Bohm Quantum Trajectories of Scalar Field in Trans-Planckian Physics

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.

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 density𝐿=𝑔^𝟏/𝟐12𝑔^𝜇𝜈(𝑥)𝜙(𝑥)_,𝜇𝜙(𝑥)_,𝜈𝜙,𝑉𝜙=𝑚−𝑉₂𝜙²2+𝜉𝑅𝜙²2, (2.1) 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 𝑔_𝜇𝜈, 𝜇,𝜈=0,1,…,𝑑. For a spatially flat (1 + d)-dimensional Robertson-Walker spacetime with scale factor a(t), we have𝑑𝑠²=𝑑𝑡²−𝑎²(𝑡)𝑑²𝑥^𝑖,𝑖=1,2,…,𝑑,𝐿=𝑎^𝑑12𝜕₀𝜙²−𝑎⁻²𝜕_𝑖𝜙²−𝑉𝜙. (2.2) In the (1 + d)-dimensional de Sitter space, we have 𝑎(𝑡)=exp(ℎ𝑡), where ℎ≡𝑎/̇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)=𝑎₀𝑡^𝑛. 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:𝜙(𝑥,𝑡)⟶𝜙_𝑗(𝑡),𝜕₀𝜙(𝑥,𝑡)⟶𝜙^•_𝑗𝜕,𝑗=1,2,…,𝑁,₁𝜙()⟶𝜙𝑥,𝑡^𝑗(𝑡)−𝜙_𝑗−1(𝑡)𝜀, (2.3) and the Lagrangian reads𝐿=^𝑁_𝑗=112^•∼𝜙_𝑗−12ℎ𝜙_𝑗²−12𝜙_𝑗−𝜙_𝑗−1²𝑎²𝜀²−12𝑚²+𝜉𝑅𝜙_𝑗², (2.4) where 𝜙_𝑗≡𝑎^1/2𝜀^1/2𝜙_𝑗, and 𝜀=𝑊/𝑁; that is, W is the overall comoving spatial size of lattice. From (2.4) we obtain the Hamiltonian𝐻=^𝑁_𝑗=112𝑝_𝑗²+12ℎ𝑝_𝑗𝜙_𝑗+12𝜙_𝑗−𝜙_𝑗−1²𝑎²𝜀²+12𝑚²+𝜉𝑅𝜙_𝑗², (2.5) where 𝑝_𝑗 is the conjugate momentum of 𝜙_𝑗,𝑝_𝑗=𝜕𝜕𝜙^•∼_𝑗L=𝜙^•∼_𝑗−12ℎ𝜙_𝑗. (2.6) Then we consider the discrete Fourier transforms𝜙_𝑗=^𝑁_𝑙=1exp−𝑖𝑗𝑙2𝜋/𝑁√𝑁𝜙_𝑙,𝜙_𝑙=^𝑁_𝑗=1exp𝑖𝑗𝑙2𝜋/𝑁√𝑁𝜙_𝑗,𝑝_𝑗=^𝑁_𝑙=1exp𝑖𝑗𝑙2𝜋/𝑁√𝑁𝑝_𝑙,𝑝_𝑙=^𝑁_𝑗=1exp−𝑖𝑗𝑙2𝜋/𝑁√𝑁𝑝_𝑗, (2.7) such that𝜙_𝑗,𝑝_𝑗=𝜙_𝑗+𝑁,𝑝_𝑗+𝑁,𝜙_𝑙,𝑝_𝑙=𝜙_𝑙+𝑁,𝑝_𝑙+𝑁,𝜙_𝑁−𝑙,𝑝_𝑁−𝑙=𝜙_−𝑙,𝑝_−𝑙=𝜙_𝑙^∗,𝑝_𝑙^∗. (2.8) Here (𝜙_𝑙,𝑝_𝑙∣𝑙=1,2,…,𝑁/2) can be chosen as the independent variables which obey the Poisson brackets {𝜙_𝑙,𝑝_𝑙}={𝜙_𝑗,𝑝_𝑗}. From (2.7) we can obtain the following identities:^𝑁_𝑗=1𝑝_𝑗²=^𝑁_𝑙=1𝑝_𝑙^∗𝑝_𝑙,^𝑁_𝑗=1𝑝_𝑗²𝜙_𝑗=^𝑁_𝑙=1𝑝_𝑙𝜙_𝑙,_𝑁𝑗=1𝜙_𝑗²=^𝑁_𝑙=1𝜙_𝑙^∗𝜙_𝑙,^𝑁_𝑗=1𝜙_𝑗−𝜙_𝑗−1²=^𝑁_𝑙=14sin²𝑙𝜋𝑁𝜙_𝑙^∗𝜙_𝑙. (2.9) Furthermore, we define𝜔_𝑙≡2𝜀sin𝑙𝜋𝑁,𝜙_𝑙=𝜙_1𝑙+𝑖𝜙_2𝑙,𝑝_𝑙=𝑝_1𝑙+𝑖𝑝_2𝑙, (2.10) where 𝑝_𝑙 is the conjugate momentum for 𝜙_𝑙, and the subscripts 1 and 2 denote the real and imaginary parts, respectively. Therefore, the Hamiltonian (2.5) becomes in momentum space𝐻=2^𝑁/2_𝑙=1²_𝑟=1𝐻_𝑟𝑙,𝐻_𝑟𝑙=12𝑝²_𝑟𝑙+12ℎ𝑝_𝑟𝑙𝜙_𝑟𝑙+12𝑎⁻²𝜔_𝑙²𝜙²_𝑟𝑙+12𝑚²+𝜉𝑅𝜙²_𝑟𝑙. (2.11)

To quantize the theory above, we note also that each pair of operators 𝜙^∧_𝑟𝑙 and 𝑝^∧_𝑟𝑙 must satisfy the equal time commutation relations [𝜙^∧_𝑟𝑙,𝑝^∧_𝑟𝑙]=𝑖. These commutation relations are realized by the representation𝜙_𝑟𝑙⟶𝜙_𝑟𝑙,𝑝_𝑟𝑙⟶−𝑖𝜕𝜕𝜙_𝑟𝑙, (2.12) 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⟩=∫(∏_𝑟𝑙𝐷𝜙_𝑟𝑙)𝜓₁^∗(𝜙)𝜓₂(𝜙). With the field basis |𝜙⟩, that is, |𝜙⟩ are the eigenstates of the operator 𝜙^∧_𝑟𝑙, 𝜙^∧_𝑟𝑙|𝜙⟩=𝜙_𝑟𝑙|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𝜕𝐻𝜓=𝑖𝜕𝑡𝜓, (2.13) with 𝐻 defined by (2.11). Writing 𝜓 as the product form𝜓𝜙_𝑟𝑙,𝑡=^𝑁/22_𝑙=1𝑟=1𝜓_𝑟𝑙𝜙_𝑟𝑙,𝑡≡_𝑟𝑙𝜓_𝑟𝑙𝜙_𝑟𝑙,𝑡, (2.14) we obtain from (2.13) the following:𝐻_𝑟𝑙𝜓_𝑟𝑙𝜕=𝑖𝜓𝜕𝑡^𝑟𝑙,𝑟=1,2. (2.15) Rewriting (2.15) as the following form: 12𝑝_𝑟𝑙+12ℎ𝜙_𝑟𝑙²−18ℎ²𝜙²_𝑟𝑙+12𝛼⁻²𝜔_𝑙²𝜙²_𝑟𝑙+12𝑚²+𝜉𝑅𝜙²_𝑟𝑙𝜓_𝑟𝑙𝜕=𝑖𝜓𝜕𝑡^𝑟𝑙, (2.16) and setting 𝑃_𝑟𝑙=𝑝_𝑟𝑙+1/2ℎ𝜙_𝑟𝑙=−𝑖𝜕/𝜕𝜙_𝑟𝑙, so that [𝜙_𝑟𝑙,𝑃_𝑟𝑙]=[𝜙_𝑟𝑙,𝑝_𝑟𝑙]=𝑖, gives the time-dependent Schrödinger equation−12𝜕²𝜓_𝑟𝑙𝜕𝜙²_𝑟𝑙+12𝑎⁻²𝜔_𝑙²+𝑚²−1+𝜉𝑅4ℎ²𝜙²_𝑟𝑙𝜓_𝑟𝑙=𝑖𝜕𝜓_𝑟𝑙𝜕𝑡. (2.17) To solve (2.17), we use the conformal time 𝜏 defined by 𝑑𝜏=𝑑𝑡/𝑎𝜏=−ℎ⁻¹exp(−ℎ𝑡)=−ℎ⁻¹𝑎⁻¹,−∞<𝜏<0. (2.18) Then, for each real mode 𝜙_𝑟𝑙, the normalized instantaneous vacuum and its excited states are found to be [24]𝜓_{(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏=𝐴_{𝑙(𝑛𝑟𝑙)}(𝜏)𝐻_{(𝑛𝑟𝑙)}𝜂_𝑟𝑙(𝜏−1)exp2𝐵_𝑙(𝜏)𝑎⁻¹𝜙²_𝑟𝑙,𝑛_𝑟𝑙=0,1,2,…,𝜂_𝑟𝑙(𝜏)≡𝑔(𝜏)𝜔_𝑙^1/2𝑎^−1/2𝜙_𝑟𝑙. (2.19) Here 𝐻_{(𝑛𝑟𝑙)} is the nth-order Hermite polynomial, and the complex function 𝐵_𝑙(𝜏), real function 𝑔(𝜏), and complex function 𝐴_{𝑙(𝑛𝑟𝑙)}(𝜏) are given by (2.20), (2.21), and (2.22), respectively𝐵_𝑙(𝜏)=2/𝜋|𝜏|𝐻_𝜈⁽¹⁾²𝜔−𝑖^𝑙2𝐻_𝜈^(1)2′𝐻_𝜈⁽¹⁾², (2.20)𝑔(𝜏)=2𝜋𝜔_𝑙|𝜏|^1/21𝐻_𝜈⁽¹⁾, (2.21)𝐴_{𝑙(𝑛𝑟𝑙)}(𝜏)1=exp−𝑖2𝐵_𝑙(𝜏)𝑑𝜏−𝑖𝑛_𝑟𝑙𝜔_𝑙𝑔²(𝜏)𝑑𝜏+const, (2.22) where 𝐻_𝜈⁽¹⁾(𝜔_𝑙|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 𝜔_𝑙|𝜏|. Note that the wave functionals for each real mode 𝜙_𝑟𝑙 can also be rewritten in the following form [25]:𝜓_{𝑟𝑙(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏=𝑅_{(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏exp𝑖Θ_{(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏,𝑛_𝑟𝑙=0,1,2,…, (2.23) with the amplitude𝑅_{(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏=2ℎ/𝜋√𝜋2^𝑛𝑛_𝑟𝑙!𝐻_𝜈^(1)1/2𝐻_{(𝑛𝑟𝑙)}𝜂_𝑟𝑙−1exp2𝜂²_𝑟𝑙,𝜂_𝑟𝑙=2ℎ/𝜋𝐻_𝜈⁽¹⁾𝜙_𝑟𝑙, (2.24) and the phase Θ_{(𝑛𝑟𝑙)}𝜙_𝑟𝑙,𝜏=−ℎ𝜔_𝑙|𝜏|2𝐻_𝜈^(1)′𝐻_𝜈⁽¹⁾𝜙²_𝑟𝑙−12+𝑛_𝑟𝑙2/𝜋|𝜏|𝐻_𝜈⁽¹⁾²𝑑𝜏. (2.25) The complete state wave functionals can be written as 𝜓_[𝑛][𝜙_𝑟𝑙,𝑡]=∏_𝑟𝑙𝜓_{(𝑛𝑟𝑙)}(𝜙_𝑟𝑙,𝑡), where 𝑛_𝑟𝑙=0,1,2,… and [𝑛]≡(𝑛_𝑖,𝑛_𝑗,…) which means that it is possible for different field modes to be in different excited states; that is, mode 𝑖 is in the 𝑛_𝑖-excited state, mode 𝑗 is in the 𝑛_𝑗 excited state, and so forth. For 𝑛_𝑟𝑙=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 𝜔_𝑙 carries labels (𝑙_𝑖,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 𝑖𝜕𝜓𝜕𝑡=_𝑟𝑙−12𝜕²𝜕𝜙²_𝑟𝑙+12𝑎⁻²𝜔_𝑙²+𝑚²−9+𝜉𝑅4ℎ²𝜙²_𝑟𝑙𝜓, (2.26) which implies the continuity equation𝜕𝜓²𝜕𝑡+_𝑟𝑙𝜕𝜕𝜙_𝑟𝑙𝜓²𝜕Θ𝜕𝜙_𝑟𝑙=0 (2.27) and the de Broglie velocities𝑑𝜙_𝑟𝑙=𝑑𝑡𝜕Θ𝜕𝜙_𝑟𝑙, (2.28) where 𝜓=|𝜓|exp[𝑖Θ]. The wave functional 𝜓_𝑟𝑙(where𝜓_𝑟𝑙=|𝜓_𝑟𝑙|exp[𝑖Θ_𝑟𝑙],andΘ=∑_𝑟𝑙Θ_𝑟𝑙) for a single mode 𝜙_𝑟𝑙 satisfies (2.17), which implies the continuity equation𝜕𝜓_𝑟𝑙²+𝜕𝜕𝑡𝜕𝜙_𝑟𝑙𝜓_𝑟𝑙²𝜕Θ_𝑟𝑙𝜕𝜙_𝑟𝑙=0, (2.29) and the de Broglie velocity field𝑑𝜙_𝑟𝑙=𝑑𝑡𝜕Θ_𝑟𝑙𝜕𝜙_𝑟𝑙. (2.30) Here 𝜓 is interpreted as a physical field in field configuration space, guiding the evolution of 𝜙_𝑟𝑙.

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 𝜙_𝑟𝑙𝑑𝜙_𝑟𝑙𝑑𝜏=𝑎𝜕Θ_(𝑛)𝜙_𝑟𝑙,𝜏𝜕𝜙_𝑟𝑙=−𝜔_𝑙𝐻_𝜈^(1)′𝐻_𝜈⁽¹⁾𝜙_𝑟𝑙. (2.31) In the continuum limit (𝜔_𝑙→𝑘), (2.31) reads𝑑𝜙_𝑟𝑘𝐻𝑑𝜏=−𝑘^𝜈(1)(𝑘|𝜏|)^′𝐻_𝜈⁽¹⁾(𝑘|𝜏𝜙|)^𝑟𝑘, (2.32) which has the solution𝜙_𝑟𝑘(𝑧)=𝐶𝐻_𝜈⁽¹⁾(𝑧), (2.33) 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 𝜙_𝑟𝑘(𝑧₀)/|𝐻_𝜈⁽¹⁾(𝑧₀)| with 𝑧₀ being some reference point. Note that the quantum trajectory (2.33) is independent of the quantum number 𝑛_𝑟𝑙 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 relations𝜔²𝑘𝑎=𝑘²1+𝑏_𝑠𝑘𝑎𝑀^2𝑠, (3.1) where 𝑀 is a cut-off scale, 𝑠 is an integer, and 𝑏_𝑠 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]𝑆=𝑑⁴𝑥√−𝑔𝐿_𝜙+𝐿_cor+𝐿_𝑢, (3.2) where 𝐿_𝜙 is the standard Lagrangian of a minimally coupled scalar field𝐿_𝜙=12𝑔^𝜇𝜈𝜕_𝜇𝜙𝜕_𝜈𝜙−𝑉𝜙, (3.3)𝐿_cor corresponds to the nonlinear part of the dispersion relation𝐿_cor𝑏=−¹𝑀²𝐷²𝜙², (3.4) and 𝐿_𝑢 describes the dynamics of a unit time-like vector field 𝑢^𝜇 which defines a preferred rest frame𝐿_𝑢=−𝜆𝑔^𝜇𝜈𝑢_𝜇𝑢_𝜈−1−𝑑₁𝐹^𝜇𝜈𝐹_𝜇𝜈, (3.5) with𝐷²𝜙=⟂^𝛼𝛽∇_𝛼∇_𝛽𝜙+𝑢^𝛼∇_𝛼𝜙∇_𝛽𝑢^𝛽,⟂_𝜇𝜈≡−𝑔_𝜇𝜈+𝑢_𝜇𝑢_𝜈,𝐹_𝜇𝜈≡∇_𝜇𝑢_𝜈−∇_𝜈𝑢_𝜇. (3.6) Here ∇_𝜇 is the covariant derivative associated with the metric 𝑔_𝜇𝜈, 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 𝐿_cor=(−𝑏₂/𝑀⁴)(𝐷²¹𝜙)(𝐷²²𝜙), where the operator 𝐷^2𝑛 is defined as 𝐷^2𝑛≡𝐷_𝜇1𝐷^𝜇1⋯𝐷_𝜇𝑛𝐷^𝜇𝑛. Then using 𝑧=𝑘|𝜏|=𝑘/𝑎ℎ, (2.26) becomes in the continuum limit (𝜔_𝑙→𝑘)𝑖𝜕𝜓𝜕𝑡=_𝑟𝑘−12𝜕²𝜕𝜙²_𝑟𝑘+12𝑧²1+𝜎²𝑧^2𝑠ℎ²−𝜈²ℎ²𝜙²_𝑟𝑘𝜓, (3.7) where 𝜎²≡𝑏_𝑠(ℎ/𝑀)^2𝑠, and the ground state wave functional of (3.7) becomes𝜓₍₀₎=_𝑟𝑘𝐴_𝑘(0)(𝜏)−1exp2𝐵_𝑘(𝜏)𝑎⁻¹𝜙²_𝑟𝑘, (3.8) where 𝐵_𝑘(𝜏) satisfies𝐵_𝑘²(𝜏)−𝑖𝐵_𝑘^′(𝜏)+𝐵_𝑘(𝜏)𝜏−𝑘²1+𝜎²𝑧^2𝑠−𝜈²𝜏²𝐵=0,_𝑘′(𝜏)≡𝑑𝐵_𝑘(𝜏),𝐴𝑑𝜏_𝑘(0)(𝜏)1=exp−𝑖2𝐵_𝑘(𝜏)𝑑𝜏+const. (3.9)(3.10) In region I where 𝑘_phys≡𝑘/𝑎>𝑀, that is, 𝑧>𝑀/ℎ, the dispersion relations can be approximated by 𝜔²(𝑘/𝑎)≈𝑘²𝜎²𝑧^2𝑠, and (3.9) becomes𝐵_𝑘^I2(𝜏)−𝑖𝐵_𝑘^I′(𝜏)+𝐵_𝑘^I(𝜏)𝜏−𝑘²𝜎²𝑧^2𝑠−𝜈²𝜏²=0. (3.11) To obtain the solution of (3.11), we define 𝐵_𝑘^I(𝜏)≡−𝑖𝑓_𝑘^I′/𝑓_𝑘^𝐼, where 𝑓_𝑘^I′(𝜏)≡𝑑𝑓_𝑘^I(𝜏)/𝑑𝜏, and transform (3.11) into𝑑²𝑓_𝑘^I𝑑𝑧²+1𝑧𝑑𝑓_𝑘^I+𝜎𝑑𝑧²𝑧^2𝑠−𝜈²𝑧²𝑓_𝑘^I=0. (3.12) The general solution of (3.12) is𝑓_𝑘^I(𝑧)=𝐶₁^I𝑧₀𝐻_𝜈⁽¹⁾𝜎𝑧1+𝑠^1+𝑠+𝐶₂^I𝑧₀𝐻_𝜈^∗(1)𝜎𝑧1+𝑠^1+𝑠, (3.13) where the Hankel function is of order 𝜈=𝜈/1+𝑠 with 𝜈²=9/4−(𝑚²+𝜉𝑅)/ℎ², and the 𝑘-dependent constants 𝐶₁^I and 𝐶₂^I are to be fixed by choosing suitable initial condition at an arbitrary initial time 𝜏₀ for each of the modes and satisfying |𝐶₁^I|²−|𝐶₂^I|²=1 from the Wronskian of 𝑓_𝑘^I and 𝑓_𝑘^I∗. We can choose 𝐶₁^I=1 and 𝐶₂^I=0 for the initial Bunch-Davies vacuum state and obtain that𝐵_𝑘^I(𝜏)=2/𝜋|𝜏|𝐻_𝜈⁽¹⁾²()𝜔1+𝑠−𝑖^𝑙2𝐻_𝜈^(1)2′𝐻_𝜈⁽¹⁾²𝜎𝑧^𝑠, (3.14) where the prime in (3.14) denotes the derivative with respect to (𝜎/(1+𝑠))𝑧^1+𝑠. The corresponding wave functional is𝜓₍₀₎^I=_𝑟𝑘𝐴_𝑘(0)^I(𝜏)−1exp2𝐵_𝑘^I(𝜏)𝑎⁻¹𝜙_𝑟𝑘^I2,𝐴_𝑘(0)^I(𝜏)1=exp−𝑖2𝐵_𝑘^I(𝜏)𝑑𝜏+const. (3.15)

In region II where 𝑘_phys≡𝑘/𝑎<𝑀, that is, 𝑧<𝑀/ℎ, the dispersion relations recover the standard linear relations 𝜔²≅𝑘², and (3.9) becomes𝐵_𝑘^II2(𝜏)−𝑖𝐵_𝑘^II′(𝜏)+𝐵_𝑘^II(𝜏)𝜏−𝑘²−𝜈²𝜏²=0. (3.16) Again, the solution of (3.16) can be obtained by defining 𝐵_𝑘^II(𝜏)≡−𝑖𝑓_𝑘^II′/𝑓_𝑘^II and transforming (3.16) into𝑑²𝑓_𝑘^II𝑑𝑧²+1𝑧𝑑𝑓_𝑘^II+𝜈𝑑𝑧1−²𝑧²𝑓_𝑘^II=0. (3.17) The general solution of (3.17) is𝑓_𝑘^II(𝑧)=𝐶₁^II𝑧₀𝐻_𝜈⁽¹⁾(𝑧)+𝐶₂^II𝑧₀𝐻_𝜈^∗(1)(𝑧), (3.18) where 𝐶₁^II and 𝐶₂^II satisfy |𝐶₁^II|²−|𝐶₂^II|²=1 from the Wronskian of 𝑓_𝑘^II and 𝑓_𝑘^II∗. Therefore, we have𝐵_𝑘^II(𝜏)=2/𝜋|𝜏|𝐶₁^II2+𝐶₂^II2𝐻_𝜈⁽¹⁾²+2Re𝐶₁^II𝐶₂^II∗𝐻_𝜈⁽¹⁾²𝑘−𝑖2𝐶₁^II2+𝐶₂^II2𝐻_𝜈⁽¹⁾²+2Re𝐶₁^II𝐶₂^II∗𝐻_𝜈^(1)2′𝐶₁^II2+𝐶₂^II2𝐻_𝜈⁽¹⁾²+2Re𝐶₁^II𝐶₂^II∗𝐻_𝜈⁽¹⁾², (3.19) 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 |𝐻_𝜈⁽¹⁾|in (2.20) according to [25]𝐻_𝜈⁽¹⁾⟶𝐶₁²+𝐶₂²𝐻_𝜈⁽¹⁾²+2Re𝐶₁𝐶₂^∗𝐻_𝜈^(1)21/2. (3.20) The corresponding wave functional is𝜓₍₀₎^II=_𝑟𝑘𝐴_𝑘(0)^II(𝜏)−1exp2𝐵_𝑘^II(𝜏)𝑎⁻¹𝜙_𝑟𝑘^II2,𝐴_𝑘(0)^II(𝜏)1=exp−𝑖2𝐵_𝑘^II(𝜏)𝑑𝜏+const. (3.21) Let 𝜏_𝑐 be the time when the modified dispersion relations take the standard linear form. Then 𝜎²𝑧_𝑐^2𝑠=1, where 𝑧_𝑐=𝑘|𝜏_𝑐|=(1/𝑏_𝑠^1/2𝑠)(𝑀/ℎ)≫1 for 𝑏_𝑠≪1. The constants 𝐶₁^II and 𝐶₂^II can be obtained by the following matching conditions at 𝜏_𝑐 for the two wave functionals (3.15) and (3.21):𝜓₍₀₎^I_𝑍𝐶=𝜓₍₀₎^II_𝑍𝐶, (3.22)𝑑𝜓₍₀₎^I𝑑𝑧_𝑍𝐶=𝑑𝜓₍₀₎^II𝑑𝑧_𝑍𝐶, (3.23) which can also be rewritten, respectively, asRe𝐵_𝑘^I_𝑍𝐶=Re𝐵_𝑘^II_𝑍𝐶, (3.24)𝑑Re𝐵_𝑘^I𝑑𝑧_𝑍𝐶=𝑑Re𝐵_𝑘^II𝑑𝑧_𝑍𝐶, (3.25) by using 𝐵_𝑘^I=𝐵_𝑘^II, 𝜙_𝑟𝑘^I=𝜙_𝑟𝑘^II, 𝐴_𝑘(0)^I=𝐴_𝑘(0)^II when 𝑧=𝑧_𝑐.

To find 𝐶₁^II and 𝐶₂^II, 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 𝜈=1/2, we have |𝐻_1/2⁽¹⁾((𝜎/3)𝑧³)|=(6/𝜋𝜎𝑧³)^1/2 with 𝜎=𝑧_𝑐⁻². Then from (3.14), (3.19), and (3.24), we obtain that1=𝐶₁^II2+𝐶₂^II2+2𝐶₁^II𝐶₂^IIcos2𝑧_𝑐−𝜃, (3.26) where we have used 𝐶₁^II𝐶₂^II∗=|𝐶₁^II||𝐶₂^II|exp(−i𝜃), 𝜎=𝑧_𝑐⁻², and 𝑧_𝑐≫1. (Note that since |𝐶₁^II|²−|𝐶₂^II|²=1 is one real condition on the two complex numbers 𝐶₁^II and 𝐶₂^II, we can choose 𝐶₁^II=|𝐶₁^II| and 𝐶₂^II=|𝐶₂^II|exp(𝑖𝜃), where 𝜃 is the relative phase between 𝐶₁^II and 𝐶₂^II and can be considered as a state parameter.) Therefore from (3.26) and |𝐶₁^II|²−|𝐶₂^II|²=1 we have𝐶₁^II=csc2𝑧_𝑐−𝜃,𝐶₂^II=−cot2𝑧_𝑐−𝜃, (3.27) where sin(2𝑧_𝑐−𝜃)>0, cos(2𝑧_𝑐−𝜃)<0. Substituting (3.14) and (3.19) into (3.25) and keeping terms up to order 1/𝑧_𝑐 on the right-hand side of (3.25), we obtain the following:2𝑧_𝑐=𝐶₁^II𝐶₂^IIcos2𝑧_𝑐8−𝜃𝑧_𝑐+4𝐶₁^II𝐶₂^IIsin2𝑧_𝑐−𝜃. (3.28) Using (3.27) in (3.28) givescot2𝑧_𝑐1−𝜃=−2𝑧_𝑐orcot2𝑧_𝑐𝑧−𝜃=−^𝑐2+12𝑧_𝑐. (3.29) Here we choose cot(2𝑧_𝑐−𝜃)=−1/2𝑧_𝑐, so that |𝐶₂^II| is small for 𝑧_𝑐≫1 to avoid an unacceptably large backreaction on the background geometry. Therefore, we have for 𝑧_𝑐≫1𝐶₂^II≅12𝑧_𝑐,𝐶₁^II=1+𝐶₂^II21≅1+8𝑧_𝑐²≅1, (3.30) or sin2𝑧_𝑐−𝜃≅1,cos2𝑧_𝑐1−𝜃≅−2𝑧_𝑐. (3.31)

4. Bohm Quantum Trajectories in the Trans-Planckian Physics

Note that in defining the pilot-wave scalar field theory in Section 2, 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 (𝜔_𝑙→𝑘):𝑑²𝜙_𝑟𝑘𝑑𝑡²𝜕=−𝜕𝜙_𝑟𝑘(𝑉+𝑄), (4.1) where the classical potential 𝑉 is given by𝑉=_𝑟𝑘12𝑧²1+𝜎²𝑧^2𝑠ℎ²−𝜈²ℎ²𝜙²_𝑟𝑘, (4.2) and the so-called “quantum potential” 𝑄 is given by𝑄=−_𝑟𝑘12𝜓₍₀₎𝜕²𝜓₍₀₎𝜕𝜙²_𝑟𝑘, (4.3) where 𝜓₍₀₎ is given by (3.8)–(3.10), and |𝜓₍₀₎| is given by (2.24) for 𝑛_𝑟𝑙=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 where 𝜔²(𝑘/𝑎)≈𝑘²𝜎²𝑧^2𝑠 and 𝑧>𝑀/ℎ, the classical potential 𝑉 in (4.2) becomes𝑉=_𝑟𝑘12𝜎²𝑧^2+2𝑠ℎ²−𝜈²ℎ²𝜙_𝑟𝑘^I2, (4.4) and from (4.3), (3.8), (3.10), and (3.14), the quantum potential 𝑄 becomes𝑄=_𝑟𝑘−12(2ℎ/𝜋)²𝐻_𝜈⁽¹⁾⁴(1+𝑠)²𝜙_𝑟𝑘^I2+122ℎ/𝜋𝐻_𝜈⁽¹⁾²(1+𝑠). (4.5) Substituting (4.4) and (4.5) into (4.1) and using 𝑑𝜏=𝑑𝑡/𝑎 and 𝑧=𝑘|𝜏|=𝑘/𝑎ℎ gives𝑧²𝑑²𝜙_𝑟𝑘^I𝑑𝑧²+𝑧𝑑𝜙_𝑟𝑘^I+𝜎𝑑𝑧²𝑧^2+2s−𝜈²−4𝜋²𝐻_𝜈⁽¹⁾⁻⁴(1+2𝑠)²𝜙_𝑟𝑘^I=0. (4.6) 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,𝑉=_𝑟𝑘12𝑧²ℎ²−𝜈²ℎ²𝜙_𝑟𝑘^II2,𝑄=_𝑟𝑘−12(2ℎ/𝜋)²𝐻_𝜈⁽¹⁾_md⁴𝜙_𝑟𝑘^II2+122ℎ/𝜋𝐻_𝜈⁽¹⁾_md², (4.7) where |𝐻_𝜈⁽¹⁾|_md means |𝐻_𝜈⁽¹⁾| modified according to (3.20). Then the quantum trajectory 𝜙_𝑟𝑘^II satisfies𝑧²𝑑²𝜙_𝑟𝑘^II𝑑𝑧²+𝑧𝑑𝜙_𝑟𝑘^II+𝑧𝑑𝑧²−𝜈²−4𝜋²𝐻_𝜈⁽¹⁾_md⁻⁴𝜙_𝑟𝑘^II=0. (4.8)

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) becomes𝑧²𝑑²𝜙_𝑟𝑘^I𝑑𝑧²+𝑧𝑑𝜙^I_rk−9𝑑𝑧4𝜙_𝑟𝑘^I=0, (4.9) 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) is𝜙_𝑟𝑘^I(𝑧)=𝐶₁^I𝑧^−3/2+𝐶₂^I𝑧^3/2, (4.10) where 𝐶₁^I and 𝐶₂^I are constants to be fixed by choosing suitable initial conditions at an arbitrary initial time 𝜏₀ for the quantum trajectory 𝜙_𝑟𝑘^I. Here we can choose 𝐶₁^I≠0 and 𝐶₂^I=0 so that the first term 𝐶₁^I𝑧^−3/2of 𝜙_𝑟𝑘^I corresponds to (2.33) with |𝐻_1/2⁽¹⁾((𝜎/3)𝑧³)|=(6/𝜋𝜎𝑧³)^1/2. On the other hand, in the region II |𝐻_3/2⁽¹⁾|_md becomes𝐻_3/2⁽¹⁾_md=𝐻_3/2⁽¹⁾𝐶₁^II2+𝐶₂^II2+2𝐶₁^II𝐶₂^II×()𝑧cos2𝑧−𝜃²−1𝑧²()+1−sin2𝑧−𝜃2𝑧1+𝑧². (4.11) For 𝑧→𝑧_𝑐≫1 (4.11) reduces to |𝐻_3/2⁽¹⁾| by using (3.26), and the quantum trajectory 𝜙_𝑟𝑘^II satisfies𝑧²𝑑²𝜙_𝑟𝑘^II𝑑𝑧²+𝑧𝑑𝜙_𝑟𝑘^II+𝑧𝑑𝑧²−94−4𝜋²𝐻_3/2⁽¹⁾⁻⁴𝜙_𝑟𝑘^II=0. (4.12) The most general asymptotic series solution of (4.12) is [26]𝜙_𝑟𝑘^II(𝑧)=𝐶₁^II𝑧^−1/211+2𝑧⁻²−18𝑧⁻⁴+⋅⋅⋅+𝐶₂^II𝑧^1/231+2𝑧⁻²+1𝑧24⁻⁴+⋅⋅⋅. (4.13) For 𝑧→𝑧_𝑐≫1, the solution (4.13) reduces to𝜙_𝑟𝑘^II(𝑧)≈𝐶₁^II𝑧^−1/2+𝐶₂^II𝑧^1/2. (4.14) Substituting (4.10) and (4.14) into the following matching conditions at 𝑧_𝑐 for the two quantum trajectories 𝜙_𝑟𝑘^I and 𝜙_𝑟𝑘^II:𝜙_𝑟𝑘^I_𝑍𝐶=𝜙_𝑟𝑘^II_𝑍𝐶,𝑑𝜙_𝑟𝑘^I𝑑𝑧_𝑍𝐶=𝑑𝜙_𝑟𝑘^II𝑑𝑧_𝑍𝐶, (4.15) we obtain𝐶₁^II=2𝐶₁^I1𝑧_𝑐,𝐶₂^II=−𝐶₁^I1𝑧_𝑐². (4.16) From (4.14) and (4.16) we see that as 𝑧 decreases from 𝑧_𝑐 to 1, 𝐶₁^II𝑧^−1/2 becomes the dominant term, that is, 𝜙_𝑟𝑘^II(𝑧)≈𝐶₁^II𝑧^−1/2. (4.17) On the other hand, for 𝑧≪1 (well after horizon exit), (4.11) also reduces to |𝐻_3/2⁽¹⁾| by using (3.30) and 𝑧_𝑐≫1, and the most general power series solution of (4.12) is [26]𝜙_𝑟𝑘^II(𝑧)=𝐶₁^II𝑧^−3/211+2𝑧²−18𝑧⁴+⋅⋅⋅+𝐶₂^II𝑧^3/2×11−𝑧10²+1𝑧280⁴+⋅⋅⋅. (4.18) 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 solution𝜙_𝑟𝑘(𝑧)=𝐶𝑁_3/2(𝑧), (4.19) where 𝑁_3/2(𝑧) is the Neumann function which has the following expression:𝑁_3/2(𝑧)Γ()⟶−3/2𝜋2𝑧^3/2,𝑧≪1, (4.20) 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 to𝜙_𝑟𝑘^II(𝑧)≈𝐶₁^II𝑧^−3/2, (4.21) which corresponds to (4.19) and (4.20). Requiring 𝜙_𝑟𝑘^II 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)𝐶₁^II=𝐶₁^II=2𝑧_𝑐𝐶₁^I. (4.22) Since in the case of d = 3, 𝜙_𝑟𝑘 contains a factor 𝑎^3/2 which is proportional to 𝑧^−3/2, we can define a new field variable 𝑢_𝑟𝑘≡𝑎^−3/2𝜙_𝑟𝑘 and use 𝑎=(𝑘/ℎ)𝑧⁻¹ to rewrite (4.10) and (4.21) as𝑢_𝑟𝑘^I=𝑘ℎ^−3/2𝐶₁^I,𝑢_𝑟𝑘^II=2𝑧_𝑐𝑘ℎ^−3/2𝐶₁^I. (4.23) Thus, we see from (4.23) that for fixed 𝑘 and𝑧_𝑐≫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 as 𝑧 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 constant Λ, 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<𝑘_max is 𝜌_vac≅𝑘⁴_max~𝑀_Pl⁴, and 𝜌_vac=𝑀_Pl²Λ/8𝜋 (here 𝑀_Pl=1.22×10¹⁹Gev is the Planck mass). For the simplest potential driving inflation 𝑉(𝜙)=𝑚²𝜙²/2, from (4.23) and 𝜌_vac=𝑉(𝜙=const), we expect that as 𝑧 decreases from 𝑧_𝑐 to 0, the cosmological constant Λ decreases from 8𝜋𝑀_Pl² to 16𝜋𝑚²𝑧_𝑐⁻². Here, we have 𝑧_𝑐=𝑘|𝜏_𝑐|=(1/𝑏₂^1/4)(𝑀_Pl/ℎ), 𝑀_Pl 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 𝜌_vac⁰∼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.

Since 𝑧_𝑐≫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 𝜙_𝑟𝑘^II the quantum force (4ℎ²/𝜋²)|𝐻_3/2⁽¹⁾|⁻⁴𝜙_𝑟𝑘^II arising from the quantum potential 𝑄 through −𝜕𝑄/𝜕𝜙_𝑟𝑘^II approximately cancels the classical force −(𝑧²ℎ²−9/4ℎ²)𝜙_𝑟𝑘^II arising from the classical potential 𝑉 through −𝜕𝑉/𝜕𝜙_𝑟𝑘^II, while for 𝑧≪1 the quantum force becomes negligible with respect to the classical force. Therefore, as 𝑧 decreases from 𝑧_𝑐 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≅𝑀_Pl. 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 Section 2, we note that the quantum trajectory (2.33) is independent of the quantum number 𝑛_𝑟𝑙. 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 𝑛_𝑟𝑙. 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, |𝐻_3/2⁽¹⁾|_md in (4.11) reduces to |𝐻_3/2⁽¹⁾| as 𝑧 decreases continuously from 𝑧_𝑐 to 0 (not only for 𝑧≫1or 𝑧≪1 or 𝑧≅1) due to (3.30) and 𝑧_𝑐≫1. Therefore, the quantum trajectory 𝜙_𝑟𝑘^II(𝑧) satisfies (4.12)𝑧²𝑑²𝜙_𝑟𝑘^II𝑑𝑧²+𝑧𝑑𝜙_𝑟𝑘^II+𝑧𝑑𝑧²−94−4𝜋²𝐻_3/2⁽¹⁾⁻⁴𝜙_𝑟𝑘^II=0, (5.1) where |𝐻_3/2⁽¹⁾(𝑧)|=𝑧^−3/2√1+𝑧² for 0≤𝑧≤𝑧_𝑐. Note also that the solution of (2.32).

𝜙_𝑟𝑘(𝑧)=𝐶|𝐻_3/2⁽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 as𝐶₁^II𝑧^−1/211+2𝑧⁻²−18𝑧⁻⁴+⋅⋅⋅=𝐶₁^II𝑧^−1/211+𝑧²=𝐶₁^II𝑧^−3/2√1+𝑧², (5.2) while the first power series solution of (4.18) can be rewritten as𝐶₁^II𝑧^−3/211+2𝑧²−18𝑧⁴+⋅⋅⋅=𝐶₁^II𝑧^−3/2√1+𝑧². (5.3) Therefore, we have 𝐶₁^II=𝐶₁^II.

(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].

Acknowledgments

The author would like to thank M.-J. Wang for fruitful discussions on trans-Planckian physics, and his colleagues at Ming Chi University of Technology for useful suggestions. The author also would like to thank the anonymous referees for helpful comments.