Chapter 6
Dynamics of Quantum Mechanical
Macroscopic Systems
6.1 General Considerations
6.1.1 The formalism developed in Chap. 5 will be used in this chapter for a determination of a microscopic time evolution of an infinite quantum system from the
macroscopic (classical) evolution. It is clear that such an unusual determination of
microscopic dynamics is possible for a very special type of interactions only. We shall
show that this is the case of a wide class of quantum mean-field theories,
1 at least
in the time invariant subset S
g of the set S(A
) of all the microscopic states on the
quasilocal algebra A
, cf. Sect. 5.1, esp. 5.1.32; cf. also ‘classical states’ in [155].
The systems of the considered type are determined by the couple (A; σ G ) consisting
of a C
∗ -algebra A (:= A
, e.g.; the upper indices will be usually omitted in this
chapter) and of a representation σ(G) := σ G ⊂
∗ - Aut A, cf. 5.2.2, as well as by a
G-measure E g , 5.2.3, and by a classical Hamiltonian function Q ∈ C
∞
(g
∗
, R). A
subclass of these systems consists of thermodynamic limits N → ∞ of systems of the
total number N of quantal (mutually equal) subsystems with dynamics described by
local Hamiltonians Q
N . These local Hamiltonians are invariant with respect to any
permutations of N subsystems and the k-body interaction constants (i.e. coefficients
at products of k operators corresponding to k different subsystems) are proportional to
N
1−k . We can construct such a sequence of the ‘local time evolutions’ τ
N
⊂
∗ - Aut A
in the following way:
Let us keep the notation of Sect. 5.1, and let a basis ξ j ( j = 1, . . . n) of g be
fixed, the dual basis being { f j : j = 1, 2, . . . n} ⊂ g
∗ . Let X j ( j = 1, 2, . . . n) be
the selfadjoint generators of the one parameter unitary groups t → U (exp(tξ j ))
on H, 5.1.3. Let Q be a polynomial in n variables and with a prescribed order of
1 For some history, general meaning and technical construction of dynamics (given by full and
correctly solved microscopic evolutions—without any approximations) of “Quantum mean-field
theories” see also [40], and for some of its applications look in [41].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_6
113
Dynamics of Quantum Mechanical
Macroscopic Systems
6.1 General Considerations
6.1.1 The formalism developed in Chap. 5 will be used in this chapter for a determination of a microscopic time evolution of an infinite quantum system from the
macroscopic (classical) evolution. It is clear that such an unusual determination of
microscopic dynamics is possible for a very special type of interactions only. We shall
show that this is the case of a wide class of quantum mean-field theories,
1 at least
in the time invariant subset S
g of the set S(A
) of all the microscopic states on the
quasilocal algebra A
, cf. Sect. 5.1, esp. 5.1.32; cf. also ‘classical states’ in [155].
The systems of the considered type are determined by the couple (A; σ G ) consisting
of a C
∗ -algebra A (:= A
, e.g.; the upper indices will be usually omitted in this
chapter) and of a representation σ(G) := σ G ⊂
∗ - Aut A, cf. 5.2.2, as well as by a
G-measure E g , 5.2.3, and by a classical Hamiltonian function Q ∈ C
∞
(g
∗
, R). A
subclass of these systems consists of thermodynamic limits N → ∞ of systems of the
total number N of quantal (mutually equal) subsystems with dynamics described by
local Hamiltonians Q
N . These local Hamiltonians are invariant with respect to any
permutations of N subsystems and the k-body interaction constants (i.e. coefficients
at products of k operators corresponding to k different subsystems) are proportional to
N
1−k . We can construct such a sequence of the ‘local time evolutions’ τ
N
⊂
∗ - Aut A
in the following way:
Let us keep the notation of Sect. 5.1, and let a basis ξ j ( j = 1, . . . n) of g be
fixed, the dual basis being { f j : j = 1, 2, . . . n} ⊂ g
∗ . Let X j ( j = 1, 2, . . . n) be
the selfadjoint generators of the one parameter unitary groups t → U (exp(tξ j ))
on H, 5.1.3. Let Q be a polynomial in n variables and with a prescribed order of
1 For some history, general meaning and technical construction of dynamics (given by full and
correctly solved microscopic evolutions—without any approximations) of “Quantum mean-field
theories” see also [40], and for some of its applications look in [41].
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_6
113