Premilinary definition of physical space comes from our mathematical concept of "axis". Or, our coordinatory system allows us it to consider at particular coordinate.

When we compare theoretical physics with mathematics, then a dot

also have meaning as per its physical properties. While , mathematically

it is nothing up to when it is not according to its already defined number line.

For e.g. if we consider it with polygons then we can apply only

those which atleast can provide more than one axis. But when its about probability of particles then we have two consider minimum 3 axis.

While, preliminary definition of time is like fixing the mathematical operators

for particular variables in any equations and not applying them as such further. Why? because this way it defines the change. But for any particle things makes sense which way it is correct, while our formal logic here maximum can say its like "uncertainity". Or, we can say we understand

and consider the reality differently in both cases, indirect reason here is

theoretical physics do not consider and declare it abstract way direclty.

Although mathematics also do not say so but things change for them who do not understand.

Main thing here is probability will be defined according to space itself. Instead of this if we use space and time both then total probability mathematically and for theoretical physics both increases where we use it either in functions or equations it depends.

Nothing’s missing, beyond the lack of knowledge of technical topics that are now standard courses. The definition of probability hasn’t changed since Kolmogorov and the definition in terms of frequencies is just a special case.

What is missing is the definition of time as a caracteristic of a corpuscle and which allow to treat corpuscles and waves as the same thing .

The theory of the interacting gases as the electrons in solids is still

in infancy. Many areas of physics need more development. Cosmology is still under uncertainties. Particle physics is hard, and in infancy.

Yes Juan, and among others... a purge of people who claim to know that Einstein was right...

In mathematics and theoretical physics, there is always room for improvement and advancement, and there may be areas that researchers are currently investigating. The definition of probability that you propose is an interesting idea, and it may be an area of active research. However, it's important to note that the definition of probability used in current mathematical and theoretical physics models has been well-tested and validated through experimentation.

While the quantum mechanical definition of probability density you mentioned may have limitations, it has been successful in explaining many phenomena in physics. In addition, there are other areas of mathematics and theoretical physics that are currently being explored, such as the development of new mathematical models for complex systems or the investigation of fundamental particles and forces.

It's also important to note that mathematics is a powerful tool for understanding the physical world, and it's often used to make predictions and develop new technologies. While new definitions and approaches may be proposed, they need to be validated through experimentation and observation before they can be widely accepted.

Technically, these answers seem to vary considerably: From: "almost everything is missing" (PG) or "Nothing’s missing..." (SN), to ... "there is always room for improvement..." (JMuA). However, let me propose something more fundamental for each field:

1) Our math is missing a "heart" -- to balance its/our intellect, and to offer some "feeling". We mathematicians are quite sensitive to the mind beauty of a theorem, but we feel quite neutral to homogenous 3-D space. On the other hand, an artist cannot enter a decorated 3-D room without feeling its aethetic beauty, or lack thereof. Our Math needs to become "more human" if is to offer Science better tools to solve human needs. Statistics, fuzzy logic, other logics, inconsistencies (Goedel, et al), fractals, ... are all indications of math's evolution toward sensitivities of "human heart". So, let's "improve" our math with a call for "a Mathematics with/of Heart".

2) Our theoretical physics is missing a dose of common sense! -- and it needs to give more credence to the role of consciousness (in influencing physical results). Common sense -- in freeing itself of a major unfruitful obsession: string theory. And Credence -- to recurring revelations (empirical data) that show the study of consciousness is worthy of far greater attention than it is presently given. Obsessions can be helpful, if and only if, they support wise choices, in directions that yield fruitful results, in the Present or in the not-too-distant Future. Economics matters, also.

Heart + Consciousness = New Ways = New World

Markov was part of the great tradition of mathematics in Russia and never missed his goal.

He designed and passed on a non-physical stochastic chain and never claimed it could solve physical problems.

We present the following brief comparison between classical stochastic Markov chains and recent B-matrix chains for the statistical transition imposed via the Cairo technique in the year 2020,

Markov defined his original nxn square transition matrix M(i,j) via two assumptions:

(1) the transition probabilities Mi,j are equal to or greater than zero.

Sigma Mi,j=1, for all the columns j=1 to n, (called the vertical matrix) OR the sum of the transition probabilities Sigma Mi,j=1, for all the rows i =1 to n , (called horizontal matrix)(2) the sum of the transition probabilities

Obviously the Markov stochastic chains can be converted into B-matrix chains and vice versa.

On the other hand, the well-founded B-matrix and B-transition matrix chains rely on four assumptions. The first two assumptions are those of Markov with a slight modification of the second to allow the insertion of boundary conditions BC.

The other two assumptions are physical, namely,

(3) the matrix B is symmetric in the geometry of 3D space like nature itself,

i.e. Bij = Bji.

condition (3) represents two of the physical facts,

i- detailed balance rule.

ii- reciprocity rule.

Note that while Mi,j comes as a set of individual inputs, Bi,j comes as a set of pairs.

We recall the following,

a-Boltzmann's statistics referring to "Stosszahlansatz" gas molecules or the molecular chaos hypothesis uses a binary or pairwise collision.

b-The classical derivation of the time-independent classical Boltzmann distribution for

discrete or continuous energy levels via the cell-and-box method rely on the mutual variability between two levels.

In short, nature's own response is like a kind of pairwise symmetry.

(4) the input elements of the main diagonal (RHS) are all equal to a constant input RO.

where RO is an element of [0,1]

Physically, RO represents the residue or remanence of the energy density at a point in space-time.

It is clear that RO must be zero when solving the Poisson and Laplace PDE and somewhere between zero and one when solving the heat diffusion/conduction PDE.

The question arises as to why stochastic Markov chains cannot solve physical problems (heat diffusion/conduction PDEs, Poisson and Laplace PDEs, derivation of integration and differentiation formulas, etc.) which are simply solved by the transition-matrix B (B i , j) of the Cairo Technique.

What is missing from mathematics and theoretical physics in general and Markov transition chains in particular?

The answer is that the physical conditions (3) and (4) are missing.

The decisive difference is that in Markov chains which ignore x-space, the time interval dt is arbitrarily chosen and not necessarily space-relevant whereas in B-matrix chains the time is merged or woven into a 4D x-t unit space, which means that the time interval dt is imperial.

It should be mentioned that the B-matrix strings are relativistic invariant as they should be.

Moreover, what are the limits of Markov analysis?

If the time interval is too short, Markov models are inappropriate because the individual movements are not random, but rather deterministically related in time. This case suggests that Markov models are generally inappropriate over sufficiently short time intervals.

The exact definition of time, the essence of time. What is time, what is it connected with, what does it depend on?

Stochastic Markov chains or Markov processes can't solve physical problems because these considers system deterministic while symmetric transition Matrix solves it because these provides limited randomness as their own. Symmetric transition matrix assumes space and time as combined in most of the studies. Alternately, we can say observers reality aspects differ because system is dynamic due to kind of processes it deals.

In researchers daily reality, what is felt « missing » is researcher’s capability for « appeals » to questionnaire this kind of subjects. Thank you for questioning.

Ismail Abbas Cyprien Saito Anil Kumar Jain Niazbeck Useinov John A. Shuster

The probabilistic interpretation of quantum mechanics, based on the wave function and the Born rule, provides a well-established and successful framework for calculating probabilities in quantum systems. It has been extensively studied, tested, and verified in experiments, and has led to significant advancements in our understanding of the microscopic world.

In quantum mechanics, the probability density function is defined as p(r, t) = |ψ(r, t)|^2, where ψ(r, t) is the wave function that describes the quantum state of a system. The probability density function gives the probability of finding the particle at a particular position in space, at a specific time.

The time dependence in quantum mechanics is inherent in the evolution of the wave function over time, governed by the Schrödinger equation. The wave function describes the state of the system at any given time, and by calculating its time evolution, one can determine how the probabilities change over time.

The probabilistic interpretation of quantum mechanics has been highly successful in describing and predicting the behavior of microscopic particles and their interactions. It has been extensively tested and verified experimentally.

While alternative interpretations or approaches to quantum mechanics have been proposed, such as hidden variable theories, they have not gained consensus among the scientific community and are not widely accepted.

The development of mathematical tools and techniques within quantum mechanics, such as integration, differentiation, and special functions like the Gamma function, is crucial for accurately describing and solving quantum mechanical problems. They provide a rigorous and precise framework for understanding and analyzing quantum phenomena.

Hope it helps: partial credit AI

Experiment!

When we can determine the geometric origin of the gravitationnal constant than we come to put the missing stone in the theoritical physics.