IFUM 651/FT-99 Bicocca-FT-99-41 An improved time–dependent Hartree–Fock approach for scalar φ4 QFT C. Destri (a,b) and E. Manfredini (b) (a) Dipartimento di Fisica G. Occhialini, Universita` di Milano–Bicocca and INFN, sezione di Milano1,2 0 0 (b) Dipartimento di Fisica, Universita` di Milano 0 and INFN, sezione di Milano1,2 2 (November 1999) n a J 8 Abstract 1 1 v The λφ4 model in a finite volume is studied within a non–gaussian Hartree– 8 7 Fock approximation (tdHF) both at equilibrium and out of equilibrium, with 1 particular attention to the structure of the ground state and of certain dy- 1 0 namical features in the broken symmetry phase. The mean–field coupled 0 time–dependent Schroedinger equations for the modes of the scalar field are 0 / derived and the suitable procedure to renormalize them is outlined. A fur- h ther controlled gaussian approximation of our tdHF approach is used in order p - to study the dynamical evolution of the system from non–equilibrium initial p e conditions characterized by a uniform condensate. We find that, during the h slowrollingdown,thelong–wavelength quantumfluctuationsdonotgrowtoa : v macroscopic size butdo scale with the linear size of the system, in accordance i X with similar results valid for the large N approximation of the O(N) model. r Thisbehavior underminesinapreciseway thegaussian approximation within a our tdHF approach, which therefore appears as a viable mean to correct an unlikelyfeatureofthestandardHFfactorization scheme,suchastheso–called “stopping at the spinodal line” of the quantum fluctuations. We also study the dynamics of the system in infinite volume with particular attention to the asymptotic evolution in the broken symmetry phase. We are able to show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential. 1mail address: Dipartimento di Fisica, Via Celoria 16, 20133 Milano, ITALIA. 2e-mail: [email protected], [email protected] 1 I. INTRODUCTION A great effort has been devoted in the last few years in order to develop a deeper qualita- tive and quantitative understanding of systems described by interacting quantum fields out of equilibrium. There is a class of physical problems that requires the consistent treatment of time dependent mean–fields in interaction with their own quantum or thermal fluctua- tions. We may mention, among others, the problem of reheating of the universe after the inflationary era of exponential growth and cooling, and the time evolution of the scalar order parameter through the chiral phase transition, soon to be probed in the forthcoming heavy– ionexperiments atCERN–SPS, BNL–RHICandCERN–LHC. Inthese situations, adetailed description of the time–dependent dynamics is necessary to calculate the non–equilibrium properties of the system. Indeed, the development of practical general techniques and the advent of faster and cheaper computers have made possible the discovery of novel and un- expected phenomena, ranging from dissipative processes via particle production to novel aspects of symmetry breaking [1–4]. From the technical point of view, it should be pointed out, first of all, that a perturbative treatment of this dynamical problem is meaningful only when the early time evolution is considered. The presence of parametric resonant bands or spinodal instabilities (in the case, respectively, of unbroken or spontaneously broken symmetries) rapidly turns the dynamics completely non–linear and non–perturbative. Thus, the asymptotic evolution at late time can be consistently studied only if approximate non–perturbative methods are applied to the problem [1]. Quite recently one of these schemes, namely the large N expansion at leading order (LN) [5,6], has been used in order to clarify some dynamical aspects of the φ4 theory in 3 spatial dimensions, reaching the conclusion that the non–perturbative and non–linear evolution of the system might eventually produce the onset of a form of non–equilibrium Bose–Einstein condensation (BEC) ofthelong–wavelength Goldstonebosonsusually present in the broken symmetry phase [3,4,7]. Another very interesting result in [7] concerns the dynamical Maxwell construction, which reproduces the flat region of the effective potential in case of broken symmetry as asymptotic fixed points of the background evolution. In a companion work [8] we have addressed the question of whether a standard BEC could actually take place as time goes on, by putting the system in a finite volume (a periodic box of size L) and carefully studying the volume dependence of out–of–equilibrium features in the broken symmetry phase. We summarize here the main result contained in [8]. The numerical solution shows the presence of a time scale τ , proportional to the linear L size L of the system, at which finite volume effects start to manifest, with the remarkable consequence that the zero-mode quantum fluctuations cannot grow macroscopically large if they start with microscopic initial conditions. In fact, the size of low–lying widths at time τ is of order L, to be compared to order L3/2 for the case of standard BEC. In other words L we confirmed that the linear growth of the zero mode width, as found also by the authors of [3,4,7], really signals the onset of a novel form of dynamical BEC, quite different from the standard one described by equilibrium finite–temperature field theory. This interpretation is reinforced by the characteristics of the long–wavelength fluctuations’ spectrum. Since after all the large N approximation is equivalent to a Gaussian ansatz for the time–dependent density matrix of the system [2,9], one might still envisage a scenario in 2 which, while gaussian fluctuations would stay microscopic, non–gaussian fluctuations would grow in time to a macroscopic size, leading to an occupation number for the zero mode proportional to the volume L3 of the system. Therefore, in order to go beyond the gaussian approximation, we will consider in this work a time–dependent HF approach capable in principle of describing the dynamics of some non–gaussian fluctuations of a single scalar field with φ4 interaction. Before going into the details of the analysis, let us briefly summarize the main limi- tations and the most remarkable results of the study of a scalar field out of equilibrium within the gaussian HF scheme [1,10–12]. First of all, this scheme has the advantage of going beyond perturbation theory, in the sense that the (numerical) solution of the evolu- tion equations will contain arbitrary powers of the coupling constant, corresponding to a non–trivial resummation of the perturbative series. For this reason, the method is able to take into account the quantum back–reaction on the fluctuations themselves, which shuts off their early exponential growth. This is achieved by the standard HF factorization of the quartic interaction, yielding a time dependent self–consistently determined mass term, which stabilizes the modes perturbatively unstable. The detailed numerical solution of the resulting dynamical equations clearly shows the dissipation associated with particle produc- tion, as a result of either parametric amplification in case of unbroken symmetry or spinodal instabilities in case of broken symmetry, as well as the shut off mechanism outlined above. However, the standard HF method is really not controllable in the case of a single scalar field, while it becomes exact only in the N limit. Moreover, previous approaches → ∞ to the dynamics in this approximation scheme had the unlikely feature of maintaining a weak (logarithmic) cut–off dependence on the renormalized equations of motion of the order parameter and the mode functions [1]. In this article we consider the case of a single scalar field (i.e. N = 1). With the aim of studying the dynamics of the model with the inclusion of some non–gaussian contributions, weintroduceanimprovedtime–dependent Hartree–Fockapproach. Evenifitisstillbasedon a factorized trial wavefunction(al), it has the merit to keep the quartic interaction diagonal in momentum space, explicitly in the hamiltonians governing the evolution of each mode of the field. In this framework, issues like the static spontaneous symmetry breaking can be better understood, and the further gaussian approximation needed to study the dynamics can be better controlled. In particular, questions like out–of–equilibrium “quantum phase ordering” and “dynamical Bose–Einstein condensation” can beproperly posed andanswered within a verifiable approximation. We also perform a detailed study of the asymptotic dynamics in infinite volume, with the aim of clarifying the issue of Maxwell construction in this approximation scheme. In fact, in the O(N)Φ4 model at leading order, the asymptotic dynamical evolution of the mean field completely covers the spinodal region of the classical potential, which coincides with the flatness region of the effective potential. This is what is called dynamical Maxwell construction [7]. When we use the HF approximation for the case of N = 1, we find that the spinodal region and the flatness region are different and the question arise of whether a full or partial dynamical Maxwell construction still takes place. In section II we set up the model in finite volume, defining all the relevant notations and the quantum representation we will be using to study the evolution of the system. We introduce in section III our improved time–dependent Hartree–Fock (tdHF) ap- 3 proximation, which generalizes the standard gaussian self-consistent approach [13] to non–gaussian wave–functionals; we then derive the mean–field coupled time–dependent Schroedinger equations for the modes of the scalar field, under the assumption of a uniform condensate, see eqs (3.5), (3.6) and (3.7). A significant difference with respect to previous tdHF approaches [1] concerns the renormalization of ultraviolet divergences. In fact, by means of a single substitution of the bare coupling constant λ with the renormalized one b λ in the Hartree–Fock hamiltonian, we obtain cut-off independent equations (apart from corrections in inverse powers, which are there due to the Landau pole). The substitution is introduced by hand, but is justified by simple diagrammatic considerations. Oneadvantageofnotrestrictingaprioritheself-consistent HFapproximationtogaussian wave–functionals, is in the possibility of a better description of the vacuum structure in case of broken symmetry. In fact we can show quite explicitly that, in any finite volume, in the ground state the zero–mode of φ field is concentrated around the two vacua of the broken symmetry, driving the probability distribution for any sufficiently wide smearing of the field into a two peaks shape. This is indeed what one would intuitively expect in case of symmetry breaking. On the other hand none of this appears in a dynamical evolution that starts from a distribution localized around a single value of the field in the spinodal region, confirming what already seen in the large N approach [8]. More precisely, within a further controlled gaussian approximation of our tdHF approach, one observes that initially microscopic quantum fluctuations never becomes macroscopic, suggesting that also non– gaussian fluctuations cannot reach macroscopic sizes. As a simple confirmation of this fact, ˙ consider the completely symmetric initial conditions φ = φ = 0 for the background: in h i h i this case we find that the dynamical equations for initially gaussian field fluctuations are identical to those of large N (apart for a rescaling of the coupling constant by a factor of three; cfr. ref. [8]), so that we observe the same asymptotic vanishing of the effective mass. However, this time no interpretation in terms of Goldstone theorem is possible, since the broken symmetry is discrete; rather, if the width of the zero–mode were allowed to evolve into a macroscopic size, then the effective mass would tend to a positive value, since the mass in case of discrete symmetry breaking is indeed larger than zero. Anyway, alsointhegaussianHFapproach, we dofindawhole classofcaseswhich exhibit the time scale τ . At that time, finite volume effects start to manifest and the size of the L low–lying widths is of order L. We then discuss why this undermines the self–consistency of the gaussian approximation, imposing the need of further study, both analytical and numerical. In section IV we study the asymptotic evolution in the broken symmetry phase, in infinite volume, when the expectation value startswithin theregion between the two minima of the potential. We are able to show by precise numerical simulations, that the fixed points of the background evolution do not cover the static flat region completely. On the contrary, the spinodal region seems to be absolutely forbidden for the late time values of the mean field. Thus, as far as the asymptotic evolution is concerned, our numerical results lead to the following conclusions. We can distinguish the points lying between the two minima in a fashion reminiscent of the static classification: first, the values satisfying the property v/√3 < φ¯ v are metastable points, in the sense that they are fixed points ∞ ≤ of the background(cid:12)evo(cid:12)lution, no matter which initial condition comprised in the interval (cid:12) (cid:12) ( v,v) we choose (cid:12)for t(cid:12)he expectation value φ¯; secondly, the points included in the interval − 4 0 < φ¯ < v/√3 are unstable points, because if the mean field starts from one of them, ∞ after(cid:12)an(cid:12)early slow rolling down, it starts to oscillate with decreasing amplitude around (cid:12) (cid:12) a poi(cid:12)nt i(cid:12)nside the classical metastable interval. Obviously, φ¯ = v is the point of stable ¯ equilibrium, and φ = 0 is a point of unstable equilibrium. Actually, it should be noted that our data do not allow a precise determination of the border between the dynamical unstable and metastable regions; thus, the number we give here should be looked at as an educated guess inspired by the analogous static classification and based on considerations about the solutions of the gap equation [see eq. (3.36)] Finally, in section VI we give a brief summary of the results presented in this article and we outline some interesting open questions that need more work before being answered properly. II. CUTOFF FIELD THEORY Let us consider the scalar field operator φ and its canonically conjugated momentum π in a D dimensional periodic box of size L and write their Fourier expansion as customary − φ(x) = L−D/2 φ eik·x , φ† = φ k k −k k X π(x) = L−D/2 π eik·x , π† = π k k −k k X with the wavevectors k naturally quantized: k = (2π/L)n, n ZD. ∈ (D) The canonical commutation rules are [φ ,π ] = iδ , as usual. The introduction of a k −k′ kk′ finite volume should be regarded as a regularization of the infrared properties of the model, which allows to “count” the different field modes and is needed especially in the case of broken symmetry. Tokeepcontrolalsoontheultravioletbehaviorandmanagetohandletherenormalization procedure properly, we restrict the sums over wavevectors to the points lying within the D dimensional sphere of radius Λ, that is k2 Λ2, with = ΛL/2π some large integer. − ≤ N Till both the cut–offs remain finite, we have reduced the originalfield–theoretical problem to a quantum–mechanical framework with finitely many (of order D−1) degrees of freedom. N The φ4 Hamiltonian is 1 H = dDx π2 +(∂φ)2 +m2φ2 +λ φ4 = 2 b b Z h i 1 λ = π π +(k2 +m2)φ φ + φ φ φ φ δ(D) 2 k −k b k −k 4 k1 k2 k3 k4 k1+k2+k3+k4,0 Xk h i k1,kX2,k3,k4 where m2 and λ are the bare parameters and depend on the UV cutoff Λ in such a way to b b guarantee a finite limit Λ for all observable quantities. It should be noted here that, → ∞ being the theory trivial [14] (as is manifest in the resummed one–loop approximation due to the Landau pole) the ultraviolet cut–off should be kept finite and much smaller than the renormalon singularity. In this case, we must regard the φ4 model as an effective low–energy 5 theory (here low–energy means practically all energies below Planck’s scale, due to the large value of the Landau pole for renormalized coupling constants of order one or less). We shall work in the wavefunction representation where ϕ Ψ = Ψ(ϕ) and h | i ∂ (φ Ψ)(ϕ) = ϕ Ψ(ϕ) , (π Ψ)(ϕ) = i Ψ(ϕ) 0 0 0 − ∂ϕ 0 while for k > 0 (in lexicographic sense) 1 1 ∂ ∂ (φ Ψ)(ϕ) = (ϕ iϕ )Ψ(ϕ) , (π Ψ)(ϕ) = i Ψ(ϕ) ±k k −k ±k √2 ± √2 − ∂ϕk ± ∂ϕ−k! Notice that by construction the variables ϕ are all real. k In practice, the problem of studying the dynamics of the φ4 field out of equilibrium consists now in trying to solve the time-dependent Schroedinger equation given an initial wavefunction Ψ(ϕ,t = 0) that describes a state of the field far away from the vacuum. This approach could be very well generalized in a straightforward way to mixtures described by density matrices, as done, for instance, in [10,15,16]. Here we shall restrict to pure states, for sake of simplicity and because all relevant aspects of the problem are already present in this case. We shall consider here the time-dependent Hartree–Fock (tdHF) approach (an improved version with respect to what is presented, for instance, in [13]), being the large N expansion to leading order treated in another work [8]. In fact these two methods are very closely related (see, for instance in [17]). However, before passing to any approximation, we would like to stress that the following rigorous result can be immediately established in this model with both UV and IR cutoffs. A. A rigorous result: the effective potential is convex This is a well known fact in statistical mechanics, being directly related to stability requirements. It would therefore hold also for the field theory in the Euclidean functional formulation. In our quantum–mechanical context we may proceed as follow. Suppose the field φ is coupled to a uniform external source J. Then the ground state energy E (J) is 0 a concave function of J, as can be inferred from the negativity of the second order term in ∆J of perturbation around any chosen value of J. Moreover, E (J) is analytic in a 0 finite neighborhood of J = 0, since Jφ is a perturbation “small” compared to the quadratic ¯ and quartic terms of the Hamiltonian. As a consequence, this effective potential V (φ) = eff E (J) Jφ¯, φ¯= E′(J) = φ , that is the Legendre transform of E (J), is a convex analytic 0 − 0 h i0 ¯ 0 function in a finite neighborhood of φ = 0. In the infrared limit L , E (J) might 0 ¯ ¯ → ∞ develop a singularity in J = 0 and V (φ) might flatten around φ = 0. Of course this eff possibility would apply in case of spontaneous symmetry breaking, that is for a double–well classical potential. This is a subtle and important point that will play a crucial role later on, even if the effective potential is relevant for the static properties of the model rather than the dynamical evolution out of equilibrium that interests us here. In fact such evolution is governed by the CTP effective action [18,19] and one might expect that, although non–local in time, it asymptotically reduces to a multiple of the effective potential for trajectories ¯ of φ(t) with a fixed point at infinite time. In such case there should exist a one–to–one correspondence between fixed points and minima of the effective potential. 6 III. TIME-DEPENDENT HARTREE–FOCK In order to follow the time evolution of the non–gaussian quantum fluctuations we con- sider in this section a time–dependent HF approximation capable in principle of describing the dynamics of non–gaussian fluctuations of a single scalar field with φ4 interaction. We examine in this work only states in which the scalar field has a uniform, albeit possibly time–dependent expectation value. In a tdHF approach we may then start from a wavefuction of the factorized form (which would be exact for free fields) Ψ(ϕ) = ψ (ϕ ) ψ (ϕ ,ϕ ) (3.1) 0 0 k k −k k>0 Y The dependence of ψ on its two arguments cannot be assumed to factorize in general k since space translations act as SO(2) rotations on ϕ and ϕ (hence in case of translation k −k invariance ψ depends only on ϕ2+ϕ2 ). The approximation consists in assuming this form k k −k as valid at all times and imposing the stationarity condition on the action δ dt i∂ H = 0 , Ψ(t) Ψ(t) (3.2) t h − i h·i ≡ h |·| i Z with respect to variations of the functions ψ . To enforce a uniform expectation value of φ k we should add a Lagrange multiplier term linear in the single modes expectations ϕ for k h i k = 0. The multiplier is then fixed at the end to obtain ϕ = 0 for all k = 0. Actually one k 6 h i 6 mayverify thatthisis equivalent to thesimpler approachinwhich ϕ isset to vanishforall k h i k = 0 before any variation. Then the only non trivial expectation value in the Hamiltonian, 6 namely that of the quartic term, assumes the form 1 3 2 dDx φ(x)4 = ϕ4 3 ϕ2 2 + (ϕ2 +ϕ2 )2 2 ϕ2 + ϕ2 h i LD h 0i− h 0i 2LD h k −k i− h ki h −ki Z h i kX>0(cid:20) (cid:16) (cid:17) (cid:21) 2 3 + ϕ2 (3.3) LD h ki! k X Noticethattheterms inthefirst rowwouldcancel completely outforgaussianwavefunctions ψ with zero mean value. The last term, where the sum extends to all wavevectors k, k corresponds instead to the standard mean field replacement φ4 3 φ2 2. The total h i → h i energy of our trial state now reads 1 ∂2 λ E = H = +(k2 +m2)ϕ2 + b dDx φ(x)4 (3.4) h i 2 *∂ϕ2 b k+ 4 h i k k Z X and from the variational principle (3.2) we obtain a set of simple Schroedinger equations i∂ ψ = H ψ (3.5) t k k k 1 ∂2 1 λ H = + ω2ϕ2 + b ϕ4 0 −2∂ϕ2 2 0 0 4LD 0 0 (3.6) H = 1 ∂2 + ∂2 + 1ω2(ϕ2 +ϕ2 )+ 3λb ϕ2 +ϕ2 2 k −2 ∂ϕ2 ∂ϕ2 ! 2 k k −k 8LD k −k k −k (cid:16) (cid:17) 7 which are coupled in a mean–field way only through 1 ω2 = k2 +m2 +3λ Σ , Σ = ϕ2 (3.7) k b b k k LD h qi q2X≤Λ2 q6=k,−k and define the HF time evolution for the theory. By construction this evolution conserves the total energy E of eq. (3.4). It should be stressed that in this particular tdHF approximation, beside the mean–field back–reaction termΣ ofallothermodesonω2, we keep alsothecontributionofthediagonal k k scattering through the diagonal quartic terms. In fact this is why Σ has no contribution k from the k mode itself: in a gaussian approximation for the trial wavefunctions ψ the k − Hamiltonians H would turn out to be harmonic, the quartic terms being absent in favor of k a complete back–reaction ϕ2 + ϕ2 1 Σ = Σ + h ki h −ki = ϕ2 (3.8) k LD LD h ki k X Of course the quartic self–interaction of the modes as well as the difference between Σ and Σ are suppressed by a volume effect and could be neglected in the infrared limit, k provided all wavefunctions ψ stays concentrated on mode amplitudes ϕ of order smaller k k than LD/2. This is the typical situation when all modes remain microscopic and the volume in the denominators is compensated only through the summation over a number of modes proportional to the volume itself, so that in the limit L sums are replaced by integrals → ∞ dDk Σ Σ ϕ2 k → → Zk2≤Λ2 (2π)Dh ki Indeed we shall apply this picture to all modes with k = 0, while we do expect exceptions 6 for the zero–mode wavefunction ψ . 0 The treatment of ultraviolet divergences requires particular care, since the HF approxi- mation typically messes things up (see, for instance, [20]). Following the same login of the large N approximation [1,8,6], we could take as renormalization condition the requirement that the frequencies ω2 are independent of Λ, assuming that m2 and λ are functions of Λ k b b itself and of renormalized Λ independent parameters m2 and λ such that − ω2 = k2 +m2 +3λ[Σ ] (3.9) k k finite where by [.] we mean the (scheme–dependent) finite part of some possibly ultraviolet finite divergent quantity. Unfortunately this would not be enough to make the spectrum of energy differences cutoff–independent, because of the bare coupling constant λ in front of the b quartic terms in H and the difference between Σ and Σ [such problem does not exist k k in large N because that is a purely gaussian approximation]. Again this would not be a problem whenever these terms become negligible as L . At any rate, to be ready to → ∞ handle the cases when this is not actually true and to define an ultraviolet–finite model also at finite volume, we shall by hand modify eq. (3.3) as follows: 2 λ dDx φ(x)4 =λL−D ϕ4 3 ϕ2 2 + 3 (ϕ2 +ϕ2 )2 2 ϕ2 + ϕ2 bZ h i (h 0i− h 0i 2 k>0(cid:20)h k −k i− (cid:16)h ki h −ki(cid:17) (cid:21)) P +3λ LDΣ2 (3.10) b 8 We keep the bare coupling constant in front of the term containing Σ2 because that part of the hamiltonian is properly renormalized by means of the usual cactus resummation [21] which corresponds to the standard HF approximation. On the other hand, within the same approximation, it is not possible to renormalize the part in curly brackets of the equation above, because of the factorized form (3.1) that we have assumed for the wavefunction of the system. In fact, the 4 legs vertices in the curly brackets are diagonal in momentum − space; at higher order in the loop expansion, when we contract two or more vertices of this type, no sum over internal loop momenta is produced, so that all higher order perturbation terms are suppressed by volume effects. However, we know that in the complete theory, the wavefunction is not factorized and loops contain all values of momentum. This suggests that, in order to get a finite hamiltonian, we need to introduce in the definition of our model some extra resummation of Feynmann diagrams, that is not automatically contained in this self–consistent HF approach. The only choice consistent with the cactus resummation performed in the two–point function by the HF scheme is the resummation of the 1-loop fish diagram in the four–point function. This amounts to the change from λ to λ and it is b enough to guarantee the ultraviolet finiteness of the hamiltonian through the redefinition H H + λ−λbϕ4 , H H + 3(λ−λb) ϕ2 +ϕ2 2 (3.11) 0 → 0 4LD 0 k → k 8LD k −k (cid:16) (cid:17) At the same time the frequencies are now related to the widths ϕ2 by h −ki ω2 = k2 +M2 3λL−D( ϕ2 + ϕ2 ) , k > 0 k − h ki h −ki (3.12) M2 ω2 +3λL−D ϕ2 = m2 +3λ Σ ≡ 0 h 0i b b Apart for O(L−D) corrections, M plays the role of time–dependent mass for modes with k = 0, in the harmonic approximation. 6 In this new setup the conserved energy reads 2 E = H 3λ LDΣ2 + 3λL−D ϕ2 2 + ϕ2 + ϕ2 (3.13) h ki− 4 b 4 "h 0i k>0 h ki h −ki # kX≥0 P (cid:16) (cid:17) Since the gap–like equations (3.12) are state–dependent, we have to perform the renor- malization first for some reference quantum state, that is for some specific collection of wavefunctions ψ ; as soon as m2 and λ are determined as functions Λ, ultraviolet finiteness k b b will hold for the entire class of states with the same ultraviolet properties of the reference state. Then an obvious consistency check for our HF approximation is that this class is closed under time evolution. Rather than a single state, we choose as reference the family of gaussian states parametrized by the uniform expectation value φ(x) = L−D/2 ϕ = φ¯ (recall that we 0 h i h i have ϕ = 0 when k = 0 by assumption) and such that the HF energy E is as small as k possibhle fior fixed φ¯. Th6en, apart from a translation by LD/2φ¯ on ϕ , these gaussian ψ are 0 k ground state eigenfunctions of the harmonic Hamiltonians obtained from H by dropping k the quartic terms. Because of the k2 in the frequencies we expect these gaussian states to dominate in the ultraviolet limit also at finite volume (as discussed above they should dominate in the infinite–volume limit for any k = 0). Moreover, since now 6 1 1 ϕ2 = LDφ¯2 + , ϕ2 = , k = 0 (3.14) h 0i 2ω h ±ki 2ω 6 0 k 9 the relation (3.12) between frequencies and widths turn into the single gap equation 1 1 M2 = m2 +3λ φ¯2 + (3.15) b b 2LD √k2 +M2 q2X≤Λ2   ¯ fixing the self-consistent value of M as a function of φ. It should be stressed that (3.12) turns through eq. (3.14) into the gap equation only because of the requirement of energy minimization. Generic ψ , regarded as initial conditions for the Schroedinger equations k (3.5), are in principle not subject to any gap equation. The treatment now follows closely that in the large N approximation [8], the only dif- ference being in the value of the coupling, now three times larger. In fact, in case of O(N) symmetry, the quantum fluctuations over a given background φ(x) = φ¯ decompose for each k into one longitudinal mode, parallel to φ¯, and N 1 trhansveirse modes orthogonal to it; by boson combinatorics the longitudinal mode cou−ples to φ¯ with strength 3λ /N b and decouple in the N limit, while the transverse modes couple to φ¯ with strength → ∞ (N 1)λ /N λ ; when N = 1 only the longitudinal mode is there. b b − → As L , ω2 k2 + M2 and M is exactly the physical mass gap. Hence it must → ∞ k → be Λ independent. At finite L we cannot use this request to determine m2 and λ , since, − b b unlike M, they cannot depend on the size L. At infinite volume we obtain dDk 1 M2 = m2 +3λ [φ¯2 +I (M2,Λ)] , I (z,Λ) b b D D ≡ Zk2≤Λ2 (2π)D 2√k2 +z (3.16) ¯ When φ = 0 this equation fixes the bare mass to be m2 = m2 3λ I (m2,Λ) (3.17) b − b D ¯ where m = M(φ = 0) may be identified with the equilibrium physical mass of the scalar particles of the infinite–volume Fock space without symmetry breaking (see below). Now, the coupling constant renormalization follows from the equalities M2 = m2 +3λ [φ¯2 +I (M2,Λ) I (m2,Λ)] b D D − (3.18) = m2 +3λφ¯2+3λ I (M2,Λ) I (m2,Λ) D D − finite h i and reads when D = 3 λ 3λ 2Λ = 1 log (3.19) λ − 8π2 m√e b that is the standard result of the one–loop renormalization group [22]. When D = 1, that is a 1+1 dimensional quantum field theory, I (M2,Λ) I (m2,Λ) is already finite and the D D − − dimensionfull coupling constant is not renormalized, λ = λ. b The Landau pole in λ prevents the actual UV limit Λ . Nonetheless, neglecting b → ∞ all inverse powers of the UV cutoff when D = 3, it is possible to rewrite the gap equation (3.18) as M2 m2 = +3φ¯2 (3.20) ˆ ˆ λ(M) λ(m) 10