The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G 0 under under G , they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.
The time-dependent variational principle proposed by Balian and Vénéroni is used to provide the best approximation to the generating functional for multi-time Green’s functions of a set of (bosonic) observables Qµ. By suitably restricting the trial spaces, the computation of the two-time Green’s function, obtained by a second order expansion in the sources, is considerably simplified. This leads to a tractable formalism suited to quantum fields out of equilibrium. We propose an illustration on the finite temperature Φ4-theory in curved space and coupled to gravity.
The mean-field pictures based on the standard time-dependent variational approach have widely been used in the study of nonlinear many-boson systems such as the Bose-Hubbard model. The mean-field schemes relevant to Gutzwiller-like trial states
∣F>, number-preserving states
∣ξ> and Glauber-like trial states
∣Z> are compared to evidence the specific properties of such schemes. After deriving the Hamiltonian picture relevant to
∣Z> from that based on
∣F>, the latter is shown to exhibit a Poisson algebra equipped with a Weyl-Heisenberg subalgebra which preludes to the
∣Z>-based picture. Then states
∣Z> are shown to be a superposition of
N-boson states
∣ξ> and the similarities/differences of the
∣Z>-based and
∣ξ>-based pictures are discussed. Finally, after proving that the simple, symmetric state
∣ξ> indeed corresponds to a SU(M) coherent state, a dual version of states
∣Z> and
∣ξ> in terms of momentum-mode operators is discussed together with some applications.
The form of the complex Hamiltonian evolution equations (linear and nonlinear) is deduced here from that of the classical Hamiltonian systems. Their properties and those of the associated continuous Poisson bracket are studied. It is shown that if any space-localized solution of the above type equations is expanded in any Hilbert space basis the expansion coefficients are canonically conjugate variables and obey Hamiltonian systems of complex equations. The last is proved also for sets of “canonically conjugate” functionals of general type. As an application, commutation relations which are equivalent in form to Dirac’s quantization rules (the constant in place of ℏ is undetermined) are obtained. The question of how a nonlinear field theory may contain quantum mechanics as a special case is discussed in view of this paper’s results.