# Thread: Spacetime and matter as emergent phenomena, unified field theory

1. Ans
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Originally Posted by Reality Check
I continue to note that you have not presented any derivation of anything in this thread, Ans, to answer not only my questions but to actually present your ATM idea here for others.
I already presented model with hyperplanes, which shows relativity of simultaneity. I wrote what in order to understand why usage of hyperplanes is correct and why rotation of hyperplane represent transition to another frame of reference, necessary to read article. As I already wrote, I tried to write article as short as possible. The derivation of relativity of simultaneity is on page 10, so in order to present in here necessary to copy 10 pages of text (and spend time to manually copy equations). You may read is in article, if you want.
Your approach with "present everything here" may work only for some short ideas.

2. Ans
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Originally Posted by Reality Check
IF07: Show your step by step building of "Minkowski spacetime on basis of Euclidean space"
Take Euclidean space and nothing else and end up with a spacetime that has the properties of Minkowski spacetime.
Your answer here again shows you not understand one of main ideas in theory, which is written many times in article.
You propose to Euclidean space and nothing else to derive Minkowski spacetime. The proposal is based on part of realism, materialism.
I use Euclidean space and defined on that space scalar field to derive Minkowski spacetime. Note - time is also derived from that space with scalar field, time and dynamic is absent on fundamental level in the theory. Space, observable by us, is not same space as fundamental space. As result, it allows to derive curved spacetime from Euclidean space and defined on that space scalar field. Details, as I already wrote, are in article.
I use subjective idealism in theory, and it allows to do that derivation which looks impossible.

Originally Posted by Reality Check
I can think of at least one flaw on your paper. F = ma appearing out of thin air followed by asserting this is deriving Newton's F = ma is wrong. Thus my question IF01.
Well, this part is earlier in article than relativity of simultaneity. May you wrote that exactly you see incorrect in that derivation of F=ma?

3. Ans
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Originally Posted by John Mendenhall
An, I may have missed it, but I do not see a link to your article.
It is in my first post, look for "vixra". Or you may use the link: http://vixra.org/abs/1809.0596

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## Got It

Originally Posted by Ans
It is in my first post, look for "vixra". Or you may use the link: http://vixra.org/abs/1809.0596

Thanks.

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Originally Posted by Ans
Your answer here again shows you not understand one of main ideas in theory...
Disparaging my understanding of your theory is not an answer to my question:
IF07: Show your step by step building of "Minkowski spacetime on basis of Euclidean space"
Do what you say you do: "use Euclidean space and defined on that space scalar field to derive Minkowski spacetime". That obviously includes "time is also derived from that space with scalar field" since you go from Euclidean space (no time) and a "space scalar field" (not time) to a Minkowski spacetime.
Last edited by Reality Check; 2018-Oct-08 at 10:47 PM.

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Originally Posted by Ans
I already presented model with hyperplanes, which shows relativity of simultaneity.
A story about hyperplanes and an assertion of relativity of simultaneity is not a derivation presented here in this thread of relativity of simultaneity. It is not "my approach" - it is the rules for an ATM thread. It may be a bit of work but you have to defend your ATM idea here in this thread rather than point to a PDF on the internet. I did you the courtesy of reading that PDF and all I can find is that same story and assertion.
Last edited by Reality Check; 2018-Oct-08 at 10:49 PM.

7. Ans: our rules require that you make your presentation here in the forum. If you cannot, we can close this thread. You may not argue about our rules within this thread. Further, do no disparage the reading abilities or understanding of other members.

Reality Check: do not attempt to moderate this thread again.

8. Ans
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Again nothing specific from Reality Check. Now he asking something what is even earlier in article than Newton's laws. Seems as best what can be done in such case is simply start from start. Below is "Introductory" part of article. If someone see any flaws in the part, post about it and what exactly you see wrong. After it would be discussed, I will post next part for discussion and so on.

Introductory
In this article I develop the theory of emergent space-time-matter [1-11]. An insight into previous
publications on this topic is not required, in this article I give a complete description of the current state
of this theory.
At present, known physics laws allow for the existence of singularities, for example, inside black holes.
Many view these singularities as a sign that a new physics begins next to singularities. We are looking
for new laws of physics that describe the state of space, time and matter near these singularities. A
common feature of all these searches is that the authors imply that space, time and matter still exist in
such conditions, albeit in some unusual form.
However, there is another option, which, to the best of my knowledge, is the first fully considered only
within the framework of the proposed theory. This second option is that in some neighborhood of the
singularity, space, time and matter do not transfer into something unusual, but cease to exist. In this
case, since something inside this neighborhood of singularity affects its environment in space-time, this
something can not be nothing. The question is, what can be this something?
If this something does not contain space-time and matter, then it must be something more
fundamental. But then, since it does not contain space-time-matter and fields, space itself, time and
matter must be derived from this something. From this perspective, space-time-matter and fields must
emerge from properties of this something. Moreover, they can not be defined everywhere, but only
where there are suitable conditions for this.
Time is a phenomenon, the manifestations of which we constantly observe. Physics still does not know
the nature of time, the existing description of time and its properties is phenomenological. Special and
general relativity theories have established a relationship between time, space and gravity. This shows
that time is not an independent phenomenon, and has a connection with space and matter that causes
gravity. Physics has established the properties of time. However, there is no knowledge of why there is
time, why it is unidirectional, whether there are quanta of time, why time has one dimension, whether
it is possible to travel to the past.
There are phenomena called emergent. For example, the second law of thermodynamics. The
properties of thermodynamics are based on the properties of individual atoms and molecules,
described by quantum mechanics. However, the equations of thermodynamics can be applied
practically independently of the equations describing individual atoms and molecules.
Do space, time, matter and fields exist independently or are they a manifestation of something more
fundamental?
This article presents the theory of emergent space-time-matter (hereinafter referred to as ESTMtheory).
In this theory, space, time, matter and fields are viewed as emergent properties of a more
fundamental entity.
Let us start our consideration of the theory from the theory model.

9. So far I see a lot of words that do nothing to make me take your line of thought seriously. I would recommend putting some mathematics where your words are so we might have a chance of seeing what you mean by such things as emergent space-time-matter.

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Originally Posted by Ans
Again nothing specific from Reality Check.
Again nothing scientific from you, Ans. You are asked for a derivation, but you reply with another word-salad essay.

Is it that you do not know what a mathematical derivation is? You stubbornly insist that you have derived things, and show irritation when asked for derivations again. Yet the reason for the continued requests is that we have not seen any derivations from you. From this I conclude provisionally that you do not know what constitutes an actual derivation.

If that conclusion is correct, then I recommend that you look at an actual scientific paper that contains derivations and see how the authors go about it. You will note that one differentiating characteristic is the presence of maths with equations and such, in contradistinction with your literature-based presentation.

If you do not have a derivation beyond words, I am sorry that you cannot claim to have derived anything. So far I have only seen opinions, claims and assertions from you, but no actual scientific support. If you have none, then say so. If you have actual derivations, please present them. Stop with the walls of text. They will not persuade us.

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Originally Posted by Ans
Again nothing specific from Reality Check. ....
No answers to my quite specific questions, Ans.

That "Introductory" seems to have mistakes.
It hints that new physics only applies around singularities. The extreme case of black holes and the Big Bang exposes the new physics that exists throughout the universe.
Your idea is not the first where spacetime ceases to exist. There are existing proposed unified theories where spacetime ceases to exist at Planck scales. It is replaced by the strings in string theory or loops in loop quantum gravity.
Thermodynamics is not quite "based on the properties of individual atoms and molecules, described by quantum mechanics". Historically thermodynamics was based on the need to describe the operation of steam engines. Most of thermodynamics was established before QM existed. Much of thermodynamics is "ideal" where we treat atoms and molecules as non-interacting particles. Thermodynamics and quantum mechanics are generally 2 different fields. However see quantum thermodynamics. This has mathematical derivation of the laws of thermodynamics from QM.

12. Ans
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Originally Posted by Hornblower
So far I see a lot of words that do nothing to make me take your line of thought seriously. I would recommend putting some mathematics where your words are so we might have a chance of seeing what you mean by such things as emergent space-time-matter.
There is lots of math in article, and I ready to discuss any part of article. If you think my words about lots of math in article are not correct - it is very easy to check, just open article and make quick glance. At most, it will take ~30 seconds.
If you prefer to see it here in forum - ok, I will slowly paste here one part after another.

13. Ans
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Originally Posted by Geo Kaplan
Again nothing scientific from you, Ans. You are asked for a derivation, but you reply with another word-salad essay.
Interesting, what you expected to see? Before move to derivation, it is necessary to clearly define model. There is only one case when it is possible to move immediately to derivation - it is case when model is known for everybody. It is definately not model of my theory. So, before start to use it, necessary to clearly define and desribe the model.

All mentioned derivations exists in my article. Peoples just prefer not read it, and complaining about absense of derivations in the forum. Well, as I said, I started posting parts of article here in forum.

14. Ans
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Originally Posted by Reality Check
No answers to my quite specific questions, Ans.
It prepares answers to your question about deriving spacetime from space with defined on that space scalar field. After the post, I will post another part of article.

Originally Posted by Reality Check
That "Introductory" seems to have mistakes.
It hints that new physics only applies around singularities. The extreme case of black holes and the Big Bang exposes the new physics that exists throughout the universe.
Your idea is not the first where spacetime ceases to exist. There are existing proposed unified theories where spacetime ceases to exist at Planck scales. It is replaced by the strings in string theory or loops in loop quantum gravity.
Finally, first time I see something specific from you. And, as expected, it looks wrong.
Both in string theory and in LQG spacetime exists, but in some unusual way, as I described.
String theory use strings defined in spacetime.
Loops also use spaceimt. It is unusual spacetime, where macroscopic time is derived from times on microlevel, but it still exists.
Both string theory and loop quantum gravity have space, time and dynamic on fundamental level.

Also, LQG is not pretends to by unified field theory, it is attempt to build theory quantum gravity. And, according to my theory, there is no quantum gravity.

Originally Posted by Reality Check
Thermodynamics is not quite "based on the properties of individual atoms and molecules, described by quantum mechanics". Historically thermodynamics was based on the need to describe the operation of steam engines. Most of thermodynamics was established before QM existed. Much of thermodynamics is "ideal" where we treat atoms and molecules as non-interacting particles. Thermodynamics and quantum mechanics are generally 2 different fields. However see quantum thermodynamics. This has mathematical derivation of the laws of thermodynamics from QM.
Here you descrribing hystory of thermodynamics. Well, the history part looks correct. At same time, it has no relation to my statement about emergence and thermodynamics. You may see it here: https://en.wikipedia.org/wiki/Emergence

15. Ans
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New parts of article. Anything looks wrong, any flaws?

Theory Model

The theory is based on the assumption that at the fundamental level there is only Euclidean space with
a certain still unknown amount of dimensions and a scalar field defined on this space. There is nothing
else at the fundamental level except the listed, including the time, space and matter observed by us. All
dimensions are the same, there are no any specific dimensions. I assume smoothness of the
fundamental scalar field . A scalar field is described by some unknown differential equation. It can be
written as follows:
𝑓(𝑥 ) = 𝑔(𝑥, 𝑆, 𝑓(𝑆 ) ) (1)
where x is a point in the fundamental space, 𝑓(𝑥 ) is the value of the fundamental field at the point x, S is the closed surface surrounding the point x, 𝑓(𝑆 ) is the field value on the surface S, 𝑔 is some function.
The fundamental space with a scalar field defined on it I will call "Meta-Universe".
The theory model has no time at the fundamental level and it leads to the question - what is existence in the framework of this theory?

The existence and Postulate of Theory

I will add a definition of what is existence in the framework of the proposed model:
If a timeless system leading to the emergence of effective
space-time and matter as emergent phenomena also contains
intelligent life (which can play the role of an observer), then we will state
that such space-time matter does exist, this is
the emergent space-time-matter.
Will rational beings be aware of themselves in such a model? Further consideration of the model is not
possible without a positive answer to this question.
Therefore, it is necessary to introduce a postulate.
Postulate:
In the emergent space-time-matter, a rational being can think, feel that it really exists, that it is in being.

Anthropic Principle

From the definition of existence and the postulate of the theory it follows that the observer is necessary
for the existence of the universe. Thus, from the theory follows the anthropic principle.
The anthropic principle was proposed [12] [13] in order to explain from a scientific point of view, why in
the observable universe there are a number of nontrivial correlations between the fundamental
physical parameters necessary for the existence of an intelligent life. There are different formulations;
usually weak and strong anthropic principles are singled out.
A variant of the strong anthropic principle is the anthropic participation principle, formulated by John Wheeler [14]:
Observers are necessary to bring the Universe into being
In the ESTM-theory is a direct consequence of the basic principles of the theory.
Causality Principle
All models of intelligent life known to me require fulfillment of the casuality principle. Observers are necessary for the existence of the Universe. Only a rational being can be an observer. It means that intelligent life is necessary for the existence of the Universe. For this reason, the emergent space-time-matter-field must be constructed in such a way that the causality principle is fulfilled. Thus, the causality
principle is a consequence of the anthropic participation principle.

16. Originally Posted by Ans
Interesting, what you expected to see? Before move to derivation, it is necessary to clearly define model. There is only one case when it is possible to move immediately to derivation - it is case when model is known for everybody. It is definately not model of my theory. So, before start to use it, necessary to clearly define and desribe the model.

All mentioned derivations exists in my article. Peoples just prefer not read it, and complaining about absense of derivations in the forum. Well, as I said, I started posting parts of article here in forum.
The rules of this Forum, say that you must present all data here, not in a link.

17. Ans
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Originally Posted by Noclevername
The rules of this Forum, say that you must present all data here, not in a link.
Well, I posting it. Do you read it at all?

18. Originally Posted by Ans
Well, I posting it. Do you read it at all?
I'm reading a lot of words and a little math. That's not a great ratio.

19. Frankly this reads like a philosophy essay so far, not a scientific presentation. Are there any hard numbers?

20. Originally Posted by Ans
Interesting, what you expected to see? Before move to derivation, it is necessary to clearly define model. There is only one case when it is possible to move immediately to derivation - it is case when model is known for everybody. It is definately not model of my theory. So, before start to use it, necessary to clearly define and desribe the model.

All mentioned derivations exists in my article. Peoples just prefer not read it, and complaining about absense of derivations in the forum. Well, as I said, I started posting parts of article here in forum.

you have been told TWICE by a moderator that you have to show you stuff here in this thread and not link to a paper somewhere
i don't think it gets any clearer than that
you may mlso want to proof read your posts as equation 1 does not seem to parse

21. Ans
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Tusenfem, may you explain your comment? If you read the topic, you may see it is exactly what I doing. Latest post with part of my article is 4 posts before your post.

22. Ans
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Originally Posted by Noclevername
Frankly this reads like a philosophy essay so far, not a scientific presentation. Are there any hard numbers?
That part which I posted today related to area, usually considered as phylosophy. However, this part create basis for deriving of spacetime from fundamental space with defined on that space scalar field. If my theory is correct, related area of phylosophy will become part of physics.
Reason why I need this part is because my theory is based on the part. Normally, scientific papers not contains anything phylosophical because they based on common vision (realism). My theory is based on subjective idealism, so I have to explicitly write it.
Tomorrow I would post another part of article, it is part with deriving spacetime from fundamental space with defined on that space scalar field. It contains math, many asked for it, and at same time it heavily depends on previous part which I posted today. Without understanding the part which I posted today, part with deriving spacetime from fundamental space with defined on that space scalar field would not be understandable.
So it is reason why I asking is anyone see any flaws in posted part.

23. Originally Posted by Ans
Tusenfem, may you explain your comment? If you read the topic, you may see it is exactly what I doing. Latest post with part of my article is 4 posts before your post.

Do not comment on modaration, in the thread. If you have a comment, then report it.
Furthermore, up to now you do not seem to have presented anything substantial.

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Originally Posted by Ans
That part which I posted today related to area, usually considered as phylosophy.
Please stop answering our specific requests for derivations with ever-more statements of philosophy. Blogs exist to serve the purpose for which you seem to be using this forum. Your goal should be to persuade us that your arguments are sound. So far, your strategy has been completely ineffective. Instead of changing strategies, you insist on doing much more of the same. It's a waste of time. If you are going to continue ignoring what we ask for, you will fail. Repeating unsupported assertions does not somehow confer validity.

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Originally Posted by Ans
New parts of article. Anything looks wrong, any flaws?
"From the definition of existence and the postulate of the theory it follows that the observer is necessary for the existence of the universe." is wrong because it is circular reasoning. You define a thing called "existence" as needing intelligent life and conclude that you need intelligent life ("the observer") as in the anthropic principle.

The anthropic principle is more philosophy than science with several variations.

Your "Causality Principle" paragraph is just assertions. Causality in physics happens regardless of the presence of intelligent life or even life at all.

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Thanks for the answer. I prefer discussion with specific statements instead of exchange of vague assertions.
There is no circular reasoning here. However, probably this part needs to be expanded to improve explanation. In the post, I expanded explanation.
Thing called "existence", as it was named by Reality Check, is definition of existence.
About postulate of the theory and definition of existence.

Why the definition necessary, why there is postulate of theory, and what they says?
The theory is based on timeless entity. The entity consists of Euclidean space with defined on that space continuous classic scalar field.

Let’s imagine that space-time-matter-fields, with exactly same properties and laws of physics as in our Universe, was somehow found on basis of the timeless entity.
Can such space-time-matter describe Universe, observable by us? If in such a world life is
possible, can sentient being belonging to such world feel reality of surrounding and itself? These
questions seem to refer to philosophy, as concept of Being is affected. However, different variants of
answer to these questions provide different results in physics, so those questions are related to physics
too. Postulate and main idea of this theory is positive answer to these questions. Without positive answer to the questions, there is no sense in building the theory.
Answers to the questions can be indirectly verified by verifications of results of the theory.

Occam’s Razor helps in positive answer to these questions, since this theory reduces number of
independent phenomena and reduces significantly. Instead of various unrelated physical phenomena
and independent space-time, this theory suggests that all physical phenomena can be derived from one
model with one field and suggests ways to find it.
If in such space-time-matter, emerged from timeless underlying entity, there is sentient, it will observe
following:
* Time exists, and all events have causal relationships, include relationships with probabilistic
nature.
* There is past, present and future.

Why present time would exist? It may seem that in such system, time will pass immediately. However, it
may be only from point of view of external observer. But external observer in this model cannot exist,
first because of absence of time as fundamental phenomenon and second because underlying entity
includes everything objectively existing. Observer here can be only object, capable of self-awareness,
and belonging to emergent space-time. If such observer will be in emergent time, it will observe changes
of states of surrounding. Human thought - it is some change in state of particles and fields in man over
time. Consequently, observer, who lives in emergent time, will also be able to think, provided that
relevant physics of emergent spacetime-matter allows for intelligent life to exist. The speed of its
thoughts will be determined by rate of change of its states in emergent time. In particular point in
space-time, observer will always have same thought. If this theory describes our Universe, it means that
any human is, in some sense, immortal. Everyone exists forever, but when our present does not coincide
with present of some other persons - they are not available to us. Similarly, at any point of present time
our current present mind cannot contact ourselves in past or future. Also, number of human thoughts is
limited by human lifespan. Anyone can do whatever he wants. However, desire of man to do something
is caused by state of human body at some point in time. Therefore, one cannot wish for anything other
than what was set by his state, so there is no real freewill.
In modern physics, human’s freewill is based on Heisenberg’s uncertainly principle. In this article, I
would show what the uncertainly principle is not contradicts to super-determinism of the theory; it arise
quite naturally in super-determinism of the theory.
Postulate:
If in objectively existing timeless system, which includes everything objectively existing, it is possible to
find space, time and matter as emergent phenomenon, and such space-time-matter allow existence of
sentient, such space-time-matter is exists; it is emergent spacetime-matter. In such emergent spacetimematter
sentient can feel, think, feel that it really exists, is in being.
Consequence of this postulate: for case when laws of physics of emergent spacetime-matter allow
intelligent life to exists, sentient belonging to such emergent spacetime-matter will feel like being in
space and feel the passage of time. He will feel emergent physical laws; laws of physics of fundamental
timeless system will be deeply hidden from his feelings.
Later in the article, I will discuss nature of time in more details.
It is possible to notice what postulate of the theory can be proven based on induction.
If, at some point of emergent time, human is able to think and feel reality of surrounding – he will be
able to think and feel reality of surrounding at any other following moment of emergent time.
Proof:
ESTM-theory describe world with exactly same physical laws and phenomena as we observe. If it cannot
describe some observable phenomena, it means ESTM-theory is not correct. So I assume here what
ESTM-theory is able to describe world with exactly same physical laws and phenomena as we observe.
So if human would not be able to think and feel reality of surrounding at any other following moment of
emergent time – it means physical laws and phenomena of our world prohibit to humans think and feel
reality of surrounding. It directly contradicts to observations, humans are able to think and feel reality of
surrounding.
So, the statement is proven.
Similar for first step of induction, what human (or sentient) belonging to such world can think and feel
reality of surrounding at some moment of emergent time. If it is possible to build exactly same laws of
physics in ESTM-theory as in our world, it means what human will be able to be born, learn and start to
think.
Postulate of ESTM-theory is more philosophical than physical, so the prove above can be questioned
based on some philosophical views. So I have to use postulate instead of theorem.

So, there is no circular reasoning.
Without observer, any solution with effective spacetime and matter, based on timeless entity, will be just mathematical abstraction. So, here comes anthropic participation principle, as result - observers are necessary for Universe to exists.

Causality principle is direct result of anthropic participation principle - without observers Universe cannot exists. All known to us models of intelligent life require causality. Intelligent life cannot exists without causality. Only intelligent life can be observer. Without observer, Universe cannot exists. So, Universe cannot exists without causality. So, causality principle is consequence of anthropic participation principle. Anthropic participation principle is consequence of postiulate of my theory.

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New part, with math. In the part I derive spacetime. Spacetime here is classic with long range action. Effects of SR are ahead.
I consider time, as one can see, as parameter of evolution in equations.

Symmetry to Translations of Emergent Time and Space

To fulfill the causality principle, it is necessary to understand which properties must physical laws have
with respect to translations of the emergent time and space. In case if there is no symmetry for the
translations of the emergent time and space, there is no way to fulfill the causality principle. For this
reason, it can be concluded that such a symmetry, which still can be called homogeneity, must exist.
This means that any solution with emergent space-time must contain such symmetries.

The order of statement of the theory in the article

At the fundamental level of the theory there are no such basic theoretical concepts as time, observable
space, matter, fields. For each of these concepts, it is necessary to find a correspondence in the
proposed theory. It is also necessary to find a correspondence for related concepts, such as mass,
energy, velocity, elementary particles etc. Each of these cocepts are linked with each other. One may
identify in one place what these concepts correspond to in this theory, but it might be difficult for
perception and understanding. Therefore, the theory will be presented in several iterations.
First, we will consider the plane Euclidean emergent space-time with long-range action. This part shows
how this space-time emerges. It shows how and why inertia emerges. The mass and energy are found.
Newton's laws and Galileo transformations are obtained. An elementary particle was given a definition
in terms of the theory. The Schrödinger equation is derived. An explanation is proposed for the spin of
particles.
In the next part, we again consider the emergent space-time, but more precisely. It considers a flat
pseudo-Euclidean emergent space-time with a finite maximum interaction speed. This part shows why
such speed exists and why it has to be finite, why this speed is constant and why it is the same in all
reference frames. Lorentz transformations and equations of the special relativity theory are obtained.
The Klein-Gordon-Fock and the Dirac equations are obtained.
Next we will consider interactions of elementary particles. An explanation is proposed for what is an
electric charge. The Maxwell equations are obtained. An explanation of what virtual particels are is
given. It is shown that the light speed must be exactly equal to the maximum interaction speed. It is
shown that the proposed theory is compatible with the standard model of elementary particle physics.
The next part deals with gravity and the curved emergent space-time. Equivalence of acceleration and
gravitation is derived. It is shown that the Einstein equations of the general relativity theory correspond
to the proposed theory. It is shown that the proposed theory implies the absence of a carrier particle of
gravitational interaction, i.e. the absence of a graviton is predicted.
Next we will consider cosmology. In this part, an explanation is offered for the origin of the Universe. It
is explained why the Universe has three spatial and one time dimension. An explanation is proposed for
dark matter and dark energy. The possibility of existence of parallel universes is considered. It is shown
that the Meta-Universe probably has more than four dimensions.
Next we considered the physical foundations of mathematics
In each of these parts, the correspondence of the theoretical concepts and theory model is consistently
introduced and equations of the theory are refined. The final form of equations of the theory is
obtained after the cosmology section.

-- conitnue in next post

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Fixed formatting of equations.

Flat Space-Time With Long-Range Action

Let us first consider the case of a flat emergent space-time. In order to find it, we consider a family of
hyperplanes satisfying the properties described below.
 Hyperplanes belonging to the same family must not overlap and be parallel.
 There is a continuous transformation that transforms one hyperplane of the family into another.
Besides the described properties, hyperplanes must satisfy a number of properties associated with
expansion of a fundamental scalar field on hyperplanes. First, we need to give a definition to expansion
of a scalar field on a hyperplane.
I will use the smooth scalar functions defined for the Euclidean space with the number of dimensions
equal to the number of hyperplane dimensions. Each function has some set of parameters that allow
one to definitely indicate its location on the hyperplane. One of these parameters is some point. For
symmetric functions you can use a symmetry point, and it can be a single parameter. For nonsymmetric
functions one may also need a vector indicating the function orientation.
Suppose there is some family of functions with the indicated properties.
I will call the expansion of the fundamental scalar field by the basis of functions the case when the field
values at any point on the hyperplane can be represented with sufficient accuracy as the sum of the
functions that belong to this family, with certain coefficients and with a series of conditions considered
later.
The same function can enter the expansion repeatedly. The definition to "sufficient accuracy" will be
given later. In this case, we can say that the family of functions form an almost complete functional basis
for the expansion and form a certain set of functions {𝑤}.
Suppose where is a vector in the fundamental space, this is an equation describing a certain hyperplane of the family under consideration. Suppose is a point on the hyperplane under consideration
Suppose is the value of the fundamental scalar field at the point . Then the expansion function of the field will be as follows:

where is a some constant, is a function characterizing the accuracy of the expansion. In the case when set form a complete functional basis, equals zero, otherwise it must be much less than the rest of the expansion. More detailed requirements to will be found later. is a point on the hyperplane characterizing the position of the function used for the i-th time. are the coefficients of the amplitude of the function. is a set of all other parameters that allow you to definitely identify the location and possible other characteristics of the function. It can be seen yet that this set should include a vector for specifying the direction of asymmetric functions. Further there will be shown other possible parameters. In order to calculate the field value at the point X, it is enough to sum it. The field value at any point of the hyperplane, if 𝑓𝑒𝑥𝑡 and 𝑓0 are neglected, can be calculated using only the expansion functions.
the necessary requirements to both the family of hyperplanes and to the functions of basis of
expansion.
We want to get an effective space based on hyperplanes, and effective time based on the distance
between them. For this it is necessary that the causality principle is fulfilled. The causality principle will
be fulfilled if the functions of expansion basis are the same on all the metrics of the emergent
hypersurface space of one family, and if all 𝑢𝑖𝑘 , 𝑌𝑖𝑘 , {𝑄𝑖𝑘 } on a single hyperplane of the family are known, then with sufficient accuracy we can calculate the values of the expansion on any other
hyperplane of this family. This is another requirement for the family of hypersurfaces and for the
expansion functions.
If the effective time corresponds to the distance between the hyperplanes, then we can talk about the
time vector. The quesion is, where this vector is directed to?
To answer this question, we can recall that there is no preferential direction in the fundamental space.
Thus, this vector must be directed in the most symmetrical manner towards the hyperplane. The
greatest symmetry is obtained if the time vector at each point of the hyperplane is directed
perpendicular to the hyperplane.
I will call the family of hyperplanes as emergent space-time with the indicated properties. Furthermore
the space of hyperplane belonging to the family will be the emergent space, the distance between
hyperplanes will be emergent time.
The notion of a world line for the point in the emergent space can be introduced. it is a curve in the
fundamental space, at each point of which the time vector is parallel to this curve, and which passes
through the indicated point. One can say that the point x from one hypersurface is mapped to the point
x′ on another hypersurface if the world line of the point x passes through the indicated hypersurface to
the point x′. For the case of flat space-time, the world line is straight.
Suppose there are two distinct points x1 and x2 lying on the hyperplanes of the family under
consideration, and not necessarily on one hyperplane. Using the notion of the world line, their relative
position can be described as (𝑟⃗, t) where 𝑟⃗ is a vector in the emergent space, t is the difference in the
emergent time. The vector 𝑟⃗ can be found by finding the intersection of the world line from the point x1
with a hyperplane containing x2, or vice versa. t is directly proportional to the distance between the
hyperplanes, with some constant coefficient.
Let me get back to the expansion of the scalar field. The expansion of a scalar field on a hyperplane,
according to equation 2, can be written as a vector of the state Ψ consisting of the values 𝑢𝑖𝑘, 𝑌𝑖𝑘 and
{𝑄𝑖𝑘 } for all functions 𝑤𝑖. From the requirement for the causality preservation, it follows that these
values for each subsequent hyperplane must be calculated on the basis of the previous values:
Ψ(t + dt ) = UΨ(t )
here U is some operator that transfer the state vector into another state vector at a subsequent time
point.
In order for the laws of physics to be always the same, symmetry is necessary for a time shift. This
means that the operator U preserves the scalar product, i.e., it is unitary. If in equation 3 dt = 0, then
U = I where I is the unit operator. Further, I assume that the function Ψ is differentiable, which means
continuity of space-time. Therefore, it can be written as follows:

From the other side,

Then

The equation can be shortened:

dividing by dt:

The derivative of the operator is also an operator, although not necessarily unitary. Marking it as A,
we get the final differential equation of the unitary system evolution:

The equation later will be used to derive Schrodinger equation
Last edited by Ans; 2018-Oct-11 at 06:09 PM.

29. Ans
Member
Join Date
Sep 2018
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Are any comments, any obvious errors and flaws?

30. Originally Posted by Ans
Are any comments, any obvious errors and flaws?

the obvious error is that youbdid not check your posts and your equations did not parse
you can use latex if you want wit the [ tex ] tags

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