Bohm Quantum Trajectories of Scalar Field in Trans-Planckian Physics

In lattice Schrödinger picture, we investigate the possible effects of trans-Planckian physics on the quantum trajectories of scalar field in de Sitter spacewithin 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.


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

Pilot-Wave Scalar Field in De Sitter Space
We consider the scalar field theory which has the Lagrangian density 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 have 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 12h 2 is also a constant.Throughout this paper, we use this exact de Sitter spacetime background to describe the inflationary era, which is only a special case n → ∞ limit of power-law inflation with a t a 0 t 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: and the Lagrangian reads where φ j ≡ a 1/2 ε 1/2 φ j , and ε W/N; that is, W is the overall comoving spatial size of lattice.From 2.4 we obtain the Hamiltonian where p j is the conjugate momentum of φ j ,

2.6
Then we consider the discrete Fourier transforms

Advances in High Energy Physics
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: 2.9 Furthermore, we define 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 space

2.11
To quantize the theory above, we note also that each pair of operators Then, for each real mode φ rl , the normalized instantaneous vacuum and its excited states are found to be 24 n rl 0, 1, 2, . . ., η rl τ ≡ g τ ω l 1/2 a −1/2 φ rl .

2.19
Here H n rl is the nth-order Hermite polynomial, and the complex function B l τ , real function g τ , and complex function A l n rl τ are given by 2.20 , 2.21 , and 2.22 , respectively where H ν 1 ω l |τ| is the Hankel function of the first kind and of order ν, with ν 2 1/4 − m 2 ξR /h 2 , 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 : R n rl φ rl , τ exp iΘ n rl φ rl , τ , n rl 0, 1, 2, . . ., 2.23 with the amplitude and the phase The complete state wave functionals can be written as ψ n φ rl , t rl ψ n rl 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 ν 2 9/4 − m 2 ξR /h 2 with R 12h 2 , 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 Schr ödinger equation for ψ is given by 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 rl 2.31 In the continuum limit ω l → k , 2.31 reads which has the solution 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 z 0 /|H ν 1 z 0 | with z 0 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 .

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 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 b 1 > 0 takes the form 27, 28 where L φ is the standard Lagrangian of a minimally coupled scalar field L cor corresponds to the nonlinear part of the dispersion relation and L u describes the dynamics of a unit time-like vector field u μ which defines a preferred rest frame

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 D 2 is proportional to the Laplacian operator on the same surface, λ is the Lagrange multiplier, and the parameters b 1 and d 1 with no dimensions and the dimensions of mass square, respectively, are constrained by the astrophysical observations.For the modified dispersion relation 3.
where σ 2 ≡ b s h/M 2s , and the ground state wave functional of 3.7 becomes where

3.10
In region I where k phys ≡ k/a > M, that is, z > M/h, the dispersion relations can be approximated by ω 2 k/a ≈ k 2 σ 2 z 2s , and 3.9 becomes To obtain the solution of 3.11 , we define B k , where f k I τ ≡ df k I τ /dτ, and transform 3.11 into The general solution of 3.12 is where the Hankel function is of order ν ν/1 s with ν where the prime in 3.14 denotes the derivative with respect to σ/ 1 s z 1 s .The corresponding wave functional is

3.15
In region II where k phys ≡ k/a < M, that is, z < M/h, the dispersion relations recover the standard linear relations ω 2 ∼ k 2 , and 3.9 becomes 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 into 3.17 The general solution of 3.17 is where 1 from the Wronskian of f k II and f k II * .Therefore, we have

3.20
The corresponding wave functional is 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: Here we choose cot 2z c − θ −1/2z c , so that |C 2 II | is small for z c 1 to avoid an unacceptably large backreaction on the background geometry.Therefore, we have for z c 1

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 firstorder 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 : where the classical potential V is given by Advances in High Energy Physics 13 and the so-called "quantum potential" Q is given by 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 ω 2 k/a ≈ k 2 σ 2 z 2s and z > M/h, the classical potential V in 4.2 becomes and from 4.3 , 3.8 , 3.10 , and 3.14 , the quantum potential Q becomes On the other hand, in region II where ω 2 k/a ≈ k 2 and z < M/h, the classical potential V in 4.1 and the corresponding quantum potential Q become, respectively,

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 The most general asymptotic series solution of 4.12 is 26 For z → z c 1, the solution 4.13 reduces to

4.14
Advances in High Energy Physics 15 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 : we obtain From 4.14 and 4.16 we see that as z decreases from z c to 1, C 1 II z −1/2 becomes the dominant term, that is, 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

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 solution where N 3/2 z is the Neumann function which has the following expression: 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 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 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 −3/2 φ rk and use a k/h z −1 to rewrite 4.10 and 4.21 as Thus, we see from 4.23 that for fixed k and z 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.

Conclusion and Discussion
In this paper we have considered a generically coupled free real scalar field in de Sitter space in the lattice Schr ödinger 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 ν 2 9/4 − m 2 ξR /h 2 which is relevant to the nonminimally coupled chaotic inflation with quadratic potential V m 2 φ 2 /2, the coupling to the curvature ξRφ 2 /2 leads to the additional mass-squared m 2 12ξh 2 in the case of d 3. Notice that m 2 h 2 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 , we find that the value of the parameter b 2 is very small, b 2 ∼ 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 4h 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 b 1 in 3.1 is b 1 < 5 × 10 −5 with the Planck cut-off scale M ∼ M Pl .On the other hand, the bound on the parameter b 2 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 Advances in High Energy Physics 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 as while the first power series solution of 4.18 can be rewritten as Therefore, we have 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 .
1 with s 2 and b 2 > 0, 3.4 should be replaced with L cor −b 2 /M 4 D 21 φ D 22 φ , where the operator D 2n is defined as

2 9 / 4 − 2 1 from the 10 Advances 1 I 1 and C 2 I0
m 2 ξR /h 2 , and the k-dependent constants C 1 I and C 2 I are to be fixed by choosing suitable initial condition at an arbitrary initial time τ 0 for each of the modes and satisfying |C 1 I | 2 − |C 2 I | in High Energy Physics Wronskian of f k I and f k I * .We can choose C for the initial Bunch-Davies vacuum state and obtain that

3 . 21 Let 1
τ c be the time when the modified dispersion relations take the standard linear form.Thenσ 2 z c 2s 1, where z c k|τ c | 1/b s 1/2s M/h 1 for b s 1.The constants C 1 II and C 2 II can be obtained by the following matching conditions at τ c for the two wave functionals 3.15 and 3.21 : when z z c .To find C 1 II and C 2 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 b 2 > 0 which corresponds to the ultraviolet limit of HL gravity.For this case ν 1/2, we have |H