# Bohm Quantum Trajectories of Scalar Field in Trans-Planckian Physics

Authors: Jung-Jeng Huang
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

In lattice Schrödinger picture, we investigate the possible effects of trans-Planckian physics on the quantum trajectories of scalar field in de Sitter space within the framework of the pilot-wave theory of de Broglie and Bohm. For the massless minimally coupled scalar field and the Corley-Jacobson type dispersion relation with sextic correction to the standard-squared linear relation, we obtain the time evolution of vacuum state of the scalar field during slow-roll inflation. We find that there exists a transition in the evolution of the quantum trajectory from well before horizon exit to well after horizon exit, which provides a possible mechanism to solve the riddle of the smallness of the cosmological constant.


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## Body

## 1. Introduction

The scenario of inflationary cosmology successfully provides the paradigm for generating the inhomogeneities which seed the structures of the universe we observe today [1]. In the simplest inflationary model, these inhomogeneities arise from the quantum fluctuations in a single scalar field about its vacuum state. The conventional choice of a vacuum during inflation is the Bunch-Davies (BD) vacuum [2]. However, it is well known that the notion of a vacuum state during inflation is ambiguous in quantum theory [3], and the choice of initial quantum vacuum state affects the predictions of inflation [4, 5].Recently Perez et al. [6] discussed how predictions for the cosmic microwave background (CMB) could be affected by a hypothetical dynamical collapse of the wave function. As a different possibility, the notion of quantum nonequilibrium [7, 8] was also discussed in terms of the pilot-wave theory of de Broglie and Bohm [9–13] and was later generalized to include all deterministic hidden-variables theories [14]. In the context of inflationary cosmology, a deterministic hidden-variables theory allows the existence of vacuum states with nonstandard or nonequilibrium field fluctuations, resulting in statistical predictions that deviate from those of quantum theory [15, 16].Moreover, the inflationary scenario has a serious trans-Planckian problem, which is whether the predictions of standard cosmology are insensitive to effects of the trans-Planckian physics. Since there is no successful quantum gravity theory to handle the physics around the Planck scale, one of the methods is to use the effective nonlinear dispersion relations to mimic the effects of the trans-Planckian physics. For example, the Corley-Jacobson (CJ) type dispersion relations were used to study the possible effects of the trans-Planckian physics on cosmological perturbations [17–19]. Note also that the CJ type dispersion relations can also be derived naturally from the recently proposed quantum gravity model called Horava-Lifshitz (HL) gravity [20–22].In our previous papers [23–26], we used the lattice Schrödinger picture to study the free scalar field theory in de Sitter space, derived the wave functionals for the BD vacuum state and its excited states, and found the corresponding de Broglie and Bohm quantum trajectories. The purpose of this paper is to study further the possible effects of the trans-Planckian physics on the quantum trajectories of scalar field in de Sitter space within the framework of the pilot-wave theory of de Broglie and Bohm. Throughout this paper, we will set ℏ=c=1.


## 2. Pilot-Wave Scalar Field in De Sitter Space

We consider the scalar field theory which has the Lagrangian density(2.1)L=|g|1/2{12[gμν(x)ϕ(x),μϕ(x),ν]-V(ϕ)},V(ϕ)=m2ϕ22+ξRϕ22,
where ϕ is a real scalar field,V(ϕ)is the potential,m is the mass of the scalar quanta, R is the Ricci scalar curvature, ξ is the coupling parameter, and g = det gμν, μ,ν=0,1,…,d. For a spatially flat (1 + d)-dimensional Robertson-Walker spacetime with scale factor a(t), we have(2.2)ds2=dt2-a2(t)d2xi,i=1,2,…,d,L=ad{12[(∂0ϕ)2-a-2(∂iϕ)2]-V(ϕ)}.
In the (1 + d)-dimensional de Sitter space, we have a(t)=exp(ht), where h≡ȧ/a is the Hubble parameter which is a constant. Note that in three spatial dimensions d = 3, the curvature R=12h2 is also a constant. Throughout this paper, we use this exact de Sitter space-time background to describe the inflationary era, which is only a special case (n→∞limit) of power-law inflation with a(t)=a0tn. For mathematical simplicity, we consider the case of d = 1 in the following. The extention to higher spatial dimension is straightforward without changing the nature of our results.In the lattice formalism, we have the following changes:(2.3)ϕ(x,t)⟶ϕj(t),∂0ϕ(x,t)⟶ϕ•j,j=1,2,…,N,∂1ϕ(x,t)⟶|ϕj(t)-ϕj-1(t)|ε,
and the Lagrangian reads(2.4)L=∑j=1N{12[ϕj~•-12hϕ̃j]2-12(ϕ̃j-ϕ̃j-1)2a2ε2-12(m2+ξR)ϕ̃j2},
where ϕ̃j≡a1/2ε1/2ϕj, and ε=W/N; that is, Wis the overall comoving spatial size of lattice. From (2.4) we obtain the Hamiltonian(2.5)H=∑j=1N{12p̃j2+12hp̃jϕ̃j+12(ϕ̃j-ϕ̃j-1)2a2ε2+12(m2+ξR)ϕ̃j2},
where p~j is the conjugate momentum of ϕ̃j,(2.6)p̃j=∂∂ϕj~•L=ϕj~•-12hϕ̃j.
Then we consider the discrete Fourier transforms(2.7)ϕ̃j=∑l=1Nexp⁡[-ijl2π/N]Nϕl,ϕl=∑j=1Nexp⁡[ijl2π/N]Nϕ̃j,p̃j=∑l=1Nexp⁡[ijl2π/N]Npl,pl=∑j=1Nexp⁡[-ijl2π/N]Np̃j,
such that(2.8){ϕ̃j,p̃j}={ϕ̃j+N,p̃j+N},{ϕl,pl}={ϕl+N,pl+N},{ϕN-l,pN-l}={ϕ-l,p-l}={ϕl*,pl*}.
Here (ϕl,pl∣l=1,2,…,N/2) can be chosen as the independent variables which obey the Poisson brackets {ϕl,pl}={ϕ̃j,p̃j}. From (2.7) we can obtain the following identities:(2.9)∑j=1Np̃j2=∑l=1Npl*pl,∑j=1Np̃j2ϕ̃j=∑l=1Nplϕl,∑j=1Nϕ̃j2=∑l=1Nϕl*ϕl,∑j=1N(ϕ̃j-ϕ̃j-1)2=∑l=1N4sin2(lπN)ϕl*ϕl.
Furthermore, we define(2.10)ωl≡2εsin⁡(lπN),ϕl=ϕ1l+iϕ2l,pl=p1l+ip2l,
where pl is the conjugate momentum for ϕl, and the subscripts 1 and 2 denote the real and imaginary parts, respectively. Therefore, the Hamiltonian (2.5) becomes in momentum space(2.11)H=2∑l=1N/2∑r=12Hrl,Hrl=12prl2+12hprlϕrl+12a-2ωl2ϕrl2+12(m2+ξR)ϕrl2.To quantize the theory above, we note also that each pair of operatorsϕrl∧ and prl∧ must satisfy the equal time commutation relations [ϕrl∧,prl∧]=i. These commutation relations are realized by the representation(2.12)ϕrl⟶ϕrl,prl⟶-i∂∂ϕrl,
that is, the functional Schrödinger representation, which is not so widely used as the Fock representation. These operators act on wave functional ψ(ϕ), and the inner product is given by 〈ψ1∣ψ2〉=∫(∏rlDϕrl)ψ1*(ϕ)ψ2(ϕ). With the field basis |ϕ〉, that is, |ϕ〉 are the eigenstates of the operator ϕ∧rl, ϕ∧rl|ϕ〉=ϕrl|ϕ〉, and then the wave functional ψ(ϕ,t) can be regarded as the expansion coefficient 〈ϕ∣ψ(t)〉 of the state |ψ(t)〉. The dynamics of the wave functional is, therefore, determined by the functional Schrödinger equation(2.13)Hψ=i∂∂tψ,
with H defined by (2.11). Writing ψ as the product form(2.14)ψ[ϕrl,t]=∏l=1N/2∏r=12ψrl(ϕrl,t)≡∏rlψrl(ϕrl,t),
we obtain from (2.13) the following:(2.15)Hrlψrl=i∂∂tψrl,r=1,2.
Rewriting (2.15) as the following form: (2.16){12(prl+12hϕrl)2-18h2ϕrl2+12α-2ωl2ϕrl2+12(m2+ξR)ϕrl2}ψrl=i∂∂tψrl,
and setting Prl=prl+1/2hϕrl=-i∂/∂ϕrl, so that [ϕrl,Prl]=[ϕrl,prl]=i, gives the time-dependent Schrödinger equation(2.17)-12∂2ψrl∂ϕrl2+12[a-2ωl2+(m2+ξR)-14h2]ϕrl2ψrl=i∂ψrl∂t.
To solve (2.17), we use the conformal time τ defined by dτ=dt/a(2.18)τ=-h-1exp⁡(-ht)=-h-1a-1,-∞<τ<0.
Then, for each real mode ϕrl, the normalized instantaneous vacuum and its excited states are found to be [24](2.19)ψ(nrl)(ϕrl,τ)=Al(nrl)(τ)H(nrl)(ηrl(τ))exp⁡(-12Bl(τ)a-1ϕrl2),nrl=0,1,2,…,ηrl(τ)≡g(τ)ωl1/2a-1/2ϕrl.
Here H(nrl) is the nth-order Hermite polynomial, and the complex function Bl(τ), real function g(τ), and complex function Al(nrl)(τ) are given by (2.20), (2.21), and (2.22), respectively(2.20)Bl(τ)=2/π|τ||Hν(1)|2-iωl2(|Hν(1)|2)′|Hν(1)|2,(2.21)g(τ)=(2πωl|τ|)1/21|Hν(1)|,(2.22)Al(nrl)(τ)=exp⁡[-i12∫Bl(τ)dτ-inrlωl∫g2(τ)dτ+const],
where Hν(1)(ωl|τ|) is the Hankel function of the first kind and of order ν, with ν2=1/4-(m2+ξR)/h2, and the prime in (2.20) denotes the derivative with respect to ωl|τ|. Note that the wave functionals for each real mode ϕrl can also be rewritten in the following form [25]:(2.23)ψrl(nrl)(ϕrl,τ)=R(nrl)(ϕrl,τ)exp⁡[iΘ(nrl)(ϕrl,τ)],nrl=0,1,2,…,
with the amplitude(2.24)R(nrl)(ϕrl,τ)=[2h/ππ2nnrl!|Hν(1)|]1/2H(nrl)(ηrl)exp⁡(-12ηrl2),ηrl=2h/π|Hν(1)|ϕrl,
and the phase (2.25)Θ(nrl)(ϕrl,τ)=-hωl|τ|2(|Hν(1)|)′|Hν(1)|ϕrl2-(12+nrl)∫2/π|τ||Hν(1)|2dτ.
The complete state wave functionals can be written as ψ[n][ϕrl,t]=∏rlψ(nrl)(ϕrl,t), where nrl=0,1,2,… and [n]≡(ni,nj,…) which means that it is possible for different field modes to be in different excited states; that is, mode i is in the ni-excited state, mode j is in the nj excited state, and so forth. For nrl=0, the ground state wave functional corresponds to the standard BD vacuum chosen conventionally in the literature. For the case of spatial dimension d = 3, we have ν2=9/4-(m2+ξR)/h2 with R=12h2, and the mode index l in ωl carries labels (li,i=1,2,3) which will be suppressed below.To define the pilot-wave scalar field theory, we note from (2.13)–(2.17) that, in the case of d = 3, the time-dependent Schrödinger equation for ψ is given by       (2.26)i∂ψ∂t=∑rl{-12∂2∂ϕrl2+12[a-2ωl2+(m2+ξR)-94h2]ϕrl2}ψ,
which implies the continuity equation(2.27)∂|ψ|2∂t+∑rl{∂∂ϕrl[|ψ|2∂Θ∂ϕrl]}=0
and the de Broglie velocities(2.28)dϕrldt=∂Θ∂ϕrl,
where ψ=|ψ|exp[iΘ]. The wave functional ψrl(whereψrl=|ψrl|exp[iΘrl],andΘ=∑rlΘrl) for a single mode ϕrl satisfies (2.17), which implies the continuity equation(2.29)∂|ψrl|2∂t+∂∂ϕrl(|ψrl|2∂Θrl∂ϕrl)=0,
and the de Broglie velocity field(2.30)dϕrldt=∂Θrl∂ϕrl.
Here ψ is interpreted as a physical field in field configuration space, guiding the evolution of ϕrl.Since the exact forms of the normalized instantaneous BD vacuum and its excited states are given by (2.23), it is straightforward to find the corresponding quantum trajectories of these eigenstates by solving (2.30) for each mode. Substituting the phase (2.25) into (2.30) and using the conformal time τ yields the de Broglie velocity field for ϕrl(2.31)dϕrldτ=a∂Θ(n)(ϕrl,τ)∂ϕrl=-ωl(|Hν(1)|)′|Hν(1)|ϕrl.
In the continuum limit (ωl→k), (2.31) reads(2.32)dϕrkdτ=-k(|Hν(1)(k|τ|)|)′|Hν(1)(k|τ|)|ϕrk,
which has the solution(2.33)ϕrk(z)=C|Hν(1)(z)|,
where z≡k|τ|=k/a/h is the ratio of physical wave number to the inverse of Hubble radius, and integration constant C is chosen to be ϕrk(z0)/|Hν(1)(z0)| with z0 being some reference point. Note that the quantum trajectory (2.33) is independent of the quantum number nrl and depends on the form of the potential V(ϕ) through ν [26].


## 3. Evolution of Vacuum Wave Functional in the Trans-Planckian Physics

To investigate the effect of the trans-Planckian physics, we consider the Corley-Jacobson type dispersion relations(3.1)ω2(ka)=k2[1+bs(kaM)2s],
where M is a cut-off scale, s is an integer, and bs is an arbitrary coefficient [17–19]. Note that the action for a scalar field with the modified dispersion relation (3.1) with s=1 and b1>0 takes the form [27, 28](3.2)S=∫d4x-g(Lϕ+Lcor+Lu),
where Lϕ is the standard Lagrangian of a minimally coupled scalar field(3.3)Lϕ=12gμν∂μϕ∂νϕ-V(ϕ),Lcor corresponds to the nonlinear part of the dispersion relation(3.4)Lcor=-b1M2(D2ϕ)2,
and Lu describes the dynamics of a unit time-like vector field uμ which defines a preferred rest frame(3.5)Lu=-λ(gμνuμuν-1)-d1FμνFμν,
with(3.6)D2ϕ=⊥αβ∇α∇βϕ+uα∇αϕ∇βuβ,⊥μν≡-gμν+uμuν,Fμν≡∇μuν-∇νuμ.
Here ∇μ is the covariant derivative associated with the metric gμν, the tensor ⊥μν gives the metric on a slice of fixed time while D2 is proportional to the Laplacian operator on the same surface, λ is the Lagrange multiplier, and the parameters b1 and d1 with no dimensions and the dimensions of mass square, respectively, are constrained by the astrophysical observations. For the modified dispersion relation (3.1) with s=2 and b2>0, (3.4) should be replaced with Lcor=(-b2/M4)(D21ϕ)(D22ϕ), where the operator D2n is defined as D2n≡Dμ1Dμ1⋯DμnDμn. Then using z=k|τ|=k/ah, (2.26) becomes in the continuum limit (ωl→k)(3.7)i∂ψ∂t=∑rk{-12∂2∂ϕrk2+12[z2(1+σ2z2s)h2-ν2h2]ϕrk2}ψ,
where σ2≡bs(h/M)2s, and the ground state wave functional of (3.7) becomes(3.8)ψ(0)=∏rkAk(0)(τ)exp⁡(-12Bk(τ)a-1ϕrk2),
where Bk(τ) satisfies(3.9)Bk2(τ)-i[Bk′(τ)+Bk(τ)τ]-[k2(1+σ2z2s)-ν2τ2]=0,(3.10)Bk′(τ)≡dBk(τ)dτ,Ak(0)(τ)=exp⁡[-i12∫Bk(τ)dτ+const].
In region I where kphys≡k/a>M, that is, z>M/h, the dispersion relations can be approximated by ω2(k/a)≈k2σ2z2s, and (3.9) becomes(3.11)BkI2(τ)-i[BkI′(τ)+BkI(τ)τ]-(k2σ2z2s-ν2τ2)=0.
To obtain the solution of (3.11), we define BkI(τ)≡-ifkI′/fkI, where fkI′(τ)≡dfkI(τ)/dτ, and transform (3.11) into(3.12)d2fkIdz2+1zdfkIdz+(σ2z2s-ν2z2)fkI=0.
The general solution of (3.12) is(3.13)fkI(z)=C1I(z0)Hν̅(1)(σ1+sz1+s)+C2I(z0)Hν̅(1)*(σ1+sz1+s),
where the Hankel function is of order ν̅=ν/1+s with ν2=9/4-(m2+ξR)/h2, and the k-dependent constants C1I and C2I are to be fixed by choosing suitable initial condition at an arbitrary initial time τ0 for each of the modes and satisfying |C1I|2-|C2I|2=1 from the Wronskian of fkI and fkI*. We can choose C1I=1 and C2I=0 for the initial Bunch-Davies vacuum state and obtain that(3.14)BkI(τ)=2/π|τ||Hν̅(1)|2(1+s)-iωl2(|Hν̅(1)|2)′|Hν̅(1)|2σzs,
where the prime in (3.14) denotes the derivative with respect to (σ/(1+s))z1+s. The corresponding wave functional is(3.15)ψ(0)I=∏rkAk(0)I(τ)exp⁡(-12BkI(τ)a-1ϕrkI2),Ak(0)I(τ)=exp⁡[-i12∫BkI(τ)dτ+const].In region II wherekphys≡k/a<M, that is, z<M/h, the dispersion relations recover the standard linear relations ω2≅k2, and (3.9) becomes(3.16)BkII2(τ)-i[BkII′(τ)+BkII(τ)τ]-(k2-ν2τ2)=0.
Again, the solution of (3.16) can be obtained by defining BkII(τ)≡-ifkII′/fkII and transforming (3.16) into(3.17)d2fkIIdz2+1zdfkIIdz+(1-ν2z2)fkII=0.
The general solution of (3.17) is(3.18)fkII(z)=C1II(z0)Hν(1)(z)+C2II(z0)Hν(1)*(z),
where C1II and C2II satisfy |C1II|2-|C2II|2=1 from the Wronskian of fkII and fkII*. Therefore, we have(3.19)BkII(τ)=2/π|τ|(|C1II|2+|C2II|2)|Hν(1)|2+2Re[C1IIC2II*(Hν(1))2]-ik2{(|C1II|2+|C2II|2)|Hν(1)|2+2Re[C1IIC2II*(Hν(1))2]}′(|C1II|2+|C2II|2)|Hν(1)|2+2Re[C1IIC2II*(Hν(1))2],
where the prime in (3.19) denotes the derivative with respect to z. Note that (3.19) can be simply obtained from (2.20) by replacing |Hν(1)|in (2.20) according to [25](3.20)|Hν(1)|⟶{(|C1|2+|C2|2)|Hν(1)|2+2Re[C1C2*(Hν(1))2]}1/2.
The corresponding wave functional is(3.21)ψ(0)II=∏rkAk(0)II(τ)exp⁡(-12BkII(τ)a-1ϕrkII2),Ak(0)II(τ)=exp⁡[-i12∫BkII(τ)dτ+const].
Let τc be the time when the modified dispersion relations take the standard linear form. Then σ2zc2s=1, where zc=k|τc|=(1/bs1/2s)(M/h)≫1 for bs≪1. The constants C1II and C2II can be obtained by the following matching conditions at τc for the two wave functionals (3.15) and (3.21):(3.22)ψ(0)I|ZC=ψ(0)II|ZC,(3.23)dψ(0)Idz|ZC=dψ(0)IIdz|ZC,
which can also be rewritten, respectively, as(3.24)Re(BkI)|ZC=Re(BkII)|ZC,(3.25)dRe(BkI)dz|ZC=dRe(BkII)dz|ZC,
by using BkI=BkII, ϕrkI=ϕrkII, Ak(0)I=Ak(0)II when z=zc.To findC1II and C2II, we focus on the case of massless minimally coupled (ν=3/2) scalar field in the slowroll inflation and take the CJ type dispersion relation (3.1) with s=2 and b2>0 which corresponds to the ultraviolet limit of HL gravity. For this case ν̅=1/2, we have |H1/2(1)((σ/3)z3)|=(6/πσz3)1/2 with σ=zc-2. Then from (3.14), (3.19), and (3.24), we obtain that(3.26)1=|C1II|2+|C2II|2+2|C1II||C2II|cos⁡(2zc-θ),
where we have used C1IIC2II*=|C1II||C2II|exp(-iθ), σ=zc-2, and zc≫1. (Note that since |C1II|2-|C2II|2=1 is one real condition on the two complex numbers C1II and C2II, we can choose C1II=|C1II| and C2II=|C2II|exp(iθ), where θ is the relative phase between C1II and C2II and can be considered as a state parameter.) Therefore from (3.26) and |C1II|2-|C2II|2=1 we have(3.27)|C1II|=csc(2zc-θ),|C2II|=-cot(2zc-θ),
where sin(2zc-θ)>0, cos(2zc-θ)<0. Substituting (3.14) and (3.19) into (3.25) and keeping terms up to order 1/zc on the right-hand side of (3.25), we obtain the following:(3.28)2zc=|C1II||C2II|cos⁡(2zc-θ)8zc+4|C1II||C2II|sin⁡(2zc-θ).
Using (3.27) in (3.28) gives(3.29)cot(2zc-θ)=-12zcorcot(2zc-θ)=-zc2+12zc.
Here we choose cot(2zc-θ)=-1/2zc, so that |C2II| is small for zc≫1 to avoid an unacceptably large backreaction on the background geometry. Therefore, we have for zc≫1(3.30)|C2II|≅12zc,|C1II|=1+|C2II|2≅1+18zc2≅1,
or
(3.31)sin⁡(2zc-θ)≅1,cos⁡(2zc-θ)≅-12zc.


## 4. Bohm Quantum Trajectories in the Trans-Planckian Physics

Note that in defining the pilot-wave scalar field theory in Section2, we used de Broglie’s first-order dynamics of 1927, which is defined by (2.26) and (2.30). To consider the effect of the trans-Planckian physics, we replace (2.26) with (3.7). In fact, we can also make use of Bohm’s second-order dynamics of 1952, which is defined by (3.7) and the following equation in the continuum limit (ωl→k):(4.1)d2ϕrkdt2=-∂∂ϕrk(V+Q),
where the classical potential V is given by(4.2)V=∑rk12[z2(1+σ2z2s)h2-ν2h2]ϕrk2,
and the so-called “quantum potential” Q is given by(4.3)Q=-∑rk12|ψ(0)|∂2|ψ(0)|∂ϕrk2,
where ψ(0) is given by (3.8)–(3.10), and |ψ(0)| is given by (2.24) for nrl=0. It has been pointed out in previous work [26] that Bohm’s second-order dynamics in general leads to more possible quantum trajectories than de Broglie’s first-order dynamics does, because Bohm regarded (4.1) as the law of motion, with the de Broglie guidance equation (2.32) added as a constraint on the initial momenta. This distinction between Bohm’s second-order dynamics and de Broglie’s first-order dynamics was also emphasized recently by Valentini [29].In region I whereω2(k/a)≈k2σ2z2s and z>M/h, the classical potential V in (4.2) becomes(4.4)V=∑rk12(σ2z2+2sh2-ν2h2)ϕrkI2,
and from (4.3), (3.8), (3.10), and (3.14), the quantum potential Q becomes(4.5)Q=∑rk(-12(2h/π)2|Hν̅(1)|4(1+s)2ϕrkI2+122h/π|Hν̅(1)|2(1+s)).
Substituting (4.4) and (4.5) into (4.1) and using dτ=dt/a and z=k|τ|=k/ah gives(4.6)z2d2ϕrkIdz2+zdϕrkIdz+[σ2z2+2s-ν2-4π2|Hν̅(1)|-4(1+2s)2]ϕrkI=0.
On the other hand, in region II where ω2(k/a)≈k2and z<M/h, the classical potential V in (4.1) and the corresponding quantum potential Q become, respectively,(4.7)V=∑rk12(z2h2-ν2h2)ϕrkII2,Q=∑rk(-12(2h/π)2|Hν(1)|md4ϕrkII2+122h/π|Hν(1)|md2),
where |Hν(1)|md means |Hν(1)| modified according to (3.20). Then the quantum trajectory ϕrkII satisfies(4.8)z2d2ϕrkIIdz2+zdϕrkIIdz+[z2-ν2-4π2|Hν(1)|md-4]ϕrkII=0.In particular, for the case of massless minimally coupled (ν=3/2) scalar field and Corley-Jacobson dispersion relation (3.1) with s=2, (4.6) becomes(4.9)z2d2ϕrkIdz2+zdϕrkIdz-94ϕrkI=0,
because in this case the first term in the square brackets of (4.6) is exactly cancelled by the last term. The general solution of (4.9) is(4.10)ϕrkI(z)=C̅1Iz-3/2+C̅2Iz3/2,
where C̅1I and C̅2I are constants to be fixed by choosing suitable initial conditions at an arbitrary initial time τ0 for the quantum trajectory ϕrkI. Here we can choose C̅1I≠0 and C̅2I=0 so that the first term C̅1Iz-3/2of ϕrkI corresponds to (2.33) with |H1/2(1)((σ/3)z3)|=(6/πσz3)1/2. On the other hand, in the region II |H3/2(1)|md becomes(4.11)|H3/2(1)|md=|H3/2(1)|{[cos⁡(2z-θ)z2-1z2+1-sin⁡(2z-θ)2z1+z2]|C1II|2+|C2II|2+2|C1II||C2II|×[cos⁡(2z-θ)z2-1z2+1-sin⁡(2z-θ)2z1+z2]}.
For z→zc≫1 (4.11) reduces to |H3/2(1)| by using (3.26), and the quantum trajectory ϕrkII satisfies(4.12)z2d2ϕrkIIdz2+zdϕrkIIdz+[z2-94-4π2|H3/2(1)|-4]ϕrkII=0.
The most general asymptotic series solution of (4.12) is [26](4.13)ϕrkII(z)=C̅1IIz-1/2(1+12z-2-18z-4+⋅⋅⋅)+C̅2IIz1/2(1+32z-2+124z-4+⋅⋅⋅).
For z→zc≫1, the solution (4.13) reduces to(4.14)ϕrkII(z)≈C̅1IIz-1/2+C̅2IIz1/2.
Substituting (4.10) and (4.14) into the following matching conditions at zc for the two quantum trajectories ϕrkI and ϕrkII:(4.15)ϕrkI|ZC=ϕrkII|ZC,dϕrkIdz|ZC=dϕrkIIdz|ZC,
we obtain(4.16)C̅1II=2C̅1I1zc,C̅2II=-C̅1I1zc2.
From (4.14) and (4.16) we see that as z decreases from zc to 1, C̅1IIz-1/2 becomes the dominant term, that is,
(4.17)ϕrkII(z)≈C̅1IIz-1/2.
On the other hand, for z≪1 (well after horizon exit), (4.11) also reduces to |H3/2(1)| by using (3.30) and zc≫1, and the most general power series solution of (4.12) is [26](4.18)ϕrkII(z)=Ĉ1IIz-3/2(1+12z2-18z4+⋅⋅⋅)+Ĉ2IIz3/2×(1-110z2+1280z4+⋅⋅⋅).
Note that for z≪1, the last term in the square brackets of (4.12) is negligible when compared with the first term, and (4.12) becomes a Bessel equation with the solution(4.19)ϕrk(z)=CN3/2(z),
where N3/2(z) is the Neumann function which has the following expression:(4.20)N3/2(z)⟶-Γ(3/2)π(2z)3/2,z≪1,
and C is a constant. Note also that, for z≪1, (4.19) is equivalent to (2.33). Therefore, for z≪1, the solution (4.18) reduces to(4.21)ϕrkII(z)≈Ĉ1IIz-3/2,
which corresponds to (4.19) and (4.20). Requiring ϕrkII to be continuous at z=1(for the justification of this patching condition, see the discussion (iv) in Section 5), we have from (4.17), (4.21), and (4.16)(4.22)Ĉ1II=C̅1II=2zcC̅1I.
Since in the case of d = 3, ϕrk contains a factor a3/2 which is proportional to z-3/2, we can define a new field variable urk≡a-3/2ϕrk and use a=(k/h)z-1 to rewrite (4.10) and (4.21) as(4.23)urkI=(kh)-3/2C̅1I,urkII=2zc(kh)-3/2C̅1I.
Thus, we see from (4.23) that for fixed k andzc≫1, as z decreases from z≫1 to z≪1, the scalar field evolves from one constant to another much smaller constant; that is, there is a transition in the time evolution of the quantum trajectory of the scalar field.


## 5. Conclusion and Discussion

In this paper we have considered a generically coupled free real scalar field in de Sitter space in the lattice Schrödinger picture within the framework of the pilot-wave theory of de Broglie and Bohm. In particular, we have investigated the possible effects of the trans-Planckian physics on the quantum trajectories of the vacuum state of scalar field.For the massless minimally coupled scalar field and the Corley-Jacobson type dispersion relation with sextic correction to the standard-squared linear relation, we have found that asz decreases from z≫1 (well before horizon exit) to z≪1 (well after horizon exit), there is a transition in the time evolution of the quantum trajectory of the scalar field.For the massive nonminimally coupled scalar field (ν2=9/4-(m2+ξR)/h2) which is relevant to the nonminimally coupled chaotic inflation with quadratic potential V=m2ϕ2/2, the coupling to the curvature ξRϕ2/2 leads to the additional mass-squared m2=12ξh2 in the case of d = 3. Notice that m2≪h2 and |ξ|≪1 are required for slowroll [30, 31]. Thus, in general, we have ν≅3/2, and expect that for the massive nonminimally coupled scalar field and the Corley-Jacobson-type dispersion relation with sextic correction, the evolution of the quantum trajectory of the scalar field also exhibits similar transitional behavior from z≫1 to z≪1.Finally we conclude this paper with the following discussions.(i) Since a constant scalar field is similar to a cosmological constantΛ, the transition could be interpreted as a transition of the Universe from a large Λ to a small Λ, thus providing a possible mechanism to solve the riddle of the smallness of Λ in the framework of the Bohmian approach to quantum theory of inflationary cosmology. Note that the vacuum energy density due to quantum states with k<kmax is ρvac≅kmax4~MPl4, and ρvac=MPl2Λ/8π (here MPl=1.22×1019Gev is the Planck mass). For the simplest potential driving inflation V(ϕ)=m2ϕ2/2, from (4.23) and ρvac=V(ϕ=const), we expect that as z decreases from zc to 0, the cosmological constant Λ decreases from 8πMPl2 to 16πm2zc-2. Here, we have zc=k|τc|=(1/b21/4)(MPl/h), MPl is the Planck cutoff scale, h~1015 Gev is the Hubble constant during inflation, and m=1.8×1013 Gev which can be obtained from COBE normalization (see [1], page 212). Since current observations indicate that ρvac0~10-47Gev4 and Λ0~10-83 Gev2, we find that the value of the parameter b2 is very small, b2~10-206. Therefore the smallness of the cosmological constant is intimately connected with the infinitesimal violation of Lorentz invariance at the level of sextic correction to the standard squared linear dispersion relation.Sincezc≫1, the transitional time is in the very early stage of inflation. We note that the quantum potential Q plays an important role in the evolution of the quantum trajectory of scalar field. From (4.1) and (4.12), we also see that for z≫1 and the mode ϕrkII the quantum force (4h2/π2)|H3/2(1)|-4ϕrkII arising from the quantum potential Q through -∂Q/∂ϕrkII approximately cancels the classical force -(z2h2-9/4h2)ϕrkII arising from the classical potential V through -∂V/∂ϕrkII, while for z≪1 the quantum force becomes negligible with respect to the classical force. Therefore, as z decreases from zc to 0, there is a quantum-to-classical transition. In this regard, the result is the same as that of the recent relevant work in which the quantum-to-classical transition of primordial cosmological perturbations is obtained in the context of the de Broglie-Bohm theory [32].(ii) In a Lorentz noninvariant theory with the action (3.2), the modified dispersion relation (3.1) breaks the local Lorentz invariance explicitly while it preserves rotational and translational invariance. The phenomenological bounds on the parameters of the theory come from the observations of ultra-high-energy cosmic rays [33]. Using effective field theory with higher-dimensional Lorentz violating operators, which results in the modified field theory with a dispersion relation, it was shown that for various standard model particles the numerical value of the bound on the parameter b1 in (3.1) is b1<5×10-5 with the Planck cut-off scale M≅MPl. On the other hand, the bound on the parameter b2 in (3.1) is currently not available observationally, but its numerical value of the bound is expected to be so small as we have shown in the previous discussion when the cut-off scale is near to the Planck scale.(iii) In the last paragraph of Section2, we note that the quantum trajectory (2.33) is independent of the quantum number nrl. This is due to the fact that the Hermite polynomials in energy eigenstates (2.19) or (2.23) are real. If we consider the state which is some superposition of energy eigenstates such as a wave packet or squeezed state, then we would obtain trajectories that generally depend on the quantum number nrl. Moreover, note that in general the trajectories for each mode are not independent of each other. However, since we are dealing with particularly simple state which factories, these issues do not arise here.(iv) Regarding the patching condition after (4.21), note that, in the region II, |H3/2(1)|md in (4.11) reduces to |H3/2(1)| as z decreases continuously from zc to 0 (not only for z≫1or z≪1 or z≅1) due to (3.30) and zc≫1. Therefore, the quantum trajectory ϕrkII(z) satisfies (4.12)(5.1)z2d2ϕrkIIdz2+zdϕrkIIdz+[z2-94-4π2|H3/2(1)|-4]ϕrkII=0,
where |H3/2(1)(z)|=z-3/21+z2 for 0≤z≤zc. Note also that the solution of (2.32).ϕrk(z)=C|H3/2(1)(z)| is also one solution of (4.12), which is just the same as that appearing in (4.13) and (4.18). This is because the first asymptotic series solution of (4.13) can be rewritten as(5.2)C̅1IIz-1/2(1+12z-2-18z-4+⋅⋅⋅)=C̅1IIz-1/21+(1z)2=C̅1IIz-3/21+z2,
while the first power series solution of (4.18) can be rewritten as(5.3)Ĉ1IIz-3/2(1+12z2-18z4+⋅⋅⋅)=Ĉ1IIz-3/21+z2.
Therefore, we have Ĉ1II=C̅1II.(v) The pilot-wave model treated in this paper is intrinsically nonlocal because of Bell’s theorem and not Lorentz invariant because of the formulation with respect to a preferred frame of reference. However, in quantum equilibrium, the pilot-wave models will reproduce the standard quantum theoretical predictions. In fact, there are some attempts to formulate Lorentz covariant pilot-wave models [34].



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