Decoherence, wave function collapses and non-ordinary statistical mechanics Page: 1
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Decoherence, wave function collapses and non-ordinary statistical mechanics
Mlauro Bologna', Paolo Grigolini'2" , Marco G. Pala, Luigi Palatella2
Center for Nonlinear Scienrce,. U niversitq of Nor'th Teas. P.O. Bor 311427, Denton, Te::as 76203-1427
2Dipartimento di Fisica dell'Universita di Pisa and INF A, via Baonarroti 2. 56126 Pisa. Italy
'lstituto di Biofisica CNR, Area della Ricerca di Pisa. Via Alfieri 1, San Cataldo 56010 Ghezza'no-Pisa, Italy
4 Dipartirrenrto di I ngegrnerriao dcll 'InforrrazionrC dell' Univer'sit di Pisa. via Diotisalvi 2. 56122 Pisao. Italy
(February 1, 2008)
We consider a toy model of pointer interacting with a
1/2-spin system, whose cr, variable is measured Iby the envi-
ronment, according to the prescription of decoherence theory.
If the environment measuring the variable a, fields ordinary
statistical mechanics, the pointer sensitive to the 1/2-spin sys-
tem undergoes the same. exponential, relaxation regardless of
whether real collapses or an entanglement with the environ-
ment, mnimicking the effect of real collapses, occur. In the
case of non-ordinary statistical mechanics the occurrence of
real collapses make the pointer still relax exponentially in
time, while the equivalent picture in terms of reduced (density
matrix generates an inverse power law relaxation.
Pacs: 03.65.Ta, 03.67.-a. 05.20.-y, 05.30.-d
Decohiere ice theory was bIornI in 197() with the seminal
work of Zehll  and grew iup with the work of Zurek and
others over the following decades. It is now regarded to
be a theory so robust as to make Tegmark and Wheeler
 claim that it renders obsolete the hypothesis of wave
function collapses made by the founding fathers of quan-
tuini mechanics. The main purpose of this paper is to
prove that this claim is correct only in the case when
decoherence is caused by interactions compatible with
ordinary statistical mechanics. If we move from the con-
dition of exponential relaxation, which is the key prop-
erty of ordinary statistical mechanics, to the condition of
inverse power law relaxation, the statistical equivalence
between wave-function collapses and decoherence is lost.
To show this basic property. let us consider the following
Hr, = G(:,: + H,1 (1)
H (/Io + qo:,r:7q + HB. (2)
WN e have a 1/2-spin system, characterized by the Pauli
matrix E, called pointer. interacting with a 1/2-spin sys-
tem. characterized by the Pauli matrix a, and called sys-
terrn of interest. T'he system of interest undergoes ain in-
teraction with a bath, through a variable 1r. dlriveni by the
Hamiltonlian HB. The density matrices of the pointer
and of the system of interest are called pz and pj, re-
spectively. The former is obtained from a contraction
over the degrees of freedom of the system of interest. andl
of its bath as well. The latter requires a contractionl over
the pointer degrees of freedom as well as on the bath of
the systein of interest. This bath is assumed to be much
faster than the pointer and, as a consequence, the time
evolution of o., is virtually independent of the pointer
First of all, we show that in the special case where the
correlation function of the fluctuation q is exponential.
the two pictures, wave-function collapses and decoher-
ence, yield the same statistical result. This supports the
point of view of the advocates of decoherence. Then,
we create a condition of anomalous statistical melechanl-
ics, by modlllatinig the Hamiltonian H, in such a way
as to create a significant departure from ordinary expo-
nential relaxation. In this case, we show that the two
perspectives yield quite different results. In a sense, the
decohereice theory is not contradicted. in so far as the
pointer density matrix become's diagonal in the basis set
of the pointer eigenstates. However, this happens via ci-
ther an exponential relaxation, if the system of interest
undergoes real collapses. or through an inverse power law
decay, if no real collapses occur.
The case of wave-function collapse in the Markoviain
case was already discussed in an earlier publication ,
where it was proved that the bath of the system of in-
terest. making measurements with frequency 1/T, where
T (1/2) [h2/(g2(q2 )rI)]  and T is the correlation time
of 1. resuLlts ini a seqeinice of symbols such as +.+.+. -
.-,...This is so because the observable a,: lhas the eigen-
states I+):, and I):,:. with the eigenvalues 1 and -1 re-
spectively. The inphasing term -Val/ forces the sys-
temn of interest to stay in a superposition of both states.
with the form
(t)) = -+), cost + ) -i ), sin(t + Q). (3)
If the first rmeasurerlment is done at t = 0, and the sys-
tern of interest Inakes an instantaneous collapse into +) :,
for instance, then the subsequent time evolution of the
system of interest is given by Eq.(3) with = 0. We
set the condition 7 <i 1/w (overdanlped condition). with
V / hi. This means that the next rneasurenment.
occurring at t = , will probably nmake the systeln col-
lapse into )+}:,, again. In fact, the overdamnped condition
makes it possible for us to evaluate the probability of col-
lapse into I-), by Taylor series expansion of sin(wt), and
this yields, for the probability of the system to collapse
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Bologna, Mauro; Grigolini, Paolo; Pala, Marco G. & Palatella, Luigi. Decoherence, wave function collapses and non-ordinary statistical mechanics, article, August 2003; [New York, New York]. (digital.library.unt.edu/ark:/67531/metadc174684/m1/1/: accessed October 20, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.