Response to “Comment on ‘Bohmian mechanics with complex action: A new trajectory-based formulation of quantum mechanics’ ” [J. Chem. Phys. 127, 197101 (2007)]

In their comment, Sanz and Miret-Artés (SMA) raise two issues. The first is a discussion of trajectory-based formalisms based on the quantum Hamilton-Jacobi formalism, where they invoke a number of antecedents to our paper. Most of these antecedents were, in fact, mentioned in our original article¹ (henceforth GDT), but this response gives an excellent opportunity to highlight the significant differences between our work and the antecedents. The second issue raised by SMA is how the term locality should be used in quantum mechanics. We suggest that at least certain aspects of nonlocality can depend on the method used to solve the problem.

We begin with some necessary background. Inserting the ansatz $ψ(x,t)=exp[(i∕ℏ)S(x,t)]$ $(S∊C)$ into the time-dependent Schrödinger equation (TDSE), one obtains the time-dependent complex quantum Hamilton-Jacobi (QHJ) equation,

SMA cite Leacock and Padgett² (LP) in connection with Eq. (1). In fact, the equation goes back much further—it can be found in Pauli’s 1933 article in Handbuch der Physik,³ where it is used as a starting point for a complex-phase time-dependent WKB expansion. The equation seems to have been rediscovered independently within several different scientific communities—by LP in 1983,² by John in 2002,⁴ and by SMA,⁵ also in 2002. Although Eq. (1) appears in these references, basic observations about this equation seem to have gone unnoticed. Given that Eq. (1) has been known since 1933 and is fully equivalent to the TDSE, it is remarkable how little attention it has received.

Leacock and Padgett, to whom we give reference in the GDT paper, begin with Eq. (1) but quickly revert to its time-independent version,

The focus of the work of LP is an extension of Dirac-Jordan transformation theory to Eq. (2). SMA refer to a recent work of Chou and Wyatt⁶ who have found a way to implement the approach of LP numerically. However, there are no trajectories in Ref. 6: Eq. (2) is solved by calculating action-type integrals along complex contours.

John’s work, which is also referenced in the GDT paper, is perhaps the most closely related antecedent. John starts with Eq. (1) and interprets the wavefunction as a guiding wave for complex trajectories, analogous to the Bohm–de Broglie interpretation in the case of real trajectories. However, as SMA note, John applied his ideas only to analytical examples, for example, the Gaussian in the harmonic potential and the step potential. Because all of John’s examples are analytically solvable it is easy to overlook the fact that John’s formulation assumes that the full wavefunction is known from a separate calculation. In mathematical terms, John’s Eq. (7) (in Ref. 4) for the velocity,

is written in the form of a mixed ordinary differential equation (ODE) and a partial differential equation (PDE), while a bona fide trajectory method expresses the equations of motion in terms of ODEs alone. Thus, John’s work does not provide a general prescription to replace quantum calculations with trajectory calculations but rather provides a trajectory interpretation of quantum mechanics.

SMA derive Eq. (1) in their work, but their primary focus is the ansatz $ψ(x,t)=exp[(i∕ℏ)S(x,t)]$⁠. They expand complex $S$ in a series in $ℏ$⁠, as in conventional WKB. However, the coordinate $x$ is always real in their treatment, and like John, their approach was developed as an interpretational tool, not as a means to solve the TDSE.

We derived Eq. (1) in the late 1990s, in an effort to reconcile the classical limit of the hydrodynamic formulation of quantum mechanics with Ehrenfest’s theorem.⁷ As noted in Ref. 8, the right-hand side (RHS) of Eq. (1) can be viewed as a complex quantum potential,

For Gaussian dynamics, $QC$ has no spatial dependence, hence, the associated quantum force vanishes. $QC$ therefore contributes only a time-dependent phase which is identical to the peculiar nonclassical term in the equation of motion for the phase in Gaussian wavepackets.⁹ In the context of Gaussian wavepackets, the real part of $QC$ contributes a geometric phase (a time-dependent Maslov index) and the imaginary part ensures normalization. This suggests a novel perspective on classical-quantum correspondence in which, to a first approximation, real classical trajectories are replaced by complex trajectories which are carrying a geometric phase and a normalization factor. What is most remarkable about this picture is that, to the extent that the quantum force vanishes, the complex trajectories do not communicate with their neighbors. Hence, the nonlocality of quantum mechanics, in the sense of trajectories being affected by their neighbors, has been greatly reduced and, in some cases, totally eliminated.

Based on the insights of Ref. 8, the GDT paper made three key contributions. First, the GDT paper conjectured that since the quantum force in Eq. (1) vanishes for Gaussian wavepackets in potentials up to quadratic, for anharmonic potentials the quantum force is likely to be much smaller than for the real QHJ equation. This conjecture was borne out beautifully: the GDT paper presented tunneling amplitudes for the Eckart potential down to $10−7$ calculated from complex trajectories without any quantum force. The smallness of the quantum force that follows from Eq. (1) has significant implications for the nonlocality in quantum mechanics, but it also has significant numerical implications: in the last ten years, extensive effort within the chemical physics community has gone into the use of real Bohmian dynamics as a numerical tool for solving the TDSE. The calculation of the quantum force is the bottleneck to such calculations, hence, the identification of a formalism in which the magnitude of the quantum force can be reduced by orders of magnitude is significant indeed.

The second contribution of the GDT paper is the following short logical step: that if the real QHJ equation can be solved using real trajectories, then the complex QHJ equation [Eq. (1)] can be solved using complex trajectories. John comes the closest, but as indicated above John does not propagate complex trajectories, he reconstructs them from the wavefunction.

The third main contribution of the GDT paper is how to deal with the residual complex quantum force. We developed a hierarchy of complex derivative equations in which the complex trajectories propagate without communicating with their neighbors. Although, formally, the hierarchy is infinite, in practice it has to be truncated at some finite value, $N$⁠. We showed that keeping only the lowest order term in the hierarchy was equivalent to the method known as generalized gaussian wavepacket dynamics^11–13 (not to be confused with the usual Gaussian wavepacket dynamics), while higher order truncation systematically increased the accuracy. We were led to the hierarchy of derivative equations because of the familiarity of one of us with a numerical method in the applied mathematics literature known as “jet propagation.”¹⁴ As we mention in our paper, different but structurally analogous hierarchies of derivative equations have been used in the context of real Bohmian mechanics.¹⁵ 

To summarize, the GDT paper had three central contributions: (1) to exploit the fact that the quantum force in Eq. (1) is much smaller than in the conventional (real) QHJ equation; (2) to solve Eq. (1) using complex trajectories, and (3) to provide explicit equations of motion for the complex trajectories, equations in which there is strictly no communication between trajectories. The result is a remarkable new approach which in principle is an exact formulation of quantum mechanics using complex trajectories that do not communicate with each other. In our opinion the subtitle of the GDT paper, “A new trajectory-based formulation of quantum mechanics,” is accurate and appropriate.

The second part of SMA’s comment deals with nonlocality. Here it is useful to distinguish between three kinds of nonlocality. The first is the interaction between neighboring trajectories—a “tube” around each trajectory that brings in information on the neighbors. A measure of this kind of nonlocality is the quantum force calculated from a hierarchy of local derivatives. The second kind of nonlocality is the effect of remote trajectories on each other, as, for example, in a two-slit experiment. The third type is multiparticle entanglement, as in the Einstein-Podolosky-Rosen (EPR) experiment. In Ref. 16 the first type of nonlocality is referred to as “regional nonlocality” while the second and third types may be called as “global nonlocalities.” In optics, the first type is the analog of diffraction (intraslit), while the second type is the analog of interference (interslit).

SMA claim that “nonlocality does not disappear with the particular method used to solve a quantum problem.” At least with respect to the first type of nonlocality we disagree. As noted above, for a Gaussian wavepacket in a harmonic potential in conventional Bohmian mechanics, there is a strong quantum force, while in complex Bohmian mechanics, the quantum force vanishes. Moreover, the GDT paper showed that accurate results for tunneling through an Eckart barrier could be obtained with no interaction between neighboring trajectories. This picture of tunneling is quite different from that described by Lopreore and Wyatt,¹⁰ in which later trajectories “push” earlier trajectories over the barrier, i.e., where communication between neighbors (hence nonlocality) is responsible for the tunneling. In our opinion, this shows that first type of nonlocality indeed does depend on the method used to solve the problem.

Even with regard to the second type of nonlocality, a case can be made that it depends on the particular method used to solve the problem. In Ref. 17 we show that interference effects are described naturally in Bohmian mechanics with complex action (BOMCA) via the superposition of independent trajectories, without the need for global correlation of the trajectory motion. Specifically, the trajectories are propagated strictly independently of their neighbors using only local derivative information, and global nonlocality is achieved by superposition. Although Ref. 17 is outside the scope of the comment by SMA, it illustrates that nonlocality, this time in the sense of global correlation of trajectory dynamics, can depend on the particular method used to solve a problem. In future work, we intend to explore the third type of nonlocality within the framework of BOMCA. It would be fascinating if this type of nonlocality can also be described with just local derivative information.

On the other hand, SMA claim that “the origin of the nonlocality is in the corrections to the classical equations of motion that come from higher order terms in an $ℏ$ expansion.” The reasoning of SMA is as follows: global nonlocality must be contained in their Eq. (6), since the appearance of the expansion of $S$ on the RHS presupposes that a full solution of the TDSE is known from a separate calculation, and the TDSE is global. Since the first term on the RHS of Eq. (6) is classical, global nonlocality must therefore be contained in the higher order $ℏ$ terms. We find this argument problematic. Global nonlocality implies the possibility of large nonclassical forces, for example, when nodes are present in the wavefunction, or in a two-slit interference experiment. The classical and Bohmian dynamics are qualitatively different in these cases and it is difficult to imagine that a series of “corrections” as appears in SMA’s Eq. (6) can account for such major differences. Rather, we suspect that when the series in SMA’s Eq. (6) converges, these terms will describe only regional nonlocality.

Where then has the global nonlocality of the Schrödinger equation gone? Note that the replacement of $∇S¯0∕m$ by $ẋcl$ in SMA’s Eq. (6) may not be completely trivial. $S¯0$ is a global solution of the classical HJ equation; it is known from the theory of PDEs that this equation displays shocks at certain places at certain times, from which time forward it does not agree with the behavior of classical trajectories. These are precisely the times at which the Lagrangian manifold folds over on itself, becoming multivalued, corresponding to trajectory crossings and, hence, nodes in the wavefunction. Thus, the replacement of $∇S¯0∕m$ by $ẋcl$⁠, although routinely made and valid in certain regions of space and time, breaks down precisely when global nonlocality come into play. Since $ẋcl$ is local and $∇S¯0∕m$ is not, global nonlocality may have been removed from Eq. (6) by making this substitution. It would be very interesting to evaluate the individual terms in SMA’s Eq. (6) for a case where global nonlocality is significant.

Finally, SMA claim that real trajectories have an interpretive advantage. While real trajectories are certainly more familiar than their complex counterparts, this is not the end of the story. We again consider Gaussian wavepackets, where the quantum force is large in conventional Bohmian mechanics but vanishes for Bohmian mechanics with complex trajectories. This suggests that the quantum force is to some extent a consequence of the representation. An analogy with Copernican versus Ptolemian astronomy is useful. While it is appealing to believe that the Earth is the center of the solar system, the resulting orbits of the sun and the planets, called epicycles, are quite complicated. With the sun at the center, the resulting orbits are simple ellipses. While all the correct astronomical predictions can be made with epicycles, the claim that the Ptolemian model has an interpretive advantage is certainly open to question. In future work, we hope to elaborate on the physical justification for the complex trajectories appearing in BOMCA and their classical limit.