Quantum Behavior of 10D Planck Unit: Stationary Electron, Compton Photon and Gravitational Field

Abstract

This work focuses on the origin of electrons, Compton photons and a gravitational field. Based on the discovered 10D Planck units, the physical behavior of these units was further studied in isolated systems. Investigation shows that a 10D Planck unit in the in situ state has the same properties as a stationary electron, while its non-in situ state shares the same physics with a Compton photon. Results indicate that photons’ potential exists between any two Compton photons, with their strength being determined by the distance between the two photons. Finally, the potential field was proved to be the gravitational field of a proton.

1. Introduction

For a scientist specializing in fundamental physics, today might be an era fraught with deadlocks, progress, crises and hopes. It is widely acknowledged that the two issues about gravity, namely its non-incorporation with the Standard Model and non-unification with quantum mechanics, constitute the most serious deadlocks that current theoretical physics is confronted with [1,2,3,4,5]. Meanwhile, experimental physics has developed relatively smoothly and achieved remarkable results, including the observation of new elemental particles and the detection of the gravitational waves [6,7]. But these advances do not seem to have brought decisive breakthroughs in the corresponding field of theories, in view of the increasing number of approaches coexisting here [8,9,10,11,12,13,14,15]. Furthermore, an annihilating paradox about mass and anti-mass in a gravitational field is leading to a catastrophic crisis upon modern physics for its apparently unbalanced energy. Nevertheless, during the decades when scientists are seeking solutions in fundamental physics, two key factors of geometry and symmetry have gradually stood out and become recognized as sources of hope [16,17,18,19,20].

Carrying geometry and symmetry both, 10D Plank unit ¹⁰$\widehat{p}$ has been established to open up a different path for quantization of gravity [21]. As briefed, the Planck length λ_P in the generalized Euclidean space determines that the space is a quasi-Euclidean space $\widehat{\mathit{P}}$, comprising uncertainty, UV cutoff $\overline{\lambda }$_P ($\overline{\lambda }$_P = ${\displaystyle \frac{{\lambda }_{\mathrm{P}}}{2\mathsf{\pi }}}$), IR cutoff ${\displaystyle \frac{{}_{ }{}^n\widehat{P}}{{{\overline{\lambda }}_{\mathrm{P}}}^{(n-1) }}}$ and the most important structure of Planck unit with the minimum nD measure ⁿ$\widehat{p}$= ${\overline{\lambda }}_{\mathrm{P}}^n$. Under control of symmetries, a Planck unit ⁿ$\widehat{p}$ should be in situ or on the nD flattened surface of an (n + 1)D Planck sphere (r = $\overline{\lambda }$_P), and these two states are usually separated by infinite potential barrier $V_r^{\infty }$ (which forbids an object from any radial displacement to satisfy the rotational symmetry) until the special case about a 10D Planck unit ¹⁰$\widehat{p}$ occurs in 10D space. Geometrical investigation demonstrates that ¹⁰$\widehat{p}$ presents the inaugural instance wherein these dual states achieve topological connectivity and mutual transformability. Then, quantum mechanics proves that two dimensionless constants, including α₁ = ${\displaystyle \frac{{(}^{10}{m)}^{\frac{1}{4}}}{{ \left(2\pi \right)}^4/9 }}$ = ${\displaystyle \frac{1}{137.036 082}}$ and α₂ = $\prod _1^6{\displaystyle \frac{1}{ {\left(2\pi \right)}^{n }}}·\prod _7^{10}{\displaystyle \frac{{}_{ }{}^{n}m}{{ \left(2\pi \right)}^n/(n-1) }}$ = ${\displaystyle \frac{1}{{1.628 \times 10}^{38 }}}$ (where ¹⁰m is generalized 10D volume of a 10D sphere, and ⁿm is nD measure of a generalized nD sphere in a system normalized by $\overline{\lambda }$_P = 1), dominate the transformation and result in two long-range interactions, which are consistent with the electromagnetism interaction and the gravity, respectively.

The above results distinguish the new path in three aspects. Firstly, no controversial hypotheses have been involved in the work, since its initial settings include only the Planck length and the symmetries of an nD Euclidean space. Obviously, both of the points come from experiments, instead of those coming from assumptions, such as a high-dimensional background space with a special structure, flat or curved. Secondly, no particle has been introduced into the system, since the minimum space structure behaves naturally as a particle. Besides the eternal contradiction between the background and the particles, such a presentation inherently leads to the unity of the space and the particle, guaranteeing the new method’s philosophical consistence. Thirdly, and most crucially, the electromagnetic coupling constant and the gravitational constant are obtained as two geometrical ratios, indicating that the gravity has initially been quantized and unified with the electromagnetism interaction [21].

After the above investigation on geometry and quantum mechanics, here 10D Planck unit ¹⁰$\widehat{p}$ has been further explored in isolated systems with conserved energy, and the obtained properties have been compared with those of an electron/Compton photon. Comparison identified them as the same and concluded the Compton photons’ potential unexpectedly. Then, the field of the photons’ potential was proved equal to the gravitational field of a proton.

2. Physical Behavior of 10D Planck Unit ¹⁰$\widehat{\mathit{p}}$

2.1. ¹⁰$\widehat{p}$: Doublet Candidates Due to the Incompatible Principles of RS↑ and E↓

As discovered, a 10D Planck unit ¹⁰$\widehat{p}$ with rotational symmetry exists in three states, including in situ state I, revolving state R and tangent state T [21]. Geometry determines the space structures for these three states, including P_I for state I, P_R for state R, which is the 10D surface of 11D Planck sphere (r = $\overline{\lambda }$_P), and P_T for state T, which is the flattened P_R (Figure 1).

Among these spaces, P_I and P_T are of flatness, theoretically allowing the transformation I$\rightleftharpoons$T when they are topologically bridged by state R [21]. To satisfy the two incompatible principles of the highest rotational symmetry (RS$\uparrow$) and the minimum energy (E$\downarrow$), ¹⁰$\widehat{p}$ moves in two different ways inside P_I or P_T [21]. Therefore, there are doublet states (RS$\uparrow$) and (E$\downarrow$) for ¹⁰$\widehat{p}$ (Figure 1). For ¹⁰$\widehat{p}$ dominated by principle RS$\uparrow$, it is of motion path measured as x = ${(}^{10}m{)}^{{\displaystyle \frac{1}{ 4 }}}$ along each dimension of a 4D subspace when it is in state I, while it is of motion path of IR cutoff L_T = ${\displaystyle \frac{{\left(2\pi \right)}^{4 }}{9}}$ for its corresponding state T, and the wavelength ratio is α₁ = ${\displaystyle \frac{1}{137.036 082}}$ during its transformation [21]. In addition, for ¹⁰$\widehat{p}$ satisfying principle E$\downarrow$, its IR cutoffs, which are L_I = ${\prod }_7^{10}{}_{ }{}^{n}m$ = 1.613 × 10² and L_T = ${\prod }_1^6{\left(2\pi \right)}^n·{\prod }_7^{10}{\displaystyle \frac{{ \left(2\pi \right)}^{n }}{n-1}}$ = 2.626 × 10⁴⁰, are taken respectively as its motion paths for the ground states in P_I and P_T, and it has a wavelength ratio α₂ = ${\displaystyle \frac{1}{{1.628\times 10}^{38 }}}$ for its transformation [21].

In short, under control of quantum mechanics only, 10D Planck unit ¹⁰$\widehat{p}$ exists in doublet states (RS$\uparrow$) and (E$\downarrow$) in P_I and P_T, and state T tends to be of lower energy compared to state I because it moves in a larger space.

2.2. ¹⁰$\widehat{p}$: Singlet States Determined by Conservation of Energy

Generally, ¹⁰$\widehat{p}$’s quantum doublet should be detected singly. And it should satisfy conservation of energy when acting as an isolated object.

Singlet state T. Being dominated by translational symmetry, state T with constant velocity v and action S = h are always of mass m_T = 0, which is required by ${\displaystyle \frac{\mathrm{d}v }{\mathrm{d}k}}$ = ${\displaystyle \frac{\mathrm{ћ}}{{ m}_{\mathbf{T}}}}$ = 0 when dv = 0 [21]. But T(RS$\uparrow$) and T(E$\downarrow$) have different energy ε = ${\displaystyle \frac{ hc }{2L}}$ inside P_T, where the longest motion path L of them is different [21]. According to the principle of the minimum energy, ¹⁰$\widehat{p}$ is observed in state in T(E$\downarrow$) for T(E$\downarrow$) has much longer L and much lower energy compared with T(RS$\uparrow$).

Singlet state I. Considering that ¹⁰$\widehat{p}$ has U = 0 inside specific spaces of P_I and P_T, which are surrounded by Vs^∞, let the observed I state be of total energy E_I, kinetic energy K_I, stationary energy M_I, and the singlet T(E$\downarrow$) be of corresponding E_T, K_T, M_T. Then, the conserved energy E_I = E_T during I$\rightleftharpoons$T requires

M_I + K_I = M_T + K_T

(1)

where it exists as M_T = 0 for m_T = 0, and let it be K_T = ε₀.

For transformation T$\to$I, let state I be of stationary mass m_I related to M_I, then M_I with a real value should be M_I∈ [0, ε₀], which leads to three possible scenarios.

The first is M_I = 0 and m_I = 0. Equation (1) requires K_I = K_T when M_I = M_T = 0, but λ_I << λ_T results in K_I >> K_T when m_I = m_T. So, this scenario does not hold.

The second is 0 < M_I < ε₀ and m_I ≠ 0. Given K_T = ε₀ = ${\displaystyle \frac{ hc }{2L}}$ for state T in Equation (1), there exists

$$E_{\mathbf{I}} =m_{\mathbf{I}}{c}^2 +K_{\mathbf{I}} ={\displaystyle \frac{ hc }{2L}}$$

(2)

where K_I contains n translational kinetic energies along n independent dimensions and angular kinetic energies in $c_n^2$ planes as

$$K_{\mathbf{I}} ={\displaystyle \frac{ h^2}{ 8m_{\mathbf{I}} }} ({\displaystyle \frac{1}{{ l}_1^2 }}+{\displaystyle \frac{1}{{ l}_2^2 }}+...+{\displaystyle \frac{1}{{ l}_n^2 }})+...={\displaystyle \frac{ h^2}{ 8m_{\mathbf{I}} }}·\mathit{\beta }$$

(3)

where the summation of the coefficients for the (n + $c_n^2$) terms is set to be β.

Then there exists

$$m_{\mathbf{I}}{c}^2+{\displaystyle \frac{ {\beta h}^2 }{8m_{\mathbf{I}}}}-{\displaystyle \frac{ hc }{2L}}= 0$$

(4)

which equals to

$$m_{\mathbf{I}}^{2 }{c}^2-{\displaystyle \frac{ hc }{2L}}{m}_{\mathbf{I}}+{\displaystyle \frac{{ \beta h}^2 }{8}}= 0$$

(5)

where the discriminant of a quadratic equation is

$$\mathsf{\Delta }={\displaystyle \frac{{ h}^2{c}^2 }{4}}({\displaystyle \frac{1}{ L^{2 }}}-{\displaystyle \frac{ \beta }{2}})$$

(6)

Given β = (${\displaystyle \frac{1}{{ l}_{1 }^2}}$ + ${\displaystyle \frac{1}{{ l}_2^2 }}$ + ... + ${\displaystyle \frac{1}{{ l}_n^{2 }}}$) + ... ≥ ${\displaystyle \frac{1}{{ l}_{max }^2}}$ = ${\displaystyle \frac{1}{{ \left(\alpha L\right)}^{2 }}}$ (α << ${\displaystyle \frac{1}{ \sqrt{2 }}}$), there exists ${\displaystyle \frac{1}{{ L}^{2 }}}$<< ${\displaystyle \frac{\beta }{ 2 }}$ and Δ always satisfies

$$\Delta < 0$$

(7)

which denies the real number solutions for m_I. So, the second scenario does not hold, either.

The third is M_I = ε₀ and m_I = ε₀. There exist K_I = 0 and M_I = K_T as the single solution to Equation (1), indicating that the kinetic energy of state T(E$\downarrow$) transforms completely into the stationary energy of state I during T$\to$I. And K_I = 0 requires that state I have no kinetic energy for either translation or rotation, showing that ¹⁰$\widehat{p}$ is completely stationary whether in state I(RS$\uparrow$) or I(E$\downarrow$). Consequently, I(RS$\uparrow$) and I(E$\downarrow$) are physically indistinguishable for they share the same motion (v = 0) and the same rest (m = ε₀), resulting in a completely stationary ¹⁰$\widehat{p}$ in state I.

Additionally, the conservation of space requires ¹⁰$\widehat{p}$ to remain in the same dimensional space during transformation. Obviously, ¹⁰$\widehat{p}$ with both transformability and observability should be inside the 4D space, which is the non-trivial subspace expanded by x₇, x₈, x₉, x₁₀ and also the common space for state I(RS$\uparrow$/E$\downarrow$) and T(E$\downarrow$), as shown in Figure 2.

Briefly, as an isolated object with conserved energy and space, 10D Planck unit ¹⁰$\widehat{p}$ transforms between massless T(E$\downarrow$) and stationary I in the 4D subspace.

2.3. ¹⁰$\widehat{p}$: Physical Behavior of I/T

Let stationary state I and massless state T (E$\downarrow$) be abbreviated as I/T. The physics of ¹⁰$\widehat{p}$ can be obtained in a monomer system and multi-body systems.

Monomer system. T takes normalized L = ${\displaystyle \frac{{ \left(2\pi \right)}^{7+8+9+10 }}{6\times 7\times 8\times 9}}$ = 4.545$\times$ 10²³ as its half wavelength, with the actual wavelength as λ_T = ${\displaystyle \frac{{ 2\times \left(2\pi \right)}^{34} }{3024}}·\overline{\lambda }$_P = 2.338 243$\times$ 10⁻¹² m (where $\overline{\lambda }$_P = 2.572 326 × 10⁻³⁶ m [22]) and energy ε₀ = ${\displaystyle \frac{ hc }{{\lambda }_{\mathbf{T}}}}$ = 0.530 245 MeV for its ground state. This corresponds with action m_T = 0, v_T = c and λ_T = 2.338$\times$ 10⁻¹² m for state T, and state I is m_I = 0.530 MeV and v_I = 0.

Two-body system. Given the non-conservative momentum built by state T in the above single-body system, a 2I/2T pair, which comprises two extremely close and synchronized I/T pairs and has the two T states be of opposite quantized momenta, is assumed to be one body to obtain a two-body system with legitimate transformation (Figure 3A). According to the sharing–coupling effects [21], any two 2T states are shared by the two 2I states, naturally resulting in coupling potential ΔU = ${\displaystyle \frac{ hc }{\lambda ’}}$ − ${\displaystyle \frac{ hc }{{\lambda }_0}}$ (Figure 3B).

Based on the geometrical constant α₂, which is responded to by T(E$\downarrow$) here [21], there exists

$$\mathsf{\Delta }U={\displaystyle \frac{{ \alpha }_{2}hc }{2\pi }}({\displaystyle \frac{ 1 }{r}}-{\displaystyle \frac{1}{{ r}_{0 }}})=1.942\times {{10}^{-}}^{64} ({\displaystyle \frac{1}{ r }}-{\displaystyle \frac{1}{r_0}}) \mathrm{J}$$

(8)

where α₂ = ${\displaystyle \frac{1}{{1.628008\times 10}^{38 }}}$ [21], h = 6.62606896 × 10⁻³⁴ J·s and c = 2.9979246 × 10⁸ m/s [22].

As demonstrated by Section 2, ¹⁰$\widehat{p}$ exhibits specific mass, wavelength and transformation, clearly pointing to two experimental particles.

3. I/T and Electron/Compton Photon

3.1. Comparison Between I/T of ¹⁰$\widehat{p}$ and e/γof Electron/Compton Photon

Let a photon with Compton wavelength 2.426$\times$ 10⁻¹² m (0.511 MeV) of a stationary electron be defined as a Compton photon γ, then physical properties of I/T can be compared with e/γ from an ideal annihilation of a stationary electron–positron, as listed in

Table 1. Comparison between I/T of ¹⁰$\widehat{p}$ and e/γ of electron/Compton photon [22].

I/T	e/γ	Deviation
I: v = 0	e: v << c	–
m = 0.530 MeV	m = 0.511 MeV	3.7%
interacting constant 1/137.036 082  with unknown ±	interacting constant 1/137.035 999  with certain ±	<10−6
T:$\mathrm{wavelength} 2.338 \times$ 10−12 m	γ$\text{:} \mathrm{wavelength} 2.426 \times$ 10−12 m	3.7%
m = 0	m = 0	0
constant velocity v = c  interacting constant $\text{~}1/(1.628 \times$ 1038)  with unknown ±	constant velocity c  Gravitational constant $\text{~}1/(1.693 \times$ 1038)  with certain +	– 3.8%
I/T: $2\mathbf{I}⟷$ 2T process with conserved  quantum momentum	e/γ: (e++e−)$⟷$2γ process with conserved momentum, charge, etc.	–

.

3.2. Hidden Properties of e/γ

The above comparison reveals that e/γ is equal to I/T, as they exhibit the same physics. And more has been revealed besides the origin of e/γ.

Shared matter for e/γ. The process of electron–positron annihilation is fundamentally different from that of the spooky action caused by quantum entanglement [23], since the latter is regardless of the distance while the former occurs only in situ, showing that annihilation requires more than an account book of energy, momentum, charge, etc., and strongly implying that a common part in situ is necessarily demanded. As discovered by I/T, structure ¹⁰$\widehat{p}$ itself is the shared matter bridging an electron (±) and its corresponding photon.

Photons’ potential. A potential is unveiled to exist between any two centers in a multi-body system (Figure 3B). For a state I in one of the two centers, it takes the close photon as its own state T with transformability and the far away one as its state T′ without transformability. Thus, a potential field is created by the two photons with different relative potentials depending on their distances, whereas the interacting potential brings no observable influence on a single-body system, since its state I is protected by v = 0 and dm = 0, while its state T is protected by m = 0 and dv = 0. But photons’ potential has an extremely significant effect on multi-body systems, considering the paradox about annihilating–creating in a gravitational field.

4. Compton Photons’ Field and Gravitational Field of a Proton

4.1. Annihilating Paradox and Its Solution

A dual annihilating–creating process containing two proton–antiproton (p⁺ + p⁻) pairs is shown in Figure 4A.

For an ideal annihilating–creating process,

2(p⁺ + p−) with r₀ → 2(p⁺ + p⁻) with r

(9)

where two extremely close (p⁺ + p⁻) pairs annihilate simultaneously with their photon pairs moving along the same dimension, which means the four photons are paired up closely into two groups with opposite directions. Then, the two groups of photons transform into (p⁺ + p⁻) pairs respectively at two positions farther apart. Given the local neutrality throughout the process, no electric field energy is involved. But unbalanced gravitational energy occurs in the system as

$$∆G=G{m}_p^2({\displaystyle \frac{1}{ r }}-{\displaystyle \frac{1}{{ r}_{0 }}})=1.867\times {{10}^{-}}^{64} ({\displaystyle \frac{1}{ r }}-{\displaystyle \frac{1}{r_0}}) \mathrm{J}$$

(10)

where G = 6.6743 × 10⁻¹¹ N·m²/kg², m_p = 1.672621638 × 10⁻²⁷ kg [22].

Here, ΔG, the potential variation caused by the gravity between the two (p⁺ + p⁻) pairs, is extremely slight but unacceptable, bringing about the annihilating paradox. This paradox could not be solved by current physics, and its significance cannot be overestimated within the framework of modern physics, since it leads directly to such a deduction, in which a couple of mass and anti-mass dual planets (with theoretical possibility only) could be removed from their solar system without any work and just via emitting photons. Unlike those crises regarding dark energy or dark matter that require new physics, the annihilating paradox with unbalanced energy raises questions about a missing link in our current theories.

Now the annihilating paradox is solved by the Compton photons’ potential field, which is equal to the gravitational field of a proton with deviation of 3.9%, as compared with Equation (8).

4.2. General Application of Compton Photons’ Potential

As a result, all of the Compton photons are identical ones but are different in their relative colors. This is derived from their constant action S = h, which means that they present various wavelengths λ = ${\displaystyle \frac{ 2\pi d }{{\alpha }_2}}$ as far as an observer at distance of d is concerned. Consequently, a Compton photon always undergoes a blue shift when approaching the observer or a red shift when moving away from them (Figure 4B). These red or blue shifts cannot be directly detected because any state T is forbidden to undergo transformation non-in situ, denying that any distant photons have been observed so far. In fact, what can be measured for a photon is always its energy during in situ transformation, and it has never been proved that a photon maintains a constant color or energy regardless of its distance. However, these blue/red shifts can be detected indirectly, since the allochroic effect of Compton photons is precisely equal to the gravitational field of a proton (Figure 4B).

5. Conclusions and Discussion

Besides the two dimensionless constants equal to FSC and gravitational constant, this work also obtained a stationary electron, a Compton photon and a gravitational field for a proton based on 10D Planck unit ¹⁰$\widehat{p}$.

After its properties have been revealed mainly by geometry and quantum mechanics, this work explores 10D Planck unit ¹⁰$\widehat{p}$ via putting it in an isolated system with conserved energy and conserved spacial structure. The results indicate that ¹⁰$\widehat{p}$ has a completely static in situ state with mass 0.530 MeV and a non-in situ state with wavelength 2.338$\times$ 10⁻¹² m and m = 0, which shows the consistency between ¹⁰$\widehat{p}$ and electron/Compton photon. Moreover, the Compton photons’ potential is revealed to exist between any two photons, with their intensity being determined by the distance between the two photons. Such a potential field is proved to be equal to the gravitational field of a proton in experiments. Briefly, e/γ is the quantum manifestation of a 10D Planck unit, and the potential field of Compton photons is equal to the gravitational field of a proton.

Therefore, a non-trivial scenario can be obtained in a 10D vacuum, where the physics is different from those trivial ones in lower-dimensional spaces. Inside their 4D subspace, 10D Planck units are always either completely stationary or moving at a constant speed. And the kinetic ones are always of wavelength $\overline{\lambda }$_P = 2.338$\times$ 10⁻¹² m, while the static ones are of mass ε₀ = 0.530 MeV after assuming v = c. Transformation freely occurs between a stationary one and a constant speed one when they satisfy those necessary principles, including the topological connection and the conservation of momentum. This scenario clearly depicts a universe containing electrons, Compton photons and their transformations. Additionally, it depicts a misplaced gravitational field, which arises from Compton photons of electrons, but serves for a proton experimentally. Disappointingly, it provides nothing else so far, such as the positive or negative nature of a particle, any other elemental particles, the more complex processes, let alone those complex systems, the macroscopic low-speed processes, and the high-speed processes (relativistic effects). And the ~4% deviation exists there stably and stubbornly, questioning whether it is derived from a systematic error.

Meanwhile, it almost solves the core part of our physical world, namely the origin of electrons and Compton photons. This solution has a significant implication for our understanding of some physical phenomena, such as the wave–particle duality of a photon.

So far, that is all ¹⁰$\widehat{p}$ has provided us with, including the most elemental fermion e, the most elemental boson γ, and the most fundamental gravitational field of a proton. Despite unsolved issues, we are still convinced that we are moving in the right direction for theoretical physics.