Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

Time-dependent variational principle (TDVP) provides powerful methods in solving the time-dependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum–classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314). We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for ‘energy related dynamics’ and the other for ‘dynamics of wave-flow’. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.


Introduction
We revisit the quantum mechanical time-dependent variational principle (TDVP) from the view point of the classical least action principles of Maupertuis and Hamilton. Much is already known about the quantumclassical correspondence up to the sophisticated studies on semiclassical mechanics [1][2][3][4][5][6]. Well-known relationships of quantum theories to classical counterparts are, for instance, the Schrödinger equation to the Hamilton-Jacobi equation [7], the Heisenberg equation to the Hamilton canonical equations of motion (the commutator to the Poisson bracket) [8], Wigner phase-space theory [3,9] to the classical Liouville equation, Feynman path integrals [10] to the Lagrangian dynamics through the stationary state approximation [1], Nelson's stochastic quantization [11] to the Newtonian equation, and so on. It is astonishing for both quantum and classical mechanics to have such a variety of seemingly different yet essentially equivalent forms. However, to the best of our knowledge, relationship between quantum theory and the least action principle of Maupertuis has not been well studied for a long time after the birth of quantum theory. Only in 1984, Zambrini [12] showed that Maupertuis' principle of least action in the stochastic calculus of variations leads both to the Nelson stochastic quantization equation [11] and the Newtonian equation in the two opposite extrema. (The basic theory of the stochastic calculus of variations had been developed by Yasue [13]. For very recent progress in stochastic variational principle, see [6,14,15].) Being beautiful, the Nelson stochastic quantization is not necessarily useful in applications to multi-dimensional systems.

Preliminary for time-dependent variational principles
This section is devoted to a brief review for some basic variational principles which are particularly relevant to the present work. This section therefore includes no new material, and may be skipped to the next section.
Further, using the Legendre transformation, the momentum coordinate p is introduced, and the Hamiltonian is defined as where ò pqdt  is referred to as the Maupertuis reduced (abbreviated) action [27]. Then another form of the least action principle follows as and one obtains the standard classic canonical equations of motion (with mass m=1 in the mass-weighted coordinates) ) is a canonical conjugate pair of independent dynamical variables. Knowing difference between the Maupertuis principle and Hamilton principle, we here refer to equation (5) simply as the Maupertuis-Hamilton principle to stress the presence of the Maupertuis reduced action. Among the most important consequences from them are energy conservation along (q(t), p(t)) and the relevant absolute invariances for a timeindependent Hamiltonian [28]. Thus, the classical Lagrangian and Hamilton are mutually convertible through the Legendre transformation. It is therefore noteworthy that Kan emphasized that such a Legendre transformation does not exist in his quantum TDVP [17].
We recall that the role of the term pq in the classical variational principle equation (5) is not limited to the Legendre transformation alone; It is of course needed to ensure energy conservation for time-independent Hamiltonians and is also known to be subject to the Poincaré-Cartan integral invariance [28]. We see later the similar situation in the quantum parameter space.

Dirac-Frenkel and Kan variational principles 2.2.1. Dirac-Frenkel
We start with the Schrödinger equation and try to find the solutions within a class of trial functions that satisfy variational condition In the studies on dynamics of molecules, for instance, the following Gaussian-type wavepackets are frequently used for trial functions, which contain three classes of parameters Here the linear parameters c t c t c t , , ,  are supposed to be complex values and are responsible for shape deformation of the packets. The set of nonlinear parameters q t p t , Thus it is readily confirmed that the linear variational principles generally project the true dynamical equations into a finite-dimensional functional spaces, which are generated by a finite number of basis functions c k { } such that This variational procedure is widely known as the Dirac-Frenkel principle, and not much difficulty is encountered in the ordinary practice.

A basic structure of Kan's method
Kan's theory [17] starts with the standard variational principle with a functional ò a where a is a vector of real-valued parameters. An equation of motion naturally emerges as the Euler-Lagrange equation such that Thus the time dependent Schrödinger equation has been transformed into a set of simultaneous ordinary differential equations over the space of variational parameters [17].
A serious problem involved in equation (19) is that the matrix B there, equation (18), can be singular and accordingly B −1 in the vicinity of the singular points will behave erroneous. It is expected that the number of those singular points will increase as the number of variational parameters. Also, the matrix B becomes null whenever the trial function happens to be real-valued. Kan himself exploited a method to avoid such singularities [17], but the procedure proposed seems to be too complicated to carry out at each step of numerical integration of the coupled ordinary differential equations. From the view point of numerical analysis, such singularity can be formally removed by means of the methods like singular decomposition at each point of divergence. Yet two kinds of singularities are expected; one is accidental, which may be augmented by interpolation and/or extrapolation techniques, and the other is more systematically caused, which can take place simultaneously in multiple parameter coordinates depending on physical situations under study. It is not nice anyway for any initial value problem to encounter singularities on the way of tracking its trajectories, since information prepared in the initial conditions may be lost and the accuracy can be sacrificed to a large extent.
In his study Kan came to a conclusion that there is no Legendre transformation that allows a d dt k to be treated as independent variables as momenta in classical mechanics. This is because the right hand side of equation (19) does not contain momentum-like variables like a d dt k [17].

Kan's method from the view point of the Dirac-Frenkel
In the linear equation of equation (12) one may insert a f ( ) as a trial function such that for an arbitrary set of c k { }. Note that the set c k { } mayinclude even those functions that are not used to expand the trial function. Then assuming that the following matrix A turns out to be a unit matrix when only the linear parameters are adopted as in equation (10) and those basis functions are all orthonormal. This linearization method looks very straightforward. However, equation (20) works as a good approximation only when the parameter set a j { } is large enough, at a risk of singularity of A, and the projecting functions c k { } must be good enough to cover the space of a f ( ).
As suggested by Broeckhove et al, we may choose the following particular basis functions and impose the conditions in equation (12) (without knowing the variational principle behind) assuming the parameters α j are all real as adopted in the Kan theory [17]. Summing up these two equalities we see which is exactly the same as Kan's equation (17). Thus, we may say that the Kan method is a linearization approximation in equation (13) with respect to the nonlinear parameters, and the matrix inversion is unavoidable in this context.

McLachlan method
Far before the Kan theory, McLachlan had proposed a method [16] to extract the solution of the time-dependent Schrödinger equation applying a minimum principle to An impression may suggest that this minimization procedure should give more accurate results than the stationary procedure of Kan's method and heavier numerical burden is demanded as a consequence. Broeckhove et al [19] have analyzed the interrelationship between the two variational method and found a remarkable fact that these two are essentially equivalent to each other, if the variational parameters have the pairwise complementary forms such that where they are all real-valued and satisfy individually

Dual least action principle in parameter space
In what follows we consider trial wavefunctions in a form of f u v , ( ), in which real-valued vectors u and v are supposed to have pairwise components u v , i i ( )as in equation (29). We here consider only time-independent Hamiltonian, yet most of the resultant dynamical equations of motion are valid even for time-dependent Hamiltonians.

Variational principle for energy conservation
It is widely known that the Hamilton-like canonical equations of motion for energy expectation value work to determine specific class of variational parameters. Take a simple coherent type trial function as an example with α being frozen. Putting = q u 0 and = p v 0 we can track them in time be means of the following equations of motion Obviously this dynamics is far more manageable than the Kan dynamics. Along the flow lines t t u v , ( ( ) ( )) thus determined, the energy conservation is readily ensured by the relation An advantage of this simple method is that quantum effects such as the zero-point energy along the path are partly taken into account, and yet their resultant paths can be similar to those of the classical counterpart. The question is which class of parameters can be determined only by means of equations (32) and (33). We will be back to this issue later in the next section. A variational functional that conserves the energy can be readily made up in analogy to the classical least action principle of equation (5), which is simply written as where C indicates arbitrary time interval under study. In what follows f is always assumed to be normalized and therefore the normalization constant is subject to the variational operation. The variation of S H gives Since both δu and δv in equation (36) are individually arbitrary, d = S 0 H requires equations (32) and (33) to identically hold. Note that the Maupertuis reduced action is seen as v u ·  in S H . Any trajectories t t u v , ( ( ) ( )) in parameter space should satisfy the energy conservation. Yet, it is obvious that the energy conservation alone is not enough to determine correct trajectories. In other words, δS H =0 or equations (32)-(33) are insufficient. Something is missing.

Dual least action principle
The missing factor is readily identified in the full variational principle This subtraction is mathematically valid since the variation is a linear operation with respect to variational functionals. Thus we find and define a variational functional ò ò As far as the exact solutions are concerned, the condition of equation which is responsible for the Schrödinger 'wave dynamics'. Note, however, that d are never independent from each other. Only the exact f ¢s u v , ( ) are supposed to satisfy both of these variational conditions simultaneously. We thus need to study the simultaneous variational principles d d d d where t t u v , ( ( ) ( )) in S H and S W should be common. Equation (43) or (44) is referred to as a dual least action principle.
The condition in equation (43) should be formally stronger than the total variational condition of equation (39) is, since the latter can hold under a weaker situation in which the deviation from zero in δS W and that in δS H happen to cancel each other. On the other hand, it is hard to anticipate that the errors arising from d ¹ S 0 H and d ¹ S 0 W are compensated well. Indeed, as we will see later, the physical and mathematical natures of δS H and δS W are so much different from each other that such systematic cancellation of errors is hardly expected to happen.
An obvious reason why the functional S W appears to be necessary is because S H of equation (35) alone cannot reflect the Schrödinger dynamics. For instance, in order to distinguish the dynamics arising from the explicit functional form of S W is definitely needed. Hence the condition δS W =0 identifies the way of propagation of the 'waves', while δS H =0 is mainly responsible for energetics of 'trajectory motion' of the wavepacket. Just as the δS H =0 has given equations (32) and (33), δS W =0 formally brings about which represents a 'flow conservation' (see below) along the path In this conjunction, we note that v u ·  in this stage is no longer regarded as a factor to induce the Legendre transformation but has been introduced to find the stationary paths and invariances in the parameter spaces in analogy to classical mechanics. Also, recalling the 2-form we identify v u ·  in this context as a quantity related to an absolute invariance in the parameter space [28], which will be discussed in subsection 3.5.
Once again we stress that the set equations (32)- (33) and that of equations (47)-(48) are not to be integrated separately. They are supposed to represent a tightly coupled mechanics of Hamiltonian dynamics and wave flow dynamics as a one unit.

Quantum flux for the wave dynamics in parameter space
We here survey the physical meaning of δS W =0, which highlights the characteristics of the Schrödinger wave dynamics. The study about how to treat equation (43) in practical approximations will be resumed in section 3.4.

Quantum flux in parameter space
which is real-valued, and rewrite the right hand side as Here for instance, corresponds to the gradient of the field of fluid in u-direction.
To be more precise, let us recall the definition of quantum mechanical flux (current of probability density) [30], which is defined as with the mass m=1. By analogy, we may define fluxes in the parameter space such that which is the probability current in the u direction, and likewise the flux in the v-direction is And therefore we have which gives an alternative interpretation of f f á ñ |  in terms of the flow in the parameter space. Let us briefly detour to a complex-valued flux emerging from j u and j v . Define complex coordinates as Broeckhove et al did [19] such that Back to equations (47)-(48), they are thus rewritten, respectively, as It is interesting to see the nonlinear nature of the dynamics manifest itself in the right hand sides of equations (62) and (63), where d dt u and d dt v , respectively, appear. This kind of nonlinearity is seen rather frequently in fluid dynamics.

Simple yet exceptional example of the flux
For a later purpose, we here show an example of the flux arising from the so-called coherent state wavepacket which is frequently used as an expansion basis [23]. The parameters q 0 and p 0 are often treated as dynamically conjugate parameters. Taking q 0 and p 0 as one of the u and v parameters, respectively, we readily see the following relation = p j , 6 5 q 0 0 ( ) which seems natural and is indeed consistent with the quantum momentum operator where the minus sign arises from -q t 0 ( ) in the representation of equation (64). As for the j p 0 , on the other hand, we see and These serve as our working equations, which should be integratedtogether with equations (32)- (33).
In equations (68) and (69), we see the factor which is a symplectic inner product between the Hamiltonian derivative vector and the flux vector. Hence this quantity represents a coupling between the Hamilton dynamics (so to say particle dynamics) and the motion of the spatial redistribution of probability density (wave dynamics). All the effects with respect to the deformation of wavefunctions manifest themselves through these couplings.  69)) from the wave flow dynamic, are supposed to be satisfied simultaneously, simply because they are satisfied by However, our trial functions are not exact in general, and we need to treat both equations (32)- (33) and (62)-(63) as pairwise simultaneous conditions to guide t t u v , ( ( ) ( )) toward the correct trajectories. We below consider this aspect in a little more systematic manner.
Suppose we have a short time approximate solution in equations (32)- (33), which is formally written as and likewise from equations (62)-(63) we have .
The suffices H and W indicate the Hamilton dynamics and the wave-flow dynamics, respectively. Finite time solution of each is to be formally given as