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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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