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The double-slit experiment explained with the
algebraic approach
Ed Gerck
(December 26, 2019. This version: November 26, 2023.)
Physics
Planalto Research
Mountain View, CA, USA
ORCID: 0000-0002-0128-5875
Abstract—We present a new vision on the double-slit experi-
ment (DSE), that does not use any waves or the so-called wave-
particle duality. It is based on particles, viewed as localized vibra-
tions. The DSE is explained physically as a LASER, even though
it can be illumined even by a single particle, and in vacuum.
LASER means Light Amplification by Stimulated Emission of
Radiation. Mathematically, this work uses the algebraic approach
and directed graphs to explain logically the DSE for photons, and
other particles, including electrons. Rather than being illogical,
or mysterious, the approach reproduces logically the interference
patterns observed externally, using the well-known “amplification
of stimulated emission” (ASE). The conservation of particle
number is not observed. Different particles can present ASE
with in-phase or counter-phase behavior, making the patterns
observed for electrons, other particles, even atoms. Dirac’s dictum
that “each photon then interferes only with itself” is interpreted
in terms of an equivalence class, not as an identity. Interference
then happens between indistinguishable photons, created by
ASE. The Law of the Excluded Middle (LEM) is not broken
illogically in two-states macroscopically, but broken naturally in
three or more states, microscopically, using ASE. Any particle
supporting ASE, in-phase or counter-phase, can enter a DSE,
and provide an external interference pattern. Frequency plays a
role, while amplitude is not considered. This work could replace
the emphasis in quantum computing, from hardware to software,
saving cost, time, and welcoming more participants.
Index Terms—communication, tri-state+, algebraic, quantum,
classical, coherence, law of the excluded middle, laser, wave-
particle duality
I. INTRODUCTION
Feynman stated that the double-slit experiment (DSE)
“. . . has in it the heart of quantum mechanics.” and applies
to a wide diversity of particles, even atoms [1].
The DSE is usually understood as a demonstration
that light and matter can display characteristics of both
classically defined waves and particles. We diverge from this
interpretation, which is a mixed metaphor, and consider the
DSE as a LASER, where LASER means “Light Amplification
by Stimulated Emission of Radiation”. We model the DSE
mathematically, using the algebraic approach with different
edges and updating events, that merge in an ordinary
two-dimensional direct graph [2]. We view the DSE as a
demonstration of QM, which we explore in various aspects in
the next Sections, where the DSE shows many states at once
— following the interpretation of QM by Niels Bohr [3].
II. A “ONE PH OTO N DSE” ISNOT POSSIBLE
Theoretically, the general state inside a one-dimensional
DSE where only one slit aand slit bexist, is given by Ψ,
as the one-dimensional Schr¨
odinger equation [4]:
−~2
2m
d2ψ(x)
dx2= [E−V(x)]ψ(x).(1)
with the boundary conditions ψ(0) = ψ(∞)=0. Here, Ψis
also the coherent superposition of the solutions Ψaand Ψb,
where only slit aor slit bare open at the same time:
Ψ = 1
√2(Ψa+ Ψb).(2)
Thus, the behavior of systems described by the Niels Bohr
interpretation of QM1is counter-intuitive according to [8]
to our usual observed experience, because the coherent
superposition appears to break the LEM.
According also to [8], only one photon at a time can
exist in the DSE apparatus. In a well-known book, by Dirk
Bouwmeester, Arthur Ekert, and Anton Zeilinger, (Eds.), the
reference [8] says, in page 1, that ”...the interference pattern
can be collected one by one, that is, by having such a low
intensity that only one particle interferes with itself”.
This, however would visibly break the LEM, which means
breaking choice and causality. Choice is broken because one
cannot say which of the two slits the one photon can take in
the experiment, and there are physically only two slits. Any
attempt to determine this, would need an interaction with the
particle, which would lead to decoherence, that is, loss of
interference.
In addition to this choice problem, that remains undefined
until today in the DSE, causality seems broken because
one has Dirac’s well-cited dictum that ”each photon then
1Reference [5] also does not support the Copenhagen interpretation [6], [7].
interferes only with itself”, as mentioned in [8].
These two principles have guided much of QM and the
DSE to date, are used in QC, in qubits, and in research such
as [8].
However, as this work shows, there is no “one photon DSE”.
Both hypothesis are revealed to be certainly inconsistent and
yet lots of ink has been spilled in futile attempts to resolve
the QM paradox of the GSE. There is “trouble at the lab”,
in physics. This includes Nobel laureate Lamb, [9], and has
placed extraordinary challenges even for noted lecturers,
claiming that the wave-particle duality can be proven by the
DSE; that the DSE is somehow a demonstration that light and
matter can display characteristics of both classically defined
waves and particles, a mixed metaphor that confuses, as a
succession of incongruous or ludicrous comparisons.
We herald a different future, by diverging from these two
interpretations, which are both non causal. We understand
Dirac’s dictum in terms of an equivalence class, not as
an identity, preserving causality. Interference then happens
between indistinguishable photons in an equivalence class,
created by ASE [5], [10]. This then says that there is no “one
photon DSE”. One may indeed have such low intensity as to
inject only one photon in the DSE. But, even though injected
with a single photon, one may have 0, 1, 2 or more photons
inside, using ASE.
We claim later that this can happen also with electrons,
notwithstanding Pauli’s exclusion principle, and other
particles, even with atoms/molecules/plasma (objects).
In all these cases, the LEM appears to be broken, and
the situation seems unavoidable. It seems broken directly by
ASE. In creating three or more states by stimulated emission,
stimulated absorption, and emission. Thus, it seems broken
by default, by one not being able to tell after all which slit
was used, without causing decoherence. There is no YES
or NO answer possible, for each of the two slits, even with
ASE, one could think.
We also diverge from this interpretation. The LEM is
an inescapable bedrock of logical reasoning in binary logic.
But any particle supporting ASE, with three or more possible
logical states, thus breaking naturally the LEM, can be
measured (i.e, embed) in a DSE, as a fixed interference
pattern. The two-slits act as a two-state spatial filter, obeying
the LEM. Thus, LEM is always obeyed externally, by physical
constraint of the DSE, and one has only two possibilities to
choose from.
But three or more logic level systems can exist at the
same time, as asserted by Pierce, in unpublished notes, before
1910, which is the same QM principle later formulated by
Niels Bohr [3], and used here.
Charles Sanders Pierce is well-known to have soundly
rejected the idea that all propositions must be either True or
False, as in Boolean logic, the same as Frege in semantics
[11]. Pierce developed well-understood rules where the LEM
is not valid, including some truth tables. A modern treatment
can be seen in the results by Jones [12], and [13], with our
works in publication. But, three or more logical states impose
additional care in their application, since there is no LEM
to guarantee results internally, nor a truth-reference exists,
to measure success externally. This problem is well-known
in land and sea navigation, spacecraft navigation, astronomy,
and special relativity, and has been famously studied by
Poincar´
e [14] before QM, in terms of uncertainty, the inverse
of coherence, created by ASE. With the book “Quickest
Calculus”, three or more states are possible, and discontinuous
functions can now be differentiated if measurable, which
satisfy the mathematical concerns of this work [13].
III. AMPLIFICATION BY STIM UL ATED EMISSION: ASE
Einstein in 1916-1917 [5], [10], used Bohr’s [3] model to
famously argue that, in addition to the random2processes of
absorption and emission, a third, new, and coherent process of
stimulated emission must exist microscopically for physical
bodies, as a result providing experimental evidence for the
quantum, reproducing exactly the macroscopically studied,
experimental data.
Einstein considered that each change in an electron
orbit corresponds to the absorption or emission of one light
quantum, and so even normal light from a candle, a lamp, or,
radio wave, have a stimulated emission component. Stimulated
emission provided the basis for the later invention of the
laser (light amplification by stimulated emission of radiation)3.
In steady-state, the DSE is in local thermodynamic
equilibrium (LTE), and the Planck statistics must be
equal to the Boltzmann statistics [5], but LTE is not required
in general, just for the derivation. Thus,
0 = ˙
N=−AN1−B(N1−2)U(ν)(3)
where Nrepresents the number of objects in either state, 1or
2, A is the Einstein coefficient for the emission4component,
Bis the Einstein coefficient for stimulated emission, N1
is the number of objects in state 1, in the lower level, N2
is the number of objects in the state 2, in the upper level,
U(ν)is the density of the radiation field between levels 1
and 2, corresponding to the incoming photon in sympathetic
resonance, and their separation in energy is given by the
Planck formula E=hν, where h is the Planck constant,
and Eis the energy difference between levels 1and 2. For
simplicity we only consider two levels, and no degeneracy,
2Now defined in a causal way.
3More than 55,000 laser-related patents have been granted in the United
States.
4Also called spontaneous emission, without concern about causality.
but the result is the same.
Note that the intensity of the incoming photon field
was not used, which is consistent with the photoelectric
effect, and E=hν, while a wave would require that the
amplitude be used. Thus, the wave picture is not represented.
The ratio of the emission and the stimulated emission
by the incident photon is given by A/B in Eq.(3), and is
[10]:
A/B = 2hν3/c2(4)
in which coherence is favored at low frequencies.
Eq.(3) represents, in the second and third terms, what
can be seen as a minimal collective effect, of one photon
influencing the emission of another photon (see Section IV),
as a necessary coherent channel that needs to exist in order
to balance the first term. Stimulated emission and stimulated
absorption have been extended recently, as well-known, with
collective effects by other photons, such as superradiance and
superabsorption, into 5 states, but with no essentially new
process in the symmetries observed.
Eq.(3) represents a simplified view of the photon fields
inside the DSE, with the incident photon with frequency
ν, and one stage of “amplification by stimulated emission”
(ASE), where the incident photon can be:
1) removed by stimulated absorption, and thus be annihi-
lated,
2) copied by stimulated emission (ASE), necessarily
in-phase with the incoming photon, and be, thus,
interference-capable with the incoming photon, or
3) continue, but in coherence with (2).
This is shown in the Fig.(1) next.
In Fig.(1), the photons exit the DSE to the right, as three
different photons, two coherent and interference-capable
(cases 2 and 3), see the list above, visibly breaking the Law
of the Excluded Middle (LEM) (i.e., with three states). The
LEM, however, cannot be broken in two-states (i.e., only two
slits physically exist). And breaking the LEM in two-states
seems logically inconsistent, adding to the “counter-intuitive”
lore of QM, here dismissed by using ASE to explain photon
multiplication (instead of division, of trying to split the
incoming photon).
An object can nonetheless create a photon by stimulated
emission5through the influence of an out-of-range photon
in our model, such as by vacuum fluctuations or cosmic
rays, as well-known. But the photon is produced necessarily
out-of-phase, as an incoherent photon, and is not interference-
capable with the incoming photon (this is shown with the
color black in Fig.(1), and is not caused by the incoming
5Also called “spontaneous” emission, with no causality concern.
Fig. 1. Example of a photon represented in tri-state+ merging to two-states
in the DSE, with one stage of “amplification by stimulated emission” (ASE).
photon).
Dirac’s dictum that “each photon then interferes only
with itself” is now logically understandable, preserving
causality, in terms of an equivalence class, not as an identity.
Interference then happens between indistinguishable photons
(e.g., cases 2 and 3) in an equivalence class, created by ASE.
The DSE could still be seen as counter-intuitive, from
Fig.(1), viewing from the interference in two-state logic.
Because it appears to break the LEM6, while one still has
the LEM as the inescapable bedrock of logical reasoning
in binary logic. A natural question is then satisfied, that
three-or-more valued logic systems can be embedded in a
two-state logical system, as the interference pattern of a
DSE. The states are in different dimensions, and a continuous
path in the higher dimension (tri-state) must necessarily map
into a discontinuous path in the lower dimension (two-state).
This happens due to a well-known theorem in topology and
projection [15], that we call TR, standing for Topological
Relationship.
This achieves freedom from the LEM in behavior, inside the
DSE, while outside the DSE, constrained by the two-slits
of the DSE, the two photons can come from either slit or
from both slits, and the interference pattern can obey the LEM.
This completes our initial effort of trying to open the
“black box” in the quantum state by means of the DSE, with
further, better, logical analysis of the interaction process, and
we use the Wolfram models of ordinary oriented graphs [2],
intended to be as minimal and structureless as possible. Here,
is where one can hope to find a “new hypothesis”, where
6One cannot split the photon, but one can multiply it coherently with ASE.
one can make a wider causal sense, with the promise of
bringing much higher speed, cybersecurity7, and scalability
to communication.
It is therefore highly desirable to understand the model
of photon interaction, especially in the so-far mysterious and
illogical DSE, as key to quantum computation (QC), quantum
speed, and cybersecurity.
IV. PARTICLE VIEW AND LEAST ACTION: UNIVERSALITY
A phenomenon may be understood by two very different
points of view. First, one may direct the attention towards
the internal mechanics of the event. In this approach,
starting from outside the attention is focused to consider
the internal causes, i.e. the microsystems and their complex
microexchanges, that lead to the macrobehavior. Second,
one may observe the event only from the outside. In this
alternative approach, the attention remains on the external
aspects and the macrobehavior is understood only by its
macroexchanges through the use of conservation laws and
minimization principles [such as the PLA, the principle of
least action]. For example, the propagation of light in a
medium can be described by considering electromagnetic
waves and atoms or, by considering the Fermat’s principle of
least time. Both approaches can be used in order to obtain
a complete picture of the phenomenon. Since the second
approach is usually more concise and not concerned about
the many contributing mechanisms, general trends can be
easily derived without laborious calculations in this case [18].
The analysis presented so far, in the three former Sections,
has been along the lines of the first approach. In order to
bring more insight into the DSE model proposed in this
work using ASE [5], it is interesting to consider the same
problem, i.e. Eq. (2) or Eq.(3), using the second approach,
as universality in physics [19]–[21]. A further advantage is
to allow a direct particularization to other DSE cases, from a
more general view, as done for the electron in this work.
After absorption of a photon with energy hν, the object is in
an excited state with energy surplus, where now this means
also a possible dissociation energy of any breaking bond,
as photodissociation. Introducing a reasonable minimization
principle for the energy surplus given by
”The molecule or atom will evolve in time in such a
way as to dispose of the energy surplus in the least amount
of time.”
as considered in [18], one may look for conditions that
will enhance the production of excited atoms or excited
molecules. Observing that the above minimization principle
has been tested in hundreds of compounds [22], it is
equivalent to: ”The energy surplus will be preferentially used
7See The Cook Report on Internet [16], [17].
at the fastest energy channel available” as the guiding-idea
to enhance the emission of energy in the DSE.
Before Einstein [5], [10], a photon was considered part
of a formless “fluid”, or classical wave, that can only be
emitted or absorbed. Such was the philosophy at the time
of Young in 1801 [23], with the DSE [8], confirmed by
the well-known Maxwell equations, published by the Royal
Society in 1865.
This work considers that the current description of the
DSE, however, ignores the creation and annihilation of
photons, the ASE, while the LASER is well-known. The
existence for the ASE was calculated by Einstein about
100 years after the DSE, in 1917 [10], and occurs by wall
interaction as well, therefore even in vacuum. Although it is
possible to work with such low light intensities that only one
photon enters the DSE apparatus at a time, because amplitude
is not a concern for ASE, it is not possible to consider that
the number of photons is conserved during that time, even
at such low intensities. The opposite is experimentally true,
more so at low frequency (see Eq.(4)), so that we expect
that “the energy surplus will be preferentially used at the
fastest energy channel available,” which is usually stimulated
emission, with the emission of a particle, favoring ASE.
We discuss the DSE mathematically, following a model
of the type Wolfram introduced [2], with a particular rule,
and explain it with an algebraic approach, going from GF (3p)
to GF (2m), for a suitable m>p, where mand pare prime
numbers, and GF() stands for Galois fields, here considered
in a prime extension into finite integers. This is easy and fast
to calculate, with modular arithmetic.
But if such a rule is found, one might then go on and
ask why — out of the infinite number of possibilities — it is
this particular rule, or, for example, a simple rule at all.
For example, a rule that involves more or equal to ternary
edges (like our model for light, as tri-state+ [24]) cannot apply
to a state with only binary edges (like qubits). Thus, any rule
that cannot become simple for all cases, is less favored. And
like with updating events, branches with different sequences
of rules applied may reach equivalent states, and thus merge,
as in stimulated emission providing an in-phase copy of the
incident photon, and as we found also for electrons, discussed
later, where the Pauli exclusion principle forces a “contrarian
coherence”.
Continuity or classical waves, are not needed, do not
fit into this philosophy of ASE, and are not representable
by any edges or updating events, or sink, in the oriented
graph [2]. However, taking into account the principle of
universality in physics [19]–[21], the same phenomena —
even the LASER — can be seen, although approximately
and partially, in terms of continuous waves, macroscopically.
This solves the apparent confusion given by the common
wave-particle duality interpretation of the DSE [8], where the
ontological view of the DSE becomes now, however, as ASE
dictates — more indicative of a particle in all cases — and
does not depend on amplitude.
This is also confirmed by the particle interpretation of
the well-known photoelectric effect, which does not depend
on amplitude. The energy of a particle in the case of a
photon is revealed by the photoelectric effect to be equal to
E=hν, where νis the frequency of the photon. A particle
is a localized vibration, and can be a photon, an electron, a
quark, or other manifestation. Wave theory, in contrast, says
that the wave extends to infinity, and the intensity should
be proportional to the square of the amplitude of the wave.
Continuity or classical waves are, thus, less favored as a
general description, for wider causal sense. But can be used
in universality, as in the foregoing, for the macroscopic sense.
This leads to a discontinuity, such as three or more
states transitioning to two-states, and a logical explanation
under different rules (including both the absence and presence
of the LEM), of all observations, separating behavior from
implementation. The foregoing provides for the DSE a similar
explanation for electrons, or other particles, notwithstanding
the well-known Pauli exclusion principle.
V. TH E DSE
The foregoing creates an opening to take a “new hypothesis”
here.
New Hypothesis: The DSE exemplifies a LASER.
Fig. 2. Example of a photon represented in tri-state+ merging to two-states
in the DSE, with five stages of ASE in avalanche process, creating a LASER.
This means that the LEM is not broken in two-states,
macroscopically, but naturally in three or more possible
states, microscopically. The details of the microscopic, even
breaking the LEM, should not be so relevant, though, to
the macroscopic two-state behavior and later in asserting
the LEM, considering universality [19]–[21]. The same
universality can be affecting the current DSE interpretation,
that people are not seeing the microscopic, the LASER effect,
observing it solely through what can only ever be a far-away
reference frame, the DSE output in the far-field.
Here, obeying the LEM macroscopically acts as a “Procrustean
bed”, where binary logic, an arbitrary standard, is used to
measure success externally, while completely disregarding the
obvious over-simplification that results from the effort, such
as three states or more, in a microscopic view, where the
quantum operates more decisively. But, why bother with the
quantum, when it will not matter for what is seen? Because
without it, there is no ASE, and its lessons.
Fig.(2) shows multiple stages of ASE for in-phase
multiplication in the DSE, more likely at lower frequencies
[5], due to stimulated emission prevalence, as the fastest
channel to reduce the surplus energy from the incoming
photon, before the object is even changed by absorption.
Stimulated emission can now be considered for coherent
and secure traffic. Between the entrance and the exit of a
DSE, which builds an end-to-end communication link that
one cannot disturb under penalty of changing the result,
one is not only capable of routing photons, but also of
coding incoming photons to produce coded outgoing ones,
in a coherent process, which creates the very conditions for
external interference of the additional photons created in the
process. This reduces the influence of white noise, and is
highly resilient to low power — interference is produced
by frequency, not by amplitude — contrary to classical waves.
The fundamental insight is that a photon exiting the apparatus
can be inferred,estimated, coherently by another photon from
the same or other slot, at a far away screen, and does not even
have to be an exact, verbatim copy of the incoming photon. A
similar concept is found in the well-known spread spectrum
techniques, and in cybersecurity [25], where anchors can be
used to correct the information received from the source using
various tools, e.g., using majority voting, for added coherence.
The significant aspect in QC, as the result of coherent
superposition in Eq.(2), and Eq.(3), is still that the actual
photon in one slit is one selected from a set of possible,
like photons at the same, or the other, slit. This is achieved
by coherence in Eq.(2) or Eq.(3), whereby the photon
is copied in-phase inside the apparatus, using ASE, and
become indistinguishable with each other, as both exit. As a
consequence, the photon has the proper frequency and phase
for interference, externally.
The once fuzzy concept of the “DSE” or wave-particle
duality, is now proposed in a precise way as stated above,
as a LASER, and using binary logic, with the LEM being
strictly valid. It does not matter which slit is used for exit,
the same slit, and/or the other slit, can now be used by two or
more photons, and, likely, there are plenty of similar photons
made available through ASE, as shown in Fig.(2).
Previous work has been included, selectively, in the references
given so far. However, they are necessarily inconsistent when
applied to the DSE, which can be seen as a consequence of
using the symmetries of a binary system for the photon, that
must use the LEM and binary logic. These are insufficient
— as Einstein already explained [5], [10], and led to the
LASER. The number of photons must not be constant inside
the apparatus, breaking the LEM naturally.
VI. OTH ER PARTICLES IN THE DSE
This work advances experimentally in binary logic, the
observation that, the same result for the DSE has been
observed for electrons, and other particles, even atoms, not
just photons.
We suggest that the Pauli exclusion principle acts as a
“boundary condition” restricting the very act of emission. So
that, even though emission is not formed by in-phase particles,
like in the case of photons, it is influenced, in a contrarian
fashion, by the opposite state. These created particles are,
then, particle in-phase to themselves, forming like-emitted
particles. This is a process which we can also call “stimulated
emission”, and include in ASE. Through the opposite state,
this creates three (or more) logical processes, with two
(or more) coherent with each other, in consistent behavior,
not randomly occurring. The unity in the macroscopic
spatial interference pattern points (again, as with photons)
to a coherent process that must exist microscopically, and
based on frequency, not amplitude (e.g., governed by QM).
Cooperation externally, betrays cooperation internally. There
must be a cause for cooperation, when viewed another way.
This allow us to separate behavior (e.g., the Pauli exclusion
principle) from observed interference pattern, in ternary (or
more) logic systems.
The argument, in “modus ponens”, is that, a coherent
logic state, building a “coherent channel”, should exist also
for electrons (and any other particle different from a boson),
although seen through what has to remain a distant binary
logic system. This happens in order to be able to model the
DSE pattern that is seen in the binary pattern, analogous to the
experimental fact that a physical state of stimulated emission
must microscopically exist in the quantum communication of
photons, as well-known by Einstein [5].
But this further establishes, in “modus tollens”, a physical
unity between any theories for other particles, of the DSE,
obeying different laws. We can use this to provide a model
for QM using ASE, in-phase or in counter-phase, reducing
decoherence.
The states are in different dimensions, and a continuous
path in the higher dimension (tri-state) must necessarily map
into a discontinuous path in the lower dimension (two-state).
This happens due to a well-known theorem in topology and
projection [15], that we call TR.
In QC, one can be more precise than physical QM if
one makes the model, as the behavior, be more inclusive
for coherence, measured microscopically, even though
macroscopically one should be limited to use GF (2m)and
use the LEM. Hence, QC promises to be easier to delve
deeper than QM using hardware.
One feels, with the foregoing, the need to introduce
more symmetries than GF (2), or binary logic, in modeling
particles by software. No longer should we be led in software,
to regard particles as a formless “fluid”, or classical wave,
obeying the LEM as a “Procrustean bed”.
Here, the role of an added mathematical apparatus as
discussed here for Galois fields, is not to create unnecessary
complications in a description of reality, but implies that
there exist more adequate and representative pictures of
reality where these other number fields can be used as basic
elements of the mathematical description [21], in universality.
Accordingly, one moves in the macroscopic, from a
classical Boolean analogy, valid for the LEM and a formless
and classical “fluid” model of particles, seen in the DSE as
GF (2m), to a more complex microscopic structure, with
a quantum tri-state+, where the photon (e.g., a particle) is
given by an algebraic approach with ternary object symmetry,
modeled by GF (3n).
VII. PHOT ON MO DE L: UNIVERSALITY
In consideration of the foregoing, a particle, as we shall
consider a photon, is described by its eigenfunction, as a linear
combination in the set {e−αx, xe−αx , x2e−αx}[26], where α
is calculated variationally. It is, therefore, localized, obeys the
Schr¨
odinger equation by construction [26], does not attempt
to go to infinity like a sine function, but when combined with
many other similar particles, can equal a sine function as close
as one wants, building universality. A wave, in this concept,
is a collective construct, not a primitive object like a particle.
The mathematical operators curl, laplacian, and so on, are
definable based on that constructed wave, and the Maxwell
equations as well as classical electrodynamics can follow also.
Thus, first comes the particle in QM, then the wave is
built. There is no “duality” in this model of the photon, but
universality.
VIII. TRI -STATE VERSUS TWO -STATE
This work advances experimentally in well-known binary
logic, the observation that, for the same function, computation
can be accomplished better even classically, by using three
logical states, rather than one can do with binary logic. This
observation can be useful here.
The two-state logic levels are given in Fig.(3), offering:
(1) a low-level state “0” when the lower transistor is on and
the upper transistor is off; and (2) a high-level state “1” when
the upper transistor is on and the lower transistor is off. The
situation is analogous to the double-site experiment. There
are only two options, and the LEM must be valid.
+
Fig. 3. Example of two-state levels in a circuit, 0 and 1. LEM must be valid.
To implement three state logic, a physical case seems
impossible, as in the DSE, without changing physical
conditions. But in the chip set environment, a conventional
tri-state buffer or gate8can be used, even with FPGAs [27].
This can be seen in Fig.(4), showing the three cases in
positive logic.
Fig. 4. Example of three states logic: 0, 1, Z.
Using state Z to behave as a coherent interconnect, current
information technologies using challenge-response systems, in
tri-state chip sets or even using FPGAs9, as in a SystemVerilog
design [28], [29], can have a semantics to connect to different
systems, can avoid race-conditions, handle faults, and
maintain a coherent design across different systems. These
aspects can also be programmed dynamically at operation
time, using tri −stateT M designs.
8Such as the 74LS241 octal buffer.
9An FPGA cannot use tri-state
The use of three logical states at the same time, in digital
circuits, is perhaps surprising, but well-known experimentally
in complex digital systems [29], and allow designers to
separate behavior from implementation at various levels
of abstraction, in order to achieve, routinely, million gate
chip designs while working with tri −stateT M using
SystemVerilog [28], a ternary logic system.
But using tri-state logic by such clever stratagem as in
Fig.(4), seems physically impossible in the DSE, and so it
seems that there is no logic synthesis possible, to reach it.
However, logic synthesis is possible, by recognizing three or
more states [5], inside the apparatus, using ASE, and tri-state+
behavior, allowing implementation as two-state using the
DSE, where the two-slits act as a two-state spatial filter,
which produces the common interference pattern outside, in
the far-field.
IX. DISCUSSION
By providing a well-known “Procrustean bed”, the DSE
shuts off indeterminacy, without allowing an observer to
measure it, and arrives at a apparently clean and determinate,
two-state result, that applies to many different particles, even
atoms.
This work’s “new hypothesis”, in trying to open the “black
box” of QM, is complete in the foregoing — namely, the
DSE is explained physically as a LASER. Then, any particle
supporting ASE, in-phase or counter-phase, can enter a DSE,
and provide the external interference pattern. No wave, nor
wave division is assumed, but particle multiplication, by ASE.
This is shown in Figs.(1) and (2). In other words, that the
LEM is not broken by the DSE in two-states macroscopically,
but naturally in three or more possible states, microscopically,
using ASE.
Thus, coherence effects should be used in particle interaction,
and the DSE proves it, rather than imagining a spurious
wave-particle duality, which would also be a mixed metaphor.
But wave-particle duality can be seen as another example of
universality in physics, allowing one to use continuity, albeit
continuity does not actually exist in the known universe,
being a collective construct.
As another task opened by this work, multilevel logic
and mathematics formulas, and software, need to be described
and implemented to take full advantage of tri-state+, yet
using binary, LEM computers to implement, as we already
have them today. This could replace the emphasis in QC,
from hardware to software, saving cost, time, and welcoming
more participants.
To the current insistence on using bits and qubits, Einstein,
in 1917 [5], proved that a binary signal is not enough by
reproducing the law of radiation found in 1900 by Max
Planck. That required a third, coherent channel to exist,
invalidating qubits before their concept started. It was a
historical mistake by Shannon [30], of difficult consequences
until today and highlighting the importance of correct
technical work, some 50 years later. This now led to bits
and qubits being seriously considered and spilling lots of ink
on them, with reputations and careers being lost. But facts
cannot be swayed by will.
The mistake was first that Information could be treated
as a fluid, which can only be blocked or let pass, as a
relay. This was against the theory of line formation, from
Max Planck and Einstein, that required three processes –
so-called ”spontaneous” emission, ”spontaneous” absorption,
and stimulated emission (today, all three processes are viewed
as stimulated emission, in different energy ranges). That led
Shannon to consider Boolean logic for information. This
second mistake led people to qubits, the third mistake in
treating information. One must use tri-state+ to represent
Information, classically or quantically.
This research received external funding from DCTA/ITA,
DCTA/IEAv, CAPES/CNPq, CNEN, Fapesp, Network
Manifold Associates, Inc. (NMA), Planalto Research, the
Max-Planck Institut f¨
ur Quantenoptik, the Bundesministerium
f´
ur Forschung und Technologie, and Euratom. The author
also declares no conflict of interest.
Acknowledgments: The author is indebted to Fellow of
the RAS Peter Jackson, Software Engineer Andr´
e Gerck,
Tiffany Gerck Project Manager of Planalto Research, Edgardo
V. Gerck doctorate student, and two anonymous reviewers.
ResearchGate discussions were also used, for “live” feedback,
important due to the physical isolation caused by COVID.
Ed Gerck Lead his own doctorate in quantum optics
and in postdoc research, at the MPQ (Max-Planck-
Institut f¨
ur Quantenoptik) in Munich in 1979-83,
while leading others to participate and change. Ob-
tained the Judo black-belt in 1971, the Dr.rer.nat.
degree in 1983 with maximum (”sehr gut”) thesis
grade in the Physics faculty of the LMU (Ludwig-
Maximilians Universit¨
at) in Munich, the Electronic
Engineering in 1978 and the M.Sc. degree in Physics
in 1979 from the elite ITA (Instituto Tecnol´
ogico de
Aeron´
autica). Brazilian Air Force lieutenant, reserve.
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