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Relational causality and classical probability: Grounding
quantum phenomenology in a superclassical theory
Gerhard Gr¨ossing1,∗Siegfried Fussy1,∗Johannes Mesa Pascasio1,2,∗and Herbert Schwabl1∗
1Austrian Institute for Nonlinear Studies, Akademiehof
Friedrichstr. 10, 1010 Vienna, Austria and
2Institute for Atomic and Subatomic Physics,
Vienna University of Technology
Operng. 9, 1040 Vienna, Austria
(Dated: March 14, 2014)
Abstract
By introducing the concepts of “superclassicality” and “relational causality”, it is shown here
that the velocity field emerging from an n-slit system can be calculated as an average classical
velocity field with suitable weightings per channel. No deviation from classical probability theory
is necessary in order to arrive at the resulting probability distributions. In addition, we can directly
show that when translating the thus obtained expression for said velocity field into a more familiar
quantum language, one immediately derives the basic postulate of the de Broglie-Bohm theory,
i.e. the guidance equation, and, as a corollary, the exact expression for the quantum mechanical
probability density current. Some other direct consequences of this result will be discussed, such as
an explanation of Born’s rule and Sorkin’s first and higher order sum rules, respectively.
Keywords: quantum mechanics, entanglement, interferometry, zero-point field
∗E-mail: ains@chello.at; Visit: http://www.nonlinearstudies.at/
1
arXiv:1403.3295v1 [quant-ph] 13 Mar 2014
1. INTRODUCTION
In 1951, Richard Feynman published a paper [1] claiming that classical probability theory
was not applicable for the description of quantum phenomena, but that instead separate
“laws of probabilities of quantum mechanics” were required. In describing the propagations
of electrons from a source Sto a location Xon the screen in a double-slit experiment,
Feynman wrote: “We might at first suppose (since the electrons behave as particles) that
I. Each electron which passes from Sto Xmust go either through hole 1 or hole 2. As a
consequence of I we expect that:
II. The chance of arrival at X should be the sum of two parts, P1, the chance of arrival
coming through hole 1, plus P2, the chance of arrival coming through hole 2.” However, we
apparently know from experiment that this is not so (Fig. 1). Feynman concludes: “Hence
experiment tells us definitely that P6=P1+P2or that II is false. (. . . ) We must conclude
that when both holes are open it is not true that the particle goes through one hole or the
other.”
X
PP1P2P1+P2
(a) (b) (c) (d)
Figure 1.1: Scheme of interference at a double slit: probability distributions for (a) both
slits open: P, (b) slit 2 closed: P1, (c) slit 1 closed: P2, (d) the sum of (b) and (c): P1+P2.
However, it was shown very clearly in several later papers, notably by B. O. Koopman in
1955 [2] and by L. E. Ballentine in 1986 [3], that Feynman’s reasoning was not conclusive.
In fact, Ballentine concluded that Feynman’s “argument draws its radical conclusion from
an incorrect application of probability theory”, as one has to use conditional probabilities
to account for the different experimental contexts: P1(X|C1), P2(X|C2), P(X|C3), where
2
the contexts C1,C2and C3refer to left, right and both slit(s) open, respectively. Then, it
is clear from the experimental data that P(X|C3)6=P1(X|C1) + P2(X|C2). Accordingly,
Ballentine also proved that “the quantum mechanical superposition principle for amplitudes
is in no way incompatible with the formalism of probability theory” [3]. Moreover, as one
can easily show with the inclusion of the wave picture for electrons (see Eq. (1.3) below),
one correctly obtains that P(X|C3) = P1(X|C3) + P2(X|C3).
Note that the focus purely on context-free “particles” in quantum mechanics has been the
source of much confusion. We recall the clear words of John Bell in this regard: “While the
founding fathers agonized over the question ‘particle’ or ‘wave’, deBroglie in 1925 proposed
the obvious answer ‘particle’ and ‘wave’. Is it not clear from the smallness of the scintillation
on the screen that we have to do with a particle? And is it not clear, from the diffraction
and interference patterns, that the motion of the particle is directed by a wave? De Broglie
showed in detail how the motion of a particle, passing through just one of two holes in a
screen, could be influenced by waves propagating through both holes. And so influenced that
the particle does not go where the waves cancel out, but is attracted to where they cooperate.
This idea seems to me so natural and simple, to resolve the wave-particle dilemma in such a
clear and ordinary way, that it is a great mystery to me that it was so generally ignored.” [4]
Particularly since the impressive Paris experiments in the classical domain performed by
Yves Couder, Emmanuel Fort and co-workers [5–7], it is very suggestive that some sort
of combination of “particle”- and wave-mechanics is at work also in the quantum domain.
Essentially, this amounts to obtaining particle distributions from calculating classical-like
wave intensity distributions.
With wave-field intensities Pifor each slit igiven by the squared amplitude Ri,i.e. with
Pi=R2
i, and with the phase difference ϕ, one obtains in the classical double slit scenario
for the intensity after slit 1
P(1) = P1+R1R2cos ϕ(1.1)
and for the intensity after slit 2
P(2) = P2+R2R1cos ϕ(1.2)
where the first expression on the r.h.s. refers to the intensity from the slit per se and the
second expression refers to interference with the other slit, respectively. This provides the
3
total intensity as
P=P(1) + P(2) = P1+P2+ 2R1R2cos ϕ. (1.3)
In the following we try to shed more light on these issues by combining results from the
new field of “Emergent Quantum Mechanics” [8] with concepts of systems theory which
we denote as “relational causality” [9]. Since the physics of different scales is concerned,
like, e.g., sub-quantum and classical macro physics, we denote our sub-quantum theory as
“super-classical”. (Both relational causality and superclassicality are to be defined more
specifically below in Section 2.) We consider the quantum itself as an emergent system
understood as off-equilibrium steady state oscillation maintained by a constant throughput
of energy provided by the (“classical“) zero-point energy field. Starting with this concept,
our group was able to assess phenomena of standard quantum mechanics like Gaussian
dispersion of wave packets, superposition, double slit interference, Planck’s energy relation,
or the Schr¨odinger equation, respectively, as the emergent property of an underlying sub-
structure of the vacuum combined with diffusion processes reflecting the stochastic parts of
the zero-point field [10–13].
In Section 3we contrast the well-known physics behind the double slit with an emergent
vector field representation of the observed interference field. The essential parts of our
superclassical approach are presented and the velocity field corresponding to the guiding
equation of the de Broglie-Bohm theory is derived. The explanation and validity of Born’s
rule is analyzed in Section 4by means of a three slit configuration. In Section 5we summarize
our results and give an outlook on possible limits of the validity of present-day quantum
mechanics from the perspective of our sub-quantum approach.
2. INTRODUCING “SUPERCLASSICALITY” AND “RELATIONAL CAUSAL-
ITY”
In quantum mechanics, as well as in our quantum-like modeling via an emergent quantum
mechanics approach, one can write down a formula for the total intensity distribution P
which is identical to (1.3). For the general case of nslits, it holds with phase differences
ϕij =ϕi−ϕjthat
P=
n
X
i=1 Pi+
n
X
j=i+1
2RiRjcos ϕij!,(2.1)
4
where the phase differences are defined over the whole domain of the experimental setup.
Apart from the role of the relative phase with important implications for the discussions on
nonlocality [13], there is one additional ingredient that distinguishes (2.1) from its classical
counterpart (1.3), namely the “dispersion of the wavepacket”. As in our model the “parti-
cle” is actually a “bouncer” in a fluctuating wave-like environment, i.e. analogously to the
bouncers of Couder and Fort’s group, one does have some (e.g. Gaussian) distribution, with
its center following the Ehrenfest trajectory in the free case, but one also has a diffusion to
the right and to the left of the mean path which is just due to that stochastic bouncing.
Thus the total velocity field of our bouncer in its fluctuating environment is given by the
sum of the forward velocity vand the respective diffusive velocities uLand uRto the left
and the right. As for any direction ithe diffusion velocity ui=D∇iP
Pdoes not necessarily
fall off with the distance, one has long effective tails of the distributions which contribute
to the nonlocal nature of the interference phenomena [13]. In sum, one has essentially three
velocity (or current) channels per slit in an n−slit system.
Earlier we showed that this phenomenon can also be understood as a variant of anomalous
diffusion termed “ballistic diffusion”: a Brownian-type displacement with a time-dependent
diffusivity (e.g., Dt=D2
σ2
0tin the case of a Gaussian wave packet with standard deviation
σ), leading to a “classically” obtained total velocity field
vtot (t) = v (t) + [xtot (t)−v(t)t]Dt
σ2,(2.2)
which is very practical to use in a computer simulation tool [14,15]. Moreover, ballistic
diffusion is a signature of what has in recent years become known as “superstatistics” [16].
Actually, the prototype of a phenomenon amenable to superstatistics is just a Brownian
particle moving through a thermally changing environment: Combining in one formalism
a relatively fast dynamics (e.g. particle velocity) on a small scale and slow changes (e.g.
due to temperature fluctuations) on a large scale leads to a superposition of two statistics,
i.e. superstatistics. One of the main features of superstatistics consists in emergent properties
arising on intermediate scales, which may be completely unexpected if one looks only at either
the small or the large scale.
The phenomenon just described, however, also provides us with the opportunity of solving
a terminological problem with regard to sub-quantum theories such as ours. For, although
we consider our approach as based on modern, “21st century classical physics”, the latter
5
Bouncer moving through the (thermally) changing environment of zero-point field
fast dynamics
(e.g. oscillation of bouncer)
on sub-quantum scale
slow changes
(e.g. in boundary conditions)
on macroscopic scale
superposition of two vastly separate domains of classical physics:
Superclassical Physics
…with unexpected emergent phenomena on intermediate scales: quantum phenomena!
Figure 2.1: Scheme of Superclassical Physics as applied to our bouncer model
must not be confused with the term “classical physics” as referring to the state-of-art of,
say, the early 20th century, including e.g. the time of the first experimental evidence of a
zero-point energy (viz. Mulliken’s work from 1924 [17], which itself was before the advent
of quantum theory proper). Acknowledging both that a) a zero-point field can in principle
considered a “classical” one and that b) quantum theory is characterized by some decisively
unexpected features when considered from a classical viewpoint, the phenomenon of super-
statistics thus suggests the following analogy for the modeling of quantum systems: quantum
behavior can be seen to emerge from the interplay of classical processes at very small (“sub-
quantum”) and large (macroscopic) scales. A combination of the physics on these vastly
different scales we call “superclassical” physics. The whole system under study then is char-
acterized by processes of emergence through the co-evolution of microscopic, local processes
(like the oscillations of a bouncer) and of macroscopic processes (like the time evolution of
the experimental boundary conditions). In other words, starting from physical processes
at such two vastly different classical scales, their superposition (i.e. within the framework
of superclassical physics) makes possible new types of phenomena due to emergent features
unexpected from either the very small or the large scale physics (Fig. 2.1; compare also the
related concept of “emergent relativity” as discussed, e.g., by Jizba and Scardigli, this vol-
6
Macroscopic Boundary Conditions
and
Emergent Macroscopic Structure
Local Microscopic Interactions
Figure 2.2: Scheme of Relational Causality: Co-evolution of bottom-up and top-down
processes (after [20])
ume, [18], and [19]). In our development of an emergent quantum mechanics, we shall thus
further on describe it as a superclassical approach, in order to avoid confusions when using
“classical” explanations.
As mentioned, our superclassical approach is particularly suited to account for the emer-
gent processes involved, i.e. for processes of a type that is well-known, for example, from the
physics of Rayleigh-B´enard cells. The latter appear in a fluid subjected to a temperature
gradient (like its container being heated from below) producing convection rolls, with the
emergent particle trajectories strongly depending on the boundary conditions: The form of
the convection rolls can be radically changed by changing the boundaries of the container.
Generally, emergence of this type is characterized by relational causality, i.e. the co-evolution
of bottom-up and top-down processes (Fig. 2.2).
In our emergent quantum mechanics approach, we have given a superclassical explanation
of interference effects at a double slit [12], thereby arriving at expressions for the probability
density current and the corresponding velocity field completely equivalent to the guidance
equation, which is the central postulate of the de Broglie-Bohm interpretation [21,22]. Our
7
claims, to be substantiated in this paper, consist in the assertion that the guidance equation
is of some “invisible hand” type, i.e. somehow mysteriously reaching out from configuration
space in order to guide particles in real three-space. Instead, we shall argue, the guid-
ance equation is completely understandable in real coordinate space, once the concept of
emergence is taken seriously and introduced within a superclassical approach.
Note that a basic characteristic of emergent systems can be described via the above-
mentioned relational causality (for a similar view, see [23]). Consider for example the fol-
lowing computer simulations (Figs. 2.3a and 2.3b). Similar to the experiments with droplets
by Couder and Fort’s group, we insert onto a fluid surface a droplet in the center of the square
constituted by four smaller squares of solid material, such that due to microscopic fluctua-
tions the developing bouncer/walker will propagate along one of the four narrow paths until
it finally escapes from the region of the square through one of the four slits into the open
area. With the bouncer creating waves that will be reflected from the walls (drawn in black),
after some time the whole system will develop standing waves between said walls, which act
as physically effective boundary conditions. In Fig. 2.3a the walls act as simple plane mir-
rors, whereas in Fig. 2.3b the reflecting part on the right contains a circular structure, with
the effect that the pattern of standing waves is altered. The figures display intensity distri-
butions of the wave field which, accordingly, coincide with probability density distributions
of finding the droplet at a specific location. Note that although the local physics in the
vicinity of the square would be identical for all four particle trajectories emerging from one
of the four slits if one disregarded the context, the pattern of the probability density field is
different in both cases. This is of course due to the fact that one cannot, in principle, neglect
the context, i.e. the boundary conditions create different standing wave patterns which in
turn interfere with the otherwise identical local processes. Relational causality means that
one must consider the co-evolution of processes stemming from the local slits and those from
the global environment, the latter including the macroscopic boundary conditions. One can
thus also speak of a “confluence” of different currents, i.e. those pertaining to the four slits
and those coming from the larger environment. Therefore, if one records the bouncers’ tra-
jectories and collects them in a synoptic manner, the whole velocity field will be a decisively
emergent one. It is the totality of the whole experimental arrangement, and not just the
classically local influences, that results in the behavior of the particle trajectories.
From this perspective, it is not very surprising that any relic of context-free modeling
8
(a) Illustration of relational causality under varying boundary conditions as described in the text,
with standing waves between plane mirrors (black).
(b) Same as (a), with modified standing waves due to modified geometry of mirrors. Note the
changes in the wave patterns around the squares despite the fact that the local physics there is kept
unchanged. These changes are solely due to the altered boundary conditions.
Figure 2.3
9
Figure 2.4: Quantum case streamlines (top) vs. geometrical rays (bottom). From [24].
Here it is demonstrated that by bringing together two sources, the temporal development
of the velocity field (and the corresponding “trajectories”) in the quantum case can be
understood only if considered as the result of a complex interplay of influences, updated at
each instant of the time evolution, i.e. as the result of emergent processes.
should run into difficulties. Ironically, it was an orthodox position that has argued against
the consequences of the more holistic approach acknowledging the context, i.e. when papers
on apparently “surreal trajectories” were published against the assertion of the de Broglie-
Bohm model that particles would not just follow trajectories as expected from context-free
propagation obeying simple classically-local momentum conservation (see, e.g., the assertions
of Scully [25]). The latter would apply only to geometrical rays, whereas the quantum
case streamlines are more complicated, a situation well-known from optics (see Fig. 2.4 for
illustration), and recently also confirmed via weak measurements in a double-slit experiment
involving photons [26].
Having thus introduced the notions of superclassical physics and relational causality, we
10
are now ready to reconsider interference at the double slit. Recall that to arrive at the
classical double slit formula for the intensities on the screen one had to just consider the
mean velocity v, i.e. there was just one channel (of the velocity, or the probability density
current, respectively) per slit. Because the currents represent wave propagations, this has
led to the total intensity of (1.3), P=P(1)+ P(2) = R2
1+R2
2+2R1R2cos ϕ. Now, however,
we are going to look for the superclassical interference formula. As mentioned, we now have
to deal with three channels per slit, due to the additional sub-quantum diffusion velocities
uLand uR, next to v, and all three co-evolving. Surprisingly, this is all that is needed to
arrive at the well-known quantum mechanical results which usually are obtained only via
complex-valued probability amplitudes.
3. A SUPERCLASSICAL DERIVATION OF THE GUIDANCE EQUATION
Considering particles as oscillators (“bouncers”) coupling to regular oscillations of the
vacuum’s zero-point field, which they also generate, we have shown how a quantum can be
understood as an emergent system. In particular, with the dynamics between the oscillator
on the one hand, and the “bath” of its thermal environment as constrained by the experi-
ment’s boundary conditions on the other hand, one can explain not only Gaussian diffraction
at a single slit [15], but also the well-known interference effects at a double slit [12,27]. As
already mentioned, we have also shown that the spreading of a wave packet can be exactly
described by combining the convective with the orthogonal diffusive velocity fields. The
latter fulfill the condition of being unbiased w.r.t. the convective velocities, i.e. the orthog-
onality relation for the averaged velocities derived in [15] is vu = 0, since any fluctuations
u=δ(∇S/m) are shifts along the surfaces of action S= const.
To account for the different velocity channels i= 1,...,3n,nbeing the number of slits,
we now introduce for general cases the generalized velocity vectors wi, which in the case of
n= 2 are
w1:= v1,w2:= u1R,w3:= u1L (3.1)
for the first channel, and
w4:= v2,w5:= u2R,w6:= u2L (3.2)
for the second channel. The associated amplitudes R(wi) for each channel are taken to be
11
the same, i.e. R(w1) = R(w2) = R(w3) = R1, and R(w4) = R(w5) = R(w6) = R2.
Now, relational causality manifests itself in that the total wave intensity field consists of
the sum of all local intensities in each channel, and the local intensity in each channel is
the result of the interference with the total intensity field. Thus, any change in the local
field affects the total field, and vice versa: Any change in the total field affects the local
one. In order to completely accommodate the totality of the system, we therefore need to
define a “wholeness”-related local wave intensity P(wi) in one channel (i.e. wi) upon the
condition that the totality of the superposing waves is given by the “rest” of the 3n−1
channels. Concretely, we account for the phase-dependent amplitude contributions of the
total system’s wave field projected on one channel’s amplitude R(wi) at the point (x, t)
with a conditional probability density P(wi). The expression for P(wi) thus constitutes the
representation of relational causality within our ansatz. Moreover, as usual one can define
a local current J(wi) per channel as the corresponding “local” intensity-weighted velocity
wi. Since the two-path set-up has 3n= 6 velocity vectors at each point (cf. Eqs. (3.1)
and (3.2)), we thus obtain for the partial intensities and currents, respectively, i.e. for each
channel component i,
P(wi) = R(wi)ˆwi·
6
X
j=1
ˆwjR(wj) (3.3)
J(wi) = wiP(wi), i = 1,...,6,(3.4)
with
cos ϕi,j := ˆwi·ˆwj.(3.5)
Consequently, the total intensity and current of our field read as
Ptot =
6
X
i=1
P(wi) = 6
X
i=1
ˆwiR(wi)!2
(3.6)
Jtot =
6
X
i=1
J(wi) =
6
X
i=1
wiP(wi),(3.7)
leading to the emergent total velocity
vtot =Jtot
Ptot
=
6
X
i=1
wiP(wi)
6
X
i=1
P(wi)
.(3.8)
12
Returning now to our previous notation for the six velocity components vi,uiR,uiL,
i= 1,2, the partial current associated with v1originates from building the scalar product
of ˆv1with all other unit vector components and reads as
J(v1) = v1P(v1) = v1R1ˆv1·(ˆv1R1+ˆu1RR1+ˆu1LR1+ˆv2R2+ˆu2RR2+ˆu2LR2).(3.9)
Since trivially
ˆuiRRi+ˆuiLRi= 0, i = 1,2,(3.10)
Eq. (3.9) leads to
J(v1) = v1R2
1+R1R2cos ϕ,(3.11)
which results from the representation of the emerging velocity fields, since we get the cosine
of the phase difference ϕas a natural result of the scalar product of the velocity vectors vi.
The non-zero residua of the other vector fields yield
J(u1R) = u1R P(u1R) = u1R (R1ˆu1R ·ˆv2R2) = u1RR1R2cos π
2−ϕ=u1RR1R2sin ϕ
(3.12)
and
J(u1L) = u1L P(u1L) = u1L (R1ˆu1L ·ˆv2R2) = u1LR1R2cos π
2+ϕ=−u1LR1R2sin ϕ.
(3.13)
Analogously, we obtain for the convective velocity vector field of the second channel
J(v2) = v2P(v2) = v2R2
2+R1R2cos ϕ.(3.14)
The corresponding diffusive velocity vector fields read as
J(u2R) =u2RP(u2R) = u2R(R2ˆu2R ·ˆv1R1) = u2RR1R2cos π
2+ϕ=−u2RR1R2sin ϕ,
(3.15)
J(u2L) =u2LP(u2L) = u2L (R2ˆu2L ·ˆv1R1) = u2LR1R2cos π
2−ϕ=u2LR1R2sin ϕ.
(3.16)
Note that the nontrivial sine contributions to the total current stem from the projections
between the diffusive velocities u1R(L) of the first channel on the unit vector ˆv2of the
13
convective velocity of the second channel, and vice versa. Combining all terms, we obtain
with Eq. (3.7) the result for the total current
Jtot =v1P(v1) + u1RP(u1R) + u1L P(u1L) + v2P(v2) + u2RP(u2R) + u2LP(u2L)
=R2
1v1+R2
2v2+R1R2(v1+v2) cos ϕ+R1R2([u1R −u1L]−[u2R −u2L ]) sin ϕ.
(3.17)
The resulting diffusive velocities uiR−uiLare identified with the effective diffusive velocities
uifor each channel. Note that one of those velocities, uiRor uiL, respectively, is always
zero, so that the product of said difference with sin ϕguarantees the correct sign of the last
term in Eq. (3.17). Thus we obtain the final expression for the total density current built
from the remaining 2n= 4 velocity components
Jtot =R2
1v1+R2
2v2+R1R2(v1+v2) cos ϕ+R1R2(u1−u2) sin ϕ. (3.18)
The obtained total density current field Jtot(x, t) spanned by the various velocity components
vi(x, t) and uiR(L)(x, t) we have denoted as the “path excitation field” [12]. It is built by
the sum of all its partial currents, which themselves are built by an amplitude weighted
projection of the total current.
Summing up the probabilities associated with each of the partial currents we obtain
according to the ansatz (3.3) and the relations (3.6) and (3.10)
Ptot = (R1ˆv1+R1ˆu1R +R1ˆu1L +R2ˆv2+R2ˆu2R +R2ˆu2L)2
= (R1ˆv1+R2ˆv2)2=R2
1+R2
2+ 2R1R2cos ϕ=P(v1) + P(v2).(3.19)
The total velocity vtot according to Eq. (3.8) now reads as
vtot =R2
1v1+R2
2v2+R1R2(v1+v2) cos ϕ+R1R2(u1−u2) sin ϕ
R2
1+R2
2+ 2R1R2cos ϕ.(3.20)
The trajectories or streamlines, respectively, are obtained according to ˙x =vtot in the
usual way by integration. As first shown in [12], by re-inserting the expressions for convec-
tive and diffusive velocities, respectively, i.e. vi,conv =∇Si
m,ui=−~
m
∇Ri
Ri, one immediately
identifies Eq. (3.20) with the Bohmian guidance equation and Eq. (3.18) with the quan-
tum mechanical pendant for the probability density current [28]. The latter can be seen as
follows. Upon employment of the Madelung transformation for each path j(j= 1 or 2),
ψj=RjeiSj/~,(3.21)
14
and thus Pj=R2
j=|ψj|2=ψ∗
jψj, with ϕ= (S1−S2)/~, and recalling the usual trigonomet-
ric identities such as cosϕ=1
2eiϕ+ e−iϕ, one can rewrite the total average current (3.18)
immediately as
Jtot =Ptotvtot
= (ψ1+ψ2)∗(ψ1+ψ2)1
21
m−i~∇(ψ1+ψ2)
(ψ1+ψ2)+1
mi~∇(ψ1+ψ2)∗
(ψ1+ψ2)∗
=−i~
2m[Ψ∗∇Ψ−Ψ∇Ψ∗] = 1
mRe {Ψ∗(−i~∇)Ψ},
(3.22)
where Ptot =|ψ1+ψ2|2=: |Ψ|2. The last two expressions of (3.22) are the exact formulations
of the quantum mechanical probability current, here obtained just by a re-formulation of
(3.18). In fact, it is a simple exercise to insert the wave functions (3.21) into (3.22) to
re-obtain (3.18).
Note that it is straightforward to extend this derivation to the many-particle case. As
the individual terms in the expressions for the total current and total probability density,
respectively, are purely additive also for Nparticles, a fact that is well-known also from
Bohmian theory, the above-mentioned “translation” into orthodox quantum language is
straightforward, with the effect that the currents’ nabla operators just have to be applied at
all of the locations xof the respective Nparticles, thus providing the quantum mechanical
formula Jtot (N) = 1
mRe {Ψ∗(−i~∇N)Ψ}, where Ψ now is the total N-particle wave function.
Again we emphasize that our result was obtained solely out of kinematic relations by
applying the superclassical rules introduced above on the basis of a relational causality,
i.e. without invoking complex ψfunctions or the like. Moreover, as opposed to the Bohmian
theory, we obtained our results not in configuration space but in ordinary coordinate space.
What looks like the necessity to superpose wave functions in configuration space, which
then are imagined to guide the particles by some invisible hand, can equally be obtained
by superpositions of all relational amplitude configurations of waves in real space, i.e. by
understanding the resulting system’s evolutions as processes of emergence.
Thus, with wi=J(wi)
P(wi)and the classical composition principles of Eqs. (3.6) and (3.7)
we have shown that the total velocity field is given in the simple form of a (super)classical
average velocity field :
15
vtot =Jtot
Ptot
=PiJ(wi)
PiP(wi)=PiwiP(wi)
PiP(wi).(3.23)
In other words, the guidance equation postulated by de Broglie-Bohm is here derived and
explained via relational causality, with vtot being an emergent velocity field.
4. THREE-SLIT INTERFERENCE, BORN’S RULE, AND SORKIN’S SUM RULES
The extension to three slits, beams, or probability current channels, respectively, is
straightforward. We just introduce a third emergent propagation velocity v3and its cor-
responding diffusive velocities u3L(R). The phase shift of the third beam is denoted as χ
and represents the angle between the second and the third beam in our geometric repre-
sentation of the path excitation field. According to Born’s rule the probability of even a
single particle passing any of the three slits splits into a sum of probabilities passing the
slits pairwise, i.e. going along both Aand B,Band C,or Aand C,but never passing A,
Band Csimultaneously.
Interference phenomena have recently been analyzed thoroughly for the cases of only one
open slit up to nopen slits by Sorkin [29]. For a double slit setup the interference term is
non-zero, i.e. IAB := PAB −PA−PB6= 0, with PA(B)being the detection probability with
only one slit/path Aor B, respectively, of a total of nslits/paths open, and PAB for both
slits Aand Bopen. This “first order sum rule” is to be contrasted with Sorkin’s results for
the following, so-called “second order sum rule” [29]:
IABC :=PABC −PAB −PAC −PBC +PA+PB+PC(4.1)
=PABC −(PA+PB+PC+IAB +IAC +IB C ) = 0.
This result is remarkable insofar as it can be inferred that interference terms theoretically
always originate from pairings of paths, but never from triples etc. Any violation of this sec-
ond order sum rule, i.e. IABC 6= 0, and thus of Born’s rule would have dramatic consequences
for quantum theory like a modification of the Schr¨odinger equation, for example.
Returning to our model, the total probability density current for three paths is calculated
according to the rules set up in Section 3. We adopt the notations of the two slit system
also for three slits, i.e. now employing nine velocity contributions: vi,uiR(L),i= 1,2,3.
Analogously, the three generally different amplitudes are denoted as R(vi) = R(uiR) =
16
R(uiL) = Ri,i= 1,2,3. We keep the definition of ϕas ϕ:= arccos(ˆv1·ˆv2), and we define the
second angle as χ:= arccos(ˆv2·ˆv3).Similarly to Eq. (3.10), the diffusive velocities uiR−uiL
combine to ui,i= 1,2,3, thus ending up with 2n= 6 effective velocities. Therefore we
obtain, analogously to the calculation in the previous section,
Jtot =R2
1v1+R2
2v2+R2
3v3+R1R2(v1+v2) cos ϕ+R1R2(u1−u2) sin ϕ
+R1R3(v1+v3) cos (ϕ+χ) + R1R3(u1−u3) sin (ϕ+χ)
+R2R3(v2+v3) cos χ+R2R3(u2−u3) sin χ(4.2)
and
Ptot =R2
1+R2
2+R2
3+ 2R1R2cos ϕ+ 2R1R3cos (ϕ+χ)+2R2R3cos χ(4.3)
=P(v1) + P(v2) + P(v3).
In analogy to the double slit case (cf. Eq. (3.19)) we obtain a classical Kolmogorov sum
rule for the probabilities on the one hand, but also the complete interference effects for the
double, three- and, as we have shown in [27], for the n-slit cases, on the other. However, the
particular probabilities P(vi) in Eqs. (3.19) and (4.3), do not correspond to the probabilities
of the assigned slits if solely opened, i.e. PAB (v1)=(R2
1+R1R2cos ϕ)6=PA(v1) = R2
1.
Consequently, each of the probability summands in said equations does not correspond to
an independent probability of the respective slit if solely opened, a fact that was already
clarified in our discussion of the issue of contexts in Section ?? .
Finally, we obtain for the cases of one (i.e. n=A), two and three open slits, respectively,
IA=PA(v1) = R2
1,(4.4)
IAB =PAB −PA(v1)−PB(v2) = 2R1R2cos ϕ, (4.5)
IABC =PABC −PAB −PAC −PBC +PA(v1) + PB(v2) + PC(v3)=0,(4.6)
where PAB is assigned to Ptot of Eq. (3.19) and PABC to Ptot of Eq. (4.3). In the double slit
case, e.g., with slits Aand Bopen, we obtain the results of (3.19). If Bwere closed and
Cwere open instead, we would get the analogous result, i.e. v2and ϕreplaced by v3and
ϕ1,3. If all three slits A,B, C are open, we can use the pairwise permutations of the double
slit case, i.e. A∧B,A∧C, or B∧C, respectively, with ϕ1,3identified with (ϕ+χ), etc.
17
Thus we conclude that in our model the addition of “sub-probabilities” indeed works and
provides the correct results.
Summarizing, with our superclassical model emerging out of a sub-quantum scenario we
arrive at the same results as standard quantum mechanics fulfilling Sorkin’s sum rules [29].
However, whereas in standard quantum mechanics Born’s rule originates from building the
squared absolute values of additive ψfunctions representing the probability amplitudes for
different paths, in our case we obtain the pairing of paths as a natural consequence of the
pairwise selection of unit vectors of all existing velocity components constituting the proba-
bility currents. Thus we obtain all possible pathways within an n-slit setup by our projection
method. The sum rules, Eqs. (3.3) through (3.8), guarantee that each partial contribution,
be it from the velocity contributions within a particular channel or from different channels,
accounts for the final total current density for each point between source and detector. Since
for only one slit open the projection rule (3.3) trivially leads to a linear relation between P
and R2, the asymmetry between the latter quantities, due to the nonlinear projection rule,
becomes effective for n≥2 slits open. Consequently, the violation of the first order sum
rule (4.5), i.e. IAB 6= 0, represents a natural result of our principle of relational causality.
Moreover, as we have argued above, the opening of an additional slit solely adds pairwise
path combinations. As all higher interference terms have already incorporated said asym-
metry, the result can finally be reduced to the double slit case, thus yielding a zero result as
in Eq. (4.6) according to Sorkin’s analysis.
This is a further hint that our model can reproduce all phenomena of standard quantum
theory with the option of giving a deeper reasoning to principles like Born’s rule or the
hierarchical sum-rules, respectively.
5. CONCLUSIONS AND OUTLOOK
We have previously shown in a series of papers [10–15,27,30] that phenomena of stan-
dard quantum mechanics like Gaussian dispersion of wave packets, superposition, double
slit interference, Planck’s energy relation, or the Schr¨odinger equation can be assessed as
the emergent property of an underlying sub-structure of the vacuum combined with diffu-
sion processes reflecting also the stochastic parts of the zero-point field, i.e. the zero point
fluctuations. (For similar approaches see the works of Cetto and de la Pe˜na [31, and this
18
volume], Nieuwenhuizen [this volume], or Khrennikov et al. [32].) Thus we obtain the quan-
tum mechanical dynamics as an averaged behavior of sub-quantum processes. The inclusion
of relativistic physics has not been considered yet, but should be possible in principle.
By introducing the concepts of superclassicality and relational causality, we have in this
paper shown that quantum phenomenology can be meaningfully grounded in a superclassical
approach relying solely on classical probability theory. Apart from an application for a
deeper understanding of Born’s rule, the central result of this work is a demonstration that
the guidance equation can be derived and explained within ordinary coordinate space. We
have proven the identity of our emergent velocity field vtot with the corresponding Bohmian
one, vtot(Bohm), and the orthodox quantum mechanical one, vtot(QM), respectively:
vtot(emergent) =X
i
wiP(wi)
X
i
P(wi)
=vtot(Bohm)=
R2
1v1+R2
2v2+R1R2(v1+v2) cos ϕ+R1R2(u1−u2) sin ϕ
R2
1+R2
2+ 2R1R2cos ϕ
=vtot(QM) =1
m|Ψ|2Re {Ψ∗(−i~∇)Ψ},with Ψ = Pjψj.
(5.1)
Finally, with our superclassical theory we can also enquire into the possible limits of
present-day quantum theory. For example, the latter is expected to break down at the time
scales of our bouncer’s oscillation frequency, e.g., for the electron ω≈ O (1021 Hz). As we
have seen, at the emerging quantum level, i.e. at times t≫1/ω, we obtain exact results
strongly suggesting the validity of Born’s rule, for example. However, approaching said
sub-quantum regions by increasing the time resolution to the order of t≈1/ω suggests a
possibly gradual breakdown of said rule, since the averaging of the diffusive and convective
velocities and their mutual orthogonality of the averaged velocities is not reliable any more.
In principle, this should eventually be testable in experiment. Moreover, upon the velocities
vand uL(R), introduction of a new bias, either in the average orthogonality condition, or
between the different velocity channels, the question may be of relevance whether these
would lead to the collapse of the superposition principle, as the assumed sub-quantum
nonlinearities would then become manifest. We have not discussed the important issue of
19
nonlocality and possible consequences with regard to the non-signaling principle here, and
refer the reader to the paper by Jan Walleczek (this volume) for consideration of some of
the topics in question.
ACKNOWLEDGMENTS
We thank Hans De Raedt for pointing out ref. [3] to us, Thomas Elze for the critical
reading of an earlier version of this paper, Jan Walleczek for many enlightening discussions,
and the Fetzer Franklin Fund for partial support of the current work.
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