Poisson bracket, classical

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... 

An identical expression holds classically [32] if- //) times the conmuitator is replaced by the classical Poisson bracket. 

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. 

The operator A is given by A =VpVp — VrVp= — , , where , are the Poisson brackets with respect to the classical phase-space variables... 

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. 

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.,... 

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. 

Note that the right-hand side of Eq. (325) consists of the classical Poisson bracket between Hw q,p) and W, q,p), plus a series of additional terms that depend on H. This makes it clear that the time-evolving Wigner function is the quanmm analog of the time-evolving classical Liouville density function. 

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... 

There remains but one important concept to complete our summary of the role of canonical transformations in classical mechanics, that of the Poisson bracket. Let F p,q) and G p,q) denote two mechanical properties of the system. Their Poisson bracket is defined as... 

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. 

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... 

Poisson bracket operator in classical mechanics [...] square matrix... 

The second equality in Eq. (1.99) defines the Poisson brackets and C is called the (classical) Lionville operator. Consider next the ensemble average 4(Z) = A t of the dynamical variable A. This average, a time-dependent observable, can be expressed in two ways that bring out two different, though equivalent, roles played by the function -4(r, p ). First, it is a function in phase space that gets a distinct numerical value at each phase point. Its average at time t is therefore given by... 

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. 

The similarity of these classical mechanical Hamilton s equations of motion to their quantum mechanical Ehrenfest s Theorem counterpart, (see Section 9.1.7), is an expression of the Correspondence Principle equivalence of a quantum mechanical commutator, [A, B], to a classical mechanical Poisson bracket,... 

The classical expression, Eq. (5), corresponds to the quantum mechanical one of Eq. (4). In fact, starting from the classical expression and, as suggested by Dirac [77], replacing the Poisson bracket of two classical quantities F and G by the commutator of the respective operators as... 

The Poisson bracket A, B in classical mechanics is given by the expression... 

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