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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

1de Broglie, L.: La mécanique ondulatoire et la structure

atomique de la matière et du rayonnement. J. de Phys. 8,

225-241 (1927).

2de Broglie, L., Andrade e Silva, J.L.: La Réinterprétation

de la mécanique ondulatoire. Gauthier-Villars (1971).

3de Broglie, L. : Une interprétation causale et non linéaire

de la mécanique quantique: la théorie de la double solu-

tion. Gauthier-Villars (1956); English translation, Elsev-

ior, Amsterdam (1960).

4Gondran, M., Gondran, A.: Discerned and non-discerned

particles in classical mechanics and convergence of quan-

tum mechanics to classical mechanics. Annales de la Fon-

dation Louis de Broglie, 36, 117-135 (2011).

5Gondran, M., Gondran, A.: The two limits of the

Schrödinger equation in the semi-classical approximation :

discerned and non-discerned particles in classical mechan-

11

ics. Foundations of Probability and Physics-6, AIP Conf.

Proc. 1424,111-115 (2012).

6Gondran,M.: Analyse MinPlus. C. R. Acad. Sci. Paris 323,

371-375 (1996).

7Gondran, M., Minoux, M.: Graphs, Dioïds and Semi-

rings: New models and Algorithms. Springer, Operations

Research/Computer Science Interfaces, chap.7 (2008).

8Maslov, V.P., Samborski, S.N.: Idempotent Analysis. Ad-

vancesin Soviet Mathematics, 13, American Math Society,

Providence (1992).

9Kolokoltsov, V.N., Maslov, V.P.: Idempotent Analysis and

its applications. Klumer Acad. Publ. (1997).

10 de Muynck, W.M.: Distinguishable-and Indistinguishable-

Particle; Descriptions of Systems of Identical Particles. In-

ternational Journal of Theoretical Physics 14, n◦5, 327-346

(1975).

11 de Maupertuis, P.L.: Accord de diﬀérentes lois de la na-

ture qui avaient jusqu’ici paru incompatibles. Mémoires

de l’Académie Royale des Sciences, p.417-426 (Paris,1744);

reprint in: Oeuvres, 4, 1-23 Reprograﬁscher Nachdruck der

Ausg. Lyon (1768).

12 Poincaré, H.: La Science et l’Hypothèse. Flammarion,

(1902); Translated in: The Foundations of Sciences: Sci-

ence and Hypothesis, The value of Science, Science and

Method. New York: Science Press (1913).

13 Euler, L.: Methodus Inveniendi Lineas Curvas Maximi

Minive Proprietate Gaudentes. Bousquet, Lausanne et

Geneva (1744). Reprint in: Leonhardi Euleri Opera Om-

nia: Series I vol 24 C. Cartheodory (ed.) Orell Fuessli,

Zurich (1952).

14 Lagrange, J.L.: Mécanique Analytique. Gauthier-Villars,

2nd ed., Paris (1888); translated in: Analytic Mechanics,

Klumer Academic, Dordrecht (2001).

15 Hamilton, W. R.: On a general method in dynamics, by

which the study of the motions of all free systems of at-

tracting or repelling points is reduced to the search and dif-

ferentiation of one central Relation or characteristic Func-

tion. Philos. Trans; R. Soci. PartII, 247-308 (1834).

16 Goldstein, H.: Classical mechanics. Addison-Wesley

(1966).

17 Butterﬁeld, J.: David Lewis Meets Hamilton and Jacobi.

in: Philosophy of Science Assoc; 18th Biennal Mtg-PSA

(2002).

18 Landau, L.D., Lifshitz, E.M.: Mechanics, Course of The-

oretical Physics. chap.1, Buttreworth-Heinemann, London

(1976).

19 Lions, P. L.: Generalized Solutions of Hamilton-Jacobi

Equations. Pitman (1982).

20 Evans, L. C.: Partial Diﬀerential Equations, p.123-124.

Graduate Studies in Mathematics 19, American Mathe-

matical Society (1998). .

21 Feynman, R., Hibbs, A.: Quantum Mechanics and Inte-

grals. McGraw-Hill (1965).

22 Bacry, H.; Introduction aux concepts de la Physique Statis-

tique. Ellipses, Paris (1992).

23 Landé, A.: New Foundations of Quantum Mechanics, p.

68. Cambridge (1965).

24 Leinaas, J. M.; Myrheim, J.: On the Theory of Identical

Particles. Il Nuovo Cimento, 37 B, 1-23 (1977).

25 Greiner, W., Neise, L., Stöcker, H.: Thermdynamics and

Statistical Mechanics. Springer (1999).

26 Madelung, E.: Quantentheorie in hydrodynamischer Form.

Zeit. Phys. 40, 322-6 (1926).

27 Schrödinger, E.: Der stetige bergang von der Mikro-zur

Makromechanik. Naturwissenschaften 14, 664-666 (1926).

28 Glauber, R. J.: in: Quantum Optics and Electronics,

Les Houches Lectures 1964, C. deWitt, A. Blandin and

C. Cohen-Tanoudji eds., Gordon and Breach, New York

(1965).

29 Bialynicki-Birula, I., Kalinski, M., Eberly, J. H.: Lagrange

Equilibrium Points in Celestial Mechanics and Nonspread-

ing Wave Packets for Strongly Driven Rydberg Electrons.

Phys. Rev. Lett. 73, 1777 (1994).

30 Buchleitner, A., Delande, D.: Non-dispersive electronic

wave packets in multiphoton processus. Phys. Rev. Lett.

75, 1487 (1995).

31 Buchleitner, A., Delande, D., Zakrzewski, J.: Non-

dispersive wave packets in periodically driven quantum sys-

tems. Physics Reports 368, 409-547 (2002).

32 Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Me-

chanics. Wiley, New York (1977).

33 Maeda, H., Gallagher, T.F.: Non dispersing Wave Packets.

Phys. Rev. Lett. 92, 133004-1 (2004).

34 Gondran, M., Gondran, A., Kenouﬁ, A.: Decoherence time

and spin measurement in the Stern-Gerlach experiment.

Foundations of Probability and Physics-6 (Växjö, Sweden,

Juin 2011), AIP Conf. Proc. 1424, pp.116-120 (2012).

35 Dürr, D., Golstein, S., Zanghi, N.: Quantum equilibrium

and the origin of absolute uncertainty. J. Stat. Phys. 67,

843-907 (1992).

36 Bohm, D.: A suggested interpretation of the quantum the-

ory in terms of ”hidden” variables. Phys. Rev. 85, 166-193

(1952).

37 Jönsson, C.: Elektroneninterferenzen an mehreren kün-

stlich hergestellten Feinspalten. Z. Phy. 161, 454-474

(1961), English translation: Electron diﬀraction at mul-

tiple slits. Am. J. Phys. 42, 4-11 (1974).

38 Gondran, M., Gondran, A.: Numerical simulation of the

double-slit interference with ultracold atoms. Am. J. Phys.

73, 507-515 (2005).

39 Bohm, D., Hiley, B.J.: The Undivided Universe. Rout-

ledge, London and New York (1993).

40 Holland, P.R.: The quantum Theory of Motion. Cam-

bridge University Press (1993).

41 Ehrenfest, P.: Bemerkung über die angenäherte Gültigkeit

der klassischen Mechanik innerhalb der Quantenmechanik.

Zeitschriﬂt für Physik 45 (7-8), 455-457 (1927).

42 Bacciagaluppi, G., Valentini, A.: Quantum Theory at the

Crossroads: Reconsidering the 1927 Solvay Conference.

Cambridge University Press (2009).

43 Nagournay, W., Sandberg, J., Dehmelt, H.: Shelved op-

tical electron ampliﬁer: Observation of quantum jumps.

Phys. Rev. Lett.56, 2797-2799 (1986).

44 Einstein, A.: Elementary Reﬂexion on Interpreting the

Foundations of Quantum Mechanics . in: Scientiﬁc Pa-

pers presented to Max Born. Edimbourg, Olivier and Boyd

(1953).