Preface
The work contains a description and an analysis of two different approaches
determining the connections between quantal and classical theories.
The first approach associates with any quantum-mechanical system with a finite
number of degrees of freedom a classical Hamiltonian system ‘living’ in projective
Hilbert space PðHÞ, and it is called here the ‘classical projection’.
The second approach deals with ‘large’ quantal (=quantum mechanical) systems
in the limit of an infinite number of degrees of freedom and with their corresponding ‘macroscopic limits’ described as classical Hamiltonian systems of the
system’s global (intensive) quantum observables.
The last part of this work contains a series of models describing interactions
of the “small” physical (micro) systems with the “macroscopic” ones, in which
these interactions lead to a (macroscopic) change of some “classical” parameters
of the large systems. These models connect, in a specific way, the two classes of the
systems considered earlier in this work by modeling their mutual interactions
leading to striking (i.e. theoretically impossible in the framework of finite quantum
systems) results.
The projective space PðHÞ of any complex Hilbert space H is endowed with a
natural symplectic structure, which allows us to rewrite the quantum mechanics of
systems with a finite number of degrees of freedom in terms of a classical
Hamiltonian dynamics. If a quantum-mechanical system is associated with a continuous unitary representation UðGÞ of a connected Lie group G on H, the orbits
(possibly factorized in a natural way) of the projected action of UðGÞ in PðHÞ are
naturally mapped onto orbits of the coadjoint representation Ad
Ã
ðGÞ of G. These
coadjoint orbits have a canonical symplectic structure which coincides with the one
induced from the structure of PðHÞ. For important classes of physical systems,
these symplectic spaces are either symplectomorphic to the ‘corresponding’ classical phase spaces, or they are some extensions of them (describing, e.g. particles
with ‘classical spin’). Quantal dynamics is projected onto these phase spaces in a
natural way, leading to classical Hamiltonian dynamical systems without any limit
of Planck constant h ! 0.
v
The work contains a description and an analysis of two different approaches
determining the connections between quantal and classical theories.
The first approach associates with any quantum-mechanical system with a finite
number of degrees of freedom a classical Hamiltonian system ‘living’ in projective
Hilbert space PðHÞ, and it is called here the ‘classical projection’.
The second approach deals with ‘large’ quantal (=quantum mechanical) systems
in the limit of an infinite number of degrees of freedom and with their corresponding ‘macroscopic limits’ described as classical Hamiltonian systems of the
system’s global (intensive) quantum observables.
The last part of this work contains a series of models describing interactions
of the “small” physical (micro) systems with the “macroscopic” ones, in which
these interactions lead to a (macroscopic) change of some “classical” parameters
of the large systems. These models connect, in a specific way, the two classes of the
systems considered earlier in this work by modeling their mutual interactions
leading to striking (i.e. theoretically impossible in the framework of finite quantum
systems) results.
The projective space PðHÞ of any complex Hilbert space H is endowed with a
natural symplectic structure, which allows us to rewrite the quantum mechanics of
systems with a finite number of degrees of freedom in terms of a classical
Hamiltonian dynamics. If a quantum-mechanical system is associated with a continuous unitary representation UðGÞ of a connected Lie group G on H, the orbits
(possibly factorized in a natural way) of the projected action of UðGÞ in PðHÞ are
naturally mapped onto orbits of the coadjoint representation Ad
Ã
ðGÞ of G. These
coadjoint orbits have a canonical symplectic structure which coincides with the one
induced from the structure of PðHÞ. For important classes of physical systems,
these symplectic spaces are either symplectomorphic to the ‘corresponding’ classical phase spaces, or they are some extensions of them (describing, e.g. particles
with ‘classical spin’). Quantal dynamics is projected onto these phase spaces in a
natural way, leading to classical Hamiltonian dynamical systems without any limit
of Planck constant h ! 0.
v