Wave-particle duality ; wave packet reduction by the
hamiltonian
• One of the delicate points of the quantum mechanics is
the projection of a state vector on an eigenvector corresponding to a
measurement apparatus. The paradoxes of the quantum
mechanics are generally directly connected to this.
In this connection, it is interesing to consider the
“direction change” of the time course, or more
precisely (this is only indirectly connected to the mode of variation
of this amplitude through C.P.T. transformation) : the absence of
variation for a transition amplitude like 
with respect to
the two projections, direct 
, or inverse 
.
One may indeed calculate the probability of a passed situation while
supposing the future as known (with use if required of the advanced
potentials in place of the delayed potentials) and the behaviour of the
spin allows to underline a particularity connected to this time
“reversal”.
Let us thus consider two polarizers intercalated between a
source and a detector, and oriented at 45° with respect to one
another. A photon which has passed the first polarizer and goes towards
the second is polarized according to the orientation of the first ; its
polarization changes therefore if it goes through the second.
But the interaction can be described in opposite direction by a photon
which, within the interval between the two polarizers, is
polarized according to the orientation of the second and not the one of
the first. One may thus think that the photon is perhaps actually
polarized neither according to one, nor according to the other of the
orientations, but owns some spin properties (possibly partly hidden)
which, through the wave, cause its global and symmetrical interaction
with the two polarizers. Said differently, some of the properties that
we attribute to the
particles are possibly some global properties of the interactions of
these particules with the measurement devices.
This corresponds to a relative separation of
the notions of wave and corpuscle, and one may then suppose that
part of the effects that we observe is connected to the
symmetries of the interactions which are carried by the wave
(and the particles might be statistically guided by the wave according
to hidden variables). With respect to the
relativistic instantaneousness, nothing thus forbids that the
measurement devices may be able to modify the meausurement results
through the effects of
non-locality [1].
• In these conditions, what must be thought of the relativistic
non-invariance of the principle of wave packet reduction (so much
stranger as the predictions derived from it
do possess this invariance) ? As a matter of fact, if the hamiltonian
contains an adequate term of interaction, the wave packet is
automatically (and obligatorily) reduced during the interaction,
without any necessity of intervention for something else than the
equation of field evolution (and every
modification of the apparatus that changes the hamiltonian induces a
different reduction of the wave packet).
So may be explained the paradox of Schrödinger's cat [2] : there
use is made of the wave function of a macroscopic object which
reinteractions are
impossible to forbid. There is thus always
reduction of the wave packet. So beeing, the structure of
Schrödinger's equation leads to an instantaneous
reduction, but a
relativistic invariant reduction may be considered (although not
obligatory) for the equations posseding this
invariance.
It is thus interesting to consider the example of a
source of light with spherical symmetry : the
detection of a photon in a position y0 at a time t0
implies instantaneously its non detection elsewhere. The
wave packet needing to be reduced in the whole space at the time t0,
the reduction must happen in a relativistic invariant manner on the
past cone of the
detector (and the future cone of the source). This may seem
contradictory, by some
apparent “retroactivity”, but thit comes from the fact that the photon
does not undergo the same time as the observer, and that it
already “knows” where it goes at the very moment it starts
[3].
__________________
References :
1. These aspects are not interpreted in the same way by all the
physicists, which leads in particular
R. Penrose to search elsewhere (in a gravitationnal effect, perhaps
connected to the expansion of Universe) the interpretation of the
time's arrow :
“La nature de l'espace et du temps”, S. Hawking and
R. Penrose, Princeton University Press 1996 (Gallimard 1997 for the
translation), chapter IV.
2. See as example :
“Le chat de
Schrödinger se prête à l'expérience”, S.
Haroche et al., La Recherche n° 304, september 1997 ;
M. Brune et
al., Phys. Rev. Lett. n° 77, p. 4887, 1996 ;
“Comprendre les
limites entre les mondes quantique et classique”, S. Reynaud, La
Recherche n° 375, may 2004 ;
“La nature de l'espace et du temps”, S.
Hawking and R. Penrose,
Princeton University Press 1996 (Gallimard 1997 for the translation),
chapters IV, VI and VII ;
“La frontière classique-quantique, M. Brune, Pour la Science
n° 350, december 2006.
3. The reduction of the wave packet by the unavoidable
interactions is generally known by the name of “decoherence”, it now
seems
to be accepted by an important part of the scientific community ;
curiousely, while the present concept of
decoherence has no relation with the relativistic instantaneousness, it
is the latter which, strengthening the perception I had of
interactions,
leaded me to be convinced of that (about in 1980, at a time when this
notion was still so little known as I did not heard about it).
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