## On Indeterminacy Problem in Quantum Mechanics

Tian Ma & Shouhong Wang, On indeterminacy problem in quantum mechanics, IUISC Preprint #1601

The main objectives of this Note are

1.  to demonstrate that the indeterminacy problem and its associated absurdities in interpretations of quantum mechanics are due to the wrong fundamental premise that there is no interference of other particles;
2. to reveal the fact that the Universe is filled with mediators, giving rise to the correct Mediator Sea Premise of quantum mechanics; and
3. to show that under the Mediator Sea Premise,
• causality holds true for the quantum-mechanical description of physical reality, removing the absurdities and confusions;
• the interference of the mediators to a moving particle is reminiscent to Brownian motion; and
• quantum mechanics is a correct and complete theory.

## I. Classical Statistical Interpretation and Indeterminacy Problem

Two dominant views of the interpretation of quantum mechanics are the realistic view, advocated by Albert Einstein, and the orthodox view, also called the Copenhagen interpretation, which was mainly advocated by Niels Bohr and Werner Karl Heisenberg. The main characteristic for the Einstein realistic view is that causality must hold true in the quantum-mechanical description of physical reality, and quantum mechanics is an incomplete theory–the indeterminacy is caused by hidden variables. The key point for the Copenhagen interpretation is non-causality of quantum behavior of particles, leading to various absurdities; see among many others [3, 2, 1, 4].

## II. The Fundamental Premise of Indeterminacy Problem

All scientific theories and conclusions are built upon a fundamental premise. If the fundamental premise is true, then we would expect the conclusions are true as well. To understand the confusion caused by the indeterminacy as classically formulated, one needs to examine its fundamental premise.

In fact, the fundamental premise of the indeterminacy in quantum mechanics is that one assumes there is no interference of other particles. Our viewpoint is that such fundamental premise is in fact incorrect. Consequently, under an incorrect premise, confusions and misunderstandings arise, and more importantly

the indeterminacy problem is a wrong question to be asked.

## III. Mediator Sea in the Background Space

One natural outcome of our field theory of fundamental interactions and the weakton theory of elementary particles suggests that our Universe is filled with mediators; see [5] and the reference by the authors therein. This clearly shows that the classical fundamental premise of the indeterminacy in the quantum-mechanical description of physical reality is a wrong assumption.

In other words, quantum mechanics should be understood under the following fundamental premise of the quantum-mechanical description of physical reality:

Mediator Sea Premise:

1.  The entire space is filled with a sea of mediators, including photons, gluons and the $\nu$-particles, as evidenced by the cosmic microwave background;
2. All mediators carry weak charges, participate in the weak interaction, and consequently will interact with a moving particle in proper ranges.

The interference of the mediator sea towards a moving particle resembles similar features as the Brownian motion.

## IV. Causality

In essence, the heart of the debate between Einstein and Bohr is the causality of the quantum-mechanical description of physical reality. We believe what puzzled Einstein was the non-causality conclusion of the Copenhagen interpretation, rather than the randomness in the quantum mechanics.

With the new Mediator-Sea-Premise, it is clear that the principle of causality holds true, as Einstein believed. The randomness is caused by interference of the mediators as a particle moves in the mediator sea, leading to the indeterminacy of the precise position and momentum of the particle. Instead, in the Copenhagen interpretation, randomness is innate with no causality, resulting various spurious paradoxes in the interpretation of quantum mechanics.

Also, as in the Brownian motion, precise physical law expressed by the wave equation for the wave function offers complete information about the system. In other words, quantum mechanics is a correct and complete theory for describing the physical reality under the Mediator Sea Premise.

## References

[1] John Bell, On the Einstein-Podolsky-Rosen paradox, Physics, 1, 195-200, 1964.

[2]  Niels Bohr, Can quantum-mechanical description of physical reality be considered complete?, Physical Review 48:8, 696-702, 1935.

[3] A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete?, Physical Review 47:10, 777-780, 1935.

[4] D. Griffiths, Introduction to quantum mechanics, Prentice Hall, 1995.

[5] Tian~Ma and Shouhong~Wang, Mathematical Principles of Theoretical Physics, Science Press, 524 pp, 2015.

Tian Ma and Shouhong Wang

## I.

There are four fundamental interactions in Nature–the electromagnetism, the gravity, the strong and the weak interactions. The current route of the unification of the four interactions is under the following assumption:

there is only one interaction of Nature, which, under different energy conditions, degenerates to the four interactions we observe.

Mathematically, this assumption translates to searching for unification through a large symmetry group.

Our first viewpoint is that such an assumption is imaginary and non-falsifiable. We have no reason to believe its validity, and consequently, any unification theory built upon this imaginary assumption may not reflect Nature. In fact, this assumption breaks the principle of representation invariance (PRI), discovered by (Ma-Wang, 2012), which is simply a logic requirement.

## II.

Nature tells us that when several interactions are present in a given physical system, each interaction obeys its own symmetry. These interactions are affecting each other in the physical system, and the coupling is inevitably necessary. In other words, Nature suggests us that

the coupling is the essence and a natural route for the unification of the four fundamental interactions.

This route of unification is the reflection of Nature, and is directly built upon observable natural phenomena, instead of on an imaginary assumption in the above traditional route of unification.

This is the route of unification we adopt in the PID unified field theory we have recently developed; see the previous blog  and a recent book.

## III.

Now we introduce briefly the main ideas of the PID unified field theory.

1. Given a physical system, involving all four interactions, the goal of a unified field theory is to derive physical laws of the system:
• a unified field theory is to derive field equations coupling the interactions and dynamics of the system, and
• the field equations should be derived based on a few fundamental principles.
2. Our PID unified field theory is based on the following principles:
3. The symmetry principles play a decisive role in determining laws of Nature. In other words, symmetry principles dictate the actions of the interactions.With the actions at our disposal, PID gives rise to the field equations of the interacting physical system coupling different interactions and subsystems.The PRI ensures the sources of the interaction, the mass charge for gravity, the electric charge for electromagnetic interaction, the weak charge for the weak interaction and the strong charge for the strong interaction.In deriving the field equations above, the coupling principle of symmetry breaking (PSB) validates the symmetry-breaking of certain subsystems in the coupling.
4. The unified (coupling) field equations offer solutions to a few challenging problems, which include the dark matter and dark energy phenomena, sources of interactions, asymptotic freedom and the quark confinement, formulas for the weak and strong interaction potentials and forces, and the nucleon interaction potential.

Tian Ma and Shouhong Wang

## Remarks on the four fundamental interactions

An interaction is a force or a potential energy between two different particles. So far, we know that there are four fundamental interactions in Nature– the electromagnetism, the gravity, the strong and the weak interactions. The discovery of the four interactions and the development of scientific theories of these interactions have always been an important endeavor of the mankind.

The most fundamental characteristic of a scientific theory for an interaction is that it can provide the interaction potential and force formulas, in agreement with experiments and observations.  Otherwise, the theory is at least incomplete or incorrect.  This fundamental characteristic suggests the following:

1. the Maxwell theory of electromagnetism, the Einstein theory of general relativity, and our field theory appear to be the only scientific theories of fundamental interactions, and

2. all other theories cannot provide strong and weak interaction potentials and force formulas, and are incomplete to be qualified as a scientific theory for the four fundamental interactions.

Two remarks are now in order.

1. The Maxwell theory of electromagnetism is a scientific theory, in which the interaction potential is electromagnetic potential ${A_\mu}$, dictated by the Maxwell equations. Interaction potential and force formulas for an electrically charged particle are the Coulomb formulas, which can be derived from the Maxwell equations.The Einstein theory of gravity, the theory of general relativity (GR), is certainly a scientific theory of the gravitational interaction. In GR, the Riemannian metric represents the basic interaction potentials, and the gravitational force formula including the Newtonian universal law of gravity can be derived from the Einstein gravitational field equations.
2. As mentioned in the previous posts, we have discovered three fundamental principles: 1) the principle of interaction dynamics (PID),  2) the principle of representation invariance (PRI), and 3) the coupling principle of symmetry-breaking (PSB) for unification.We have then derived a number of experimentally verifiable laws of Nature based only on the Einstein principle of general relativity, the gauge symmetry and the above three principles. In particular, among other implications, our theory offers basic interaction potential and force formulas for the weak and the strong interactions.

Tian Ma & Shouhong Wang

## New Book: Mathematical Principles of Theoretical Physics

Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, Beijing, 524pp., August 2015, ISBN: 9787030452894.

[The book can now be ordered in amazon.cn. Note:  In order to order at amazon.cn, one needs to sign up for an account there, and it does ship to the US. The total cost (book + shipping to Indiana, USA) is about \$35.]

This book is devoted to our discoveries in general relativity, particle physics, cosmology and astrophysics over the last few years. We have derived experimentally verifiable laws of Nature based only on a few fundamental principles, and have established a unified field theory, as Albert Einstein hoped. Our work solves a number of challenging problems, which include

• the dark matter and dark energy phenomena,
• the structure of black holes,
• the structure and origin of our Universe,
• the quark confinement,
• first principle approach to Higgs fields, and
• mechanism of supernovae explosion and active galactic nucleus jets (AGN).

The current discoveries began in 2012, when we made a breakthrough both on the Einstein law of gravity and on the dark matter and dark energy phenomena – two of the greatest unsolved mysteries in physics. Both dark matter and dark energy are not accounted for in the Einstein equations. We rigorously proved that the presence of dark matter and dark energy requires that the variation of the Einstein-Hilbert action be taken under energy-momentum conservation constraint. The Einstein equations, however, are derived as the variation of the Einstein-Hilbert action under no constraint. This gives rise to a new set of field equations, altering the Einstein equations with a new term analytically derived from the constraints:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu} - \nabla_{\mu}\Phi_{\nu}.$

The new law of gravity we established shows that gravity behaves like the Einstein gravity in the solar system, and it has more attraction in the galactic scale (dark matter), and it becomes repulsive over very large scale (dark energy).

Since 2012, we have discovered that the same principle of taking variation of the actions under the energy-momentum conservation constraints is required by the quark confinement and the Higgs field, and is valid for all four forces–the gravity, the electromagnetism, the nuclear weak and the strong forces. We call it the principle of interaction dynamics (PID); see the previous post here. We have then derived a number of experimentally verifiable laws of Nature based only on the Einstein principle of general relativity, the gauge symmetry and our PID. This has led to new insights and solutions to a number of unsolved challenging problems of theoretical physics and cosmology since 1950s, and led to the unified field theory as Einstein hoped.

Most attempts in modern physics and cosmology focus on artificially modifying the actions such as the Einstein-Hilbert action. This is the primary reason behind the difficulties and challenges that modern physics has faced for many decades. What distinguishes our work is that the actions are uniquely dictated by basic principles, and the laws of Nature are then derived using PID. This leads to dual fields, given for example by the new term in our gravitational field equations. The dual fields are hardly achievable by any other means.

Another discovery we made in 2012 is the principle of representation invariance (PRI); see the previous post. PRI requires that the gauge theory be independent of the choices of the representation generators. These representation generators play the same role as coordinates, and in this sense, PRI is a coordinate-free invariance/covariance, reminiscent of the Einstein principle of general relativity. In other words, PRI is purely a logic requirement for the gauge theory.

PRI suggests the introduction of  the principle of symmetry-breaking (PSB) for unification. The three sets of symmetries — the general relativistic invariance, the Lorentz and gauge invariances, as well as the Galileo invariance — are mutually independent and dictate in part the physical laws in different levels of Nature. PSB assets that for a system coupling different levels of physical laws, part of these symmetries must be broken.

These three new principles–PID, PRI and PSB– have profound physical consequences, and, in particular, provide a new route of unification for the four interactions, different from the Einstein unification route which uses large symmetry group:

• the general relativity and the gauge symmetries dictate the Lagrangian;
• the coupling of the four interactions is achieved through PID and PRI in the field equations, which obey the PGR and PRI, but break spontaneously the gauge symmetry;
• the unified field model can be easily decoupled to study individual interaction, when the other interactions are negligible; and
• the unified field model coupling the matter fields using PSB.

Tian Ma & Shouhong Wang

Posted in Field Theory | 7 Comments

## John F. Nash Jr. and Louis Nirenberg: The Abel Prize Laureates 2015

The 2015 Abel Prize is awarded to John F. Nash Jr. and Louis Nirenberg; warmest congratulations!

See here for the prize announcement,and  here for the New York Times News.

## Remarks on a New Blackhole Theorem

This post makes a few remarks on the following blackhole theorem proved in (Tian Ma & Shouhong Wang, Astrophysical Dynamics and Cosmology, J. Math. Study, 47:4 (2014), 305-378):

Blackhole Theorem (Ma-Wang, 2014) Assume the validity of the Einstein theory of general relativity, then the following assertions hold true:

1.  black holes are closed: matters can neither enter nor leave their interiors,
2. black holes are innate: they are neither born to explosion of cosmic objects, nor born to gravitational collapsing, and
3. black holes are filled and incompressible, and if the matter field is non-homogeneously distributed in a black hole, then there must be sub-blackholes in the interior of the black hole.

This theorem leads to drastically different views on the structure and formation of our Universe, as well as the mechanism of supernovae explosion and the active galactic nucleus (AGN) jets. We refer interested readers to the original paper for the detailed proof.

An intuitive observation. One important part of the theorem is that all black holes are closed: matters can neither enter nor leave their interiors. Classical view was that nothing can get out of black holes, but matters can fall into black holes. We show that nothing can get inside the black hole either.

To understand this result better, let’s consider the implication of the classical theory that matters can fall inside a black hole. Take for example the supermassive black hole at the center of our galaxy, the Milky Way. By the classical theory, this black hole would continuously gobble matters nearby, such as the cosmic microwave background (CMB). As the Schwarzschild radius of the black hole

$\displaystyle R_s = \frac{2 M G}{c^2} \ \ \ \ \ (1)$

is proportional to the mass, then the radius ${R_s}$ would increase in cubic rate, as the mass ${M}$ is proportional to the volume. Then it would not hard to see the black hole will consume the entire Milky Way, and eventually the entire Universe. However observational evidence demonstrates otherwise, and supports our result in the black hole theorem.

Singularity at the Schwarzschild radius is physical. One important ingredient is that the singularity of the space-time metric at the Schwarzschild radius ${R_s}$ is essential, and cannot be removed by any differentiable coordinate transformations. Classical transformations such as e.g. those by Eddington and Kruskal are non-differentiable, and are not valid for removing the singularity at the Schwarzschild radius. In other words, the singularity displayed in both the Schwarzschild metric

$\displaystyle ds^2= -\left(1-\frac{R_s}{r}\right)c^2dt^2+\left(1-\frac{R_s}{r}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2, \ \ \ \ \ (2)$

and the Tolman-Oppenheimer-Volkoff (TOV) metric

$\displaystyle ds^2= -e^u c^2dt^2+\left(1-\frac{r^2}{R_s^2}\right)^{-1}dr^2 +r^2d\theta^2+r^2\sin^2\theta d\varphi^2, \ \ \ \ \ (3)$

is a true singularity, and defines the black hole boundary.

Geometric Realization of a black hole. As described in Section 4.1 in the paper, the geometrical realization of a black hole, dictated by the Schwarzschild and TOV metrics, clearly manifests that the real world in the black hole is a hemisphere with radius ${R_s}$ embedded in ${R^4}$, and at the singularity ${r=R_s}$, the tangent space of the black hole is perpendicular to the coordinate space ${R^3}$.

This geometric realization clearly demonstrates that the disk in the realization space ${R^3}$ is equivalent to the real world in the black hole. If one observes from outside that nothing gets inside the black hole, nothing will get in the real world (the black hole).

Tian Ma & Shouhong Wang

## Gauge Field Theory For Strong Interactions

This post presents a brief introduction to the gauge field theory of strong interactions, developed recently by the authors, based only on the ${SU(3)}$ gauge invariance, the principle of Lorentz invariance, the principle of representation invariance (PRI) and the principle of interaction dynamics (PID). The field theory leads to strong interaction potential formulas, provides duality between spin-1 gluons and dual spin-0 gluons, and offers a theoretical explanation of the quark confinement and asymptotic freedom.

### 1. Fundamental Symmetries and Strong Charges

Strong interaction is one of the four fundamental interactions of Nature, the others being the electromagnetic, the weak and the gravitational interactions. The strong interaction is responsible for binding protons and neutrons together to form atoms, and for binding quarks together to form hadrons (baryons and mesons).

The current theory for the strong interactions is the quantum chromodynamics (QCD), described by a non-Abelian ${SU(3)}$ gauge theory. The ${SU(3)}$ gauge theory consists of

• the eight ${SU(3)}$ gauge fields, representing the strong interaction potentials:

$\displaystyle S^k_{\mu}=(S^k_0,S^k_1,S^k_2,S^k_3) \qquad \text{ for } 1\leq k\leq 8,$

• three copies of Dirac spinors, representing different flavors of quarks:

$\displaystyle \Psi=(\psi_1, \psi_2, \psi_3)^T,$

which satisfy the Dirac equations:

$\displaystyle \left[i{\rm \gamma}^{\mu}D_{\mu} - m \right] \Psi =0, \ \ \ \ \ (1)$

where

$\displaystyle D_{\mu}=\partial_{\mu}+igS^a_{\mu}\tau_a,$

where ${\{\tau_1, \cdots ,\tau_8\}}$ is a basis of the set of traceless Hermitian matrices with ${\lambda^j_{kl}}$ being the structure constants.

Elements in ${SU(3)}$ can be written as

$\displaystyle \Omega =e^{i\theta^a\tau_a}\in SU(3).$

Then the  ${SU(3)}$ gauge transformation takes the following form:

$\displaystyle (\tilde{\Psi}, \ \ \tilde{G}^a_{\mu}\tau_a, \ \ \tilde m) =\left( \Omega\Psi, \ \ G^a_{\mu}\Omega\tau_a\Omega^{-1}+\frac{i}{g}(\partial_{\mu}\Omega)\Omega^{-1}, \ \ \Omega m\Omega^{-1} \right). \ \ \ \ \ (2)$

Principle of Gauge Invariance. The strong interactions obey the gauge invariance:

• the Dirac equations are gauge covariant, and
• the Lagrangian action of the interaction fields is gauge invariant.

In classical quantum chromodynamics (QCD), the gauge invariance of the strong interactions refers to the color charge of the quarks cannot be distinguished, and consequently, the energy contribution of different flavors of quarks is invariant under the ${SU(3) }$ phase (gauge) transformation.

As presented in the previous post, we can choose different basis for the set of traceless Hermitian matrices, leading to the principle of representation invariance (PRI), first discovered and postulated by (Ma-Wang, 2012). Namely, consider the following representation transformation of ${SU(3)}$:

$\displaystyle \tilde{\tau}_a=x^b_a\tau_b, \quad X=(x^b_a) \text{ is a nondegenerate complex matrix.} \ \ \ \ \ (3)$

Then it is easy to verify that ${\theta^a}$, ${S^a_{\mu}}$, and and ${\lambda^c_{ab}}$ are ${SU(3)}$-tensors under the representation transformation (3).

PRI (Ma-Wang, 2012): The ${SU(3)}$ gauge theory of strong interactions must be invariant under the representation transformation (3):

• the Yang-Mills action of the gauge fields is invariant, and
• the corresponding gauge field equations are covariant.

By PRI, there is an ${SU(3)}$ vector ${\alpha_a}$ associated with the representation transformation (3) such that the following contraction

$\displaystyle S_\mu = \alpha_a S^a_\mu \ \ \ \ \ (4)$

defines the (total) strong interaction potential, and ${S_0}$ is the ${SU(3)}$ strong interaction charge potential, and ${F=-g_s \nabla S_0}$ is the strong force. Consequently, PRI implies that ${g_s}$ plays the role of the strong charge, as the electric charge ${e}$ in the ${U(1)}$ abelian gauge theory for quantum electrodynamics (QED).

### 2. Action

The field strengths for the ${SU(3)}$ gauge theory for strong interactions are naturally defined by

$\displaystyle S_{\mu\nu}=S^a_{\mu\nu} \tau_a = \frac{i}{g_s}\left[D_{\mu},D_{\nu}\right],$

which implies that

$\displaystyle S^a_{\mu\nu} = \partial_{\mu}S^a_{\nu}-\partial_{\nu}S^a_{\mu}+g\lambda^a_{bc}S^b_{\mu}S^c_{\nu}. \ \ \ \ \ (5)$

Then the Lagrangian action density should be functions of the contraction

$\displaystyle {\mathcal G}^s_{kl}S^k_{\mu\nu}S^{\mu\nu l},$

where ${\mathcal G_{ab}=\frac12 \text{Tr}(\tau_a \tau_b^\dagger)}$. Consequently by the simplicity of laws of physics, the action of an ${SU(3)}$ gauge theory for strong interactions takes the following standard form of the Yang-Mills action:

$\displaystyle \mathcal{L}_S=-\frac{1}{4} {\mathcal G}^s_{kl}S^k_{\mu\nu}S^{\mu\nu l}+\bar{\Psi}\left[i\gamma^{\mu}(\partial_{\mu}+ig_sS^k_{\mu}\tau_k)-m \right] \Psi. \ \ \ \ \ (6)$

Geometrically, ${D_\mu}$ and ${S_{\mu\nu}}$ are the connection and curvature tensors on the complex spinor bundle ${M\otimes_p (\mathbb C^4)^3}$. By construction, it is clear that ${\mathcal L_S}$ is Lorentz, gauge and representation invariant, and is dictated by these three symmetries.

### 3. Field Equations and Dual Gluon Fields

With the Lagrangian action at our disposal, the classical Yang-Mills equations used in the classical QCD follow from the least action principle also called the principle of Lagrangian dynamics.

However, as we have demonstrated in the previous posts, for the four fundamental interactions, the principle of Lagrangian dynamics should be replaced the principle of interaction dynamics (PID), which takes the variation of the Lagrangian action under energy-momentum conservation constraint. As indicated in the previous post, by PID and PRI, we derive the following field equations for strong interactions (Ma-Wang, 2012), with proper scaling:

$\displaystyle \partial^{\nu}S^k_{\nu\mu}-\frac{g_s}{\hbar c}f^k_{ij}g^{\alpha\beta}S^i_{\alpha\mu}S^j_{\beta}- g_s Q^k_\mu =\left[\partial_{\mu}-\frac{1}{4}k^2_sx_{\mu}+\frac{g_s\delta}{\hbar c}S_{\mu}\right]\phi^k_s, \ \ \ \ \ (7)$

$\displaystyle \partial^{\mu}\partial_{\mu}\phi^k_s-k^2\phi^k_s+\frac{1}{4}k^2_sx_{\mu}\partial^{\mu}\phi^k_s+\frac{g_s\delta}{\hbar c}\partial^{\mu}(S_{\mu}\phi^k_s) =-g_s\partial^{\mu}Q^k_{\mu}-\frac{g_s}{\hbar c}f^k_{ij}g^{\alpha\beta}\partial^{\mu}(S^i_{\alpha\mu}S^j_{\beta}), \ \ \ \ \ (8)$

$\displaystyle \left[ i\gamma^{\mu}\left( \partial_{\mu}+i\frac{g_s}{\hbar c}S^l_{\mu}\tau_l\right) -\frac{mc}{\hbar}\right] \Psi =0, \ \ \ \ \ (9)$

for ${1 \le k \le 8}$, where ${\delta}$ is a parameter, ${S_{\mu}}$ is as in (4), and

$\displaystyle Q^k_\mu= \Psi \gamma_{\mu}\tau_k\Psi.$

Here we have taken the representation basis ${\tau_k}$ to be the Gell-Mann matrices.

The right-hand side of the field equations (7) above is due to PID. In other words, PID induces a natural duality between the interaction fields ${S^k_{\mu}}$, representing the spin-1 massless gluons, and their corresponding spin-0 dual gluon fields ${\phi^S_k}$. Namely, corresponding to the eight gluon fields ${S^k_{\mu}\ (1\leq k\leq 8)}$, there are eight dual gluon fields ${\phi^k}$, which we call the scalar gluons due to ${\phi^k_s}$:

$\displaystyle \text{ gluons}\ g_k\ \leftrightarrow\ \text{ scalar\ gluons}\ g^k_0 \qquad 1\leq k\leq 8.$

In the weakton model [8], we realize that these dual gluons possess the same weakton constituents, but different spins, as the gluons.

The above duality can be viewed as a duality of the strong forces. We start with the physical significance of the parameters ${k_s}$ and ${\delta}$. Usually, ${k_s}$ and ${\delta}$ are regarded as masses of the field particles. However, when (7)-(9) are viewed as the field equations for the interaction forces, ${k^{-1}}$ represents the range of attracting force for the strong interaction, and ${\left(\frac{g_s\phi^0_s}{\hbar c}\delta \right)^{-1}}$ is the range of the repelling force, where ${\phi^0}$ is a ground state of ${\phi=\alpha_a \phi^a}$ and ${\alpha_a}$ is as given in (4).

In fact, we have shown in [5] that for a particle with ${N}$ strong charges ${g_s}$ of the elementary particles, its strong interaction potential is given by

$\displaystyle \Phi_s=Ng_s(\rho)\left[\frac{1}{r}-\frac{A}{\rho}(1+kr)e^{-kr}\right],\qquad g_s(\rho )=\left(\frac{\rho_w}{\rho}\right)^3g_s, \ \ \ \ \ (10)$

where ${\rho_w}$ is the radius of the elementary particle (i.e. the ${w^*}$ weakton), ${\rho}$ is the particle radius, ${k>0}$ is a constant with ${k^{-1}}$ being the strong interaction attraction radius of this particle, and ${A}$ is the strong interaction constant, which depends on the type of particles.

It is clear that the first part, involving ${1/r}$, of the strong interaction potential is repulsive and is due to the gluon fields, and the second main term

$\displaystyle - Ng_s(\rho)\frac{A}{\rho}(1+kr)e^{-kr}$

gives rise to attractions, responsible for quark confinement. This term is due to the dual gluon fields. This observation  leads immediately to the following consequences.

First, in order for a gauge theory to describe the quark confinement, the inclusion of the dual scalar fields are inevitably necessary. Hence the observed quark confinement phenomena can be viewed as another physical evidence for PID.

Second, strong forces display both repulsive and attractive behaviors. For example, as the distance between two quarks increases, the strong force changes from repulsive, to asymptotically free, and then to attractive.

Third, strong forces/interactions are layered, and we shall explore more physical consequences of the layered potentials in the future posts.

Tian Ma & Shouhong Wang