7.6 Radiating Finite Spin Chain
203
defined “macroscopic difference” between some of its states.
11 Of course, the infinite size of the previous models is a mathematical idealization, and there should be
some empirical possibility of distinction between “microscopic” and “macroscopic”,
resp. between “quantum” and “classical”, also in ‘large but finite systems’, as it is
perceived in our everyday life.
12
This distinction does not need to be, however, mathematically sharp. Such a possibility was sketched in [153]: In a verbal transcription it could be, perhaps, formulated
so that it would be very improbable to construct such an observation device on states
of large (however finite) system, which could “see” simultaneously sufficiently many
atoms of the system to be able to detect some interference phenomenon. This could
be considered as a rough ‘definition’ of the notion that some set of states of the
(now finite) apparatus consists of elements being pairwise ‘almost macroscopically
different’ (cf. also [153]).
13 To proceed in these considerations, one would need to
build some (more) general theory of observational devices. E.g., as far as the present
author knows, there were no published works paying attention to the fact that human
observers come into contact with measuring apparatuses by electromagnetic interactions, and probably only by them. Shortly, according to the point of view proposed
here: The formalized set of “observables” of any physical system should depend on
the existing possibilities of the construction of measuring devices in accordance with
physical laws and environmental conditions.
We have not stressed up to now, however, that the spin chain of our present model
is also coupled to a Fermi particle (resp. to the Fermi field) representing a sort of
‘environment’. The particle occurs in the initial state of the system in its vacuum
state, and afterwards it is radiated by the chain and subsequently escapes into infinity;
the state of the Fermi field containing the radiated particle is in each finite time
orthogonal to its vacuum state. This facilitates, in the intuitive sense of some sort of a
‘decoherence program’, cf. e.g. [124, 279, 343, 347], the possibility of interpretation
of the effective absence of interference between the initial and final states of the spin
chain in our model, as representing the two different ‘macroscopically’ distinguished
‘pointer positions’.
11 An exception consists in possible introduction ‘by hand’ by a theoretician some ‘superselection
rules’ representing a model of ‘macroscopic difference’ and forbidding interference between vectors
from specific subspaces of H v , cf. e.g. [167].
12 Another possibility is some, up to now not clearly specified basic change of QM, as it was most
urgently proposed by Penrose in several his publications, e.g. in [236–238]; the main motivation
for these reformulations of QM was some inclusion of the usually postulated “reduction of wave
packet” [226], called by Penrose the “process R”, into the dynamics of general QM systems.
13 Let us illustrate briefly this idea on a long but finite spin-1/2 chain of the length N with the
C ∗ -algebra A of its observables generated by the spin creation-annihilation operators a j , a ∗
j ( j =
1, 2, . . . N ) acting on the finite dimensional Hilbert space H N := (C 2 ) N : If we are able to use
apparatuses detecting the observables of this chain occurring in an arbitrary of the C ∗ -subalgebras
B ⊂ A generated by any of the fixed restricted set of operators a jm , a ∗
jm (m = 1, 2, . . . K N , 0 ≤
j m ≤ N ) only, then the states |, | from H N for which it holds |B| ≡ 0 ∀B ∈ B could be
considered as ‘almost macroscopically different’, resp. ‘empirically disjoint’. This happens, e.g., if
in the state | all the spins are ‘pointing up’, and in the state | all the spins are ‘pointing down’.
203
defined “macroscopic difference” between some of its states.
11 Of course, the infinite size of the previous models is a mathematical idealization, and there should be
some empirical possibility of distinction between “microscopic” and “macroscopic”,
resp. between “quantum” and “classical”, also in ‘large but finite systems’, as it is
perceived in our everyday life.
12
This distinction does not need to be, however, mathematically sharp. Such a possibility was sketched in [153]: In a verbal transcription it could be, perhaps, formulated
so that it would be very improbable to construct such an observation device on states
of large (however finite) system, which could “see” simultaneously sufficiently many
atoms of the system to be able to detect some interference phenomenon. This could
be considered as a rough ‘definition’ of the notion that some set of states of the
(now finite) apparatus consists of elements being pairwise ‘almost macroscopically
different’ (cf. also [153]).
13 To proceed in these considerations, one would need to
build some (more) general theory of observational devices. E.g., as far as the present
author knows, there were no published works paying attention to the fact that human
observers come into contact with measuring apparatuses by electromagnetic interactions, and probably only by them. Shortly, according to the point of view proposed
here: The formalized set of “observables” of any physical system should depend on
the existing possibilities of the construction of measuring devices in accordance with
physical laws and environmental conditions.
We have not stressed up to now, however, that the spin chain of our present model
is also coupled to a Fermi particle (resp. to the Fermi field) representing a sort of
‘environment’. The particle occurs in the initial state of the system in its vacuum
state, and afterwards it is radiated by the chain and subsequently escapes into infinity;
the state of the Fermi field containing the radiated particle is in each finite time
orthogonal to its vacuum state. This facilitates, in the intuitive sense of some sort of a
‘decoherence program’, cf. e.g. [124, 279, 343, 347], the possibility of interpretation
of the effective absence of interference between the initial and final states of the spin
chain in our model, as representing the two different ‘macroscopically’ distinguished
‘pointer positions’.
11 An exception consists in possible introduction ‘by hand’ by a theoretician some ‘superselection
rules’ representing a model of ‘macroscopic difference’ and forbidding interference between vectors
from specific subspaces of H v , cf. e.g. [167].
12 Another possibility is some, up to now not clearly specified basic change of QM, as it was most
urgently proposed by Penrose in several his publications, e.g. in [236–238]; the main motivation
for these reformulations of QM was some inclusion of the usually postulated “reduction of wave
packet” [226], called by Penrose the “process R”, into the dynamics of general QM systems.
13 Let us illustrate briefly this idea on a long but finite spin-1/2 chain of the length N with the
C ∗ -algebra A of its observables generated by the spin creation-annihilation operators a j , a ∗
j ( j =
1, 2, . . . N ) acting on the finite dimensional Hilbert space H N := (C 2 ) N : If we are able to use
apparatuses detecting the observables of this chain occurring in an arbitrary of the C ∗ -subalgebras
B ⊂ A generated by any of the fixed restricted set of operators a jm , a ∗
jm (m = 1, 2, . . . K N , 0 ≤
j m ≤ N ) only, then the states |, | from H N for which it holds |B| ≡ 0 ∀B ∈ B could be
considered as ‘almost macroscopically different’, resp. ‘empirically disjoint’. This happens, e.g., if
in the state | all the spins are ‘pointing up’, and in the state | all the spins are ‘pointing down’.