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Advances in High Energy Physics
Volume 2012 (2012), Article ID 312841, 19 pages
Bohm Quantum Trajectories of Scalar Field in Trans-Planckian Physics
Physics Division, Department of Mechanical Engineering, Ming Chi University of Technology, Taishan, New Taipei City 24301, Taiwan
Received 17 October 2011; Revised 7 December 2011; Accepted 29 December 2011
Academic Editor: Kadayam S. Viswanathan
Copyright © 2012 Jung-Jeng Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
The scenario of inflationary cosmology successfully provides the paradigm for generating the inhomogeneities which seed the structures of the universe we observe today . 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 . However, it is well known that the notion of a vacuum state during inflation is ambiguous in quantum theory , and the choice of initial quantum vacuum state affects the predictions of inflation [4, 5].
Recently Perez et al.  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 . 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 .
2. Pilot-Wave Scalar Field in De Sitter Space
We consider the scalar field theory which has the Lagrangian density where is a real scalar field,is the potential, m is the mass of the scalar quanta, is the Ricci scalar curvature, is the coupling parameter, and g = det , . 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 , where is the Hubble parameter which is a constant. Note that in three spatial dimensions d = 3, the curvature 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 () of power-law inflation with . 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 , and ; that is, W is the overall comoving spatial size of lattice. From (2.4) we obtain the Hamiltonian where is the conjugate momentum of , Then we consider the discrete Fourier transforms such that Here can be chosen as the independent variables which obey the Poisson brackets . From (2.7) we can obtain the following identities: Furthermore, we define where is the conjugate momentum for , and the subscripts 1 and 2 denote the real and imaginary parts, respectively. Therefore, the Hamiltonian (2.5) becomes in momentum space
To quantize the theory above, we note also that each pair of operators and must satisfy the equal time commutation relations . These commutation relations are realized by the representation that is, the functional Schrödinger representation, which is not so widely used as the Fock representation. These operators act on wave functional , and the inner product is given by . With the field basis , that is, are the eigenstates of the operator , , and then the wave functional can be regarded as the expansion coefficient of the state . The dynamics of the wave functional is, therefore, determined by the functional Schrödinger equation with defined by (2.11). Writing as the product form we obtain from (2.13) the following: Rewriting (2.15) as the following form: and setting , so that , gives the time-dependent Schrödinger equation To solve (2.17), we use the conformal time defined by Then, for each real mode , the normalized instantaneous vacuum and its excited states are found to be  Here is the nth-order Hermite polynomial, and the complex function , real function , and complex function are given by (2.20), (2.21), and (2.22), respectively where is the Hankel function of the first kind and of order , with , and the prime in (2.20) denotes the derivative with respect to . Note that the wave functionals for each real mode can also be rewritten in the following form : with the amplitude and the phase The complete state wave functionals can be written as , where and which means that it is possible for different field modes to be in different excited states; that is, mode is in the -excited state, mode is in the excited state, and so forth. For , 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 with , and the mode index in carries labels 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 is given by which implies the continuity equation and the de Broglie velocities where . The wave functional for a single mode satisfies (2.17), which implies the continuity equation and the de Broglie velocity field Here is interpreted as a physical field in field configuration space, guiding the evolution of .
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 yields the de Broglie velocity field for In the continuum limit (), (2.31) reads which has the solution where is the ratio of physical wave number to the inverse of Hubble radius, and integration constant is chosen to be with being some reference point. Note that the quantum trajectory (2.33) is independent of the quantum number and depends on the form of the potential through .
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 where is a cut-off scale, is an integer, and is an arbitrary coefficient [17–19]. Note that the action for a scalar field with the modified dispersion relation (3.1) with and takes the form [27, 28] where is the standard Lagrangian of a minimally coupled scalar field corresponds to the nonlinear part of the dispersion relation and describes the dynamics of a unit time-like vector field which defines a preferred rest frame with Here is the covariant derivative associated with the metric , the tensor gives the metric on a slice of fixed time while is proportional to the Laplacian operator on the same surface, is the Lagrange multiplier, and the parameters and with no dimensions and the dimensions of mass square, respectively, are constrained by the astrophysical observations. For the modified dispersion relation (3.1) with and , (3.4) should be replaced with , where the operator is defined as . Then using , (2.26) becomes in the continuum limit () where , and the ground state wave functional of (3.7) becomes where satisfies In region I where , that is, , the dispersion relations can be approximated by , and (3.9) becomes To obtain the solution of (3.11), we define , where , and transform (3.11) into The general solution of (3.12) is where the Hankel function is of order with , and the -dependent constants and are to be fixed by choosing suitable initial condition at an arbitrary initial time for each of the modes and satisfying from the Wronskian of and . We can choose and for the initial Bunch-Davies vacuum state and obtain that where the prime in (3.14) denotes the derivative with respect to . The corresponding wave functional is
In region II where , that is, , the dispersion relations recover the standard linear relations , and (3.9) becomes Again, the solution of (3.16) can be obtained by defining and transforming (3.16) into The general solution of (3.17) is where and satisfy from the Wronskian of and . Therefore, we have where the prime in (3.19) denotes the derivative with respect to . Note that (3.19) can be simply obtained from (2.20) by replacing in (2.20) according to  The corresponding wave functional is Let be the time when the modified dispersion relations take the standard linear form. Then , where for . The constants and can be obtained by the following matching conditions at for the two wave functionals (3.15) and (3.21): which can also be rewritten, respectively, as by using , , when .
To find and , we focus on the case of massless minimally coupled () scalar field in the slowroll inflation and take the CJ type dispersion relation (3.1) with and which corresponds to the ultraviolet limit of HL gravity. For this case , we have with . Then from (3.14), (3.19), and (3.24), we obtain that where we have used , , and . (Note that since is one real condition on the two complex numbers and , we can choose and , where is the relative phase between and and can be considered as a state parameter.) Therefore from (3.26) and we have where , . Substituting (3.14) and (3.19) into (3.25) and keeping terms up to order on the right-hand side of (3.25), we obtain the following: Using (3.27) in (3.28) gives Here we choose , so that is small for to avoid an unacceptably large backreaction on the background geometry. Therefore, we have for or
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 (): where the classical potential is given by and the so-called “quantum potential” is given by where is given by (3.8)–(3.10), and is given by (2.24) for . It has been pointed out in previous work  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 .
In region I where and , the classical potential in (4.2) becomes and from (4.3), (3.8), (3.10), and (3.14), the quantum potential becomes Substituting (4.4) and (4.5) into (4.1) and using and gives On the other hand, in region II where and , the classical potential in (4.1) and the corresponding quantum potential become, respectively, where means modified according to (3.20). Then the quantum trajectory satisfies
In particular, for the case of massless minimally coupled () scalar field and Corley-Jacobson dispersion relation (3.1) with , (4.6) becomes 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 where and are constants to be fixed by choosing suitable initial conditions at an arbitrary initial time for the quantum trajectory . Here we can choose and so that the first term of corresponds to (2.33) with . On the other hand, in the region II becomes For (4.11) reduces to by using (3.26), and the quantum trajectory satisfies The most general asymptotic series solution of (4.12) is  For , the solution (4.13) reduces to Substituting (4.10) and (4.14) into the following matching conditions at for the two quantum trajectories and : we obtain From (4.14) and (4.16) we see that as decreases from to 1, becomes the dominant term, that is, On the other hand, for (well after horizon exit), (4.11) also reduces to by using (3.30) and , and the most general power series solution of (4.12) is  Note that for , 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 is the Neumann function which has the following expression: and is a constant. Note also that, for , (4.19) is equivalent to (2.33). Therefore, for , the solution (4.18) reduces to which corresponds to (4.19) and (4.20). Requiring to be continuous at (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, contains a factor which is proportional to , we can define a new field variable and use to rewrite (4.10) and (4.21) as Thus, we see from (4.23) that for fixed and, as decreases from to , 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 decreases from (well before horizon exit) to (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 () which is relevant to the nonminimally coupled chaotic inflation with quadratic potential , the coupling to the curvature leads to the additional mass-squared in the case of d = 3. Notice that and are required for slowroll [30, 31]. Thus, in general, we have , 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 to .
Finally we conclude this paper with the following discussions.
(i) Since a constant scalar field is similar to a cosmological constant , the transition could be interpreted as a transition of the Universe from a large to a small , thus providing a possible mechanism to solve the riddle of the smallness of in the framework of the Bohmian approach to quantum theory of inflationary cosmology. Note that the vacuum energy density due to quantum states with is ~, and (here Gev is the Planck mass). For the simplest potential driving inflation , from (4.23) and , we expect that as decreases from to 0, the cosmological constant decreases from to . Here, we have , is the Planck cutoff scale, Gev is the Hubble constant during inflation, and Gev which can be obtained from COBE normalization (see , page 212). Since current observations indicate that Gev4 and Gev2, we find that the value of the parameter is very small, . 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 , the transitional time is in the very early stage of inflation. We note that the quantum potential 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 and the mode the quantum force arising from the quantum potential through approximately cancels the classical force arising from the classical potential through , while for the quantum force becomes negligible with respect to the classical force. Therefore, as decreases from 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 .
(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 . 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 in (3.1) is with the Planck cut-off scale . On the other hand, the bound on the parameter 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 . 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 . 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, in (4.11) reduces to as decreases continuously from to 0 (not only for or or ) due to (3.30) and . Therefore, the quantum trajectory satisfies (4.12) where for . Note also that the solution of (2.32).
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 .
(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 .
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.
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