However, the process of "It from Bit" remains to be clarified. In this work, we propose Information Mechanics. We suggest that information determines observed phenomena. The interaction of the two basic elements making up information creates everything we observe. We introduce information space and time to be the basic elements. The interaction of information space and time creates everything we observe. Based on these assumptions, we derive an action, the information action, and the information function. The information action represents the maximum possibilities, i.e., information, in a system. Information action appears to be similar to the string action in string theory and superstring theory but with a different meaning. The information function calculates the possible states in a system. We demonstrate that elementary particles and their wave-particle duality, fundamental forces, dark matter, and dark energy can emerge from the information function. We show that it is possible to derive a value of the cosmological constant consistent with astrophysical observation. We demonstrate that one may derive the hierarchy between the weak scale and the Planck scale. We point out that one may study what is inside a black hole and deduce that the entropy of a black hole to an outside observer is proportional to the area of the event horizon. Based on the various problems Information Mechanics may address, we suggest that it could lead to the grand unification theory we seek.

I. INTRODUCTION

The principles and laws of creation are sought in many disciplines, including sciences, philosophies, and ideologies. Current cosmology suggests that the creation of our universe is through a "big bang." However, the natural law that has led to the big bang waits to be explored further.

John Wheeler proposed [Refs 1, 2, 3] the idea "It from Bit." He suggested that information sits at the core of physics and every "it," whether a particle or field, derives its existence from observations. To show how everything comes to existence through observation, John Wheeler acknowledged [Ref 3] that time played an essential role, but this is not understood well enough.

The Grand Unification Theory is an attempt to use one mathematical formula to explain everything, including all elementary particles, fundamental forces, dark matter, dark energy, and the macro structure of the universe, and to unify quantum physics with Einstein's general relativity theory about gravity. So far, string theory is the only mathematically consistent theory that can unify everything [Refs 4, 5]. However, current string theory is still limited in its ability to make predictions. Something is still missing.

In this paper, we propose Information Mechanics, in which information determines and creates everything we observe. We present the basic principles and formula about how information underlies all observed phenomena, including giving the emergence to elementary particles, fundamental forces, dark matter, dark energy, black holes, and the universe. We proffer two basic laws governing Information Mechanics. The First Law of Information Mechanics comes from quantum physics, indicating the information contained in our measurement determines the observed phenomena. The Second Law of Information Mechanics comes from the ancient Chinese wisdom about yin yang. It proposes that two basic elements, yin and yang, make up everything, including information, and that the interaction of these two elements creates all the observed physical phenomena. We will show that the interaction of two pairs of yin yang elements: space-time and exclusion-exclusion, create all elementary particles, forces, dark matter, dark energy, and the universe we observe.

We show that the laws of Information Mechanics give rise to an information action. Information action represents the maximum possibilities, i.e., information, in a system. Information action appears to be similar to string action. From the information action, one can define information function. Information function calculates the possible states in a system. Information function seems to be an extension of wave function in quantum mechanics. With information action and information function, we demonstrate that elementary particles and their wave-particle duality emerge from the Poincaré symmetry, fundamental forces come about due to the diffeomorphism symmetry, and classical equations of motion come from Weyl symmetry. Observation of dark matter and dark energy at the large scale of universe can be explained in Information Mechanics. We find that it is possible to calculate the cosmological constant consistent with the current astrophysical observation. The hierarchy problem regarding cosmological constant can be possibly explained and derived. We also demonstrate that it is plausible to deduce the large hierarchy between the weak symmetry breaking scale and Planck scale. We find that information action and function provide a way to study what is inside a black hole and also to derive the entropy of a black hole as seen by an outside observer to be proportional to the area of its event horizon.

II. BASIC PRINCIPLES AND LAWS OF INFORMATION MECHANICS

a) First Law of Information Mechanics

The observed phenomenon is determined by the information of the measurement.

The First Law of Information Mechanics comes from quantum physics, which indicates that the process of making a measurement determines the observed phenomenon.

For instance, quantum physics shows us that what detector we use and where we place the detector determines what phenomenon we observe.

b) Second Law of Information Mechanics

Everything is made of two basic elements. These two elements are opposite, relative, co-created, inseparable, and co-dependent. The interaction of these two elements creates everything.

The Second Law of Information Mechanics originates from ancient Chinese wisdom about yin yang [Ref 6, 7]. We keep the Chinese words here and call these two basic elements that make up everything: yin and yang.

These two basic elements, yin and yang, make up the "Bit" of information. The Second Law of Information Mechanics suggests the interaction of yin and yang, the essential elements of information, creates everything.

III. SPACE AND TIME AS THE FUNDAMENTAL YIN-YANG PAIR

What are the basic yin yang elements that create the observed elementary particles, forces, dark matter, dark energy, and the universe?

We suggest that space and time are one of the fundamental pairs of yin-yang elements that create everything we observe.

We propose another meaning of space-time, which we call information space and time. Information space and time are related to two basic measurements we conduct. Information time relates to the measurement of movement and change. Information space relates to the measurement of stillness and solidity. For instance, the flow of sand in an hourglass and the movements of the sun and the moon have all been used as measurements of time. The duration of a day is based on the measurement of the rotation of the earth. The measurements of space, such as the length, height, and width of an object, are the measurement of its fixedness and stillness.

Information space and time are a yin-yang pair. They are opposite and relative. Change and stillness are opposites and relative. Space and time are co-created because whenever one measures change, one refers to something unchanged. Whenever one measures something as unchanged, one compares it with something one considers changing. Therefore, information space and time are inseparable and co-dependent.

According to quantum physics, it takes energy and momentum to measure time and space. How accurately time and space can be measured depends on the amount of energy and momentum used in the measurement. More specifically, to measure the time of duration $\mathrm{\Delta }\tau$, it takes an energy of $\mathrm{\Delta }E\sim \hslash /\mathrm{\Delta }\tau$. To measure a space of size $\mathrm{\Delta }\sigma$, it takes the momentum $\mathrm{\Delta }p\sim \hslash /\mathrm{\Delta }\sigma$.

If one takes gravity into consideration, according to general relativity, energy curves space-time. When one measures time interval $\mathrm{\Delta }\tau$, the energy $\mathrm{\Delta }E$ used for a time measurement will curve space-time. It will create a black hole with the horizon on the order of $G\mathrm{\Delta }E/c^4$. When $\mathrm{\Delta }\sigma$ is smaller than $G\mathrm{\Delta }E/c^4$, no information can escape. Therefore, the measurable causal region is:

Therefore, there exists the uncertainty relation between the measurable space $\mathrm{\Delta }\sigma$ and measurable time $\mathrm{\Delta }\tau$,

Here $I_p$ is the Planck length and $t_p$ is the Planck time. This uncertainty relation suggests that information space and time affect each other. They are not independent. They are a yin-yang pair.

Next, we propose that inclusion and exclusion are the other basic yin-yang pair in measurement. This is because, when one makes a measurement, one needs to give the information about what is included and what is excluded.

We propound, all measurements are based on these two basic yin-yang pairs: information space-time measurement and inclusive-exclusive measurement. To see this, one can examine all possible measurements, such as measurement of velocity, acceleration, energy, momentum, temperature, spin, electricity, magnetic field, mass, charge, and force. One can see that these measurements are various combinations of space and

Here the symbol $\sigma$ represents information space and the symbol $\tau$ represents information time. The integral symbol $\int$ represents the summation over information space and time, and $\alpha$ is a constant. From the uncertainty relation between information space and time (1), we get

To introduce the second yin-yang pair into the action, we need to include the inclusion and exclusion elements. To do this, we realize that corresponding to the inclusion and exclusion yin-yang pair, in nature there exist two types of particles, fermions and bosons. Fermions have half (1/2) spin. They repel each other. They cannot be in the same state. Bosons have integer spin. They tend to clump. The normal time and space coordinates $\tau$ and $\sigma$ are of bosonic nature. If we assume each information space or time coordinate has both the fermion (yang, repulsive) and boson (yin, clumping) parts, each information time and space coordinate become two elements:

Here we use ${\theta }_{\sigma }$ and ${\theta }_{\tau }$ to represent the fermion part of information space and time coordinates $\sigma$ and $\tau$. The ${\theta }_{\sigma }$ and ${\theta }_{\tau }$ can only take on the value of 0 or 1 because they are repulsive and refuse to stay at the same place. The four elements of the two yin-yang pairs are represented by $\sigma$, $\tau$, ${\theta }_{\sigma }$, and ${\theta }_{\tau }$.

The simplest action created by these two yin-yang pairs is:

We will call the action $A_1$ and $A_2$ the information action.

One can see that the action $A_1$ is basically the action of bosonic string and the action $A_2$ is the action of the super string[Ref 4, 5]. It is interesting to see that from the basic laws of Information Mechanics we can derive string action [Ref 4, 5].

The information actions $A_1$ and $A_2$ in equations (2) and (5) give the amount of the information in a system with the observation scale (0, T) and (0, L). Information action expresses the possible states that can be observed in a system. Seth Lloyd derived a similar result in his paper [Ref 8], viz. that the maximum observable information in a system is represented by (2).

V. TWO TYPES OF SPACE-TIME

To derive the observable phenomena in Information Mechanics, it is necessary to realize that there exist two types of space time. One is the information space time associated with the fundamental yin-yang pair $(\tau ,\sigma )$. It is related to the basic elements of information. We call it information space and time. The other is the physical measurement of space distance and time duration or physical location of space time $X^{\mu }$. Let's call $X^{\mu }$ physical space and time.

The physical space and time $X^{\mu }$ is a projection from the information space and time $(\tau ,\sigma )$,

Here,

Suppose in this projection, the total information is unchanged. The action $A_1$ becomes:

In the following, for the sake of the simplicity of illustration, we will work with the "bosonic string,"

One can follow and extend the same discussion to the general case of "superstring,"

VI. DEFINITION OF INFORMATION FUNCTION

Now let's define the information function $\mathrm{\Psi }$:

Here A is the information action. We can see that the information function $\mathrm{\Psi }$ is related to the amount of information I in a system through the formula:

Here $\int \mathcal{D}X$ represents summing over all possible $X$, similar to Feynman's path integral definition [Ref 9].

Compare to the wave function in quantum physics:

Here $S={\int }_0^{T}dt\mathcal{L}(x(t),\dot{x}(t))$.

One can see that the information function is a natural extension of the wave function in quantum physics. The action in Information Mechanics integrates over both time and space while the action in quantum physics integrates over time.

Information Mechanics is also different from quantum field theory. The action of Information Mechanics integrates over two-dimensional information space and time, while in quantum field theory the action integrates over four-dimensional physical space time. The main task of quantum field theory is to calculate the correlation function and scattering cross-sections. In Information Mechanics, the correlation function and, thus, scattering cross-section can be obtained through the information function in the equation (9). Note that the wavelength and frequency in quantum field theory are now replaced with the measurement scales T and L in Information Mechanics. We will discuss the correspondence between the calculations of Information Mechanics with those of quantum field theory in more detail in future work.

VII. EMERGENCE OF ELEMENTARY PARTICLES AND FUNDAMENTAL FORCES

As discovered in particle physics, the basic constituents of everything are elementary particles and fundamental forces. It is found that elementary particles have wave-particle duality, meaning that they behave like a wave but each elementary particle has its own specific mass and spin, no matter where and when one makes the measurement. The wave-particle duality of elementary particles and fundamental forces was proposed by Einstein and Niels Bohr and indicated by experiments [Ref10, 11]. However, it is never derived from the first principle in theoretical physics.

The action $A_1^{\prime \prime }$ is invariant under the following three transformations:

1. D-dimensional Poincaré transformation

2. Diffeomorphism transformation

3. Two-dimensional Weyl transformation

Information action has three symmetries: Poincaré symmetry, diffeomorphism symmetry, and Weyl symmetry.

VIII. EMERGENCE AND OBSERVATION OF ELEMENTARY PARTICLES DUE TO POINCARÉ SYMMETRY

In Information Mechanics, the observed world is made of a certain amount of information, which represents different possibilities in a system. The observation of the same elementary particles regardless of physical space and time is due to the Poincaré symmetry. The observed elementary constituents should be invariants of Poincaré transformation. From group theory, one knows that mass and spin are the two invariants under Poincaré transformation. Because of this, the basic constituents, elementary particles, are specified by mass and spin.

The wave aspect of elementary particles is represented by the information function. It comes from the basic assumption that everything we observe comes from information. Information represents different possibilities. In this way, the wave-particle duality of

The action $A_1^{{}^{‴}}$ [X, $\gamma$, $G_{\mu \nu }$, $B_{\mu \nu }$ ] has the diffeomorphism invariance in physical space-time $X^{\mu }$:

The introduction of physical space-time tensor metric $G_{\mu \nu }(X^{\mu })$ and anti-symmetric tensor $B_{\mu \nu }(X^{\mu })$ can induce gravity and gauge interaction in physical space-time.

The fact that $G_{\mu \nu }(X^{\mu })$ and $B_{\mu \nu }(X^{\mu })$ describe the gravity and gauge interaction in physical space-time can be further confirmed by the equations of motion associated with $G_{\mu \nu }(X^{\mu })$ and $B_{\mu \nu }(X^{\mu })$. In the following, we will show that, from the Weyl invariance, one can obtain the equations of motion regarding $G_{\mu \nu }$ and $B_{\mu \nu }$, which shows that $G_{\mu \nu }$ and $B_{\mu \nu }$ follow the equations of motion associated with gravity and gauge force.

X. WEYL INVARIANCE, HOLOGRAPHY, AND CLASSICAL EQUATIONS OF MOTION

The Weyl invariance (13) is automatically preserved at the first order in information action. However, higher-order corrections could possibly violate it. For instance, in [Refs 4, 5], it is shown that there are the following second-order corrections to the information action:

The preservation of Weyl Invariance at the higher orders requires that:

Notice that ${\beta }_{\mu \nu }^{\mathrm{G}}=0$ leads to the generalized Einstein's equation with the source terms from the anti- In Information Mechanics, all the physical phenomena are projected from a two-dimensional information space time. The two-dimensional information space time has Weyl invariance. This means that the two-dimensional information space time is a hologram from which all observed phenomena, including physical space time, elementary particles, gravity, and gauge interactions emerge. Classical equations of motion come from the holographic nature, the Weyl invariance of information space time.

XI. EMERGENCE OF DARK ENERGY AND DARK MATTER

The observed accelerated expansion and large structure of our universe [Refs12, 13, 14, 15, 16] indicates an unknown source of energy, dark energy, and an unknown source of matter, dark matter. There are many proposals about the potential candidates for dark matter and dark energy.

In the following, we will show that Information Mechanics may explain the observation of dark energy and dark matter. Dark matter and dark energy can emerge from information function.

Dark energy and dark matter are phenomena observed in the large structure of the universe. To see what matter and energy emerges in the large structure of universe, we need to study the information function:

Here

where T is the age of the universe and L is the length of the horizon of the universe.

One may notice that in the information function $\mathrm{\Psi }$, there exist vibrations in the energy state (n, m) with the frequency $v_n=n/T$ and wavelength ${\lambda }_m=L/m$. The frequency and wavelength of some of these vibrations have a frequency and wavelength on the order of 1/T and L. These vibrations are almost impossible to detect at this moment. This is because, to observe them, it takes a detector as large as the universe or it takes time as long as the age of the universe. These vibrations are very "dark" due to this innate difficulty to be observed by detectors. They can only be observed on the large structure of the universe. They are natural possible candidates for dark matter and dark energy.

XII. CALCULATION OF COSMOLOGICAL CONSTANT

The cosmological constant is the simplest possible form of dark energy. The current standard model of cosmology, the Lambda-CDM model, assumes a non-zero cosmological constant as the source of dark energy. It is found that, in terms of the Planck unit, and as a natural dimensionless value, the cosmological constant is calculated to be on the order of ${10}^{-122}$ [Ref 15, 16]. The large discrepancy between the natural energy scale, Planck scale, and the observed value of ${10}^{-122}$ in terms of Planck scale is the so called cosmology constant problem.

In the following, we will estimate the vacuum energy of the information function. Surprisingly, we are able to obtain a value for the cosmological constant consistent with the observation.

To calculate the vacuum energy of the universe, we use the fact that the energy at state $(n,m)$ with the frequency $v_n=n/T$ and wavelength ${\lambda }_m=L/m$ is:

The lowest energy of the vacuum fluctuation is $E_{0n,m}=\hslash /2T$.

In Information Mechanics, with the space and time measurement scale at T and L, the total number of possible states N is:

If we assume each of the possible states can contribute to vacuum fluctuation energy of $E_{0n,m}=\hslash /2T$, then the total vacuum energy is:

The vacuum energy density in three-dimensional observed space is:

Here ${\rho }_p$ is the Planck energy density,

and $E_p$ is the Planck energy $E_p=\hslash /t_p=1.956\times {10}^9$ Joule.

The cosmological constant ${\mathrm{\Lambda }}_c$ is: ${\mathrm{\Lambda }}_c=8\pi {\rho }_{vac}$. Therefore

We know that:

Here $t_u$ is the age of the universe. We use:

$t_u=13.799$ billion years = $4.35\times {10}^{17}$ seconds.

From this, we obtain:

According to results published by the Planck Collaboration in 2018 [Refs 15,16], the cosmological constant is $2.888\times {10}^{-122}$ in Planck units. The result in (17), $\mathrm{\Lambda }=3\times {10}^{-122}{\rho }_{\mathrm{p}}$, is consistent with the data from the Planck Collaboration in 2018 [Refs 15, 16].

Information Mechanics seems to be able to address the large hierarchy problem regarding the cosmological constant.

XIII. EMERGENCE OF THE ELECTROWEAK SCALE AND PLANCK SCALE HIERARCHY

There are two outstanding hierarchy problems in physics. One is the cosmological constant problem. The other is to derive the large difference between weak force and gravity, or equivalently between Higgs mass and Planck mass [Ref 17, 18, 19, 20]. We have shown above that Information Mechanics may help address the cosmological constant problem; next, we will explore how it may help cope with the second hierarchy problem.

To derive the weak scale and Higgs mass in Information Mechanics, we study the information action $A_2$ and write it in terms of complex coordinates, z, $\overline{z}$, $\theta$, and $\overline{\theta }$ [Ref 4, 5]:

We introduce observable space-time $X^{\mu }(z,\overline{z},\theta ,\overline{\theta })$ in superspace, which includes both bosonic spacetime $X^{\mu }(Z,\overline{z})$ and its fermionic counterpart ${\psi }^{\mu }(Z,\overline{z})$ and ${\tilde{\psi }}^{\mu }(z,\overline{z})$:

The term $F^{\mu }$ is the auxiliary field, which can usually be eliminated through equations of motion. We suggest that bosonic space time $X^{\mu }(z,\overline{z})$ corresponds to observable spacetime coordinates and the fermionic spacetime ${\psi }^{\mu }(z,\overline{z})$ and ${\tilde{\psi }}^{\mu }(z,\overline{z})$ correspond to elementary particles. In superspace $X^{\mu }(z,\overline{z},\theta ,\overline{\theta })$, there is supersymmetry, which is the invariance under the transformation between space-time bosonic coordinates and fermion coordinates. After integrating over fermion coordinates $(\theta ,\overline{\theta })$, the action $A_2$ including metric tensor $G_{\mu \nu }$ and anti symmetric tensor $B_{\mu \nu }$ becomes [Ref 4, 5]:

Here, covariant derivatives are: