Quantum-classical bracket

V. V. Kisil. A quantum-classical bracket from p-mechanics. Europhys. Lett., 72(6) 873-879, DEC 2005. 

O. V. Prezhdo. A quantum-classical bracket that satisfies the Jacobi identity. J. Chem. Phys., 124(20) 201104, May 2006. 

However, the quantum-classical brackets ( , ) introduced in eq.(7) are not Lie brackets [19], because they do not satisfy properties that are instead satisfied by the commutator and the Poisson brackets (respectively, quantum and classical Lie brackets), e.g., the Jacobi identity. 

An immediate and undesirable consequence of the violation of the Jacobi identity is that the quantum-classical brackets between two constants of motion... 

The second line has been obtained by making the action of Ai and A2 in the first line explicit, as shown in eq.(29) and eq.(34). Eq.(49) defines the quantum-classical brackets as... 

However, the Heisenberg group formalism is a very useful tool to represent quantum and classical dynamical quantities, such as observables and equations of motion, only when a prescription on the generator of the time evolution exists. The comparison with the fully quantum or fully classical dynamics allows us to deduce only the formal properties that the mixed quantum-classical brackets have to satisfy in order to generate a consistent evolution, but does... 

Kisil, V.V. Quantum-classical brackets. Internat. J. Theoret. Phys. 41 63... 

In contrast to quantum mechanical and classical brackets, the quantum-classical bracket does not satisfy the Jacobi identity since [5]... 

The plan of the paper is the following. In section 2 we introduce the elements at the basis of the Heisenberg group representation representation theory [12-14,20] that are needed to understand the alternative group-theoretical formulation of quantum mechanics. In section 3 the Heisenberg representation of quantum mechanics (with the time dependence transferred from the vectors of the Hilbert space to the operators) is used to introduce quantum observables and quantum Lie brackets within the group-theoretical formalism described in the previous section. In section 4 classical mechanics is obtained by taking the formal limit h —> 0 of quantum observables and brackets to obtain their classical counterparts. Section 5 is devoted to the derivation of... 

In the present and in the following section we discuss the application of the group-theoretical formalism to the formulation of quantum-classical mechanics. Our purpose is to determine evolution equations for two coupled subsystems, with two different degrees of quantization. We have shown in the previous sections that the classical behaviour of a system is formally obtained as a limiting case of the quantum behaviour, when the Planck constant h tends to zero. In this section we will associate two different values of the Planck constant, say hi and /12, to the two subsystems and introduce suitable Lie brackets to determine the evolution of the two subsystems [15]. The consistency, e.g., with respect to Jacobi identity, is guaranteed by the very definition of the... 

The property of the new (Lie) brackets (44) of being correct in the known full quantum and full classical limits may reasonably convince ourselves that the intermediate situation, in which hi —> h and h2 —> 0, generates quantum-classical dynamics. If the assumedly quantum-classical limit is performed on Ph1,h2(9i, 92), we obtain... 

We have thus reconstructed the derivation and interpreted the results of Ref [15], The first two terms, i.e., the commutator and the Poisson brackets, are already present in a theory based on the quantum-classical Liouville representation discussed in section 1. The new term, which appears within the Heisenberg group approach, needs to be explained. In the attempt to provide a physical interpretation to this term we have shown, in Ref. [1], that the new equation of motion is purely classical. This will be illustrated in the following section. 

An attempt to solve the difficulties and inconsistencies arising from an approximated derivation of quantum-classical equations of motion was made some time ago [15] to restore the properties that are expected to hold within a consistent formulation of dynamics and statistical mechanics, and are instead missed by the existing approximate methods. We refer not only to the properties that the Lie brackets, which generate the dynamics, satisfy in a full quantum and full classical formulation, e.g., the bi-linearity and anti-symmetry properties, the Jacobi identity and the Leibniz rule12, but also to statistical mechanical properties, like the time translational invariance of equilibrium correlation functions [see eq.(8)]. 

The Wigner form of the quantum evolution operator iLw X ) in (62) for the equation of motion for W X, X2,t) can be rewritten in a form that is convenient for the passage to the quantum-classical limit. Recalling that the system may be partitioned into S and S subspaces, the Poisson bracket operator A can be written as the sum of Poisson bracket operators acting in each of these subspaces as A Xi) = A xi) + A Xi). Thus, we may write... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

We have bracketed the potential energy U with vib because both quantities depend on the coordinates of the nuclei relative to one another, whereas Trot averaged over the vibrations does not. The quantum-mechanical Hamiltonian operator is obtained by replacing the classical quantities with operators ... 

We have thus obtained the expected result classically, the time derivative of a function of the phase-space (that does not depend explicitly on time) is given by the Poisson brackets between the function itself and the hamiltonian. This result has been obtained by taking the formal limit h — 0 on the quantum expression in eq.(30), i.e.,... 

For heavy particles or high temperatures, the quantum correction (square brackets) converge against 1, and the classical partition function is recovered. 

Before concluding this section, let us just mention that, while all of the equations above refer explicitly to the Van Vleck quantum procedure [10-12,15], they are most straightforwardly adapted to the classical procedure based on Lie algebra [16-18] by replacing quantum commutators with Poisson brackets. Most of the concepts remain also valid for the classical... 

The analoguous behiaviour of the Poisson bracket and the commutator has been used to establishcorrespondence between classical and quantum mechanics. It is, however/shown in the next section following the derivation of Schwinger s quantum action principle that the correspondence goes deeper and that the analogous behaviour of the Poisson bracket and commutator is a consequence of the properties of infinitesimal canonical transformations which are common to both mechanics. 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

In classical mechanics the abstract Hermitian operator L, defined as operating on the Hilbert space of distributions, is Lc = -i, H), whereas it is Lq = h l [, H] in quantum mechanics. Here, denotes a Poisson bracket and [, ] denotes a commutator. 

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