Dark Matter Has Already Been Discovered

The energy-momentum relationship in the special theory of relativity (STR) holds in an isolated system in free space. However, this relationship is not applicable to an electron in a hydrogen atom where there is potential energy. Using three types of methods, the author has already derived an energy-momentum relationship applicable to an electron in a hydrogen atom. In the past, Dirac asserted that Einstein’s relationship has negative solutions. This paper too obtains negative solutions (energy) from the derived relationship using Dirac’s reasoning. However, the discontinuity peculiar to the micro world is not incorporated into that solution. Thus discontinuity is incorporated into the solution by using a new quantum condition already derived by the author. Next, the orbital radius of an electron with negative energy in an absolute sense is found, and that radius is compared with the orbital radius of an electron in an ordinary hydrogen atom. A search is conducted for experiments supporting the DM model advocated by this paper. A hydrogen atom at this ultra-low energy level is formed from an atomic nucleus (proton) with positive mass, and a single electron with negative mass existing near that. In this paper, this unknown type of matter will be called a dark hydrogen atom (DHA). This paper also points out that DHA accounts for part of the true nature of dark matter (DM), the mysterious material whose true nature is currently unknown. Although this differs from the conventional interpretation, this paper holds that the experiment which demonstrates the existence of DHA is triplet production. The paper points out that one of the two electrons produced in a triplet production experiment is the electron which forms a DHA. If the DM model advocated by this paper is correct, then DM has already been discovered.


Introduction
The existence of dark matter (DM), a mysterious material which exists in space, was first noticed by the astronomer Fritz Zwicky (Zwicky, 1933). In the 1930s, he observed groups of galaxies called galaxy clusters. At that time, he noticed that the galaxies of the Coma Cluster were moving at a speed sufficient to fly away from the cluster. Nevertheless, the galaxy cluster did not fly apart. Zwicky predicted that something is present, besides visible matter, which increases the gravity of the galaxy cluster. In 1933, Zwicky named this something "missing mass" and recognized its importance. This was a revolutionary theory, but no one paid attention to his hypothesis at the time. At that time, full-fledged observation of space had just begun. It was not possible to determine whether this unknown type of matter was just invisible galaxies, or gas, or something completely different. Scientists first began paying attention to Zwicky's claims 40 years later.
In the 1970s, the astronomer Vera Rubin observed galaxies and discovered that the outside of a galaxy has the same rotation speed as the inside. According to Newtonian mechanics, the rotation speed of the outside of the galaxy should be slower than the rotation speed of the inside. In fact, in the case of the solar system, planets on the inside rotate faster than planets on the outside. Rubin concluded that something invisible with a large mass acts as a gravitational source and affects the rotation of a galaxy. After that, astronomers all over the world began verifying these observations. The first candidates for DM, this unknown matter, were dark, invisible objects, neutrinos, black holes, and so forth. However, these candidates were ruled out by experiments.
. m c p c mc  (1) This relationship is not applicable to an electron in a hydrogen atom where there is potential energy. Therefore, from the beginning we should have looked for an energy-momentum relationship applicable to an electron in a hydrogen atom. However, there was no search for such a relationship. This is because it has been thought that quantum mechanics is the theory describing the behavior of an electron in a hydrogen atom.
Using three types of methods, the author has already derived an energy-momentum relationship applicable to an electron in a hydrogen atom (Suto, 2011;Suto, 2020a;Suto, 2020b). This paper presents the method derived in the second paper.
Since Formulas (1) and (3) are equal, the following relationship must hold when Formula (4) is taken into account.
The following formula is obtained from this.
Next, the relativistic kinetic energy of an electron in a hydrogen atom is defined as follows by referring to Formula (6). Here, n m is the relativistic mass of the electron. Also, re,n p indicates the relativistic momentum of the electron.
Incidentally, the energy of an electron at rest in an isolated system in free space is 2 e . mc Here, we consider the case where this electron is drawn in by the electrical attraction of the proton, and forms a hydrogen atom. At this time, the electron emits a photon to the outside. Therefore, the relativistic energy of an electron in a hydrogen atom 2 n mc becomes smaller than the rest mass energy The behavior of an electron inside an atom, where there is potential energy, cannot be described with the relationship of Einstein (1). Caution is necessary because it is completely overlooked in Formula (8).
Sommerfeld thought the kinetic energy of an electron in a hydrogen atom could be described with Formula (4). However, this is an error.
Now, referring to Formula (4), it is natural to define the relativistic kinetic energy of an electron in a hydrogen atom as follows (Suto, 2019).
This paper defines re,n E as the relativistic energy levels of the hydrogen atom derived at the level of classical quantum theory. (The quantum number used here is just the principal quantum number. Therefore, re,n E is not a formula which predicts all the relativistic energy levels of the hydrogen atom.) However, the term "relativistic" used here does not mean based on the STR. It means that the expression takes into account the fact that the mass of the electron varies due to velocity. According to the STR, the electron's mass increases when its velocity increases. However, inside the hydrogen atom, the mass of the electron decreases when the velocity of the electron increases. Attention must be paid to the fact that, inside the hydrogen atom, the relativistic mass of the electron n m is smaller than the rest mass e . m In this way, two formulas have been obtained for the relativistic kinetic energy of the electron in a hydrogen atom (Formulas (7), and (9)). Incidentally, the following equation can be derived from Formulas (7) and (9).
Formula (11) is the energy-momentum relationship applicable to the electron in a hydrogen atom. Now, in the past, Dirac derived the following negative solution from Formula (1).
If the same logic is applied to Formula (11), then the following formula can be derived. However, Formula (13) does not incorporate the discontinuity peculiar to the micro world. Therefore, Formula (13) must be rewritten into a relationship where energy is discontinuous.

A New Quantum Condition for Incorporating Discontinuity in Formula (13)
This quantum condition has already been derived in another paper (Suto, 2021).
Incidentally, the energy levels derived by Bohr are given by the following formulas (Bohr, 1913: Bohr, 1952 Here, BO,n E signifies the energy levels derived by Bohr. However, Formula (14) cannot be regarded as beautiful. Intrinsically, the electron has a rest mass energy of 2 e . mc In Formula (14), the relationship with the rest mass energy of an electron is not clear. Thus, Formula (14) Here, α is the following fine-structure constant.
From the author's perspective, Formula (15) containing 2 e mc is superior to Formula (14) from a physical standpoint.
Also, Bohr's orbital radius BO,n r is normally described with the following formula.  (17) is not very impressive. This is because, in Formula (17), the relationship between the electron's orbital radius and the classical electron radius e r is not clear. Bohr thought the following quantum condition was necessary to find the energy levels of the hydrogen atom.
In Bohr's theory, the energy levels of the hydrogen atom is treated non-relativistically, and thus here the momentum of the electron is taken to be e .
mv Also, the Planck constant h can be written as follows (Suto, 2020c Due to Formula (26), it is possible to identify discontinuous states that are permissible in terms of quantum mechanics in the continuous motions of classical theory.

Relativistic Energy Levels of the Hydrogen Atom
If Formula (26) derived in Section 3 is substituted into Formula (13), the relativistic energy E of an electron becomes as follows. gives the relativistic energy of the electron, but this is also the absolute energy of the electron. The "ab" subscript of ab,n E stands for "absolute." Also, 2 n mc is the relativistic energy of the electron when the principal quantum number is in the state n.
The positive solution of Formula (28) From this, it is evident that Formula (15) derived by Bohr is an approximation of Formula (29).
Next, the following table summarizes the energies of a hydrogen atom obtained from Formulas (15) and (29). (Table 1) (Suto, 2020d) Table 1. Comparison of the energies of a hydrogen atom predicted by Bohr's classical quantum theory and this paper The following values of CODATA were used when calculating energies.     (11) can not only predict the energy levels of an ordinary hydrogen atom, derived by Bohr, but can also predict the existence of ultra-low energy levels.

Orbital Radius of an Electron at the Negative Energy Level in Formula (28)
Incidentally, there are also positive and negative solutions for A hydrogen atom in the ab,n E  state will henceforth be called a "dark hydrogen atom" (DHA) in this paper.
Incidentally, the energy of the hydrogen atom can also be written as follows.
The radius r where ab 0 E  is e /2 r due to Formula (37). Dirac predicted that the vacuum energy E satisfies the relation 2 e , E m c  but actually ab 0 E  is the energy of the virtual electron-positron pair which make up the vacuum (Figure 2). (Suto, 2017a).   In Dirac's hole theory, when the γ-ray gives all of its energy to the virtual particles ( 2 e 2 E m c  ) comprising the vacuum around the atomic nucleus, a virtual particle acquires rest mass, and is emitted as an electron into free space, while the hole opened in the vacuum is the positron (Figure 2a).
In the author's interpretation, an electron-positron pair is created because a γ-ray with an energy of 1.022 MeV gives rest mass to a virtual electron-positron pair at the position e /2 rr  (Figure 2b). Dirac pointed out that there is a negative solution to Formula (1). Adopting the same viewpoint, there is a negative solution to Formula (11). To find the negative solution, it is necessary to create a quadratic equation for r. Thus, from Formulas (31b) and (37), (Suto, 2017a Also, Formulas (42) and (43) can be written as follows. (Suto, 2017a)   The next compares the orbital radii of an electron in a hydrogen atom n r  and the orbital radii of an electron with a negative mass . n r  The following ratio is obtained from Formulas (42) Suto pointed out that an electron with negative mass forming DHA exists near the atomic nucleus (proton) (Suto, 2015;Suto, 2017b). A single DHA has about the same mass as an ordinary hydrogen atom. However, a DHA is far smaller than an ordinary hydrogen atom, and thus dark hydrogen can attain high density. Therefore, dark matter can be an important source of gravity.
If the above is indicated graphically, the result is as follows (Figure 3). (Suto, 2018).

Experiments Demonstrating the Existence of DHA
How can hydrogen atoms in this energy state be verified? This paper looks at triplet production.
It is generally assumed that in triplet production, in which 2 electrons and 1 positron are created, electron-pair creation occurs not near the atomic nucleus, but near the electron in the outer shell orbital. A total of three particles are created in this case: one outer shell electron forming the atom, and a positron and electron created through pair production. However, in this model, (1.022MeVn E  ) should be sufficient as the necessary photon energy.
If an energy of 2.044MeV ( 2 e 4mc ), is needed for triplet production, then the recoiled electron should be regarded as being at an ultra-low energy level. Now, consider the case where an incident γ-ray has the energy corresponding to the mass of 4 electrons (2.044 MeV). If this is discussed classically, the γ-ray can create an electron and positron near e /2 rr  (Figure 4). (Suto, 2017b). Consider the case where a γ-ray with the energy of 4 electrons (2.044 MeV) is incident on an atomic nucleus (proton).This γ-ray will give 1.022 MeV of energy to the virtual particles at e / 2, rr  and an electron-positron pair will be created (↑①). When this γ-ray approaches closer to the atomic nuclear, and the electron in the orbital around the proton absorbs this energy, the electron will be excited and appear in free space (↑②). As a result, 2 electrons and 1 positron will appear in free space.
This paper points out that one of the two electrons which appears is an electron in the ab,n E  state.

Conclusion
The hydrogen atom has ultra-low energy levels. In this state, there is an electron with negative mass near the atomic nucleus (proton) which has positive mass. Matter formed from this proton and an electron is DHA. This paper has focused on the triplet production experiment as the experiment demonstrating the existence of DHA. This paper concludes that one of the two electrons produced in triplet production is an electron which forms DHA. If the DM model advocated by this paper is correct, then DM has already been verified by experiment.
Is it possible to experimentally determine the correctness of previous explanations and the interpretation of this paper (Figure 4) regarding triplet production?
If four electrons can be produced in experiments using γ-rays with an energy of 2.044 MeV ( 2 e 4mc ), the interpretation of this paper is definitely superior. What causes this quadruplet production is an orbital electron, a dark electron which forms DHA, and a virtual electron and virtual positron which make up the vacuum. That is, the breakdown of the four produced particles is: 3 electrons and 1 positron.
Even if a quadruplet production phenomenon is rarely mixed in among the innumerable triplet production phenomena, the probability is likely to be extremely low. However, if the orbital electron is made to recoil at the stage before the γ-ray produces the electron-positron pair, then it should be possible to detect the quadruplet production phenomenon.