Decoherence, wave function collapses and non-ordinary statistical mechanics Page: 4
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Using the Tauberian theorem [10], we arrive at the main
conclusion that in the time asymptotic limit (t - oo)
R(t) oa (32)
What about the case when wave-function collapses oc-
cur? If they do occur, the important property to evaluate
becomes the waiting time distribution ~,(t). According
to the modulation prescription, this important function
turns out to be given by
(t) = dy _ 17-2 exp(-7T)7exp(-7t). (33)
Notice that it is straightforward to prove that this waiting
time distribution reads
r (t)= (p -1)(t+ ' (34)
which is related to the correlation function of Eq.(25) by
the renewal theory prescription of Eq.(8). This means
that the correlation function J4(t) of the fluctuations f(t)
created by the wave-function collapses is identical to the
quantum mechanical correlation, evaluated without us-
ing the collapse assumption. This is apparently a very
reassuring condition for the advocates of the decoherence
theory. In fact, it generates the impression that even in
this case the direct use of the wave-function collapses is
unnecessary.
It is not so. The fact that the existence of trajectories
enforces the use of (t) rather than that of (, (t) = (I(t),
has impressive consequences, in this case. The important
relation of Eq.(14), if wave-function collapses occur, be-
comes identical to the characteristic function (exp(ikx))t
of the corresponding diffusion process, determined by the
the waiting time distribution 4(t). In this case, the wait-
ing time distribution has the form of Eq. (34). We re-
mind the reader that we are referring ourselves to the case
where the condition 2 < < 3 applies. This means that
the second moment of the waiting time distribution 0(t)
is divergent. We cannot use the ordinary central limit
theorem to evaluate this characteristic function. We can,
however, use the generalized central limit theorem [11]
and, following Ref. [12-14], we obtainR(t) = exp(-blK -lt),
(35)
where K = 2G/ and b = T -2 sin[wr(u - 2)/2]F(3 - ,u).
In conclusion, in the case of anomalous statistical me-
chanics, the pointer relaxation turns out to be exponen-
tial or an inverse power law, according to whether wave
-function collapses do or do not occur.
This is a striking result since it implies that a real ex-
periment might make it possible to assess if wave-function
collapses occur or not. If the experiment assessed thepower law nature of the pointer relaxation, it would con-
firm the validity of quantum mechanics in a condition
where the equivalence between decoherence theory and
wave-function collapses is broken. If, on the contrary,
the experiment yielded for the pointer an experimental
relaxation, this would support the existence of real wave-
function collapses, and these wave-function collapses, in
turn, could not be entirely attributed to an environmen-
tal effect. This is an issue of fundamental importance
that might be resolved by means of real experiments,
since there are currently realistic projects, for instance
the cantilever experiment of Ref. [15] with pointers (the
cantilever) sensitive to the fluctuations of also a single
spin.
Financial support from ARO, through Grant
DAAD19-02-0037, is gratefully acknowledged.[1] H.D. Zeh, Foundation of Physics 1, 69 (1970).
[2] M. Tegmark, J.A. Wheeler, Scientific American, Feb 2001
issue, p. 68-75.
[3] P. Grigolini, M.G. Pala, L. Palatella, Phys. Lett. A 285,
49 (2001).
[4] As in [3], the heuristic arguments here adopted are based
on the assumption that the time distance between one
measurement and the next is fixed, with no consequence
on the theoretical prediction.
[5] G. Lindblad, Commun. Math. Phys. 48, 119 (1976).
[6] T. Geisel, J. Nierwetberg, and A. Zacherl, Phys. Rev.
Lett. 54, 616 (1985).
[7] G. Zumofen, J. Klafter, Phys. Rev. E 47, 851 (1993).
[8] C. Beck, Phys.Rev. Lett. 87, 180601 (2001).
[9] M. Bologna, P. Grigolini, J. Riccardi, Phys. Rev. 60,
6435 (1999).
[10] G.H. Weiss, Aspects and Applications of the Random
Walk, North-Holland, Amsterdam (1994), p. 56.
[11] B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions
for Sums of Independent Random Variables , Addison-
Wesley (1954).
[12] M. Annnunziato, P. Grigolini, Phys. Lett. A 269. 31
(2000).
[13] P. Allegrini, P. Grigolini, B.J. West, Phys. Rev. E 54,
4760 (1996).
[14] Note that the authors of Ref. [13] proved that a slowly
decaying oscillatory contribution to the pointer relax-
ation exists, due to the important fact that the diffusion
process is characterized by a finite propagation front.
However, this term becomes irrelevant with increasing
the time scale separation between system of interest and
pointer.
[15] G.P. Bermann, D.T. Kamenev, and V.I. Tsirinovich,
arXiv: quant-ph/0203013.
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Bologna, Mauro; Grigolini, Paolo; Pala, Marco G. & Palatella, Luigi. Decoherence, wave function collapses and non-ordinary statistical mechanics, article, February 1, 2008; [New York, New York]. (https://digital.library.unt.edu/ark:/67531/metadc174684/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.