DISCRETE MODEL OF SPACE AND TIME

A.I. Cherepanov

Institute of Automation and Electrometry

Siberian Branch of the Russian Academy of Sciences

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“”What we need is imagination, but imagination in a terrible strait-jacket. We have to find a new view of the world that has to agree with everything that is known, but disagree in its predictions somewhere, otherwise it is not interesting. And in that disagreement it must agree with nature. If you can find any other view of the world which agrees over the entire range where things have already been observed, but disagrees somewhere else, you have made a great discovery. It is very nearly impossible, but not quite, to find any theory which agrees with experiments over the entire range in which all theories have been checked, and yet gives different consequences in some other range, even a theory whose different consequences do not turn out to agree with nature. A new idea is extremely difficult to think of. It takes a fantastic imagination”.
“What is it about nature that lets this happen, that it is possible to guess from one part what the rest is going to do? That is an unscientific question: I do not know how to answer it, and therefore I am going to give an unscientific answer. I think it is because nature has a simplicity and therefore a great beauty.

Richard P. Feynman.1

ABSTRACT

A basic descriptive physical model based on a discrete representation of space and time is proposed. This model can be considered a paradigmatic foundation for discrete physics. Both the consequences of this model and hypotheses concerning elementary particles, the nature of certain fundamental interactions (including gravity), relativity, and the evolution of the universe are presented.

 

INTRODUCTION

When we invent physical theories, we describe certain classes of physical phenomena using systems of mathematical equations that link certain characteristic parameters of these phenomena to each other and to certain general parameters (for example, spatial and temporal).

In other words, we create a mathematical model of a physical phenomenon that is its analogy. And, like any analogy, it is fundamentally limited and cannot describe the phenomenon in its entirety. For example, the equations of continuum physics are obviously inapplicable at sizes where the medium can no longer be considered conditionally continuous.

On the other hand, it’s possible to have several different mathematical models of the same class of phenomena. For example, we can use the wave function formalism, or we can use the path integration formalism. The “correct” relativistic theory did not abolish the “incorrect” Newtonian mechanics. The choice of the theory used for calculations is simply a matter of convenience, simplicity, and the required accuracy of the result.

Thus, the point is not to invent a “mathematical copy” of reality (this is impossible), but rather for the new theory to expand (or supplement) the boundaries of knowledge.

However, how do we know how we should write our system of equations? Scientific intuition comes to the rescue. But it (intuition) is not based on simple guesswork. Some general, non-mathematical considerations are necessary. When we inquire about the physical meaning of a formula, we are interested in precisely these considerations.

For “fundamental” formulas, such general considerations are basic descriptive concepts expressed in a language of physical concepts, ideas, and terms specific to these concepts. We will call this “language” a physical model. Generally speaking, the term “physical model” has various interpretations, but we will use it in this narrow sense—as a language for describing basic physical concepts.

Physical principles are a generalization of all human observation of nature and are therefore the most reliable and consistent. Significant, revolutionary progress in physical science is most often associated with a revision (sometimes radical) not only (and often not so much) of the principles themselves, but of the language used to describe them. When new experimental facts “refuse to fit” into the old theory, and the old physical model no longer contains any “hints” for corrections or the development of a new physical theory, the need arises to revise the descriptive physical model. Einstein pointed this out, saying that “serious problems cannot be solved at the same level on which they arose.”

Lee Smolin wrote quite convincingly in his book2 that modern physical science is in crisis. “The trouble with physics: the rise of string theory, the fall of a science, and what comes next”in 2006. It’s hard to say that anything has changed radically since then. Moreover, even the “Copenhagen” explanation of the physical meaning of quantum mechanics, proposed back in 1927, remains controversial.

If The General Theory of Relativity is based on the idea of ​​four-dimensional space-time as a descriptive model, then it would be a great exaggeration to call the Copenhagen interpretation of quantum mechanics a descriptive physical model.

Although some progress has been made in the process of unifying theories describing different classes of phenomena, the situation can generally be described as a dead end.

There is a belief that all observable physical phenomena do not exist independently, in isolation, but are manifestations of certain general fundamental properties of our universe. In other words, the fundamental principles of physics are manifestations of the properties of a single fundamental law.

Perhaps some of the problems with such “unifying” physical theories as “string theory” or “loop quantum gravity theory” would be easier to overcome if they were based not on the idea of ​​some elementary volumes with properties convenient for the theory, but on a more general descriptive model.

This work is devoted to an attempt to construct a basic (unifying) descriptive physical model based on a discrete representation of space and time. In doing so, we will appeal to fundamental physical principles, which will serve both as a support and as the “straitjacket for the imagination” of which Feynman spoke.

Of course, we cannot count on discovering the “ultimate truth,” nor even on a complete description. Because (and we have already noted this), every descriptive physical model, like every physical theory, is nothing more than an analogy of reality. And analogies are always limited. Nevertheless, we can hope that our analogy will be useful.

 

HISTORY OF THE ISSUE

The reasons why scientists arrived at the idea of ​​discreteness vary.

Richard Feynman, himself one of the founders of quantum electrodynamics, was well aware that the problems with infinities in relativistic quantum mechanics have much deeper roots than the imperfections of the theory. That the renormalization procedure is nothing more than an attempt to introduce formal corrections to “fit” the theory to experimental results (or, as he himself called it, “sweeping the dust under the rug”). That the closer we get to the smaller characteristic sizes and times of physical processes, the more difficult it will be to “sweep the dust under the rug.” That, ultimately, the point will come when this will simply become meaningless. Feynman formulated the reasons why he came to the idea of ​​the discreteness of space as follows:

“At the same time, the theory that space is continuous seems wrong to me, because it leads to infinitely large quantities and other difficulties. Moreover, it does not answer the question of what determines the sizes of all particles. I strongly suspect that simple representations of geometry, extended to very small regions of space, are incorrect.”1

With the advent of computers and the ability to “digitally” simulate many physical processes, the idea arose (perhaps first proposed by Konrad Zuse3) that the real universe is in some way like a giant processor. A consistent proponent of this idea is Edward Fredkin.4.

Julian Barbour wrote that his impetus for thinking about the nature of time came from an article about Paul Dirac’s attempt to combine Einstein’s theory of relativity with quantum mechanics. Dirac’s doubts about the functionality of four-dimensional space seemed well-reasoned to Barbour. Further reflection led him to the conclusion that time, as we know it, simply does not exist.4.

Work in the field of cellular automata (especially those dealing with self-reproducing structures) has given the “digital worldview” a new lease on life. Stephen Wolfram spent over a decade working in this direction. The result was the book “A New Kind of Science” 6.

Finally, Nobel laureate Gerard ‘t Hooft appears to have been guided by the same considerations as Richard Feynman. He also believes that nature, in some sense, can be structured like “cellular automata.” He expounded on these views in detail in his book, “The Cellular Automaton Interpretation of Quantum Mechanics.”7

This (more than incomplete) list could be continued, but it is sufficient to illustrate that the idea of ​​discreteness did not arise out of nowhere.

As for the author, his main argument in favor of the discrete model was his deep conviction in the validity of the locality principle.

 

THE PRINCIPLE OF LOCALITY AS A BASIS FOR DISCRETE REPRESENTATION

Einstein attached fundamental importance to the principle of locality: “If this axiom were completely rejected, then the idea of ​​the existence of quasi-closed systems, and therefore the postulation of laws that can be verified empirically in the generally accepted sense, would become impossible“8. So far, no reasonable objections to Einstein’s argument have been raised.

The classical principle of locality states that any physical object is directly influenced only by its immediate environment. The concept of environment (or nearest neighbor) is extremely important in this case. If, as Einstein believed, space is not an empty “nothing” but possesses certain local properties, then we must certainly assume that spatial elements can influence one another and, therefore, obey a general principle of locality.

However, what specific region of space can we consider the immediate environment? Within the framework of the concept of spatial continuity, such a question itself seems meaningless. Thus, we come to the understanding that the principle of locality without the concept of spatial discreteness would be meaningless.

When we ask the question: how long does it actually take for one spatial element to influence another element that is part of its “immediate environment,” we must conclude that if there is a certain maximum speed of influence propagation, then there is also a certain minimum time during which one spatial element can influence another. According to modern concepts, confirmed by experience, there is a maximum speed of influence propagation. Therefore, assuming space to be discrete, we must also conclude that time is also discrete.

The following should be noted. The concept of an elementary (local) volume is internally contradictory, since it presupposes a specific geometry, that is, the presence of elements whose dimensions are smaller than the dimensions of the volume itself. Furthermore, the “cellular” model of metric space suffers from an extremely significant and fundamentally irreparable defect: it is not isotropic. Which leads us to the need to consider an element of discrete space not as a certain volume, but as a point that can be in one state or another, but does not have its own dimensions.

 

FOUNDATIONS OF THE DISCRETE MODEL. CONCEPTS AND TERMS

Of course, the basic concepts of the discrete model cannot be “derived” in any way and must be postulated intuitively. However, intuition must also be based on something. We already wrote above that we will focus on basic physical principles. In addition, we will also “adopt” some of the most general (philosophical) principles. Namely: the principle of the ontological primacy of matter, the dialectical principle of the “unity of opposites,” and the principle of “Occam’s razor.” Moreover, we will interpret the principle of “Occam’s razor” to mean that nature at its core is as simple as can be imagined.

  • Space

The concept of space as an empty, unproper container for physical bodies contradicts the principle of locality. “Nothing” can transmit influence from one body to another. Something is needed to mediate such interaction. It is from this understanding that the concepts of ether and field were born. The main difference is that ether is conceived as something existing in itself, while a field is conceived as something generated by material objects.

Einstein formulated his concept of space as follows:

“Space-time does not claim to exist in itself, but only as a structural quality of the field. Thus, Descartes was not so far from the truth when he believed that he must exclude the existence of empty space.”8

What they have in common is that both the ether and field models, in one sense or another, require the concept of space as a separate entity—a container (the fact that, according to Einstein, this container is itself generated by the field essentially changes nothing). However, it is worth noting that toward the end of his life, Einstein began to doubt the field model as such (see the section Fundamental Interactions).

Our point of view will be that physical bodies themselves and the processes of their interaction are a property of physical space evolving over time. In this sense, the concept of “physical space” will be identical to the concept of “physical universe.”

In other words, we will consider space to be a set of elements that can be in various states and are in adjacent relationships with one another. We will treat the elements of space as point objects that have no intrinsic dimensions.

More formally, we will consider physical space to be a non-metric, connected, discrete space, for whose elements an adjacency relationship is established and a set of states in which these elements can be found is defined.

  • Element of space

One ​​can make an infinite number of assumptions about the possible states of a spatial element, without any concrete basis. Therefore, here too, we will follow Occam’s razor (and, to a certain extent, the dialectical principle of the “unity of opposites”):

A spatial element can only exist in two distinct states, differing from each other in some sense.

By distinguishing “in some sense,” we mean the kind of differences that have a logical “Yes” and “No.” That is, concepts that are opposite and make no sense without each other. Thus, we cannot ascribe any other properties, including mass, size, etc., to a single spatial element.

  • Iterative “Time”

Within this model, we will assume that the evolution of the universe is an ordered sequence of discrete states. This sequence is the basis for defining the concept of physical time, but is not identical to it. There are reasons for this, which we will discuss later. So, if we use the term “time layer,” it will be purely for convenience, keeping in mind its conventional nature.

  • Physical Laws and the “Basic Law”

“Particular” or“physical” laws are subjective rules that allow us to describe the evolution over time of a limited set of parameters of the state of the universe in a certain region, within a certain approximation, and under certain conditions.

Within this model, we will assume that there exists a single, objectively defined, “observer-independent” rule that precisely determines the ordered sequence of states of the universe. We will refer to this rule as the “fundamental law.”

According to the principle of locality, the fundamental law determines the state of each element of space, depending on its state and the state of its nearest neighbors in the previous “time layer.”

In a philosophical sense, this means that we accept the point of view that the universe exists objectively, i.e., includes all entities (including the observer) and obeys a single law.

  • Physical object (physical body)

When we speak of a physical object, we mean that there exists something with a specific set of properties that persist for a certain sufficient period of time under certain conditions.

Based on the fact that physical objects are properties of physical space evolving over time, we can say that the properties of a physical object are determined solely by the relationships between neighbors and the nature of the evolution of the states of elements in a certain part of space, since (within the framework of the model) there is simply nothing else to determine them.

We will call the set of neighborhood relations and states of elements of a certain group of elements a spatial structure (or simply a structure).

If, with consistent application of the fundamental law, the structure is reproduced a sufficient number of times and with sufficient accuracy, we will call such a structure self-reproducing.

If the structure is reproduced after N≥2 steps, then we will call such a structure a self-reproducing periodic structure (SRP structure), which is characterized by an iterative period N (i.e., the number of different phases of this structure) and a size (a certain characteristic number of spatial elements structure).

Thus, we can formulate the concept of a physical object as follows:

By a physical object we mean a finite self-reproducing periodic (spatial) structure or some set of such structures.

  • Discrete “Distance”

Since space in our model is not metric, we can only speak of the “distance” between two spatial elements conditionally, implying the degree (or order) of proximity of these elements.

We can also define the “conditional distance” between physical bodies as the result of the averaged set of degrees of proximity between elements belonging to the SRP structures that form these bodies.

Of course, the “conditional distance,” like all other quantities that we will introduce in our model for convenience, is a dimensionless quantity.

 

PHYSICAL CONCEPTS AS PROPERTIES OF THE BASIC LAW

  • Locality

The principle of locality underlies the discrete model, and therefore its interpretation within the framework of the discrete representation is obvious, does not require any special reasoning on this topic, and can be formulated as follows:

The state of a spatial element can only be determined by the state of its neighborhood (including the element itself) in the previous “time layer.”

Note that this definition can also be interpreted as the (first) property of the “fundamental law,” from which it is clear that the “fundamental law” is both universal and local at the same time. This explains “why nature allows us to guess what’s happening everywhere by observing one part of it.”

There is, however, an opinion that the principle of locality is incorrect, “since it contradicts the Copenhagen interpretation of quantum mechanics,” and the results of experiments with “quantum entangled” particles supposedly confirm this.

We’ll explore this point later.

  • Irreversibility of Time

a) Formulation

Any consideration of the phenomenon of time irreversibility comes down to a simple question: how, exactly, does the past differ from the future in relation to the present?

Applied to our model, this question should be: is the rule that determines the sequence of “temporal layers” reversible, and what is the direction of its action?

We formulate the principle of irreversibility of time as follows:

The “fundamental law” of nature is a deterministic, irreversible law, the direction of which is opposite to the subjectively perceived direction of time.

Note that this formulation defines the second (“paradoxical”) property of the “fundamental law.”

It should also be noted that this formulation means that any “time layer” (from which the “current” “time layer” is obtained after a single application of the “fundamental law”) is a “neighboring future” relative to it. That is, the number of alternative “neighboring future” states depends only on the state of the “current” layer. There may be several of them, or there may be only one. But there’s always only one adjacent “past” for each “present.”

We called the second property of the “fundamental law” paradoxical. Indeed, at first glance, it contradicts common sense. But that’s only “at first glance.” In fact, it’s precisely the opposite. From the perspective of common sense, based on experience, the past is absolutely deterministic and “predictable”: if we don’t know something about the past, it’s not fundamental and only means we haven’t discovered enough “evidence” that undoubtedly exists in the present. The future, on the contrary, seems vague and inherently difficult to predict, that is, fundamentally alternative.

All that remains is to understand the “cause and effect.” But we’ll do that later, in the “Discussion” section.

b) The Concept of Probability

From the perspective of the time irreversibility principle we formulated, all possible “future” states are equal. However, experience tells us that some future events have a higher chance of occurring than others. How can this be explained?

Let’s imagine a dice. We all agree that the probabilities of landing on any side are equal. Now imagine that we’ve painted two sides yellow and the rest red. Clearly, the probability of landing on the red side is higher than the yellow side. Although the “future” states of the dice are equal, we “combined” different states, the properties of which are insignificantly different for us, into one.

Thus, we can say that the “probability weight” of a future event is determined by the number of different states that are perceived by the observer as the same.

c)The Principle of Time Irreversibility and Quasi-Closed Systems

The principle of locality allows us to isolate a certain region of the universe and study the evolution over time of a certain set of state parameters of this local region.

Depending on what set of parameters is available to us for observation (or is perceived by us as sufficient), we formulate certain physical “laws” that are fulfilled with varying degrees of accuracy.

For some sets of parameters, we can ignore the principle of time irreversibility and describe, for example, the flight of a cannonball, determined by reversible functions (Newton’s laws).

Including core temperature in the list of essential parameters necessitates introducing the laws of thermodynamics and partially accounting for the principle of time irreversibility (the law of entropy increase).

However, there are sets of parameters and local (quasi-closed) systems for which we cannot use deterministic functions to describe them. For example, the process of electron “scattering” by a collimator slit.

What should we do in this case? First, we don’t know the “fundamental law” of nature. Second, even if we did know it, we would have to, using the “fundamental law” as a criterion, “test” sequentially the entire set of states of our local system across all “time layers.” Even without taking into account that the boundaries of our local system will expand, following the “event horizon,” the task seems unrealistic.

And yet, there is a way out here too. We can use not all parameters, but only some (only those we consider “essential”). In this case, states with the same values ​​of the “essential” parameters can be considered identical. Consequently, we can use the concept of the probability of a future state. And these state probabilities, it turns out, can be described (with some accuracy) by deterministic functions. Moreover, the parameters themselves can take both continuous and discrete values. An example is the Schrödinger wave function.

As follows from the above reasoning, this does not at all mean that the electron itself, whose probability of being found in a particular location in space is described by the wave function, is in some sense smeared across that very space. Here we are very close to the Copenhagen interpretation of quantum mechanics. And this question should be considered separately.

d) The Copenhagen Interpretation of Quantum Mechanics

In 1927 in Copenhagen, Niels Bohr and Werner Heisenberg proposed an explanation of the physical meaning of quantum mechanics, which can be summarized as follows:

– In quantum mechanics, the result of a measurement is fundamentallynon-deterministic.
– The act of measurement causes an instantaneous collapse of the wave function.

In this case (in some interpretations), the wave function is considered a real “field,” not just a “probability amplitude.”

The need for an “observer,” as well as the “instantaneousness” (superluminal speed) of the wave function’s collapse, are still controversial. Einstein effectively (and not without reason) accused adherents of this interpretation of solipsism: “Do you really think the moon exists only when you look at it?”

It is easy to see that the principle of time irreversibility, as formulated above, does not require any special interpretations for quantum mechanics, does not require an “observer” and does not violate the principle of locality.

Indeed, the need for a probabilistic or deterministic reversible description of the evolution of the states of a quasi-closed system, as the evolution of a select set of parameters, depends both on the number of admissible future states and on the parameters we choose to describe these states. Or, in other words, which states we agree to consider “indistinguishable.”

Therefore, in some cases, it is possible (or necessary) to apply a probabilistic description in the “macroworld” (the case of the dice), and a deterministic description in the “quantum microworld.”

The presence of a rule deterministic in the direction opposite to the “arrow of time” not only does not exclude some (even if not unambiguous) correlation between the current and future states of the system, but also implies such a correlation. There is no reason to interpret this correlation as “quantum entanglement.”

A sophisticated reader will ask: what about John Bell’s inequalities and the experiments of John Clauser, Alain Aspect, and Anton Zeilinger? The answer is actually very simple. These experiments did indeed prove the absence of hidden variables. In other words, it has been experimentally proven that there is no deterministic law (rule) that would precisely determine the future state of a system, given its current state. But this doesn’t necessarily mean the “Copenhagen interpretation” is correct.

Thus, the experiment with “quantum entangled” particles (combined with the recognition that information transfer by influencing one of the “quantum entangled” particles is impossible) only adds to our reason to reject the “Copenhagen interpretation” of quantum mechanics in favor of the “discrete interpretation” of the principles of locality and irreversibility of time, which are universal with respect to spatiotemporal scales.

  • The Relativity of Physical Time

Let us ask ourselves: will the period of one structure always correspond to the period of another, exactly identical, structure?

The question is not as meaningless as it seems at first glance. Since, from the observer’s point of view, the time of physical processes in an observed quasi-closed physical system depends on its velocity and direction of motion relative to the observer, then the periods of the basic structures (from the observer’s point of view) must also depend on the velocity and direction of motion of the observed system. Since the time of all physical processes is determined by these basic cyclic processes.

For illustration, consider the dependence of a photon’s frequency on the relative velocity and direction of motion of the local system that generated it.

After Einstein’s explanation of the photoelectric effect, there is no doubt that the photon is indeed a particle. And that it is in this capacity that it propagates and interacts. Therefore, the explanation of photon frequency changes by the Doppler effect seems, at the very least, strange. At the same time, the relativistic explanation of both red and blue shifts in cosmology, as well as the dependence of a single photon’s frequency on the velocity of the electron that “emitted” it, do not contradict the concept of the photon as a periodic structure (or, in other words, a discrete oscillator).

 

DISCUSSION

As you read the previous section, you will inevitably have questions. Therefore, before moving on to the next section, the author considered it appropriate to answer some (what he considers the most probable) imaginary questions.

  • How does the proposed model differ from models that represent the universe as a giant computer?

Firstly, the very idea that all physical phenomena are based on certain general basic algorithms, rules, or laws (the term is unimportant) is inherent in both discrete and more traditional concepts of the nature of space and time. Some may be irritated by the uninformative concept of “information” or programming terms (iteration, algorithm, etc.), but this doesn’t change the essence and doesn’t turn the universe into a “giant computer.”

Secondly, so-called digital models of the universe are based on deterministic, reversible algorithms. To somehow explain the probabilistic nature of future states, authors must introduce various artificial mechanisms that “corrupt” these algorithms (and unilaterally—in the direction of the “arrow of time”). In the proposed model, the variability of future states follows from the model itself, despite the fact that the “fundamental law” is deterministic (though acting only in one direction—opposite the “arrow of time”). An important distinction is that our model ensures spatial isotropy.

It should also be noted that the underlying idea behind various “digital models of the universe” is often to somehow select (find) some initial algorithms and then rebuild the entire edifice of physical science on this foundation. This is essentially a claim to attaining some ultimate knowledge and some kind of “new science.”

The motives of the proposed model are different. The point is that, without encroaching (moreover, relying) on ​​the known basic physical principles, introduce some new, additional ideas, concepts and terms, on the basis of which it would be possible to “understand” physical phenomena that no longer fit into the framework of current ideas or to rethink those that “seem to” fit. In fact, we are talking about a certain “basic unifying language” in which we can talk about a wide variety of natural phenomena. Which more or less fits into the tradition of the development of physical science.

  • Isn’t this interpretation of the “principle of irreversibility of time” equivalent to the assertion that the effect precedes the cause?

Or, more simply, isn’t everything turned on its head? First, let’s clarify what we mean by the terms “cause” and “effect” in different contexts.

Let’s start with a simple model in which the future is completely predetermined, and the past is without alternative. In this case, time is simply the fourth coordinate (the fourth dimension). And the “present” is any point on the time axis, relative to which everything “to the left” is the past, and everything “to the right” is the future. In this model, all laws of nature must be time-reversible. That is, the assertion that the future (the effect) is determined by the past (the cause) and the converse are completely equivalent. In this version, “cause” and “effect” differ only in their relative position on the time axis and are simple synonyms for “past” and “future.” There’s simply nothing to “turn on its head” here.

Now consider the “probabilistic future” model. It assumes that the laws of nature are such that from any past state of the universe we can, by applying certain rules (the laws of nature), obtain a set of possible future states. In this case, we call the “cause” the state to which the rules “apply”, and the “effect” is the resulting result (in this case, a set of possible future states).

This model implies the indeterminacy of the laws of nature. Which does not exclude the possibility that some “future” state may be a consequence, generally speaking, of various “causes”. The fundamental possibility of an “alternative” past is the “Achilles heel” of these models. And in this sense, the Copenhagen interpretation and Everett’s interpretation (as well as its later variations) differ little from each other. Discussions about whether alternative universes “really” exist or whether potential alternatives “collapse” at every moment of time may be interesting from a philosophical point of view, but are unlikely to have any practical value.

In the proposed interpretation of the “principle of irreversibility of time,” the absence of alternatives to the past is ensured by the fact that possible future states can only be those states from which all previous states can be unambiguously “calculated” using a deterministic “basic law.”

In other words, this interpretation implies that the laws of nature regarding possible “futures” act as a “selection criterion.” It can be said that any future state has the right to exist only if all past states can be unambiguously “calculated” from it. This means that the presence of these past states is the cause of “futures.”

As we see, the terms “cause” and “effect” cannot be applied in the same sense to different models.

  • Okay, we’ve described physical principles in the language of a discrete model. Can this be of any use to those developing physical theories? In other words, can we consider our mission complete?

Not quite. And here’s why. Physical theories deal with equations that relate the values ​​of physical parameters, such as mass, charge, momentum, energy, time, distance… Attempting to completely replace these parameters with some integer, dimensionless “basic” quantities and construct some kind of “new science” seems counterproductive. To put it mildly.

Therefore, the next step should not be an attempt to completely replace classical physical terms, concepts, and ideas, but to clarify their meaning in the language of a discrete model. This can already be considered useful. Of course, to the extent and in the area in which this discrete model proves more or less adequate.

  • So, at this stage we can’t draw any practical conclusions?

We can draw some general conclusions.

For example:

  • A direct consequence is the absence of “bad infinities.” In particular, the existence of “black holes,” as objects with infinite density, is impossible. For the simple reason that discrete space and elementary periodic structures are “incompressible” and “non-expandable.”
  • For the same reason, the proposed model has no place for the “Big Bang Theory.”
  • A consequence of the fact that spatial elements can only exist in one of two states is that each SRP structure has its own “negative” version. This explains the existence of antiparticles. Conversely, the fact that particles exist in two versions (particle/antiparticle) confirms the duality of the elements of discrete space.
  • Generally speaking, “mirror” versions of SRP structures should also exist.
  • Since the periodic structures that form particles can be in different phases at the moment of collision, the result of the interaction (for example, the scattering angle) can depend on their phase states. This allows us to abandon the “wave-particle” oxymoron in favor of a more comprehensible representation of particles as discrete oscillators. In this light, the formation of a “diffraction” pattern during the scattering of a sequence of single electrons by a collimator slit seems quite natural.
  • Perhaps it is worth mentioning another consequence: so-called quantum communication is not fundamentally secure.

The content of the next chapter should be viewed rather as a set of hypotheses that, although based on our discrete model, the statements it contains are not necessarily its direct consequences.

On the other hand, the usefulness of any new physical model is largely determined by the ability to put forward various (including alternative) hypotheses that provide both a new perspective on physical phenomena and offer explanations for those phenomena whose explanations within the framework of the “old” model are questionable or are considered “miraculous.”

 

DISCRETE INTERPRETATION OF CERTAIN PHYSICAL TERMS, CONCEPTS, AND CONCEPTS

  • Physical Time. “Time Quanta.”

There is some reason to believe that Henri Poincaré’s article “On the Measurement of Time”9 inspired Einstein to develop his special theory of relativity. And this is far from coincidental. Any measurement of the parameters of a physical object (speed, momentum, energy… including, if you think about it, dimensions) is associated with measuring the time of certain processes and establishing the order or simultaneity of certain events.

This means that in order to judge certain physical processes, we must have a standard of time. We have nowhere else to get such a standard except from nature.

How do we measure time? We use periodic processes. We count how many complete cycles a “reference” physical process has completed (be it the number of revolutions of the Earth around its axis or the periodic processes of an atomic clock).

The fact that in nature, the number of periods of one physical process corresponds to a certain number of periods of other physical processes allows us to judge the progress of other processes by observing some physical processes. For example, without even looking out the window, we can judge whether it is day or night by the number of revolutions of the clock hand. This is precisely the practical benefit of measuring time. On the other hand, this gives us reason to believe that certain basic cyclical processes underlie a wide variety of physical phenomena, the periods of which ultimately determine the duration of certain events in nature. Which brings us back to our periodic structures.

In other words, we should associate the concept of a “quantum of physical time” not with the iteration “step” between adjacent “time layers,” but with the periods of elementary structures whose interactions determine the nature of certain physical processes.

Furthermore, “time quanta” corresponding to the same physical processes may (in accordance with the principle of relativity in the discrete interpretation) differ from the perspective of a “stationary observer” if the structures defining them move at different speeds relative to this observer and/or experience “sufficiently strong” external influences.

Therefore, we will use the term “physical time” or simply “local system time” to refer to a quantity calculated in system-averaged “time quanta.” In other words, we, in a sense, share the opinion that physical time is “something like temperature.”

  • Physical Distance. “Distance Quanta.” The Number Pi.

In the discrete model, there is no concept of distance. The degree of proximity is an absolute value and cannot be directly “scaled” into physical distance in all cases (except for a stationary inertial system), since physical distance is a relative quantity.

For physical space, we must define the distance between two points using the formula ∆T/c. Where “∆T” is the maximum time of influence propagation (i.e., the duration of this event), measured in the observer’s frame of reference, and the constant “c” is the speed of light. It is generally accepted that the maximum time of influence propagation between two points is the time it takes a photon to travel the distance between these points in a vacuum. That is, the physical distance in the observed system depends on the rate of time (in other words, on the “temporal temperature”) in this observed system and, generally speaking, can be calculated in units of time (in astronomy, for example, this is what they do – they measure distance in light years).

Таким образом, мы можем определить “квант расстояния” формулой: ∆x=∆t/c.

Where ∆t is the “quantum of time,” and c is the “speed of light.” Being a derivative of the “quantum of time,” the “quantum of distance” is also “something like temperature.”

Besides distance, the most important basic characteristic of physical space is the number π. To understand the physical meaning of the number π from the perspective of the discrete model, let’s consider a simple special case—a sphere in a stationary inertial frame. The analogue of a sphere in the discrete model is a set of spatial elements whose degree of proximity to a certain element (the center) is equal to or less than a certain (integer) number n (which is the analogue of the sphere’s radius). Accordingly, the analogue of the surface of a “discrete sphere” is a set of spatial elements whose degree of proximity to the “central” element of the discrete “sphere” is equal to n. We denote the number of these elements as S∆(n). In this case, the discrete analogue of the number π should be the sequence πn = S∆(n)/4n2.

It follows that:

  • The coordination number of a discrete space must be such that the function S∆(n) ensures the convergence of the πn sequence to the “classical” number π.

  • From the point of view of the discrete model, “physical π” is not a constant and, at “sufficiently small” radii, can differ significantly from its limiting value.

This last conclusion, you must admit, is quite unexpected.

*By the way, in this regard, the article10 “Quantum Mechanical Derivation of the Wallis Formula for π” by Tamar Friedmann and Karl Hagen seems quite interesting.

  • Vacuum. Vacuum Density. Particles.

Based on the discrete model, spatial regions can be divided into three main types:

  • Non-periodic structure.
  • Self-reproducing periodic (SRP) structure.
  • Quasi-periodic forced-reproducing structure.

By non-periodic structure we mean a set of discrete space elements that is not reproduced after N≥2 iteration steps. The spatial elements of such a structure can be in either the same or different states. Such spatial regions (static in the first case, and chaotically evolving in time in the second) are analogous to the physical vacuum.

Conventionally, we denote the opposite states of space elements with the symbols b and w (black and white). We denote the number of “black” elements in a certain region of space as nb, and the number of “white” elements as nw. Let us define the “vacuum density” as ρv = (nb-nw)/(nb+nw).

As we can see, from the point of view of the discrete model, at least one property can be determined for a vacuum – its density.

By quasiperiodic forced reproducing structure (or shell) we will understand a region of discrete space that experiences a “sufficiently strong” influence of the “parent” SRP structure. Of course, the boundary between the shell and the vacuum is quite arbitrary. There must be a certain zone with a gradient of vacuum density from the boundary of the shell to a certain characteristic (average) vacuum density for a given region of space.

We will call the set of SRP structure and its shell an elementary particle.

A collection of several SRP structures with intersecting shells is called a composite particle. One can also imagine a scenario in which some (or all) of the SRP structures of a composite particle are individually unstable.

Generally speaking, particle stability obviously depends on the vacuum density. Particles stable at a positive ρv value become unstable at a negative value. And vice versa. This explains the imbalance in the number of free particles and antiparticles in free chaos (vacuum). On the other hand, particles unstable in free chaos may well be stable in a shell of composite particles.

  • Fundamental Interactions.

According to the “Standard Model” of physics, the particles responsible for a given fundamental interaction are the corresponding quanta of the corresponding field (with the exception of gravitational interaction, which has not yet been defined).

In the discrete model, the concept of a field (as well as ether) is absent. Therefore, the question naturally arises: what are fundamental interactions from the standpoint of the discrete model?
From the perspective of the discrete model, interactions can be divided into the following types:

a) Interactions of elementary SRP structures of composite particles

This includes both interactions “inside” stable particles and collisions that form an unstable particle followed by its decay. In this case, new SRP structures can form within the shell, metamorphoses of the original structures can occur, etc. However, all of this can be viewed as the result of interactions between structures sharing a common shell. That is, from the perspective of the discrete model, “classical” fundamental interactions (with the exception of gravity), despite their extreme complexity, have a common underlying nature.

In particular, the electromagnetic interaction can be viewed as the result of a photon colliding with an electron (proton), producing an unstable composite particle and subsequent decay.

Nevertheless, it would be interesting to discuss this interaction separately and in more detail.

Already towards the end of his life, in a letter11 (1954) to Michele Besso, Einstein wrote:
“All these fifty years of conscious reflection have not brought me any closer to an answer to the question: ‘What are light quanta?’ Today, every Tom, Dick, and Harry thinks they know, but they are mistaken… I consider it quite possible that physics cannot be based on the concept of a field, i.e., on continuous structures. In that case, nothing will remain of my entire castle in the air, including the theory of gravity, [and] of the rest of modern physics.”

To consider electromagnetic interaction as mirror-symmetric, we will formally represent the proton as a composite particle containing an antielectron. And the interaction of a photon (antiphoton) with a proton will be considered an interaction with this antielectron. That is, from the perspective of our model, it is electrons (and their negative versions, positrons) that directly interact with photons.

In this case, we must ask: why does a photon transfer a positive momentum to an electron/positron in some cases, and a negative momentum in others?

This can be explained by making the following assumptions:

  • Not only the photon but also its “negative” pair, the antiphoton, acts as a “carrier” of electromagnetic interaction.

  • An electron can form a composite particle upon collision with either a photon or an antiphoton, but this unstable particle always decays into an electron and a photon. Similarly, a positron can form a composite particle upon collision with either a photon or an antiphoton, but this particle decays into a positron and an antiphoton.

  • As a result of the interaction, a photon transfers a positive momentum to an electron and a negative momentum to a positron. Similarly, an antiphoton transfers a positive momentum to a positron and a negative momentum to an electron. In other words, we formally assign positive inertial mass to particles and negative mass to antiparticles.

The assumptions we’ve made explain why “like charges” repel each other, while “opposite charges” attract each other.

It should also be noted that any SRP structure has its own fixed iteration period. Therefore, the observed (effective) frequency of a photon (antiphoton) during the decay of a composite particle, and consequently its momentum, in accordance with the principle of relativity, will depend only on the rate of time in the system of the composite particle relative to the observer’s system. That is, in our case, from its momentum vector in the system of the “receiving” electron (positron).

In this case, the radial component of the momentum of our unstable composite particle is responsible for the “Coulomb interaction,” and the tangential component is responsible for the “magnetic interaction.”

The attentive reader will notice that this last statement means that we do not consider the rate of time in the observed system to be a scalar quantity. This is indeed true. We simply considered it more appropriate and illustrative to make this statement here.

And not as a hypothesis, but rather as a fun speculation:

Are the photon and antiphoton shell-less versions of the electron and positron? And perhaps the so-called annihilation of an electron and positron isn’t annihilation at all, but simply the disintegration of shells as a result of the collision? Could this explain the “special” relationship between these particles? And isn’t the photon’s unique ability to travel at maximum speed related to the absence of a shell?

b) Interaction of particle shells with vacuum (chaos). Gravity.

The universal nature of gravitational interaction, the lack of serious candidates for the role of an intermediary particle, certain oddities in the behavior of astrophysical objects, and the known problems in measuring the gravitational constant lead us to assume that the mechanism of gravitational interaction is fundamentally different from the mechanisms described above.

In our model, the only candidate for the gravitational intermediary is the vacuum itself. More precisely, the vacuum gradient (see the section “Vacuum Vacuum Density Particles”). The more massive the physical object (i.e., the more particles it contains and the larger the shells of these particles), the greater the distance the vacuum density gradient extends. A collection of physical bodies creates a certain distribution of vacuum density in space.

We define the gravitational force as a force proportional to some average value of the vacuum density gradient in the region of space occupied by this body.

It should be noted that the vacuum density in a region of space is obviously not a simple sum of the influence of the bodies. This is likely the reason for the formation of the so-called cosmic web threads.

Furthermore, antiparticles in our universe should “fall upward” since the density gradient (from their point of view) would have the opposite sign. Or, in other words, antiparticles should have negative gravitational mass.

  • The Evolution of the Universe.

Let’s imagine that the universe has the following structure:

  • The universe is infinite and divided into two parts – the inner and the outer.

  • All spatial elements in the outer part are in the same state (it doesn’t matter which – the important thing is that they are all in one of two possible states). This can be called a dead zone or absolute emptiness – whichever you prefer. Alternatively, it can be called a vacuum zone with a density of -1. The sign in this case is a matter of taste.

  • The interior is the space described in the “Vacuum. Vacuum Density. Particles” section of this chapter. Let’s call this part the living zone. Or the zone in which the vacuum density has a positive value closer to +1.

This construction seems rather exotic at first glance. However, it is precisely this construction (as we will see later) that allows us to explain, within the framework of a discrete model, certain phenomena that, within the classical paradigm, either cannot be explained at all, or that require “unnecessary multiplication of entities.”

Let’s consider the evolution of such a universe from two perspectives—iterative “time” and physical time.

a) Iteration Time

We will assume that the nature of the fundamental law is such that:

  • The iteration process does not lead to a change in the state of a spatial element if the element itself and its nearest neighbors are in the same state.

  • The iteration process leads to a shift in the boundary between the “dead” and “live” zones. Moreover, in the direction of the “live” zone.

Thus, in the process of iterative evolution, the “living” zone will be absorbed by the “dead” zone and, at some point, will disappear completely. At this point, all evolution ends.

b) Physical Time

Since the arrow of physical time has a direction opposite to that of iterative time, the evolutionary process appears different.

From a physical point of view, the “living” zone is our universe, and the “dead” zone is not even a void. It is simply absolute NOTHING.

The universe arose and is constantly expanding its boundaries. And, it should be noted, this does not require any “initial push” or explosion. Because the moment of the universe’s death in iterative “time” corresponds to the moment of its birth in physical time.

In the zone of seething border chaos, elementary particles and atoms are born and die. The receding border leaves behind clouds of atomic gas from which stars and planets are born.

In this light, the phenomenon of so-called “relic radiation” takes on a completely opposite meaning. And perhaps it should be renamed “frontier radiation” or something similar.

If we accept the gravity hypothesis presented in the previous chapter, then the observed phenomenon of the “accelerating expansion of the universe” does not seem surprising. Indeed, the vacuum density varies from -1 at the boundary to values ​​closer to +1 at the center. That is, we have a certain vacuum density gradient from the periphery to the center (which, of course, is somehow superimposed on the vacuum density distribution generated by physical bodies). So, the stars simply “fall into the sky,” so to speak. And they will never fall, because the sky (aka the frontier of the universe) is receding faster.

As the reader has probably already noticed, here too we did not have to go beyond the model and invoke magical entities like dark matter and dark energy.

It is worth noting that this hypothesis is very optimistic. Yes, stars eventually cool and fade. But a growing universe creates new stars. And there will be no end to this. Perhaps this explains the Fermi Paradox, and our fellow intelligent beings aren’t coming to us because the conditions for their emergence haven’t yet been created on these new worlds? Or maybe we’re the first? And will we ever visit them? Agree, it’s not so bad to be the older race. ☺.

On this optimistic note, we reach the final part.

 

CONCLUSION

Lee Smolin formulated the “five great problems of theoretical physics”:

  • The problem of quantum gravity: Unify general relativity and quantum theory into a single theory that can claim to be a complete theory of nature.

  • Fundamental problems of quantum mechanics: Solve the problems underlying quantum mechanics, either by making sense of the theory as it currently stands or by inventing a new theory that does make sense.

  • Unification of particles and forces: Determine whether the various particles and forces can be unified in a theory that explains them all as manifestations of a single fundamental entity.

  • Tuning Problem: Explain how the values ​​of the free constants in the Standard Model of particle physics are chosen in nature.

  • Cosmological Riddles Problem: Explain dark matter and dark energy. Or, if they don’t exist, determine how and why gravity varies on large scales. More generally, explain why the constants of the Standard Model of cosmology, including dark energy, have the values ​​they do.

Of course, Lee Smolin formulated the problems of theoretical physics, or in our terms, physical theories. But, to repeat Einstein once again, “serious problems cannot be solved at the same level where they arose.” In other words, it is impossible to solve serious problems of physical theories while remaining within the same paradigm of physical models.

Therefore, the proposed physical model is an attempt at a first, but absolutely necessary, basic step in solving the “great problems.”

Time will tell how successful this attempt is.

Literature:

1. Richard Feynman, “The character of physical law”. A series of lectures recorded by the ВВС at Cornell University USA, Cox and Wyman Ltd., London, 1965

2. Lee Smolin. The trouble with physics: the rise of string theory, the fall of a science, and what comes next. Houghton Mifflin, Boston, 2006.

3. Konrad Zuse, “Rechnender Raum”. Braunschweig: Friedrich Vieweg & Sohn, 1969

4. Fredkin, E. An Introduction to Digital Philosophy. International Journal of Theoretical Physics 42, 189–247 (2003).

5. Julian Barbour, “The End of Time: The Next Revolution in our Understanding of the Universe”, Oxford Univ. Press. 1999

6. Stephen Wolfram “A New Kind of Science”, Wolfram Media, Inc., Champaign, IL, 2002

7. Gerard ‘t Hooft “The Cellular Automaton Interpretation of Quantum Mechanics” https://arxiv.org/abs/1405.1548

8. Albert Einstein “Quanten-Mechanik und Wirklichkeit”, Dialectica 2:320-324, 1948.

9. Poincaré A. La mesuré du témps. Revue de Metaphysique et de Morale. 1898. V.6. Pp.1-13.

10. Tamar Friedmann C. R. Hagen “Quantum Mechanical Derivation of the Wallis Formula for π”, Cornell University, arXiv.org 2015.

11. The Science and the Life of Albert Einstein”, Oxford University Press, (1982) 467

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