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The theory of the double preparation: discerned and indiscerned particles
Michel Gondran
University Paris Dauphine, Lamsade, 75 016 Paris, France∗
Alexandre Gondran
École Nationale de l’Aviation Civile, 31000 Toulouse, France†
In this paper we propose a deterministic and realistic quantum mechanics interpretation which
may correspond to Louis de Broglie’s "double solution theory". Louis de Broglie considers two solu-
tions to the Schrödinger equation, a singular and physical wave u representing the particle (soliton
wave) and a regular wave representing probability (statistical wave). We return to the idea of two
solutions, but in the form of an interpretation of the wave function based on two different prepa-
rations of the quantum system. We demonstrate the necessity of this double interpretation when
the particles are subjected to a semi-classical field by studying the convergence of the Schrödinger
equation when the Planck constant tends to 0. For this convergence, we reexamine not only the
foundations of quantum mechanics but also those of classical mechanics, and in particular two im-
portant paradox of classical mechanics: the interpretation of the principle of least action and the
the Gibbs paradox. We find two very different convergences which depend on the preparation of the
quantum particles: particles called indiscerned (prepared in the same way and whose initial density
is regular, such as atomic beams) and particles called discerned (whose density is singular, such as
coherent states). These results are based on the Minplus analysis, a new branch of mathematics that
we have developed following Maslov, and on the Minplus path integral which is the analog in classi-
cal mechanics of the Feynman path integral in quantum mechanics. The indiscerned (or discerned)
quantum particles converge to indiscerned (or discerned) classical particles and we deduce that the
de Broglie-Bohm pilot wave is the correct interpretation for the indiscerned quantum particles (wave
statistics) and the Schrödinger interpretation is the correct interpretation for discerned quantum
particles (wave soliton). Finally, we show that this double interpretation can be extended to the
non semi-classical case.
I. INTRODUCTION
For Louis de Broglie, the correct interpretation of
quantum mechanics was the "theory of the double so-
lution" introduced in 19271and for which the pilot-wave
was just a low-level product2:I introduced as a ’double
solution theory’ the idea that it was necessary to distin-
guish two different solutions but both linked to the wave
equation, one that I called wave uwhich was a real phys-
ical wave but not normalizable having a local anomaly
defining the particle and represented by a singularity, the
other one as the Schrödinger Ψwave, which is normal-
izable without singularities and being a probability repre-
sentation.
Louis de Broglie distinguishes two solutions to the
Schrödinger equation, a singular and physical wave urep-
resenting the particle (soliton wave) and a regular wave
Ψrepresenting probability (statistical wave). But, de
Broglie don’t have never find a consistent "double solu-
tion theory". We return to the idea of two solutions,
but in the form of a double interpretation of the wave
function based on different preparations of the quan-
tum system. We demonstrate the necessity of this dou-
ble interpretation when the particles are subjected to a
semi-classical field by studying the convergence of the
Schrödinger equation when the Planck constant tends to
04,5. This convergence of quantum to classical mechanics
poses three types of difficulty that seem insurmountable:
•a physical difficulty, because h is a constant and
therefore its convergence to 0 is not physical;
•a conceptual difficulty: in quantum mechanics,
particles are regarded as indistinguishable whereas
they are considered to be distinguishable in classi-
cal mechanics;
•mathematical difficulties of convergence of equa-
tions.
The physical difficulty is the easiest to solve: it is only
mathematically, not physically, that we decrease the
Planck constant to 0; numerically we obtain the same
results if we increase the mass mof the particle to infin-
ity.
The conceptual difficulty forces us into reexamining
not only the foundations of quantum mechanics but also
those of classical mechanics. It is necessary to understand
and solve two important paradoxes of classical mechanics:
the interpretation of the principle of least action where
the "final causes" seem to be substituted for the "efficient
causes"; the Gibbs paradox where the entropy calculation
of a mixture of two identical gases by classical mechanics
with distinguishable particles leads to an entropy twice
as big as expected. We solve the conceptual difficulty
by showing that it is natural to introduce the concepts
of discerned and indiscerned particles both in classical
mechanics and in quantum mechanics.
The mathematical difficulties will be greatly simpli-
fied by considering two types of initial conditions (dis-
cerned and indiscerned particles) which yield very differ-
arXiv:1311.1466v1 [quant-ph] 6 Nov 2013
2
ent mathematical convergences. They are also simplified
by the Minplus analysis6,7, a new branch of mathematics
that we have developed following Maslov8,9. The paper is
organized as follows. In section II, we will show that the
difficulties of interpretation of the principle of least ac-
tion concerning the "final causes" come from the "Euler-
Lagrange action" (or classical action) Scl(x, t;x0), which
links the initial position x0and its position xat time t
and not from the "Hamilton-Jacobi action" S(x, t), which
depends on an initial action S0(x). These two actions
are solutions to the same Hamilton-Jacobi equation, but
with very different initial conditions: smooth conditions
for the Hamilton-Jacobi action, singular conditions for
the Euler-Lagrange action.
In section III, we show how Minplus analysis, a new
branch of nonlinear mathematics, explains the differ-
ence between the Hamilton-Jacobi action and the Euler-
Lagrange action. We obtain the equation between these
two actions, which we call the Minplus path integral: it
is the analog in classical mechanics of the Feynman path
integral in quantum mechanics. We show that it is the
key to understanding the principle of least action.
In section IV, we introduce in classical mechanics the
concept of indiscerned particles through the statistical
Hamilton-Jacobi equations. The discerned particles in
classical mechanics correspond to a deterministic action
S(x, t;x0,v0), which links a particle in initial position x0
and initial velocity v0to its position xat time tand ver-
ifies the deterministic Hamilton-Jacobi equations. And
the Gibbs paradox is solved by the indiscerned particles
in classical mechanics.
In section V, we study the convergence of quantum me-
chanics to classical mechanics when the Planck constant
tends to 0 by considering two cases : the first corresponds
to the convergence to an indiscerned classical particle,
and the second corresponds to the convergence to a clas-
sical discerned particle.4,5 Based on these convergences,
we propose a new interpretation of quantum mechanics,
the "theory of the double preparation", a response that
corresponds to Louis de Broglie’s "theory of the double
solution".
In section VI, we generalize this interpretation when
the semi-classical approximation is not valid. Following
de Muynck10, we show that it is possible to construct
a deterministic field quantum theory that extends the
previous double semi-classical interpretation to the non
semi-classical case.
II. THE EULER-LAGRANGE AND
HAMILTON-JACOBI ACTIONS
The intense debate on the interpretation of the wave
function in quantum mechanics for eighty years has in
fact left the debate on the interpretation of the action
and the principle of least action in classical mechanics in
the dark, since their introduction in 1744 by Pierre-Louis
Moreau de Maupertuis11 : "Nature, in the production of
its effects, does so always by the simplest means [...] the
path it takes is the one by which the quantity of action
is the least". Maupertuis understood that, under certain
conditions, Newton’s equations are equivalent to the fact
that a quantity, which he calls the action, is minimal. In-
deed, one can verify that the trajectory realized in Nature
is that which minimizes (or renders extremal) the action,
which is a function depending on the different possible
trajectories.
However, this principle has often been viewed as puz-
zling by many scholars, including Henri Poincaré, who
was nonetheless one of its most intensive users12: "The
very enunciation of the principle of least action is objec-
tionable. To move from one point to another, a material
molecule, acted on by no force, but compelled to move
on a surface, will take as its path the geodesic line, i.e.,
the shortest path. This molecule seems to know the point
to which we want to take it, to foresee the time it will
take to reach it by such a path, and then to know how to
choose the most convenient path. The enunciation of the
principle presents it to us, so to speak, as a living and
free entity. It is clear that it would be better to replace
it by a less objectionable enunciation, one in which, as
philosophers would say, final effects do not seem to be
substituted for acting causes."
We will show that the difficulties of interpretation
of the principle of least action concerning the "final
causes" or the "efficient causes" come from the existence
of two different actions: the "Euler-Lagrange action"
Scl(x, t;x0)and the "Hamilton-Jacobi action" S(x, t).
Let us consider a system evolving from the position
x0at initial time to the position xat time twhere the
variable of control u(s) is the velocity:
dx(s)
ds =u(s),∀s∈[0, t](1)
x(0) = x0,x(t) = x.(2)
If L(x,˙
x, t)is the Lagrangian of the system, when the
two positions x0and xare given, the Euler-Lagrange ac-
tion Scl(x, t;x0)is the function defined by:
Scl(x, t;x0) = min
u(s),0≤s≤tZt
0
L(x(s),u(s), s)ds, (3)
where the minimum (or more generally an extremum) is
taken on the controls u(s),s∈[0, t], with the state x(s)
given by equations (1) and (2). This is the principle of
least action defined by Euler13 in 1744 and Lagrange14
in 1755.
The solution (e
x(s),e
u(s)) of (3), if the Lagrangian
L(x,˙
x, t)is twice differentiable, satisfies the Euler-
Lagrange equations on the interval [0, t]:
d
ds
∂L
∂˙
x(x(s),˙
x(s), s)−∂L
∂x(x(s),˙
x(s), s) = 0,∀s∈[0, t]
(4)
x(0) = x0,x(t) = x.(5)
3
For a non-relativistic particle in a linear potential field
with the Lagrangian L(x,˙x, t) = 1
2m˙x2+K.x, equation
(4) yields d
ds (m˙
x(s))−K= 0. The trajectory minimizing
the action is e
x(s) = x0+s
t(x−x0)−K
2mts +K
2ms2, and
the Euler-Lagrange action is equal to
Scl(x, t;x0) = m(x−x0)2
2t+K.(x+x0)
2t−K2
24mt3.(6)
Figure 1 shows different trajectories going from x0at
time t= 0 to xat final time t. The parabolic trajectory
ex(s)corresponds to this which realizes the minimum in
the equation (3).
FIG. 1. Different trajectories x(s)(0≤s≤t) between
(x0,0) and (x, t)and the optimal trajectory ex(s)with e
v0=
x−x0
t−Kt
2m.
Equation (3) seems to show that, among the trajecto-
ries which can reach (x, t) from the initial position x0,
the principle of least action allows to choose the veloc-
ity at each time. In reality, the principle of least action
used in equation (3) does not choose the velocity at each
time sbetween 0and t, but only when it arrives at x
at time t. The knowledge of the velocity at each time s
(0≤s≤t) requires the resolution of the Euler-Lagrange
equations (4,5) on the whole trajectory. In the case of
a non-relativistic particle in a linear potential field, the
velocity at time s(0≤s≤t) is e
v(s) = x−x0
t−Kt
2m+Ks
m
with the initial velocity
e
v0=x−x0
t−Kt
2m.(7)
Then, e
v0depends on the position xof the particle at
the final time t. This dependence of the "final causes"
is general. This is the Poincaré’s main criticism of the
principle of least action: "This molecule seems to know
the point to which we want to take it, to foresee the time
it will take to reach it by such a path, and then to know
how to choose the most convenient path."
One must conclude that, without knowing the initial
velocity, the Euler-Lagrange action answers a problem
posed by an observer, and not by Nature: "What would
be the velocity of the particle at the initial time to arrive
in xat time t?" The resolution of this problem implies
that the observer solves the Euler-Lagrange equations
(4,5) after the observation of xat time t. This is an
a posteriori point of view.
But from 1830, Hamilton15 proposes to consider the
action Sas a function of the coordinates and of the
time S(x, t). It is customary to call it Hamilton’s prin-
cipal function16–18. In the following, we refer to it as
the Hamilton-Jacobi action. Indeed, for the Lagrangian
L(x,˙x, t) = 1
2m˙x2−V(x, t), this action satisfies the
Hamilton-Jacobi equations:
∂S
∂t +1
2m(OS)2+V(x, t)=0 (8)
S(x,0) = S0(x).(9)
The initial condition S0(x)is essential to defining
the general solution to the Hamilton-Jacobi equations
(8,9) although it is ignored in the classical mechanics
textbooks such as those of Landau18 chap.7 § 47 and
Goldstein16 chap. 10. However, the initial condition
S0(x)is mathematically necessary to obtain the general
solution to the Hamilton-Jacobi equations. Physically,
it is the condition that describes the preparation of the
particles. We will see that this initial condition is the key
to understanding the principle of least action.
The main property of the Hamilton-Jacobi action is
that the velocity of a non-relativistic classical particle is
given for each point (x,t)by:
v(x,t) = ∇S(x,t)
m.(10)
In the general case where S0(x)is a regular function,
for example differentiable, equation (10) shows that the
solution S(x,t)of the Hamilton-Jacobi equations yields
the velocity field for each point (x, t) from the veloc-
ity field ∇S0(x)
mat the initial time. In particular, if at
the initial time, we know the initial position xinit of a
particle, its velocity at this time is equal to ∇S0(xinit )
m.
From the solution S(x,t)of the Hamilton-Jacobi equa-
tions, we deduce with (10) the trajectories of the particle.
The Hamilton-Jacobi action S(x,t)is then a field which
"pilots" the particle.
There is another solution to the Hamilton-Jacobi equa-
tion; it is the Euler-Lagrange action. Indeed, Scl (x, t;x0)
satisfies the Hamilton-Jacobi (8) with the initial condi-
tion
S(x,0) = {0if x=x0,+∞if not}(11)
which is a very singular function. Mathematical analysis
will help us to interpret the solution to the Hamilton-
Jacobi equations and the principle of least action.
III. MINPLUS ANALYSIS AND THE MINPLUS
PATH INTEGRAL
There exists a new branch of mathematics, Minplus
analysis, which studies nonlinear problems through a lin-
4
ear approach, cf. Maslov8,9 and Gondran6,7. The idea
is to substitute the usual scalar product RXf(x)g(x)dx
with the Minplus scalar product:
(f, g) = inf
x∈X{f(x) + g(x)}(12)
In the scalar product we replace the field of the real
number (R,+,×)with the algebraic structure Minplus
(R∪{+∞},min,+), i.e. the set of real numbers (with the
element infinity {+∞}) endowed with the operation Min
(minimum of two reals), which remplaces the usual addi-
tion, and with the operation + (sum of two reals), which
remplaces the usual multiplication. The element {+∞}
corresponds to the neutral element for the operation Min,
Min({+∞}, a) = a∀a∈R. This approach bears a close
similarity to the theory of distributions for the nonlinear
case; here, the operator is "linear" and continuous with
respect to the Minplus structure, though nonlinear with
respect to the classical structure (R,+,×). In this Min-
plus structure, the Hamilton-Jacobi equation is linear,
because if S1(x, t)and S2(x, t)are solutions to (8), then
min{λ+S1(x, t), µ +S2(x, t)}is also a solution to the
Hamilton-Jacobi equation (8).
The analog to the Dirac distribution δ(x)in Min-
plus analysis is the nonlinear distribution δmin(x) =
{0if x=0,+∞if not}. With this nonlinear Dirac dis-
tribution, we can define elementary solutions as in clas-
sical distribution theory. In particular, we obtain:
The classical Euler-Lagrange action Scl(x, t;x0)is the
elementary solution to the Hamilton-Jacobi equations
(8)(9) in the Minplus analysis with the initial condition
S(x,0) = δmin(x−x0) = {0if x=x0,+∞if not}.
(13)
The Hamilton-Jacobi action S(x, t)is then given by the
Minplus integral
S(x, t) = inf
x0{S0(x0) + Scl(x, t;x0)}.(14)
that we call the Minplus path integral. It is an equation
similar to the Hopf-Lax formula19,20. This equation is in
analogy with the solution to the heat transfer equation
given by the classical integral:
S(x, t) = ZS0(x0)1
2√πt e−(x−x0)2
4tdx0,(15)
which is the product of convolution of the initial con-
dition S0(x)with the elementary solution to the heat
transfer equation e−x2
4t.
This Minplus path integral yields a very simple relation
between the Hamilton-Jacobi action, the general solution
to the Hamilton-Jacobi equation, and the Euler-Lagrange
actions, the elementary solutions to the Hamilton-Jacobi
equation. We can also consider that the Minplus integral
(14) for the action in classical mechanics is analogous
to the Feynmann path integral for the wave function
in quantum mechanics. Indeed, in the Feynman paths
integral21 (p. 58), the wave function Ψ(x, t)at time tis
written as a function of the initial wave function Ψ0(x):
Ψ(x, t) = ZF(t, ~) exp i
~Scl(x, t;x0Ψ0(x0)dx0
(16)
where F(t, ~)is an independent function of xand of x0.
For a particle in a linear potential V(x) = −K.xwith
the initial action S0(x) = mv0·x, we deduce from equa-
tion (14) that the Hamilton-Jacobi action is equal to
S(x, t) = mv0·x−1
2mv2
0t+K.xt−1
2K.v0t2−K2t3
6m.
Figure 2 shows the classical trajectories (parabols) go-
ing from different starting points xi
0at time t= 0 to
the point xat final time t. The Hamilton-Jacobi action
is compute with these trajectories in the Minplus path
integral (14).
FIG. 2. Classical trajectories ex(s)(0≤s≤t) between the
different initial positions xi
0and the position xat time t. We
obtain e
vi
0=x−xi
0
t−Kt
2m.
Finally, we can write the Minplus paths integral as
follows:
S(x, t) = min
x0;u(s),0≤s≤tS0(x0) + Zt
0
L(x(s),u(s), s)ds
(17)
where the minimum is taken on all initial positions x0
and on the controls u(s),s∈[0, t], with the state x(s)
given by equations (1) and (2). This is possible because
S0(x0)does not play a role in (17) for the minimization
on u(s).
Equation (17) seems to show that, among the trajecto-
ries which can reach (x, t) from an unknown initial posi-
tion and a known initial velocity field, Nature chooses the
initial position and at each time the velocity that yields
the minimum (or the extremum) of the Hamilton-Jacobi
action.
Equations (10), (8) and (9) confirm this interpretation.
They show that the Hamilton-Jacobi action S(x, t)does
not solve only a given problem with a single initial condi-
tion x0,∇S0(x0)
m, but a set of problems with an infinity
of initial conditions, all the pairs y,∇S0(y)
m. It answers
5
the following question: "If we know the action (or the
velocity field) at the initial time, can we determine the
action (or the velocity field) at each later time?" This
problem is solved sequentially by the (local) evolution
equation (8). This is an a priori point of view. It is
the problem solved by Nature with the principle of least
action.
For a particle in a linear potential V(x) = −K.xwith
the initial action S0(x) = mv0·x, the initial velocity field
is constant, v(x,0) = ∇S0(x)
m=v0and the velocity field
at time tis also constant, v(x, t) = ∇S(x,t)
m=v0+Kt
m.
Figure 3 shows these velocity fields.
t
0
x
FIG. 3. Velocity field that corresponds to the Hamilton-
Jacobi action S(x, t) = mv0·x−1
2mv2
0t+K.xt−1
2K.v0t2−
K2t3
6m(v(x, t) = ∇S(x,t)
m=v0+Kt
m) and three trajectories of
particles piloted by this field.
IV. DISCERNED AND INDISCERNED
PARTICLES IN CLASSICAL MECHANICS
We show that the difficulties interpreting the action
and the wave function result from the ambiguity in the
definition of the conditions for the preparation of par-
ticles, which entails an ambiguity concerning the initial
conditions. This ambiguity is related to the notion of
indiscernibility which has never been well defined in the
literature. It is responsible in particular for the Gibbs
paradox: when calculating the entropy of a mixture of
two identical gases in equilibrium, calculation by classi-
cal mechanics with distinguishable particles leads to an
entropy twice as big as expected. If we replace these
particles with indistinguishable particles, then the factor
related to the indiscernibility yields the correct result.
In almost all textbooks on statistical mechanics, it is
considered that this paradox stated by Willard Gibbs in
1889, was "solved" by quantum mechanics over thirty-
five years later, thanks to the introduction of the indis-
tinguishability postulate for identical quantum particles.
Indeed, it was Einstein who, in 1924, introduced the
indistinguishability of molecules of an ideal gas at the
same time as the Bose-Einstein statistics. Nonetheless,
as pointed out by Henri Bacry, "history might have fol-
lowed a different path. Indeed, quite logically, we could
have applied the principle of indiscernibility to save the
Gibbs paradox. [...] This principle can be added to the
postulates of quantum mechanics as well as to those of
classical mechanics".22
This same observation has been made by a large num-
ber of other authors. In 1965 Landé23 demonstrated that
this indiscernability postulate of classical particles is suf-
ficient and necessary in order to explain why entropy van-
ished. In 1977, Leinaas and Myrheim24 used it for the
foundation of their identical classical and quantum par-
ticles theory. Moreover, as noted by Greiner et al.25, in
addition to the Gibbs paradox, several cases where it is
needed to consider indistinguishable particles in classi-
cal mechanics and distinguishable particles in quantum
mechanics can be found: "Hence, the Gibbs factor 1
N!
is indeed the correct recipe for avoiding the Gibbs para-
dox. From now on we will therefore always take into
account the Gibbs correction factor for indistinguishable
states when we count the microstates. However, we want
to emphasize that this factor is no more than a recipe to
avoid the contradictions of classical statistical mechanics.
In the case of distinguishable objects (e.g., atoms which
are localized at certain grid points), the Gibbs factor must
not be added. In classical theory the particles remain dis-
tinguishable. We will meet this inconsistency more fre-
quently in classical statistical mechanics."25 p.134
We propose an accurate definition of both discernabil-
ity and indiscernability in classical mechanics and a way
to avoid ambiguities and paradoxes. Here, we only con-
sider the case of a single particle or a system of identical
particles without interactions and prepared in the same
way.
In classical mechanics, a particle is usually considered
as a point and is described by its mass m, its charge if it
has one, as well as its position x0and velocity v0at the
initial instant. If the particle is subject to a potential field
V(x), we can deduce its path because its future evolution
is given by Newton’s or Lorentz’s equations. This is why
classical particles are considered distinguishable. We will
show, however, that a classical particle can be either non-
discerned or discerned depending on how it is prepared.
We now consider a particle within a stationary beam
of classical identical particles such as electronic, atomic
or molecular beams (CO2or C60). At a very macroscopic
level, one can consider a tennis ball canon. Let us note
that there is an abuse of language when one talks about a
classical particle. One should instead speak of a particle
that is studied in the framework of classical mechanics.
For a particle of this beam, we do not know at the ini-
tial instant the exact position or the exact velocity, only
the characteristics describing the beam, that is to say, an
initial probability density ρ0(x)and an initial velocity
field v0(x)known through the initial action S0(x)by the
6
equation v0(x) = ∇S0(x)
mwhere mis the particle mass.
This yields the following definition:
Definition 1 (Indiscerned prepared Particle) - A
classical particle is said to be indiscerned prepared when
only the characteristics of the beam from which it comes
(initial probability density ρ0(x)and initial action S0(x))
are defined at the initial time.
In contrast, we have:
Definition 2 (Discerned prepared Particle) - A
classical particle is said to be discerned prepared, if one
knows, at the initial time, its position x0and velocity
v0.
The notion of indiscernibility that we introduced does
not depend on the observer’s knowledge, but is related
to the mode of preparation of the particle.
Let us consider Nindiscerned particles, that is to say
Nidentical particles prepared in the same way, each with
the same initial density ρ0(x)and the same initial action
S0(x), subject to the same potential field V(x)and which
will have independent behaviors. This is particularly the
case of identical classical particles without mutual inter-
action and prepared the same way. It is also the case
of identical classical particles such as electrons, prepared
in the same way and which, although they may interact,
will have independent behaviors if they are generated one
by one in the system.
We called these particles indiscerned, and not indistin-
guishable, because if we knew their initial positions, their
trajectories would be known.
The difference between discerned and indiscerned par-
ticles depends on the preparation style. A device pre-
pares either discerned or indiscerned particles. By way of
example a tennis ball machine randomly launches balls
in different directions. Therefore it prepares some in-
discerned particles; only the characteristics of the balls’
beams are known: probability of presence and velocity
(action). A tennis player plunged into complete darkness
that uses this machine knows only the presence proba-
bility of balls. However it is possible to discern indis-
cerned particles, if we knew their initial positions. This
is what happens during the day: the tennis player is able
to make successive measurements of the ball position by
watching it. In this case, the player is able to plan the
trajectory. It is important to note that without mea-
surements, the balls remain indiscerned. In this specific
case, the position measurement changes neither the state
of the particle nor its trajectories. This is not always
the case in quantum mechanics. It is therefore easy for
the observer to identify the indistinguishability of indis-
cerned particles. However the tennis ball machine still
produces indiscerned particles. A shotgun that fires a
number of small spherical pellets also produces a beam
of indiscerned particules. The positions of the pellets are
unknown, only their probability densities are known as
well as their velocities. If the precision of the shotgun
is very high and if one uses a bullet (instead of pellets),
the initial position x0of the bullet and its velocity v0
are known with exactitude. Therefore the bullet is a dis-
cerned prepared particle. The trajectories of the bullets
will be always the same. How the particles are prepared
is fundamental.
Based on the previous definitions, we may state the
following:
1. An indiscerned prepared particule whose initial po-
sition x0is also known is a discerned prepared par-
ticule.
2. An indiscerned prepared particule whose initial
probability density ρ0(x)is equal to a Dirac dis-
tribution ρ0(x) = δ(x−x0)is a discerned prepared
particule.
This means that the indiscerned particules can be dis-
tinguishable. Furthermore, in their enumerations indis-
cerned particules have the same properties that are usu-
ally granted to indistinguishable particles. Thus, if we
select Nidentical particles at random from the initial
density ρ0(x), the various permutations of the Nparti-
cles are strictly equivalent and correspond, as for indis-
tinguishable particles, to only one configuration. In this
framework, the Gibbs paradox is no longer paradoxical as
it applies to Nindiscerned particles whose different per-
mutations correspond to the same configuration as for
indistinguishable particles. This means that if Xis the
coordinate space of an indiscerned particle, the true con-
figuration space of Nindiscerned particles is not XNbut
rather XN/SNwhere SNis the permutation group.
For indiscerned particles, we have the following theo-
rem:
THEOREM 1 - The probability density ρ(x, t)and the
action S(x,t)of classical particles prepared in the same
way, with initial density ρ0(x), with the same initial ac-
tion S0(x), and evolving in the same potential V(x), are
solutions to the statistical Hamilton-Jacobi equa-
tions:
∂S (x, t)
∂t +1
2m(∇S(x, t))2+V(x) = 0 (18)
S(x,0) = S0(x)(19)
∂ρ (x, t)
∂t +div ρ(x, t)∇S(x, t)
m= 0 (20)
ρ(x,0) = ρ0(x).(21)
Let us recall that the velocity field is v(x, t) = ∇S(x,t)
m
and that the Hamilton-Jacobi equation (18) is not cou-
pled to the continuity equation (20).
The difference between discerned and indiscerned par-
ticles will provide a simple explanation to the "recipes"
denounced by Greiner et al.25 that are commonly pre-
sented in manuals on classical statistical mechanics.
However, as we have seen, this is not a principle that
must be added. The nature of this discernability of the
7
particle depends on the preparation conditions of the par-
ticles, whether discerned or indiscerned.
Can we define an action for a discerned particle in a
potential field V(x)? Such an action should depend only
on the starting point x0, the initial velocity v0and the
potential field V(x).
THEOREM 2 - If ξ(t)is the classical trajectory in the
field V(x)of a particle with the initial position x0and
with initial velocity v0, then the function
S(x, t;x0,v0) = mdξ(t)
dt ·x+g(t)(22)
where dg(t)
dt =−1
2m(dξ(t)
dt )2−V(ξ(t)) −md2ξ(t)
dt2·ξ(t), is
called the deterministic action, and is a solution to
deterministic Hamilton-Jacobi equations:
0 = ∂S (x, t;x0,v0)
∂t |x=ξ(t)+1
2m(∇S(x, t;x0,v0))2|x=ξ(t)
+V(x)|x=ξ(t)(23)
dξ(t)
dt =∇S(ξ(t), t;x0,v0)
m(24)
S(x,0; x0,v0) = mv0xand ξ(0) = x0.(25)
The deterministic action S(x, t;x0,v0)satisfies the
Hamilton-Jacobi equations only along the trajectory ξ(t).
The interest of such an action related to a single localized
trajectory is above all theoretical by proposing a mathe-
matical framework for the discerned particle. This action
will take on a meaning in the following section where we
show that it corresponds to the limit of the wave func-
tion of a quantum particle in a coherent state when one
makes the Planck constant htend to 0.
As for the Hamilton-Jacobi action, the deterministic
action only depends on the initial conditions (x0,v0), the
"efficient causes". In the end, we have three actions in
classical mechanics, an epistemological action (the Euler-
Lagrange action S(x, t;x0)) and two ontological actions,
the Hamilton-Jacobi action S(x, t)for the indiscerned
particles and the deterministic action S(x, t;x0,v0)for
the discerned particles.
V. THE TWO LIMITS OF THE SCHRÖDINGER
EQUATION
Let us consider the case semi-classical where the wave
function Ψ(x, t)is a solution to the Schrödinger equation
:
i~∂Ψ
∂t =−~2
2m4Ψ + V(x, t)Ψ (26)
Ψ(x,0) = Ψ0(x).(27)
With the variable change Ψ(x, t) =
pρ~(x, t) exp(iS~(x,t)
~), the Schrödinger equation
can be decomposed into Madelung equations26 (1926):
∂S~(x, t)
∂t +1
2m(∇S~(x, t))2+V(x, t)−~2
2m4pρ~(x, t)
pρ~(x, t)= 0
(28)
∂ρ~(x, t)
∂t +div ρ~(x, t)∇S~(x, t)
m= 0 (29)
with initial conditions:
ρ~(x,0) = ρ~
0(x)and S~(x,0) = S~
0(x).(30)
We consider two cases depending on the preparation of
the particles4,5 .
Definition 3 (Semi-Classical indiscerned particle)
- A quantum particle is said to be semi-classical indis-
cerned prepared if its initial probability density ρ~
0(x)
and its initial action S~
0(x)are regular functions ρ0(x)
and S0(x)not depending on ~.
It is the case of a set of non-interacting particles all
prepared in the same way: a free particle beam in a linear
potential, an electronic or C60 beam in the Young’s slits
diffraction, or an atomic beam in the Stern and Gerlach
experiment.
Definition 4 (Semi-Classical discerned particle) -
A quantum particle is said to be semi-classical discerned
prepared if its initial probability density ρ~
0(x)converges,
when ~→0, to a Dirac distribution and if its initial
action S~
0(x)is a regular function S0(x)not depending
on ~.
This situation occurs when the wave packet corresponds
to a quasi-classical coherent state, introduced in 1926 by
Schrödinger27. The field quantum theory and the sec-
ond quantification are built on these coherent states28.
The existence for the hydrogen atom of a localized wave
packet whose motion is on the classical trajectory (an
old dream of Schrödinger’s) was predicted in 1994 by
Bialynicki-Birula, Kalinski, Eberly, Buchleitner and De-
lande29–31, and discovered recently by Maeda and Gal-
lagher33 on Rydberg atoms.
A. Semi-Classical indiscerned quantum particles
THEOREM 3 4,5 For semi-classical indiscerned quan-
tum particles, the probability density ρ~(x, t)and the
action S~(x, t), solutions to the Madelung equations
(28)(29)(30), converge, when ~→0, to the clas-
sical density ρ(x, t)and the classical action S(x, t),
solutions to the statistical Hamilton-Jacobi equations
(18)(19)(20)(21).
We give some indications on the demonstration of this
theorem and we propose its interpretation. Let us con-
sider the case where the wave function Ψ(x, t)at time
8
tis written as a function of the initial wave function
Ψ0(x)by the Feynman paths integral21 (16). For a semi-
classical indiscerned quantum particle, the wave function
is written Ψ(x, t) = F(t, ~)Rpρ0(x0) exp( i
~(S0(x0) +
Scl(x, t;x0))dx0. The theorem of the stationary phase
shows that, if ~tends towards 0, we have Ψ(x, t)∼
exp( i
~minx0(S0(x0) + Scl(x, t;x0)), that is to say that
the quantum action Sh(x, t)converges to the function
S(x, t) = minx0(S0(x0) + Scl(x, t;x0)) (31)
which is the solution to the Hamilton-Jacobi equation
(18) with the initial condition (19). Moreover, as the
quantum density ρh(x, t)satisfies the continuity equation
(29), we deduce, since Sh(x, t)tends towards S(x, t), that
ρh(x, t)converges to the classical density ρ(x, t), which
satisfies the continuity equation (20). We obtain both
announced convergences.
−35 −30 −20 −10 0 10 20 30 35
−4
−3
−2
−1
0
1
2
3
4
cm
µm
FIG. 4. 100 electron trajectories for the Jönsson experiment.
For a semi-classical indiscerned quantum particle, the
Madelung equations converge to the statistical Hamilton-
Jacobi equations, which correspond to indiscerned classi-
cal particles. We use now the interpretation of the statis-
tical Hamilton-Jacobi equations to deduce the interpre-
tation of the Madelung equations. For these indiscerned
classical particles, the density and the action are not suffi-
cient to describe a classical particle. To know its position
at time t, it is necessary to know its initial position. It is
logical to do the same in quantum mechanics. We con-
sider this indiscerned quantum particle as the classical
particle.
We conclude that a semi-classical indiscerned quan-
tum particle is not completely described by its wave
function. It is necessary to add its initial position and
it becomes natural to introduce the de Broglie-Bohm
interpretation1,36. In this interpretation, the two first
postulates of quantum mechanics, describing the quan-
tum state and its evolution, must be completed. At ini-
tial time t= 0, the state of the particle is given by the
initial wave function Ψ0(x)(a wave packet) and its initial
position X(0); it is the new first postulate. The second
new postulate gives the evolution on the wave function
and on the position. For a single, spin-less particle in
a potential V(x), the evolution of the wave function is
given by the usual Schrödinger equation (26)(27) and the
evolution of the particle position is given by
dX(t)
dt =1
m∇S~(x, t)|x=X(t).(32)
In the case of a particle with spin, as in the Stern and
Gerlach experiment, the Schrödinger equation must be
replaced by the Pauli or Dirac equations.
The other quantum mechanics postulates which de-
scribe the measurement are not necessary. One can
demonstrate that the three postulates of measurement
can be explained on each example; see the double-slit,
Stern-Gerlach and EPR-B experiments34. These postu-
lates are remplaced by a single one, the "quantum equi-
librium hypothesis", that describes the interaction be-
tween the initial wave function Ψ0(x)and the probabilty
distribution of the initial particle position X(0):
P[X(0) = x] = |Ψ0(x)|2.(33)
One deduces that for all times
P[X(t) = x] = |Ψ(x, t)|2.(34)
This is the "equivariance" property of the |Ψ(x, t)|2prob-
ability distribution35 which yields the Born probabilis-
tic interpretation. Let us note the minimality of the de
Broglie-Bohm interpretation.
Figure 4 shows a simulation of the de Broglie-Bohm
trajectories in the double slit experiment of Jönsson37
where an electron gun emits electrons one by one through
a hole with a radius of a few micrometers. The electrons,
prepared similarly, are represented by the same initial
wave function, but not by the same initial position. In the
simulation, these initial positions are randomly selected
in the initial wave packet. We have represented only the
quantum trajectories through one of two slits.
Figure 5 shows the 100 previous trajectories when the
Planck constant is divided by 10, 100, 1000 and 10000
respectively. We obtain, when h tends to 0, the conver-
gence of quantum trajectories to classical trajectories.
B. Semi-Classical discerned quantum particles
The convergence study of the semi-classical discerned
quantum particle is mathematically very difficult. We
only study the example of a coherent state where an ex-
plicit calculation is possible.
For the two dimensional harmonic oscillator, V(x) =
1
2mω2x2, coherent states are built32 from the initial
wave function Ψ0(x)which corresponds to the den-
sity and initial action ρ~
0(x) = (2πσ2
~)−1e−(x−x0)2
2σ2
~and
S0(x) = S~
0(x) = mv0·xwith σ~=q~
2mω . Here,
9
−30 −20 −10 0 10 20 30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
cm
µm
h/10
−30 −20 −10 0 10 20 30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
cm
µm
h/100
−30 −20 −10 0 10 20 30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
cm
µm
h/1000
−30 −20 −10 0 10 20 30
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
cm
µm
h/10000
FIG. 5. .Convergence of 100 electron trajectories when h is
divided by 10, 100, 1 000 and 10 000.
v0and x0are constant vectors and independent from
~, but σ~will tend to 0as ~. With initial condi-
tions, the density ρ~(x, t)and the action S~(x, t), solu-
tions to the Madelung equations (28)(29)(30), are equal
to 32:ρ~(x, t) = 2πσ2
~−1e−(x−ξ(t))2
2σ2
~and S~(x, t) =
+mdξ(t)
dt ·x+g(t)−~ωt, where ξ(t)is the trajectory
of a classical particle evolving in the potential V(x) =
1
2mω2x2, with x0and v0as initial position and velocity
and g(t) = Rt
0(−1
2m(dξ(s)
ds )2+1
2mω2ξ(s)2)ds.
THEOREM 4 4,5- For the harmonic oscillator, when
~→0, the density ρ~(x, t)and the action S~(x, t)con-
verge to
ρ(x, t) = δ(x−ξ(t)) and S(x, t) = mdξ(t)
dt ·x+g(t)(35)
where S(x, t)and the trajectory ξ(t)are solutions to the
deterministic Hamilton-Jacobi equations (23)(24)(25).
Therefore, the kinematic of the wave packet converges
to the single harmonic oscillator described by ξ(t), which
corresponds to a discerned classical particle. It is then
possible to consider, unlike for the semi-classical indis-
cerned particles, that the wave function can be viewed
as a single quantum particle. Then, we consider this dis-
cerned quantum particle as the classical particle. The
semi-classical discerned quantum particle is in line with
the Copenhagen interpretation of the wave function,
which contains all the information on the particle. A
natural interpretation was proposed by Schrödinger27 in
1926 for the coherent states of the harmonic oscillator:
the quantum particle is a spatially extended particle, rep-
resented by a wave packet whose center follows a classical
trajectory. In this interpretation, the first two usual pos-
tulates of quantum mechanics are maintened. The others
are not necessary. Then, the particle center is the mean
value of the position (X(t) = Rx|Ψ(x, t)|2dx) and satis-
fies the Ehrenfest theorem41. Let us note the minimality
of the Schrödinger interpretation.
VI. THE NON SEMI-CLASSICAL CASE
The de Broglie-Bohm and Schrödinger interpretations
correspond to the semi-classical approximation. They
correspond to the two interpretations proposed in 1927
at the Solvay congress42 by de Broglie and Schrödinger.
But there exist situations in which the semi-classical ap-
proximation is not valid. It is in particular the case
of state transitions in a hydrogen atom. Indeed, since
Delmelt’experiment43 in 1986, the physical reality of in-
dividual quantum jumps has been fully validated. The
semi-classical approximation, where the interaction with
the potential field can be described classically, is no
longer possible and it is necessary to use electromagnetic
field quantization since the exchanges occur photon by
photon. In this situation, the Schrödinger equation can-
not give a deterministic interpretation and the statistical
Born interpretation is the only valid one. It was the third
interpretation proposed in 1927 at the Solvay congress.
These three interpretations are considered by Einstein in
one of his last articles (1953), "Elementary Reflexion on
Interpreting the Foundations of Quantum Mechanics" in
a homage to Max Born44:
"The fact that the Schrödinger equation associated with
the Born interpretation does not lead to a description of
the "real states" of an individual system, naturally incites
one to find a theory that is not subject to this limitation.
Up to now, the two attempts have in common that they
conserve the Schrödinger equation and abandon the Born
interpretation. The first one, which marks de Broglie’s
comeback, was continued by Bohm.... The second one,
which aimed to get a "real description" of an individ-
ual system and which might be based on the Schrödinger
equation is very late and is from Schrödinger himself.
The general idea is briefly the following : the function
ψrepresents in itself the reality and it is not necessary to
add it to Born’s statistical interpretation.[...] From previ-
ous considerations, it results that the only acceptable in-
terpretation of the Schrödinger equation is the statistical
interpretation given by Born. Nevertheless, this interpre-
tation doesn’t give the "real description" of an individual
system, it just gives statistical statements of entire sys-
tems."
Thus, Einstein retained de Broglie’s and Schrödinger’s
attempts to interpret the "real states" of a single system:
these are our two semi-classical quantum particles (indis-
cerned and discerned). But as de Broglie and Schrödinger
retained the Schrödinger equation, Einstein, who consid-
ered this equation as fundamentally statistical, refuted
each of their interpretations. We will see that he went
too far in his rebuttal.
The novelty of our approach is to consider that each
of these three interpretations depends on the prepara-
10
tion of the particles. The de Broglie-Bohm interpretation
concerns the semi-classical indiscerned quantum parti-
cles, the Schrödinger interpretation concerns the semi-
classical discerned quantum particles, and the Born in-
terpretation concerns the statistic of the semi-classical
indiscerned quantum particles, but also the statistic of
the transitions in a hydrogen atom.
This does not mean that we should abandon deter-
minism and realism, but that at this scale, Schrödinger’s
statistical wave function is not the effective equation to
obtain an individual behavior of a particle, in particular
to investigate the instant of transition in a determinis-
tic manner. An individual interpretation needs to use
the creation and annihilation operators of quantum field
theory, but this interpretation still remains statistical.
We assume that it is possible to construct a determin-
istic quantum field theory that extends to the non semi-
classical interpretation of the double semi-classical prepa-
ration. First, as shown by de Muynck10, we can con-
struct a theory with discerned (labeled) creation and an-
nihilation operators in addition to the usual indiscerned
creation and annihilation operators. But, to satisfy the
determinism, it is necessary to search, at a lower scale,
the mechanisms that allow the emergence of the creation
operator.
VII. CONCLUSION
The introduction into classical mechanics of the con-
cepts of indiscerned particles verifying the statistical
Hamilton-Jacobi equations and of discerned particles ver-
ifying the deterministic Hamilton-Jacobi equations can
provide a simple answer to the Gibbs paradox of classical
statistical mechanics. Furthermore, the distinction be-
tween the Hamilton-Jacobi and Euler-Lagrange actions,
based on the Minplus path integral makes it easier to un-
derstand the principle of least action. The study of the
convergence of the Madelung equations when htends to
0leads us to consider the following two cases:
1. Semi-classical indiscerned quantum particles,
which are prepared in the same way and without
mutual interaction, for which the evolution equa-
tions converge to the statistical Hamilton-Jacobi
equations of indiscerned classical particles. The
wave function is therefore not sufficient to rep-
resent quantum particles and it is mandatory to
add their initial positions, just as for indiscerned
classical particles. Subsequently, the interpretation
of the de Broglie-Bohm pilot wave is necessary.
2. The semi-classical discerned quantum particle for
which the evolution equations converge to the de-
terministic Hamilton-Jacobi equations of a dis-
cerned classical particle. The interpretation of the
Broglie-Bohm pilot wave is no longer necessary be-
cause the wave function is sufficient to represent
the particles as in the Copenhagen interpretation.
Subsequently, the Schrödinger interpretation where
the wave function represents an extended particle,
is the most natural.
We can consider the previous interpretation which de-
pends on a double preparation of the quantum particle
(discerned or non-discerned) as a response to the "theory
of the double solution" that Louis de Broglie was seeking
in 1927. We call it "the theory of the double prepara-
tion".
In the case where the semi-classical approximation is
no longer valid, the interpretation needs to use the cre-
ation and annihilation operators of the quantum field.
W. M. de Muynck10 shows that is possible to construct
a theory with discerned (labeled) creation and annihi-
lation operators in addition to the usual non-discerned
creation and annihilation operators. But, to satisfy the
determinism, it is necessary to search, at a lower scale,
the mechanisms that allow the emergence of the creation
operator.
This interpretation of quantum mechanics following
the preparation of the system can explain the discussions
between the founding fathers, in particular the discus-
sion of the Solvay Congress of 1927. Indeed, one may
consider that the misunderstanding between them may
have come from the fact that they each had an element
of truth: Louis de Broglie’s pilot-wave interpretation for
the semi-classical indiscerned particle, Schrödinger’s in-
terpretation for the semi-classical discerned particle and
Born’s interpretation for the non-semi-classical case. But
each applied his particular case to the general case and
they consequently made mutually incompatible interpre-
tations.
∗michel.gondran@polytechnique.org
†alexandre.gondran@enac.fr
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