. Lifshitz, Vol. 10 (Pergamon Press, Oxford, 1975).[21] A. I. Nikishov, “S matrix in quantum electrodynamics with external field,” Teor. Mat. Fiz. , 48(1974)[22] N. B. Narozhnyi and A. I. Nikishov, “Solutions of the Klein-Gordon and Dirac equations for a particlein a constant electric field and a plane electromagnetic wave propagating along the field,” Teor. Mat.Fiz. , 16 (1976).[23] A. A. Grib, S. G. Mamaev, V. M. Mostepanenko, “Quantum effects in strong external fields” (At-omizdat, Moscow 1980, 296).A. A. Grib, S. G. Mamaev, V. M. Mostepanenko, “Vacuum quantum effects in strong fields” (St. Pe-tersburg: Friedmann Laboratory, 1994).[24] N. D. Birrell and P. C. W. Davies, “Quantum Fields in Curved Space,” Cambridge, UK: Univ. Pr.(1982). 5425] M. Bertola, F. Corbetta and U. Moschella, “Massless scalar field in two-dimensional de Sitter uni-verse,” Prog. Math. , 27 (2007) [math-ph/0609080].[26] T. S. Evans and D. A. Steer, “Wick’s theorem at finite temperature,” Nucl. Phys. B , 481 (1996)[hep-ph/9601268].[27] E. M. Lifshitz and L. P. Pitaevskii, Vol. 9 (Butterworth-Heinemann, Oxford, 1980).[28] H. Bateman, “Higher transcendental functions. Vol. 2” (MC. Graw-Hill, 1953)[29] E. T. Whittaker and G. N. Watson, “A course of modern analysis” (Cambridge University Press,1996).[30] E. T. Akhmedov, P. V. Buividovich and D. A. Singleton, “De Sitter space and perpetuum mobile,”Phys. Atom. Nucl. , 525 (2012) [arXiv:0905.2742 [gr-qc]].[31] E. T. Akhmedov and E. T. Musaev, “Comments on QED with background electric fields,” New J.Phys. , 103048 (2009) [arXiv:0901.0424 [hep-ph]].[32] E. T. Akhmedov and P. V. Buividovich, “Interacting Field Theories in de Sitter Space are Non-Unitary,” Phys. Rev. D , 104005 (2008) [arXiv:0808.4106 [hep-th]].[33] L. D. Landau and E. M. Lifshitz, Vol. 3 (Pergamon Press, Oxford, 1977).[34] F. W. J. Olver, “Uniform asymptotic expansions for Weber parabolic cylinder functions of largeorder”, J. Research NBS , no. 2, 131 (1959).[35] L. Kofman, A. D. Linde and A. A. Starobinsky, “Reheating after inflation,” Phys. Rev. Lett. ,3195 (1994) [hep-th/9405187].[36] L. Kofman, A. D. Linde and A. A. Starobinsky, “Towards the theory of reheating after inflation,”Phys. Rev. D , 3258 (1997) [hep-ph/9704452].[37] S. R. Coleman, “The Fate of the False Vacuum. 1. Semiclassical Theory,” Phys. Rev. D , 2929(1977) Erratum: [Phys. Rev. D , 1248 (1977)].[38] C. G. Callan, Jr. and S. R. Coleman, “The Fate of the False Vacuum. 2. First Quantum Corrections,”Phys. Rev. D , 1762 (1977).[39] A. D. Linde, “Decay of the False Vacuum at Finite Temperature,” Nucl. Phys. B , 421 (1983)Erratum: [Nucl. Phys. B , 544 (1983)].[40] D. S. F. Crothers, “Asymptotic expansions for parabolic cylinder functions of large order and argu-ment,” J. Phys. A: Gen. Phys. Vol. 5 1680 (1977).[41] S. R. Coleman and E. J. Weinberg, “Radiative Corrections as the Origin of Spontaneous SymmetryBreaking,” Phys. Rev. D , 1888 (1973).E. J. Weinberg, “Radiative corrections as the origin of spontaneous symmetry breaking,” hep-th/0507214.[42] J. Zinn-Justin, “Quantum field theory and critical phenomena,” Int. Ser. Monogr. Phys. , 1(2002).[43] A. P. C. Malbouisson, B. F. Swaiter and N. F. Svaiter, “Analytic regularization of the Yukawa modelat finite temperature,” J. Math. Phys. , 2210 (1997) [hep-th/9611030].[44] U. Mosel, “Path integrals in field theory: An introduction,” Berlin, Germany: Springer (2004) 213 p5545] F. Cooper, S. Habib, Y. Kluger, E. Mottola, J. P. Paz and P. R. Anderson, “Nonequilibrium quantumfields in the large N expansion,” Phys. Rev. D50