A procedure of linearizing quantum brackets (commutator of a dynamic operator with the Hamiltonian function of the system) is considered as an algorithm of terminating an infinite hierarchical system of linked equations of motion. Quantum brackets are introduced as a commutator of a dynamic operator with the Hamiltonian function of the system. This procedure is applied to the study of a configurational model of a binary atomic system using dynamical symmetry of the system. It was revealed that dynamical variables form a closed space with respect to commutation with the Hamiltonian function. This is a necessary and sufficient condition for invariance of a physical system relative to the group of dynamical symmetry of Hamiltonian.
A procedure of linearizing quantum brackets (commutator of a dynamic operator with the Hamiltonian function of the system) is considered as an algorithm of terminating an infinite hierarchical system of linked equations of motion. Quantum brackets are introduced as a commutator of a dynamic operator with the Hamiltonian function of the system. This procedure is applied to the study of a configurational model of a binary atomic system using dynamical symmetry of the system. It was revealed that dynamical variables form a closed space with respect to commutation with the Hamiltonian function. This is a necessary and sufficient condition for invariance of a physical system relative to the group of dynamical symmetry of Hamiltonian.
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.
The behavior of symmetries of classical equations of motion under quantization is studied from a new point of view. GL(3, R), which is an invariance group of the linear equations of motion for the nonrelativistic free particle as well as the isotropic harmonic oscillator, is imposed as a group of automorphisms on acceptable Poisson brackets, and the consequences are examined in detail. The six independent variables of the classical system arrange themselves, in each acceptable bracket, into one canonical pair and four neutral elements. Consequences of this for the equations of motion, existence of a Hamiltonian, breakup of the states of motion into superselected sectors due to existence of neutral elements, and determination of the canonically realized subgroup of GL(3, R) are all discussed. The possible relevance of this manner of symmetry breakdown for solid state and particle physics is pointed out.
In the first part of this series we have shown how the Schrodinger equation for a single particle and the corresponding non relativistic quantum observables can be obtained from a purely classical phonon model through the Newtonian equations of motion. In this work we focus instead on how the classical Hamiltonian formalism applied to the same phonon system allows to recover the feature of relativistic quantum mechanics for a single spinless particle. Using the classical nature of the phonon model, we naturally define continuous Poisson brackets between classical observables, which allows to recover the dynamics of such observables, i.e. the Ehrenfest relations associated to real-valued Klein-Gordon fields. The Poisson brackets also permits to obtain the generic form of constants of motions, thus generalizing the concept of inner products and momentum on Klein-Gordon fields. We then connect the formalism of real-valued classical functionals with that of hermitian operators and complex-valued wave functions. This is done through the introduction of a non-local complex-valued change of variables which allows to rewrite the real-valued Klein-Gordon equation in a form akin to the Schrodinger equation, and the classical observables as quantum expectation values. Then, we show how this change of variables allows to rewrite the classical Poisson brackets as commutators of hermitian operators. This points out the strict equivalence between the Heisenberg formalism and the formalism of classical Poisson bracket. Eventually, we illustrate how the Poisson brackets allows to recover the transformations of Poincare group in 1+1 dimension together with its algebra. The latter makes the link between the Lorentz invariant inner product of Mostafazadeh and the Casimir invariant associated to the mass of particle.
This thesis is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
Quantum mechanics takes various forms. For atomic and nuclear physics, the Schrödinger picture is convenient. For quantum field theory and for correspondence with classical mechanics through Poisson brackets, the Heisenberg picture is useful. The interaction picture is useful in calculations. The Wigner phase-space is convenient for studying the uncertainty relations and their symmetries. The density matrix and probability distribution are necessary for extracting measurable dynamic quantities and are essential for ensemble averages. Mathematical instruments, such as the Lie algebra of the rotation group, the creation and annihilation operators of the Fock space and the Lie algebra governing Dirac’s two oscillators are shown to be derivable from Heisenberg’s commutation relations.
Statistical reformulation of quantum mechanics in terms of phase-space distribution functions as given by Moyal using Weyl's correspondence rule between classical functions and operators has been extended to various different correspondence rules. The dynamical bracket in the Weyl correspondence (the ``Moyal'' or the ``sine'' bracket) is shown to be a Lie bracket. It is further shown that if the theory is restricted to Lie brackets of the form [u(p, q),υ(p, q)]={f(∂2∂q1 ∂p2−∂2∂q2 ∂p1)u(p1, q1)υ(p2, q2)},evaluated for p1 = p2 = p; q1 = q2 = q after differentiation, then the only admissible functional form of f is f(x) = β[(sin αx)/α], where α and β are constants. A law of multiplication which is associative and distributive with respect to addition is also introduced in each case. It gives a correct correspondence between operator multiplication and the multiplication of classical functions. The dynamical brackets obtained in each case are also found to be Lie brackets. Conditions on the phase-space distribution functions to describe pure states are also given.