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correlations. The tridecompositional uniqueness theorem nate property of our experience can then be explained
then guarantees the uniqueness of the expansion of the by referring to the dependence of most interactions on
final state |È = cn|sn |an |en (where no constraints distance (Zurek, 1981, 1982, 1991).
n
on the cn have to be imposed) and thereby the uniqueness
This holds in particular for mesoscopic and macro-
of the preferred pointer basis.
scopic systems, as demonstrated for instance by the pi-
Besides the commutativity requirement, Eq. (3.21),
oneering study by Joos and Zeh (1985) where surround-
other (yet similar) criteria have been suggested for the
ing photons and air molecules are shown to continuously
selection of the preferred pointer basis because it turns
measure the spatial structure of dust particles, leading
out that in realistic cases the simple relation of Eq. (3.21)
to rapid decoherence into an apparent (i.e., improper)
can usually only be fulfilled approximately (Zurek, 1993;
mixture of wavepackets that are sharply peaked in po-
sition space. Similar results sometimes even hold for
microscopic systems (that are usually found in energy
eigenstates, see below) when they occur in distinct spa-
9
For fundamental limitations on the precision of von Neumann
tial structures that couple strongly to the surrounding
measurements of operators that do not commute with a glob-
medium. For instance, chiral molecules such as sugar
ally conserved quantity, see the Wigner Araki Yanase theorem
are always observed to be in chirality eigenstates (left-
(Araki and Yanase, 1960; Wigner, 1952).
10
handed and right-handed) which are superpositions of
For simplicity, we assume here that the environment E interacts
directly only with the apparatus A, but not with the system S. different energy eigenstates (Harris and Stodolsky, 1981;
14
Zeh, 1999a). This is explained by the fact that the spatial that are energy eigenstates of HS (Paz and Zurek,
structure of these molecules is continuously monitored 1999).
by the environment, for example, through the scattering
of air molecules, which gives rise to a much stronger cou- 3. In the intermediate case, when the evolution of the
pling than could typically be achieved by a measuring
system is governed by HSE and HS in roughly equal
device that was intended to measure, e.g., parity or en-
strength, the resulting preferred states will repre-
ergy; furthermore, any attempt to prepare such molecules
sent a compromise between the first two cases;
in energy eigenstates would lead to immediate decoher-
for instance, the frequently studied model of quan-
ence into environmentally stable ( dynamically robust )
tum Brownian motion has shown the emergence
chirality eigenstates, thus selecting position as the pre-
of pointer states localized in phase space, i.e., in
ferred basis.
both position and momentum, for such a situation
On the other hand, it is well-known that many systems, (Eisert, 2004; Joos et al., 2003; Unruh and Zurek,
especially in the microsopic domain, are typically found 1989; Zurek, 2003a; Zurek et al., 1993).
in energy eigenstates, even if the interaction Hamilto-
nian depends on a different observable than energy, e.g.,
position. Paz and Zurek (1999) have shown that this sit-
3. Implications for the preferred basis problem
uation arises when the frequencies dominantly present in
the environment are significantly lower than the intrinsic
The idea of the decoherence program that the pre-
frequencies of the system, that is, when the separation
ferred basis is selected by the requirement that corre-
between the energy states of the system is greater than
lations must be preserved in spite of the interaction with
the largest energies available in the environment. Then,
the environment, and thus chosen through the form of
the environment will be only able to monitor quantities
the system environment interaction Hamiltonian, seems
that are constants of motion. In the case of nondegener-
certainly reasonable, since only such robust states will
acy, this will be energy, thus leading to the environment-
in general be observable and after all we solely demand
induced superselection of energy eigenstates for the sys-
an explanation for our experience (see the discussion in
tem.
Sec. II.B.3). Although only particular examples have
Another example for environment-induced superselec-
been studied (for a survey and references, see for example
tion that has been studied is related to the fact that only
Blanchard et al., 2000; Joos et al., 2003; Zurek, 2003a),
eigenstates of the charge operator are observed, but never
the results thus far suggest that the selected properties
superpositions of different charges. The existence of the
are in agreement with our observation: for mesoscopic
corresponding superselection rules was first only postu-
and macroscopic objects the distance-dependent scatter-
lated (Wick et al., 1952, 1970), but could subsequently
ing interaction with surrounding air molecules, photons,
be explained in the framework of decoherence by refer-
etc., will in general give rise to immediate decoherence
ring to the interaction of the charge with its own Coulomb
into spatially localized wave packets and thus select po-
(far) field which takes the rôle of an environment , lead-
sition as the preferred basis; on the other hand, when
ing to immediate decoherence of charge superpositions
the environment is comparably slow , as it is frequently
into an apparent mixture of charge eigenstates (Giulini,
the case for microsopic systems, environment-induced su-
2000; Giulini et al., 1995).
perselection will typically yield energy eigenstates as the
In general, three different cases have typically been
preferred states.
distinguished (for example, in Paz and Zurek, 1999) for
The clear merit of the approach of environment-
the kind of pointer observable emerging from the inter-
induced superselection lies in the fact that the preferred
action with the environment, depending on the relative
basis is not chosen in an ad hoc manner as to simply
strengths of the system s self-Hamiltonian HS and of the
make our measurement records determinate or as to
system environment interaction Hamiltonian HSE:
plainly match our experience of which physical quanti-
ties are usually perceived as determinate (for example,
1. When the dynamics of the system are dominated
position). Instead the selection is motivated on physi-
by HSE, i.e., the interaction with the environment,
cal, observer-free grounds, namely, through the system
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