1.1 Motivation and Summary
3
by density matrices (called there “elementary mixtures”), and others are called “genuine mixtures” (or. also “proper mixtures”)—these correspond to the states which
arose by a real ‘mixing’ of different quantal states, as it appears in classical statistical
mechanics in ensembles of systems occurring in different states—different points of
the phase space of the described system; they are introduced in [37, Sect. 2.1-e] and
difference of these two kinds of mixtures is illustrated e.g. in [37, Sect. 3.3-e].
In the remaining sections of this introductory chapter it is specified briefly what
we mean here by QM, CM and by the ‘quantum theory of large systems’. The second
chapter is devoted to a detailed study of geometry of the projective Hilbert space
P(H), where H is the Hilbert space used in description of a quantal system. We
emphasize there the natural symplectic structure on P(H), cf. e.g. [7, 37, 214, 231].
This structure is used in Sect. 2.3 to description of QM in terms of infinite dimensional
CM, i.e. of classical Hamiltonian field theory with, however, the standard quantum
statistical interpretation.
1.1.5 The Chap. 3 “Classical Mechanical Projections” is devoted to a general construction of Hamiltonian CM from a given quantal system (provided that an interpretation of its ‘basic quantities’ is specified by a unitary representation U (G) of
a Lie group G; for Lie groups see e.g. [13, 50, 209, 247]). The scheme of this
construction is very simple: Take the orbit O := G. through a point ∈ P(H)
of the action of U(G) on P(H) corresponding to the action of U (G) on H and
restrict the natural symplectic form on P(H) onto O . For properly chosen the
orbit O is an immersed (and regularly embedded, cf. [37, Proposition 2.1.5(iv)]
completed by [47]) submanifold of P(H) ⊂ T(H) (cf. 1.2.3), hence the restriction
is well defined. The obtained two-form on the manifold O might be degenerate, but
after a natural factorization of the orbit we obtain a symplectic manifold which is
symplectomorphic to an orbit of the coadjoint representation Ad
∗
(G). Symplectic
manifolds obtained in this way are interpreted as classical phase spaces. In some
cases, if the generator of time evolution (the Hamiltonian operator) belongs to the
generators of U (G), we can obtain from the symplectic structure of P(H) a contact
structure on O which reproduces an ‘extended phase space’ (odd dimensional) of
classical mechanics. If the Hamiltonian is not a generator of U (G) (i.e. if G is only a
‘kinematical group’ without representing any time evolution), the quantal dynamics
might be in some cases naturally projected onto the obtained classical phase space as
a globally Hamiltonian complete vector field; this situation is analyzed in Sect. 3.3.
Although such a construction of CM from QM is equally applicable to any quantal
system (specified by some U (G)), the interpretation of the obtained classical system
depends on the specific physical system, and also on the physical quantal state
from which the orbit O is constructed. In any case, it is obtained a formal procedure
for construction of ‘classical projections’ from arbitrary (finite) quantal systems.
Chapter 4 provides some simple examples of this formal procedure. In the Subsect.
4.1.6, we obtain from a simple nonrelativistic quantal system with the potential energy
V the corresponding classical system (in the conventional sense) with a modified
potential energy, where the modification depends on the choice of the ‘initial state’
3
by density matrices (called there “elementary mixtures”), and others are called “genuine mixtures” (or. also “proper mixtures”)—these correspond to the states which
arose by a real ‘mixing’ of different quantal states, as it appears in classical statistical
mechanics in ensembles of systems occurring in different states—different points of
the phase space of the described system; they are introduced in [37, Sect. 2.1-e] and
difference of these two kinds of mixtures is illustrated e.g. in [37, Sect. 3.3-e].
In the remaining sections of this introductory chapter it is specified briefly what
we mean here by QM, CM and by the ‘quantum theory of large systems’. The second
chapter is devoted to a detailed study of geometry of the projective Hilbert space
P(H), where H is the Hilbert space used in description of a quantal system. We
emphasize there the natural symplectic structure on P(H), cf. e.g. [7, 37, 214, 231].
This structure is used in Sect. 2.3 to description of QM in terms of infinite dimensional
CM, i.e. of classical Hamiltonian field theory with, however, the standard quantum
statistical interpretation.
1.1.5 The Chap. 3 “Classical Mechanical Projections” is devoted to a general construction of Hamiltonian CM from a given quantal system (provided that an interpretation of its ‘basic quantities’ is specified by a unitary representation U (G) of
a Lie group G; for Lie groups see e.g. [13, 50, 209, 247]). The scheme of this
construction is very simple: Take the orbit O := G. through a point ∈ P(H)
of the action of U(G) on P(H) corresponding to the action of U (G) on H and
restrict the natural symplectic form on P(H) onto O . For properly chosen the
orbit O is an immersed (and regularly embedded, cf. [37, Proposition 2.1.5(iv)]
completed by [47]) submanifold of P(H) ⊂ T(H) (cf. 1.2.3), hence the restriction
is well defined. The obtained two-form on the manifold O might be degenerate, but
after a natural factorization of the orbit we obtain a symplectic manifold which is
symplectomorphic to an orbit of the coadjoint representation Ad
∗
(G). Symplectic
manifolds obtained in this way are interpreted as classical phase spaces. In some
cases, if the generator of time evolution (the Hamiltonian operator) belongs to the
generators of U (G), we can obtain from the symplectic structure of P(H) a contact
structure on O which reproduces an ‘extended phase space’ (odd dimensional) of
classical mechanics. If the Hamiltonian is not a generator of U (G) (i.e. if G is only a
‘kinematical group’ without representing any time evolution), the quantal dynamics
might be in some cases naturally projected onto the obtained classical phase space as
a globally Hamiltonian complete vector field; this situation is analyzed in Sect. 3.3.
Although such a construction of CM from QM is equally applicable to any quantal
system (specified by some U (G)), the interpretation of the obtained classical system
depends on the specific physical system, and also on the physical quantal state
from which the orbit O is constructed. In any case, it is obtained a formal procedure
for construction of ‘classical projections’ from arbitrary (finite) quantal systems.
Chapter 4 provides some simple examples of this formal procedure. In the Subsect.
4.1.6, we obtain from a simple nonrelativistic quantal system with the potential energy
V the corresponding classical system (in the conventional sense) with a modified
potential energy, where the modification depends on the choice of the ‘initial state’