CC BY
19
DOI: 10.33693/2313-223X-2021-8-3-13-18
On the Physical Representation of Quantum Systems
E. Gerck ©
Planalto Research, Mountain View, CA, USA
E-mail: [email protected]
Abstract. The Schrödinger equation for bound states depends on a second derivative, that only exists if the solution is continuous, which is - by itself - contradictory, and cannot be digitally calculated. Photons can be created in-phase by stimulated emission or annihilated by spontaneous absorption, and break the LEM, more likely at lower frequencies, and even in vacuum. Thus, the number of particles is not conserved, e.g., in the double-slit experiment, even at low-light intensity. Physical representations of quantum computation (QC), cannot, thus, follow some customarily assumed aspects of quantum mechanics. This is solved by considering the Schrödinger equation depending on the curvature, which is expressed exactly as a difference equation, works for any wavelength, and is variationally solved for natural numbers, representing naturally the quantum energy levels. This leads to accepting both forms in a universality model. Further, one follows the Bohr model in QC, in a software-defined QC, where GF (2m) can be used with binary logic to implement in software Bohr's idea of "many states at once", without breaking the LEM, in the macro, without necessarily using special hardware (e.g. quantum annealing), or incurring in decoherence, designed with today's binary computers, even a cell phone.
Key words: bound states, qubit, qutrit, qudit, tri-state+, information, algebraic, quantum, classical, coherence
Acknowledgments. The author is indebted to Edgardo V. Gerck doctorate student, and four anonymous reviewers. Research Gate discussions were also used, for "live" feedback, important due to the physical isolation caused by COVID.
f -\
FOR CITATION: Ed Gerck. On the Physical Representation of Quantum Systems. Computational Nanotechnology. 2021. Vol. 8. No. 3. Pp. 13-18. DOI: 10.33693/2313-223X-2021-8-3-13-18
v J
1. INTRODUCTION
In trying to open the "black box" in the quantum state, with further analysis of the interaction process where the data can make a wider causal sense, we hope to better understand the limitations about quantum processes. The development of nanoelec-tronics devices, when nanoprocesses needs to involve quantum computing, also needs prediction of the structure of matter. But, if the second derivative is to be included in quantum mechanics (QM), even though the second derivative only exists if the solution is continuous, which is - by itself - contradictory [1]. The standard justification for using derivatives with the wave function describing a discrete behavior on/off without even continuity, is that ^ represents an average behavior, in the Bohr interpretation.
But this leaves out many behaviors that are not continuous, and they do not have to be continuous for ^ to be an average, even for a continuous function interpolating isolated data points. A number of contradictions then arise from the use of infinitesimal analysis in QM, in particular, for the core Schroding-er equation, the very applicability of which turns out also to be limited by non-compliance with the conservation of the number of particles. We offer ways to overcome these contradictions. In particular, on the basis of replacement of the Schrodinger equation by an exact difference scheme, which results in a curvature representation, Eq.(4), that does not use continuity and yet can reproduces all known behaviors. It then reveals a remarkable behavior of all energy levels En, that they all scale as
En = an + b + — + O [-1 |, n \ n J
where n is the quantum number. This allows scaling laws to be calculated, even for very complicated potentials, such as for Rydberg levels.
Besides, a continuous solution, even as the average cannot be digitally calculated, so that any digital code must be seen necessarily as an approximation of some expected (although mythical) "analog", continuous code. These two contradictions, and more, directly impact quantum computing (QC) and has diverse manifestations. In particular, in the need for renormaliza-tion in quantum electrodynamics.
The second derivative is indeed included, e.g., in the expression of the one-dimensional Schrodinger equation for bound states in QM [2]:
h2 d2 y(x) 2m dx2
[E - V (x )] x ),
(1)
where E is the energy and V(x) is the potential, with the boundary conditions ^(0) = = 0.
Besides the continuity question (i.e, with 'lack of continuity' and 'code is not continuous'), Eq. (1) brings in the Law of the Excluded Middle (LEM) as a third contradiction, in 'breaking the LEM'.
The LEM is broken in the double-slit experiment - reported in [3], because one can't say which of the two slits a particle takes in the experiment. Any attempt to determine this, would need an interaction with the particle, which would lead to decoherence, and consequent loss of interference. This as was concluded in [3], and one must also consider the well-known particle creation and annihilation. Photons can be created in-phase
КВАНТОВЫЕ СТРУКТУРЫ И КВАНТОВОЕ МОДЕЛИРОВАНИЕ
QUANTUM AND MOLECULAR COMPUTING AND QUANTUM SIMULATIONS
by stimulated emission or annihilated by spontaneous absorption, more likely at lower frequencies, even in vacuum, as predicted by Einstein [4; 5]. Thus, which effect must depend on wavelength, this represents a fourth contradiction - 'number of particles is not conserved'.
E.g., in the double-slit experiment, this means that one cannot say that only one photon exists in the apparatus, contrary even to [6]. There, in a well-known book, by Dirk Bouwmeester, Arthur Ekert, and Anton Zeilinger (eds.), they say, 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".
Why this would break the LEM? Because we can't say which of the two slits a particle takes in the experiment. Any attempt to determine this, would need an interaction with the particle, which would lead to decoherence, that is, loss of interference.
This reasoning is correct, but relies on having only one particle in the experiment, which is questionable. One may indeed have such low intensity as to inject only one particle in the experiment, but the particle can multiply in-phase, indistinguish-ably, by stimulated emission, or be annihilated, by spontaneous absorption. This happens more likely at low frequencies, and can happen in vacuum, as the experiment has at least one wall with two slits. Thus, even though injected with 1 particle, one may have 0, 1, 2 or more particles inside. In these cases, the LEM is also broken, and the situation cannot be avoided. The number of photons is not conserved. The reference perhaps considers, naively, that the number of particles is constant. But the LEM is, nonetheless, broken. It is broken by stimulated emission, which can produce an extra particle, and broken by default, by not being able to tell which slit was used. There is no YES or NO answer possible, for each of the two slits. That the reference would "forget" about stimulated emission is, nonetheless, incorrect. The reader is advised, though, that the LEM is indeed broken anyway.
Now, it becomes more forceful, as breaking the LEM is further helped by photon multiplication, producing two or more photons out of one, in the same phase space, using stimulated emission. The LEM is broken without any doubleslit, by the very existence of a third, coherent state, as found by Einstein in 1917 [5; 6]. These two or more "identical" photons, as Einstein found out in the B coefficients, are more likely in lower frequency. Albeit, the external field can be theoretically calculated, as follows.
For light (i.e., a photon) interacting with a double-slit, the general external state as understood by Eq. (1) is given by Here, ^ is also the coherent superposition of the solutions ^ and , where only slit a or slit b are open at the same time:
1
Y = (С0 + Y "
(2)
Thus, the behavior of systems described by the Niels Bohr interpretation of QM [7] does not reproduce classical physics in the limit of small quantum numbers, although it reproduces for high quantum numbers, being counter-intuitive [6] to our usual observed experience, in those small numbers (see Universality discussion). With the Copenhagen interpretation [8; 9] in lieu of the Bohr interpretation, this would contradict also the observation of a particle to have matter or charge, such as electrons, protons and neutrons, or subatomic, and consider a spurious special role by one arbitrary, subjective observer producing an objectively, important to all, solipsistic "collapse". This leads us to consider, instead, the model of spatial averages in the Bohr interpretation, of the quantum level probability
density function, as representing the particle in classical physics, although the former four contradictions remain.
The reason the double-slit experiment is counter-intuitive is because it breaks the LEM.1 One cannot split the photon at the double-slit experiment, notwithstanding Huygens and all classical considerations, such as the Maxwell equations (ME) in any form, even when represented by relativistic equations for the field strength tensor, with B and E using the same units [3].
On these considerations, some computational aspects of quantum mechanics are to be hopefully improved. This will lead us to consider an equivalent form of Eq. (1) in an extended algebra approach, depending on the curvature, and valid for any frequency, which can be expressed exactly as a first-order difference equation, and is variationally solved for natural numbers, representing naturally the quantum energy levels.
While many are considering a far-future and expensive hardware solution with quantum annealing for QC, this work on QC sees the noted four contradictions of:
• 'lack of continuity',
• 'code is not continuous',
• 'breaking the LEM', and
• 'number of particles is not conserved'.
These are four openings to consider a "new hypothesis" here, promising a new, coherent basis for QC.
A shorter version was used in the actual presentation, and is available online at [10].
2. UNIVERSALITY AND INFINITESIMALS
Although QM unquestionably requires discrete values and breaking the LEM, it has been presumably accurate when using continuity and LEM to calculate them. But both sides of an equation representing a physical relationship with discrete values and no LEM, such as A = B, QM must be kept discrete and breaking the LEM when the frame of reference changes and the so-called continuum condition is denied.
To change the reference frame, a well-known theorem of topology [11], which we call Topological Relationship (TR), says that a generic one-to-one mapping between spaces of different dimensionality must be discontinuous, in that a continuous path in one space must map into a broken path in the other. The consequences here are multiple, and this is being explored in [3] as well. Thus, the mathematical condition seems to imply the physical condition, and continuity (e.g., including all four contractions, with LEM), denied in one frame, is to be denied in all.
It would be desirable, therefore, to isolate those aspects of the current QM theory that involve such continuous quantities in Eq. (1), if they exist, and are subject to modification by a more satisfactory theory, from aspects that involve only discrete values that do not obey the LEM, and are thus relatively more trustworthy.
This is a candidate for such a more satisfactory theory of QM.
Historically, however, it is well-known that Eq. (1), can correctly describe the evolution of bound states of a quantum physical system for high frequencies, and also work in terms of quantum, discrete variables, which results would need to be preserved under the "new hypothesis". This item argues that this is possible under a difference in scale as universality,
1 Without photon creation or annihilation, effects more likely at lower frequencies, as well known. The same interference pattern would be observed with just one particle at a time (e.g., one photon) in the experiment, so that the photon must interfere with itself in this case.
and one thus may not be even able to bother with finer details when only seen at a distance.
If the world is quantum or not, anyway Newton's calculus and real numbers lead to infinitesimals and infinities, which were shown by Brillouin [12] to be non physical. No one can physically push spacetime arbitrarily close to zero, without limit, or exclude particle creation or annihilation at low frequencies. Galois fields and a finite difference approach can be used, to build an alternative to conventional calculus, without infinitesimals or the limit concept, so that the conventional approach of [1] need not to be the only approach to analysis, for physicists, as we shown in Section 3 on the curvature representation. The derivative and integral formulas, however, remain the same. The new finite difference approach can be exactly accurate and yet there is always a space between integers, which represent different points in spacetime.
However, while physics shows that nothing is continuous in nature and, although non physical, one can keep using infinitesimals, continuity, and irrationals in mathematics, aka continuity. No one needs to change the traditional treatment [1] of analysis (i.e., calculus) or limits. This is because of TR, or Topological Relationship, a well-known theorem in topology where a generic one-to-one mapping between spaces of different dimensionality must be discontinuous. Therefore, a higher dimensional state can embed in a lower-dimensional state, as well-known in topology, projection, and physics, although it is subject to TR.
Thus, the universe can have singularities, be quantum at the core, and yet reality is the consequence of a continuous-looking universality as discussed in Section 2, where we observe this through what can only be an ever far-away reference frame [13; 14], where we observe this through what can only be an ever far-away reference frame. Universality as the reason for the "black box" in the quantum state. The details of the microscopic, even breaking the conservation of the number of particles, and the LEM, should not be so relevant to the macroscopic behavior and asserting the LEM, in universality [13; 14]. Can the same be affecting QC, and that is why we do not see a strong microscopic effect resulting from a, even though coherent, microscopic cause?
Different from the interaction with a double-slit, as seen above, when a photon interacts with matter as in [4], one needs to consider not just passing or not passing an aperture in a wall, but further consider the equilibrium of the photon field with the material in the wall, even in vacuum. In 1916-1917, Einstein took this latter case and [5; 6] famously argued that, in addition to the random processes of spontaneous absorption and spontaneous emission in Eq. (1), a third, new, and coherent process of stimulated emission must exist microscopically for physical bodies, as a result providing experimental evidence for the quantum, allowing photon creation in-phase2 and annihilation.
Einstein contradicted, thus, the well-known Maxwell equations, and reproduced exactly the experimental studies of the thermal, statistical radiation of bodies in quantum communication, which provided the basis for the later invention of the laser (light amplification by simulated emission of radiation).3 This is the so-called black-body radiation law,
2 Stimulated emission, so that a photon still only interacts with a photon in the same phase space, i.e., 'identical'.
3 More than 55 000 laser-related patents have been granted in the United States.
macroscopic, and even normal light from a candle, a lamp, or, a radio wave, have a stimulated emission component. This has been extended recently, as well-known, with collective effects, such as superradiance and superabsorption, into 5 states, but with no essentially new process.
This work's "new hypothesis" is introduced here, where one moves from the classical Shannon Boolean analogy of circuits with relays, valid for the LEM and a formless and classical "fluid" model of information, with a syntactic expression called 'bit', to a quantum tri-state+, where information is given by an abstract, algebraic approach with ternary object symmetry and extensions, modeled by GF(3n) and implementable as GF (2m) [3].
For interaction with matter, we found [4] that Einstein's "stimulated emission" provides coherence in universality, and applies not only to bodies that we must use to transmit and receive information, but also to how we communicate.
This is based on the topological projection [4] properties, as GF(3n) ^ GF(2m), for suitable m > n, with GF(3n), GF(2m) e Z/Zp, where p is a prime number, meaning that any three-valued logic system, breaking thus the LEM, can be represented (i.e, embed) in a binary logical system, obeying the LEM, although subjected to TR.
For interactions with matter, we are to apply GF(3n) in behavior, but use GF(2m) [4] for practical implementation, using binary logic. For the double-slit experiment, and accepting the natural processes of particle in-phase creation and annihilation, we model behavior through the binary decision process in two stages as GF(22). We use the first binary tree, by taking measurements from the front of the apparatus. After the double-slit, we take measurements again, from the immediate side after the apertures, and use a second binary tree. This makes it possible to use standard binary logic in QC, and always obey the LEM in the aggregate. Finer cases, with particle in-phase creation and annihilation, can be analysed by further binary trees, by using GF (2m).
Also, as known [15-20], the exact representation of Eq. (4) can be easily, analytically calculated for common and any potentials, instead of masked in high frequency by WKB [2] or numerical methods. The scaling law reported [16] would not had been revealed by WKB or numerical methods.
Although calculus requires continuity for the existence of derivatives, that is based not only on the four operations of arithmetic but also on the definition of real numbers. It does not seem necessary to require continuity, in general, as Cauchy did in analysis in the field of real numbers. In the field of finite integers, Z/Zp, such as in Galois fields, calculus can be defined exactly, as well-known, while not requiring continuity.
Continuity can be thought here as a collective construct from universality, not as a result from the use of infinitesimals (as Cauchy thought) that, are revealed, do not exist, and no such example can be shown [12; 21]. Code is also not continuous, and cannot create infinitesimals. Continuity, though, happens collectively also using Galois fields, finite integers, and code -through a collective construct also viewed from universality.
NATURE IS QUANTUM IN THE MICRO, CONTINUOUS IN THE MACRO
Mathematics can reflect nature, as Computer Science does today with digital only - where binary calculations belong to the reality we see, not as mythical "approximations" of an ideal analogue signal that does not exist or can be made.
However, for more than 50 years, as explained first by Brillouin, nothing was an infinitesimal to the physicist [12] and one cannot calculate or code it - the topology is digital, not continuous - thus, continuity has had no topological meaning (as well as no Cauchy epsilon-deltas) in the physical sense. Computers have no "continuity chip" - it is all done by binary code. Even so-called "floating point processors" such as the old coprocessor are using binary data in calculating mantissas and exponents. Operations performed by the coprocessor may be floating point arithmetic, graphics, signal processing, string processing, cryptography or I/O interfacing with peripheral devices. Yet, computers can emulate continuous results, and CDs can play apparently continuous music, any rhythm, better than analogue. Analogue and continuity are the approximation, while digital is the true result. Code is the exact result we see everywhere, while analogue has been, more and more, deprecated. This is now justified, in universality, but leads one to consider what may be an artificial renormalization in quantum field theories [22], and an artificial continuity in general relativity, (GR) all of which should be quantum.
This comment then, opens mathematics and physics to a new understanding (e.g., opposing Brillouin), as follows:
• one keeps infinitesimals, Cauchy results, and continuity, even though they are not observable and are not able to be constructed ever, and proceeds with these hypothesis as IF true; or,
• one takes the side of the opposite hypothesis, uses the algebraic approach with only finite integer fields and extension fields (e.g., Galois fields), that obey all four mathematical operations (+ - x/), and pursues further its consequences, such as the spurious continuity requirements in GR, and in the Schru,dinger equation.
The latter is a universality solution we propose, also to overcome a "sacred cow" feeling on infinitesimals [21] or even on the double-slit experiment, that are not obeyed classically (i.e., with infinitesimals, as one divides a volume to reach the supposed infinitesimal, soon one passes molecules, atoms, and even particles) nor in quantum physics, (i.e., number of particles is not conserved; indistinguishable photons can be created in-phase, or annihilated).
On the particle view of nature, the latter view imposes natural limits also on the coherence of collective effects not only on the isolated particle itself, such, as e.g., in the length of the particle. The issue calls attention to the resulting universality in the first case, as leading to continuity in the macrocosm, although no particle is ever continuous, in the microcosm, and can be observed only through what can ever be a far-away reference frame. This is in spite of the noted four contradictions of 'lack of continuity', 'code is not continuous', 'breaking the LEM', and 'number of particles is not conserved'.
Thus, the concept of universality allows us to use nice, known formulas such as derivatives in Eq. (1), and also the curvature method of Eq. (3), when only discrete values should be used, without conflict.
3. CURVATURE REPRESENTATION
OF BOUND STATES IN QUANTUM MECHANICS
The second-derivative in Eq. (1) can be represented exactly as a first-order difference equation, in any function spanned by linear combinations in the set U:
U = {e~ax, xe ~ax, x2e"ax}, (3)
with a suitable a > 0 [16; 19; 20; 23; 24], which already obeys the known boundary conditions of the Eg. (1). This motivates
us to eliminate the second derivative in Eq. (1), eschewing the hypothesis of continuity (needed by the derivative d2^(x)/dx2). Then, Eq. (1) becomes an equivalent first-order difference equation, normalizing in atomic units,
v(xk-i) = Jr[E-a -V(xk)]v(xk), (4)
ak
where the index k = 1, 2, 3, ... , K refers to the partition of the coordinate space, as usual, ak is a piece-wise variational parameter, and the boundary conditions are ^(0) = = 0, where ~ represents a large enough, finite, separation in spacetime [19]. Eq. (3) is hereafter called the curvature representation of Eq. (1), and is discrete.
Based on these potentials and the spectra of all the other potentials tested at high and low frequencies [20; 23], including the logarithmic, power-law, and square-root, we expect that with the curvature representation in Eq. (3) we can exactly represent Eq. (1), although not in vice-versa. Eq. (3) provides insights not reachable by Eq. (1), such as the scaling law in [16]. The usual functional dependence of the eigenvalues is to be obtained by using the quantum number for a generic potential V (x), and we expect it to reproduce all the eigenvalues of the usual Eq. (1).
We conjecture that Eq. (3) is an exact discrete representation of Eq. (1), although not in vice-versa, and without using any implicit or explicit continuity. Universality was not important at the micro level either - it happens at what can only be an ever far-away reference frame.
4. UNIVERSAL ITY MODEL
In this paper, we hope to use a clear difference in the regimes of small versus high quantum numbers to improve significantly upon QC using physical systems or computers, and the QC interpretation of calculated physics as a consequence. Is code result to be considered part of reality? How about continuity? Universality answers both questions.
In this regard, we consider what can be called the "universality model" of QC, in two cases:
1) Without the Heisenberg Principle. Also called the Bohr model. The quantum particles have well-defined location and velocity, but we are just not able to know them precisely, as it happens at what can only be an ever faraway reference frame. Niels Bohr [7] described that a quantum particle does not exist in one state or another, but in all of its possible states at once.4 Here, Eq. (1) or Eq. (3), can then be used to determine the probability density distributions for a particle location and velocity.
2) With the Heisenberg Principle. All quantum particle states co-exist but, as exemplified by the "Schrodinger's cat" mental experiment, only becoming a well-defined location and velocity when collapsing in the macro, upon measurement or observation by an observer.
The main difference between the two views in the "universality models" is collapsing the wave function, which is not a matter in the Bohr model. Here, we find the first model, without the Heisenberg Principle, to be more useful.
Experimentally, the behavior of systems described by the first "universality model", the Niels Bohr theory of QM, does not reproduce classical physics in the limit of small quantum numbers, although it reproduces for high quantum
4 Not to be confused with the complementarity principle, also formulated by Bohr.
numbers, being often counter-intuitive [6] to our usual observed experience (see Introduction).
This, which may be surprising at first, can be clarified by examples from complex analysis [25]. The fact that the product of two negative numbers is a positive number, also seems surprising at first. In the debit model, where the negative number is a debit, how to explain that the product of two debits is a credit? However, in a complex number model, a negative number is a 180° degree rotation, so the product of two such numbers is a 360° rotation, positive therefore.
Results begin and end in real number theory, but have a path through the complex plane, which influences the result, but remains hidden.
As Edward Titchmarsh [26] observed, V-1 is a much simpler concept than , which is an irrational number, essentially unknowable.
There are certainly people that regard V2 as something perfectly obvious, but sneer at V-1 . This is because they think they can visualize the former as something in physical space, but not the latter.
This investigation uses the complement: one can not really visualize V2, but one can visualize V-1, as a 90° rotation, and apply it in physics and engineering of real systems. One could do this using the real-line only, but one will benefit from the complex plane [25], as shown here.
Complex numbers are not part of the reality that we can measure, as we can only measure finite integers (e.g., Galois numbers) and their ratio.
Mutatis mutandis, quantum numbers offer a similar opportunity in QC. The original "universality model" of QC is due to Bohr [7], that a quantum particle does not exist in one state or another, but in all of its possible states at once. This "universality model" does not need to have an analogue in real systems, nor in language, nor even in our mathematics, nor code, nor that one can necessarily realize it in a physical system, as "quantum annealing", nor that can avoid decoherence.
In QM, the energy field must also have a discontinuous, discrete structure,5 where only the mathematical nature is evident as a description of reality, while a physical description of "continuity" is to be denied.
According to Khrennikov [27], the role of a mathematical apparatus in a description of reality implies that there exist other pictures of reality where other number fields are used as basic elements of a mathematical description.
According to Ozhigov [21], the problem of scalability in quantum computation represents the old question of the description of the measurements and decoherence in QM.
The problem of scalability in QC is made worse when using the Copenhagen interpretation [8] in lieu of the Bohr interpretation [7], whereas the collapsing of the wave function is not a matter in the Bohr model.
According to Bohr [7], there is no quantum world: "This is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature". We see this as an universality view of nature, where different abstract micro descriptions can correspond to what one can say about the same nature, macroscopically.
5 The discontinuity is often described to mean that between two points there is a nothing - no objects, no atoms, no molecules, no particles, just nothing, where even the word 'nothing' is maybe too much. as basic elements of a mathematical description.
This work does not move, though, to a post-Bohr reality, in trying to open the "black box" of QM. But we support the Bohr model in a software-defined QC, where GF (2m) can be used with binary logic to implement Bohr's "many states at once" model, without breaking the LEM in the macro, in universality. This is our "new hypothesis", closing the former four contradictions, where information is given by an abstract, algebraic approach with ternary object symmetry and extensions, modeled by GF (3n) and implementable as GF (2m).
In particular, this is important today, when it is wellknown that a shadow has fallen over the race to detect a new type of quantum particle, the Majorana fermion, that could power quantum computers. The Nature retraction is a setback for Microsoft's approach to quantum computing, as researchers continue to search for the exotic quantum states. While the evidence of elusive Majorana particle dies - computing hope lives on, and is now made possible by using tri-state+ in software with standard binary hardware, while enabling the use of spintronic methods and other novel approaches using integers.
5. DISCUSSION
In standard QM, the Schrodinger Equation for bound states is well-known. Eq. (1) is one of only a few solvable models in QM, and shares many qualitative features with physically important models, e.g., tuning of quantum-well lasers by long wavelength radiation, and in the scaling of magnetic fields using Rydberg atoms.
We showed that the double-slit experiment is often wrongly seen, including in cited references. Using low light fields so as to consider only one photon in the apparatus at a time, because only one photon came in, is not valid since the number of particles is not conserved, which becomes more important at lower frequencies. We formulate here a consistent QM framework using universality, albeit without any continuity hypothesis. This was clarified by examples in complex analysis.
We showed that it is desirable, therefore, to isolate those aspects of the current QM theory that involve continuous quantities, and are subject to modification by a more satisfactory theory, from aspects that involve only discrete values and are thus relatively more stable, and trustworthy.
This framework reduces to an equivalence of Eq. (1), without using second derivatives, which eschews continuity, and was validated in specifically four major potential models: harmonic, Coulomb, linear, and Rydberg states, at any frequency.
One cannot split the photon at the double-slit experiment, Huygens and all considerations. It would not be one particle anymore. This is not just semantics, this is the semantics, and the ME cannot explain continuity or the coherence term either. Although coherence gives origin to the laser, and stimulated emission. Everything has a stimulated emission component, with in-phase particle creation, even the light from an ordinary candle. But the ME fail to express the stimulated emission component. There are other well-known examples, like diamagnetism and superconductivity, which might seem at first disturbing, where the ME fails but QM explains. However, keeping the ME and Huygens' principle are right and valuable to Physics in universality, in the macro, as considered here. But one cannot use the macro to explain the micro, while the reverse seems possible. As we provided with the "new hypothesis", so we can support the LEM, the ME, and the Huygens' principle, all in the macro, and doing away with the four contradictions. Information, in universality, is given by an abstract, algebraic approach with ternary object symmetry and extensions, modeled by GF (3n) in the micro and implementable as GF (2m) in the macro.
One can also use universality to support the Bohr model of QC in the micro, with the photon as a particle in the micro, breaking the LEM with no conservation of the number of particles, in a software-defined QC. Here, GF (2m) can be used with binary logic to implement "many states at once" in the field Z/Zp, without breaking the LEM in the macro.
This should all be possible without necessarily using special hardware (e.g. quantum annealing), or incurring in decoherence at all, and wholly designed in software, with today's binary computers, even with a cell phone. This should provide not only exceptional speed, but also much needed cybersecurity, and a fresh approach with wide new opportunities for anyone.
References
1. Courant R., Hilbert D. Methods of mathematical physics. Vol. 1. New York: Wiley, 1989.
2. Schrodinger E. Collected papers on wave mechanics. International Series of Monographs on Physics. Book 27. Clarendon Pess, 1982.
3. Bouwmeester D. The physics of quantum information: Quantum cryptography, quantum teleportation, quantum computation. A. Ekert, A. Zeilinger (eds.). Springer Publishing Company, Incorporated, 2010.
4. Gerck E. Presentation: Tri-State+ (or more) quantum information model. Quantum Informatics 2021. Faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University. 2021. Current, longer version at https://www. researchgate.net/publication/347563918/
5. Einstein A. Strahlungs-Emission und Absorption nach der Quantentheorie. Deutsche Physikalische Gesellschaft. Jan. 1916. Vol. 18. Pp. 318-323,
6. Einstein A. Zur Quantentheorie der Strahlung. Physikalische Zeitschrift. Jan. 1917. Vol. 18. Pp. 121-128,
7. Petersen A. The philosophy of Niels Bohr. Bulletin of the Atomic Scientists. 1963. Vol. 19. No. 7.
8. Grib A., Rodrigues W.A.Jr. Copenhagen interpretation. In: Nonlocality in quantum physics. Boston, MA: Springer, 1999. https://doi. Org/10.1007/978-1-4615-4687-0.5
9. Howard D. Who invented the "Copenhagen interpretation"? A study in mythology. Philosophy of Science. 2004. Vol. 71. No. 5. Pp. 669-682.
10. Gerck E. Presentation: On the physical representation of quantum systems. Quantum Informatics 2021. Faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University. 2021.
11. Carlson B.A. Communication Systems. McGraw Hill Kogakusha, Ltd., 1968.
12. Brillouin L. Science and information theory. N.Y.: Academic Press, 1956.
13. Feigenbaum M.J. Universality in complex discrete dynamics. Los Alamos Theoretical Division Annual Report 1975-1976. 1976.
14. Barzel B. Barabasi A.-L. Universality in network dynamics. Nature Physics. 2013. Vol. 9. Pp. 673-768,
15. Gerck E., d'Oliveira A.B. Matrix-Variational Method: An efficient approach to bound state eigenproblems. Report number: EAV-12/78. Laboratorio de Estudos Avancados, IAE, CTA. Brazil: S.J. Campos, SP, 1978. Copy online at https://www.researchgate. net/publication/286625459/
16. Gallas J.A.C., Gerck E., O'Connell R.F. Scaling laws for Rydberg atoms in magnetic fields. 1983.
17. Gerck E., d'Oliveira A.B. The non-relativistic three-body problem with potential of the form K1rn + K2/r + C. Report number: EAV-11/78. Laboratorio de Estudos Avancados, IAE, CTA. Brazil: S.J. Campos, SP, 1978. Copy online at https://www.researchgate. net/publication/286640675/
18. Gerck E., d'Oliveira A.B. Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schrödinger equation. Journal of Computational and Applied Mathematics. 1980. No. 6 (1). Pp. 81-82. Copy online at https://www.researchgate.net/ publication/242978992/
19. Gerck E., Gallas J.A.C., d'Oliveira A.B. Solution of the Schrödinger equation for bound states in closed form. Physical Review. 1982. No. A 26. P. 1 (1).
20. Gerck E., d'Oliveira A.B., Gallas J.A.C. New approach to calculate bound state eigenvalues. Revista Brasileira de Ensino de Fisica. 1983. No. 13 (1). Pp. 183-300.
21. Ozhigov Y.I. Constructive physics (physics research and technology). Ed. UK: Nova Science Pub. Inc, 2011. ISBN 1612095534.
22. Wilson K.G. The renormalization group: Critical phenomena and the Kondo problem. J. Physique Lett. 1975. No. 43. Pp. 211-216,
23. Gerck E. The exponential difference. Private communication, cited in report number: EAV-12/78. Laboratorio de Estudos Avancados, IAE, CTA, Brazil: S.J. Campos, SP, 1978. Copy online at https://www. researchgate.net/publication/286625459/
24. Gerck E., Miranda L. Quantum well lasers tunable by long wavelength radiation. Applied Physics Letters. 1984. No. 44 (9). Pp. 837-839.
25. Ahlfors L. Complex analysis. McGraw-Hill, Inc., 1979.
26. Havil J. The irrationals. Princeton University Press, 2012.
27. Khrennikov A.Y. Universality in network dynamics. Springer Science Business Media. 2013. Vol. 427.
Статья поступила в редакцию 11.06.2021, принята к публикации 16.07.2021 The article was received on 11.06.2021, accepted for publication 16.07.2021
ABOUT THE AUTHOR
Ed Gerck, PhD (Physics), Planalto Research. Mountain View, CA, USA. ORCID: 0000-0002-0128-5875D. E-mail: [email protected]
СВЕДЕНИЯ ОБ АВТОРЕ
Эд Герк, PhD (Physics) Planalto Research, Маунтин-Вью, Калифорния, США. ORCID: 0000-0002-0128-5875D. E-mail: [email protected]
