6
1 Introduction
‘apparatus’ radiates a Fermi particle escaping to infinity and, contrary to the other
above mentioned models of Chap. 7, it converges very quickly to the final ‘almost
macroscopic’ state.
1.1.10 Bibliographical notes.
The canonical symplectic structure (in the case of finite dimensions) on complex
projective Hilbert spaces is described in [7]; in context of QM it appeared, e.g. in [17,
37, 69, 268]. Orbits of U (G) in the Hilbert space were introduced in the special case
of Heisenberg group G in [125], and in general case in [176, 239] under the name
‘(generalized) coherent states’. John Klauder obtained CM on such orbits (or even
on more general submanifolds of Hilbert space) from the quantal Hamilton principle
restricted to corresponding orbits (resp. to ‘overcomplete sets of unit vectors’), see
[176]. The orbits G. in P(H), and the functions ν → f A (ν) := T r(ν A) (ν ∈ G.),
named (in the case of one-dimensional ) ‘covariant symbols’ by Berezin [18] or
‘lower symbols’ by Simon [291], were used for determination of bounds for quantum
partition functions (see [199, 291]), in time dependent Hartree-Fock theory [268],
and also for description of specific types of unitary representations of Galilean and
Poincaré groups [2]. Some essential properties of generalized coherent states are
described in [84]. The natural symplectic orbits of coadjoint representations was
introduced in [174].
A further development of these (mathematical, as well as physical) ideas is also
contained in the work [37, 47], which contains also a nonlinear extension of the
formulation of QM. This nonlinear extension is also compared in [37, Sect. 3.6] with
the Weinberg attempt [328] to formulate a nonlinear version of QM.
Some of the main ideas on connections between QM and CM leading to the
present work are implicitly contained already in the classical work [330]. The idea
and techniques used for transition to infinite systems was gained mainly from works
by Haag, Hepp, Lieb, Neumann, Ruelle and others (see e.g. [139, 155, 227, 271], and
for a review compare [53, 54, 106]). A transition to macroscopic limit (‘statistical
quasiclassics’) is described in [17] for a specific choice of the group G and a meanfield type interaction. A review of works on macroscopic limits (‘large N limits’) is
given in [342]. An attempt of the description of classical quantities of large quantal
systems analogous to the here presented one is described in works by Rieckers with
collaborators [101, 265], and by Morchio with Strocchi [221, 222]; see also the
works [317–320] by Thomas Unnerstall. A preliminary outline of a part of this work
is contained in [32], and also in [40, 41]. The necessary mathematics can be found
in the cited monographs, cf. also Appendices in [37].
An alternative way of description of thermodynamics and dynamics of quantum
mean-field systems was later proposed in the work of the group around R.F.Werner,
see e.g. [100].
A new approach to the theoretical description of classical (macroscopic) systems
in the framework of quantum theory in a unique mathematical formalism is presented
in a series of papers by Jean-Bernard Bru and collaborators [60].
1 Introduction
‘apparatus’ radiates a Fermi particle escaping to infinity and, contrary to the other
above mentioned models of Chap. 7, it converges very quickly to the final ‘almost
macroscopic’ state.
1.1.10 Bibliographical notes.
The canonical symplectic structure (in the case of finite dimensions) on complex
projective Hilbert spaces is described in [7]; in context of QM it appeared, e.g. in [17,
37, 69, 268]. Orbits of U (G) in the Hilbert space were introduced in the special case
of Heisenberg group G in [125], and in general case in [176, 239] under the name
‘(generalized) coherent states’. John Klauder obtained CM on such orbits (or even
on more general submanifolds of Hilbert space) from the quantal Hamilton principle
restricted to corresponding orbits (resp. to ‘overcomplete sets of unit vectors’), see
[176]. The orbits G. in P(H), and the functions ν → f A (ν) := T r(ν A) (ν ∈ G.),
named (in the case of one-dimensional ) ‘covariant symbols’ by Berezin [18] or
‘lower symbols’ by Simon [291], were used for determination of bounds for quantum
partition functions (see [199, 291]), in time dependent Hartree-Fock theory [268],
and also for description of specific types of unitary representations of Galilean and
Poincaré groups [2]. Some essential properties of generalized coherent states are
described in [84]. The natural symplectic orbits of coadjoint representations was
introduced in [174].
A further development of these (mathematical, as well as physical) ideas is also
contained in the work [37, 47], which contains also a nonlinear extension of the
formulation of QM. This nonlinear extension is also compared in [37, Sect. 3.6] with
the Weinberg attempt [328] to formulate a nonlinear version of QM.
Some of the main ideas on connections between QM and CM leading to the
present work are implicitly contained already in the classical work [330]. The idea
and techniques used for transition to infinite systems was gained mainly from works
by Haag, Hepp, Lieb, Neumann, Ruelle and others (see e.g. [139, 155, 227, 271], and
for a review compare [53, 54, 106]). A transition to macroscopic limit (‘statistical
quasiclassics’) is described in [17] for a specific choice of the group G and a meanfield type interaction. A review of works on macroscopic limits (‘large N limits’) is
given in [342]. An attempt of the description of classical quantities of large quantal
systems analogous to the here presented one is described in works by Rieckers with
collaborators [101, 265], and by Morchio with Strocchi [221, 222]; see also the
works [317–320] by Thomas Unnerstall. A preliminary outline of a part of this work
is contained in [32], and also in [40, 41]. The necessary mathematics can be found
in the cited monographs, cf. also Appendices in [37].
An alternative way of description of thermodynamics and dynamics of quantum
mean-field systems was later proposed in the work of the group around R.F.Werner,
see e.g. [100].
A new approach to the theoretical description of classical (macroscopic) systems
in the framework of quantum theory in a unique mathematical formalism is presented
in a series of papers by Jean-Bernard Bru and collaborators [60].