Mutual energy principle

Considering the principle action-at-a-distance (A.D. Fokker) (K. Schwarzschild) (H. Tetrode), the Wheeler–Feynman absorber theory (J. A. Wheeler) (J. A. Wheeler), the transactional interpretation of quantum mechanics by John Cramer (John Cramer), the time domain reciprocity theorem proposed by W. J. Welch in 1960 (W. J. Welch), The mutual energy theorem of Shuang-ren Zhao published in early 1987 (Shuang-ren Zhao) (Shuang-ren Zhao) (Shuang-ren Zhao) and the reciprocity theorem of time domain correlation published at the end of 1987 by Adrianus T. de Hoop (Adrianus T. de Hoop), The mutual energy principle is proposed by Shuang-ren·Zhao (Shuang-ren Zhao)

The mutual energy principle is introduced to solve the conflict among (1) The Maxwell equation for single charge, (2) the superposition principle and (3) The energy conservation of the charges. This principle can be applied as a united theory for electromagnetic field and light (or photon).

The mutual energy principle of $N$ charges

In the language description, this principle describes the balance of energy. In a spatial region (within a volume $V$), there are $N$ charges, and the mutual energy flow into this volume is equal to the increase in mutual energy stored in this volume and the mutual power loss in this volume. Mathematically, with differential form summarized as follows:

$-\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}\nabla \cdot \mathbf {S} _{ij}=\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}{\frac {\partial u_{ij}}{\partial t}}+\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}w_{ij}$

where,

$\mathbf {S} _{ij}=\mathbf {E} _{i}\times \mathbf {H} _{j}$

$u_{ij}=\mathbf {E} _{i}\cdot \mathbf {D} _{j}+\mathbf {D} _{i}\cdot \mathbf {E} _{j}+\mathbf {H} _{i}\cdot \mathbf {B} _{j}+\mathbf {B} _{i}\cdot \mathbf {H} _{j}$

$w_{ij}=\mathbf {E} _{i}\cdot \mathbf {J} _{j}$

where $\mathbf {D} =\epsilon _{0}\mathbf {E}$ is vector of electric displacement and $\mathbf {E}$ is electric field, $\mathbf {B} =\mu _{0}\mathbf {H}$ is magnetic B-field. $\mathbf {H}$ is magnetic H-field, $\epsilon _{0}$ is vacuum permittivity, $\mu _{0}$ is Permeability. $\mathbf {J}$ is the intensity of current. Subscript $i,j$ is the index of current element.

UsingDivergence theorem, the above formula can be written as the integral:

$-$ $\scriptstyle \partial V$ $\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}\mathbf {S_{ij}} \cdot d\mathbf {A} ={\frac {\partial }{\partial t}}\int _{V}\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}u_{ij}dV+\int _{V}\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}\mathbf {w_{ij}} dV$

Where $\partial V\!$ is boundary of V. The shape of the volume is arbitrary but fixed for the calculation.

The above formula can be derived from the Maxwell equation, or it can be derived from the Poynting theorem, so it can be called mutual energy formula. Called it a formula, lower than the theorem. But in next section we will notice that there is something wrong either in the Maxwell equations for single charge or in the superposition principle. If the problem is at the superposition principle, we cannot obtained the Maxwell equation for N charges even the Maxwell equations for single charge is still OK. In this case, the Poynting theorem for $N$ charges have also the problem. However, in this situation, if the mutual energy formula is still correct, it should be seen as an axiom of the electromagnetic field and light(or photon) theory. In the next section, we will explain why the mutual energy formula should be chosen as a principle or an axiom for electromagnetic field theory.

Deriving the mutual energy principle by the conflict

The mutual energy principle can be derived by the conflict among Maxwell equation, superposition principle and energy conservation condition.

The total energy of the interaction of the $N$ charge is,

$\int _{t=-\infty }^{\infty }\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}w_{ij}dt=0\,\,\,\,\,\,\,\,\,\,\,(1)$

$w_{ij}=\mathbf {E} _{i}\cdot \mathbf {J} _{j}$

In the above equation each charge is subjected to the force of other $N-1$ charges. The charge can not give itself any force.

Assume each charge satisfy Maxwell equations. Assume the principle of superposition principle is correct. Hence the Maxwell equation also correct for N charges. The Poynting theorem of $N$ charges can be obtained from the Maxwell equations of the $N$ charges,

$-{\frac {\partial u}{\partial t}}=\nabla \cdot \mathbf {S} +w$

where

$\mathbf {S} =\mathbf {E} \times \mathbf {H}$

$u=\mathbf {E} \cdot \mathbf {D} +\mathbf {H} \cdot \mathbf {B}$

$w=\mathbf {J} \cdot \mathbf {E}$

$\mathbf {D} =\epsilon _{0}\mathbf {E}$,

$\mathbf {B} =\mu _{0}\mathbf {H}$

Considering that there are $N$ charges, the electromagnetic fields of N charges can be obtained by the superposition principle,

$\mathbf {E} =\sum _{i=1}^{N}\mathbf {E} _{i},\,\,\,\,\,\,\,\,\mathbf {H} =\sum _{i=1}^{N}\mathbf {H} _{i},\,\,\,\,\,\,\,\,\mathbf {J} =\sum _{i=1}^{N}\mathbf {J} _{i}\,\,\,\,\,\,\,\,\,\,\,(2)$

From which, the Poynting theorem for $N$ charges can be obtained as follows,

$-\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial u_{ij}}{\partial t}}=\sum _{i=1}^{N}\sum _{j=1}^{N}\nabla \cdot \mathbf {S} _{ij}+\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}\,\,\,\,\,\,\,\,\,\,\,(3)$

It is not possible to obtained the energy conservation condition Eq.(1) from the above Poynting theorem of $N$ charges Eq.(3). There is a conflict.

If the energy conservation of $N$ charges Eq.(1) is correct, then the power term of the $N$ charge in the Poynting theorem, $\sum _{i=1}^{N}\sum _{j=1}^{N}w_{ij}$ is too high to estimate this power. If this power item is overestimated, other items may be also overestimated too. These overestimation, in fact, means that,

${\frac {\partial u_{ii}}{\partial t}}=0,\,\,\,\,\,\,\,\,\,\,\,w_{ii}=0,\,\,\,\,\,\,\,\,\,\,\,\nabla \cdot \mathbf {S} _{ii}=0\,\,\,\,\,\,\,\,\,\,\,(4)$

This formula can be referred as the self-energy condition. From the $N$ charge of the Poynting theorem to remove the above zero items, the mutual energy formula in the electromagnetic field theory is abotained as following,

$-\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}\nabla \cdot \mathbf {S} _{ij}=\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}{\frac {\partial u_{ij}}{\partial t}}+\sum _{i=1}^{N}\sum _{j=1,j\neq i}^{N}w_{ij}\,\,\,\,\,\,\,\,\,\,\,(5)$

It can be proved in the latter section the energy conservation condition Eq.(1) can be derived from above mutual energy formula. Hence the above mutual formula is still correct even the Maxwell equation for single charge, superposition principle and the energy conservation condition Eq.(1) are conflict. The mutual energy formula does satisfy all the 3 conditions:(A) Maxwell equation for single charge, (B) superposition principle and (C) energy conservation condition Eq.(1). According to these Shuang-ren Zhao considered that the above mutual energy formula Eq.(5) should be used as a principle for electromagnetic field and light (or photon) theory.

On the other hand, if the above three zero-type formulas Eq.(4) are established, then the corresponding single charge Poynting theorem

$-{\frac {\partial u_{ii}}{\partial t}}=\nabla \cdot \mathbf {S} _{ii}+\mathbf {J} _{ii}\cdot \mathbf {E} _{ii}\,\,\,\,\,\,\,\,\,\,\,(6)$

On both sides of the equation should be zero. In fact, this is also consistent with Wheeler and Feynman's absorber theory, in which a single moving charge does not produce electromagnetic field. In absorber theory the electromagnetic field is an interaction between two charges.

It also consistent with quantum mechanics in which single charge sends the probability wave, which means for single charge, some time the wave is sent out and some time the wave is not sent out. At least when some time the wave is not sent out that the both side of the above formula Eq.(6) equal to zero is correct.

The Maxwell equations for single charge is only correct partially

If in the system there is only two charges one is emitter and the other is the absorber, the mutual energy principle Eq.(5) can be written as following

$-\nabla \cdot (\mathbf {S} _{12}+\mathbf {S} _{21})=({\frac {\partial u_{12}}{\partial t}}+{\frac {\partial u_{21}}{\partial t}})+(w_{12}+w_{21})$

or

$-\nabla \cdot (\mathbf {E} _{1}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1})$

$=(\mathbf {E} _{1}\cdot {\frac {\partial \mathbf {D} _{2}}{\partial t}}+\mathbf {E} _{2}\cdot {\frac {\partial \mathbf {D} _{1}}{\partial t}}+\mathbf {H} _{1}\cdot {\frac {\partial \mathbf {B} _{2}}{\partial t}}+\mathbf {H} _{2}\cdot {\frac {\partial \mathbf {B} _{1}}{\partial t}})$

$+(\mathbf {E} _{1}\cdot \mathbf {J} _{2}+\mathbf {E} _{2}\cdot \mathbf {J} _{1})$

The above mutual energy principle for two charges can be rewritten as,

$-(\nabla \times \mathbf {E} _{1}+{\frac {\partial \mathbf {B} _{1}}{\partial t}})\cdot \mathbf {H} _{2}+(\nabla \times \mathbf {H} _{1}-\mathbf {J} _{1}-{\frac {\partial \mathbf {D} _{1}}{\partial t}})\cdot \mathbf {E} _{2}$

$-(\nabla \times \mathbf {E} _{2}+{\frac {\partial \mathbf {B} _{2}}{\partial t}})\cdot \mathbf {H} _{1}+(\nabla \times \mathbf {H} _{2}-\mathbf {J} _{2}-{\frac {\partial \mathbf {D} _{2}}{\partial t}})\cdot \mathbf {E} _{1}=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$

In the above electric field $\mathbf {E} _{1}$ and magnetic field $\mathbf {H} _{1}$ are not linear related. The electric field $\mathbf {E} _{2}$ and the magnetic field $\mathbf {H} _{2}$ are also not linearly related. Hence the above formula require,

$\nabla \times \mathbf {E} _{1}+{\frac {\partial \mathbf {B} _{1}}{\partial t}}=0,\,\,\,\,\,\,\,\nabla \times \mathbf {H} _{1}-\mathbf {J} _{1}-{\frac {\partial \mathbf {D} _{1}}{\partial t}}=0$

$\nabla \times \mathbf {E} _{2}+{\frac {\partial \mathbf {B} _{2}}{\partial t}}=0,\,\,\,\,\,\,\,\nabla \times \mathbf {H} _{2}-\mathbf {J} _{2}-{\frac {\partial \mathbf {D} _{2}}{\partial t}}=0$

Thus we get two sets of Maxwell equations. It must be pointed out that the Maxwell equation derived from the principle of mutual energy is fundamentally different from the traditional Maxwell equation theory. Here, the solution of the Maxwell equation must be a solution of two groups at the same time.

If $\mathbf {J} _{1}=0$, the electric field $\mathbf {E} _{1}=0$ and magnetic field $\mathbf {H} _{1}=0$, electric field $\mathbf {E} _{2}$ and magnetic field $\mathbf {H} _{2}$ are not zero, according to mutual energy principle Eq.(1) should have,

$\nabla \times \mathbf {E} _{2}+{\frac {\partial \mathbf {B} _{2}}{\partial t}}=AnyThing<\infty ,\,\,\,\,\,\,\,\nabla \times \mathbf {H} _{2}-\mathbf {J} _{2}-{\frac {\partial \mathbf {D} _{2}}{\partial t}}=AnyThing<\infty$

If $\mathbf {J} _{2}=0$, $\mathbf {E} _{2}=0$ and megnetic field $\mathbf {H} _{2}=0$,$\mathbf {E} _{1}$ and $\mathbf {H} _{1}$ are all not zero, according to the mutual energy principle Eq.(1) should have,

$\nabla \times \mathbf {E} _{2}+{\frac {\partial \mathbf {B} _{2}}{\partial t}}=AnyThing<\infty ,\,\,\,\,\,\,\,\nabla \times \mathbf {H} _{2}-\mathbf {J} _{2}-{\frac {\partial \mathbf {D} _{2}}{\partial t}}=AnyThing<\infty$

Where $AnyThing$ is any function that is not infinity. These two situation are not the accept solution for the mutual energy principle.

Hence, if $\mathbf {J} _{1}=0,$ $\mathbf {E} _{1}=0$ and $\mathbf {H} _{1}=0$ or if $\mathbf {J} _{2}=0$, $\mathbf {E} _{2}=0$ and $\mathbf {H} _{2}=0$ all are all not effective solution for the mutual energy principle.

This means the solution of the two group Maxwell equation must nonzero in the same time, i.e. the two solutions must synchronized.

The Maxwell equation has two kinds of solutions, the solution of the retarded wave, the solution of the advanced wave. So there are three cases,

1. Both solutions are retarded waves.

2. Both solutions are advanced waves.

3. For the two solutions, one is a retarded wave another is an advanced wave.

It is only possible to synchronize the two waves which are two solution of Maxwell equation with the case 3. It is not possible to synchronize two retarded waves or two advanced waves.

Hence the Maxwell equation for single charge is still correct partially. If a retarded wave can find an advanced wave to match it and hence synchronized together, this retarded wave is a solution of the Mutual energy principle. Otherwise, if there is no any advanced wave just synchronized with this retarded wave, it is not a solution of the mutual energy principle.

Now the mutual energy principle is taken as axiom of electromagnetic field theory, if the mutual energy principle is not satisfied, even Maxwell equation is satisfied, it is still not a physic solution.

This is also consistent with quantum mechanics, in which the waves are probability waves. That is because this wave only satisfy Maxwell equations. If the waves satisfy the mutual energy principle, there must be a pair waves one is retarded wave, another is advanced wave and these two waves are synchronized. In this case the waves are real physical waves, and not the probability waves.

This is also the reason Shuang-ren Zhao gave up to apply the Maxwell equations as the axiom of electromagnetic field and light (photon) theory.

The principle of the self-energy

The conflict between the Maxwell equation for single charge and the self-energy condition

Last section shows that the Maxwell equations for single charge are at least partially correct. If the Maxwell equations are correct, the corresponding items cannot be zero. This has a conflict with the self-energy condition Eq.(2). In order to solve this conflict, Shuang-ren Zhao introduced the self-energy principle as following,

The self-energy principle of and the return of self-energy flow

All of the self-energy terms are zero in the self-energy condition Eq.(2), which tells us that all self-energy items of the single charge of an accelerated or deceleration movement are zero. It is said the single charge in the acceleration or deceleration movement can produce radiation, is this has a contradictory to the self-energy condition? In fact, there is no contradiction, radiation is done by mutual energy flow. the radiation occurs between an emitter charge and an absorber charge, which is done by mutual energy, and mutual energy flow. The self-energy, self-energy flow are zero. However even the Maxwell equations for single charge is only partially correct, it require the items of corresponding Poynting theorem also nonzero. Or with the Poynting theorem that these self-energy of a single charge is not zero. Hence, Maxwell's equations contradicts with the self-energy condition. In order to solve this contradiction, Shuang-ren Zhao argues that the wrong side lies in Maxwell's theory and the Poynting theorem. The principle of self-energy condition must be adhered to.

Shuang-ren Zhao's guess that these self-energy terms may have existed the beginning, but they are automatically returned. This return process is not a normal return process. For a normal return process for a retarded wave is still a retarded wave. This return process is a time reversal process. Satisfies the following time reversal condition,

$\mathbf {r} (\mathbf {E} (t),\mathbf {H} (t),\mathbf {D} (t),\mathbf {B} (t),\mathbf {x} (t),t)=(\mathbf {E} (-t),\mathbf {H} (-t),\mathbf {D} (-t),\mathbf {B} (-t),\mathbf {x} (-t),-t)$

In the above formula $\mathbf {r}$ is the time reversal transformation. The Maxwell equations before to apply the time reversal transform are

$\nabla \times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}},\,\,\,\,\,\,\,\,\,\,\,\,\,\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$

After the time reversal transformation, the transformed equation is called the time reversal Maxwell equation. The time reversal of the Maxwell equation is not a Maxwell equations which is shown as following,

$\nabla \times \mathbf {H} =-\mathbf {J} -{\frac {\partial \mathbf {D} }{\partial t}},\,\,\,\,\,\,\,\,\,\,\,\,\,\nabla \times \mathbf {E} =+{\frac {\partial \mathbf {B} }{\partial t}}$

In the above

$\mathbf {J} =q\delta (\mathbf {x} -\mathbf {x'} ){\frac {d\mathbf {x} (t)}{dt}}$

is considered.

Similar to the Maxwell equations, but they are not the Maxwell equations. As a result of this time reversal process, if there is self-energy flow outflow, there is a self-energy flow inflow which cancels the outflow. All other self-energy items are offset by the corresponding reversal process of the time. So all the self-energy terms vanish. These guarantee that the energy transfer for photon can only be done through the mutual energy flow which will be introduced the following section.

The mutual energy theorem and the mutual energy flow theorem

Mutual energy theorem

Consider the instantaneous electromagnetic field process of a photon, where the energy of the electromagnetic field in the space is zero at the beginning and at the end of the electromagnetic process. So the energy of the space is from zero to a value, and finally down to zero. So obviously there should be,

$\int _{t=-\infty }^{\infty }\int _{V}(\mathbf {E} _{1}\cdot {\frac {\partial \mathbf {D} _{2}}{\partial t}}+{\frac {\partial \mathbf {D} _{1}}{\partial t}}\cdot \mathbf {E} _{2}+\mathbf {H} _{1}\cdot {\frac {\partial \mathbf {B} _{2}}{\partial t}}+{\frac {\partial \mathbf {B} _{1}}{\partial t}}\cdot \mathbf {H} _{2})dVdt=0$

From this the mutal energy theorem can be obtained,

$-\int _{t=-\infty }^{\infty }$ $\scriptstyle \partial V$ $(\mathbf {E} _{1}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot d\mathbf {A} dt=\int _{t=-\infty }^{\infty }\int _{V}(\mathbf {E} _{1}\cdot \mathbf {J} _{2}+\mathbf {E} _{2}\cdot \mathbf {J} _{1})dVdt$

If the two electromagnetic fields $\mathbf {E} _{1}$, $\mathbf {H} _{1}$ and $\mathbf {E} _{2}$, $\mathbf {H} _{2}$ One of the retarded waves, another is the advanced wave. The advanced wave arrives at the big sphere at some point in the past, and the retarded wave reaches the large sphere at a certain moment in the future, so that the two fields are not nonzero at the same time in the large Sphere. So there are,

$\int _{t=\infty }^{\infty }$ $\scriptstyle \partial V$ $(\mathbf {E} _{1}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot d\mathbf {A} dt=0$

and hence,

$\int _{t=-\infty }^{\infty }\int _{V}(\mathbf {E} _{1}\cdot \mathbf {J} _{2}+\mathbf {E} _{2}\cdot \mathbf {J} _{1})dVdt=0$

Suppose $\mathbf {J} _{1}$ is in volume $V1$. $\mathbf {J} _{2}$ Within volume $V2$ So there are the following forms of electromagnetic field mutual theorem,

$-\int _{t=-\infty }^{\infty }\int _{V2}\mathbf {E} _{2}\cdot \mathbf {J} _{1}dVdt=\int _{t=-\infty }^{\infty }\int _{V1}\mathbf {E} _{1}\cdot \mathbf {J} _{2}dVdt$

The mutual energy theorem is the receiprocity theorem in time domain which is proposed by W. J. Welch in 1960. (W. J. Welch) and the mutual energy theorem pubulished by shuang-ren Zhao in early 1987. (Shuang-ren Zhao) (Shuang-ren Zhao) (Shuang-ren Zhao) and time-domain corelation reciprocity theorem published by Adrianus T. de Hoop at the end of 1987. (Adrianus T. de Hoop). It can be proved that the reciprocal theorem of time domain correlation and the mutual theorem are only one Fourier transform, so we can see them as one theorem. Time domain reciprocity theorem (W. J. Welch) is an apecial case of the time-domain correlation reciprocity theorem (Adrianus T. de Hoop)

So these three theorems can all be seen as one theorem. We call it the mutual energy theorem, emphasizing that the theorem should be the energy theorem in the theory of electromagnetic fields. Status should go beyond the Poynting energy theorem.

It should be clear that the original time-domain reciprocity theorem (W. J. Welch). and the mutual energy theorem (Shuang-ren Zhao) are tell us there is energy sent from transmitting antenna to the receiving antenna. However the receiving antenna is only received part of the energy of the transmitting antenna.

Now the mutual energy is re-derived from the mutual energy principle and self-energy principle, the formula is same, but the meaning is different. There is big difference in the meaning of the theorems. The new mutual energy theorem tell us that all energy sends from the emitter are received by the absorber. There is no any energy sends to the whole space. Corresponding to the retarded wave, the energy sends to the whole space by the emitter is referred as self-energy flow which is canceled by the corresponding time reversal energy flow. It is same to the advanced wave, the self-energy flow of the advanced wave sent by the absorber is also canceled by the corresponding time-reversal process. It should be notice according this theory, there are 4 waves: (A) retarded wave, (B) advanced wave, (C) the time-reversal wave corresponding to the retarded wave, (D) the time-reversal wave corresponding to the advanced wave. There are 4 energy flow corresponding to these 4 waves. These 4 energy flow cancel each other completely. However the retarded wave and the advanced wave also produce a mutual energy flow. The mutual energy flow is responsible for the energy transfer from the emitter to the absorber.

Mutual energy flow theorem

The mutual energy theorem, when originally proposed, tends to classify it as a reciprocity theorem. The reciprocity theorem is a mathematical theorem by some sort of mathematical relationship. Mutual energy theorem is different, it is an energy theorem. Shuang-ren Zhao, first of all realized that this is an energy relationship and called it as the mutual energy theorem. From the proposition of mutual energy to the real proof that it is an energy theorem Shuang-ren Zhao spent almost 30 years. After understanding the energy relationship, so it will ask the energy flow from emitter to the absorber. From the mutual theorem it is not difficult to prove the mutual energy flow theorem,

$-\int _{t=-\infty }^{\infty }\int _{V2}\mathbf {E} _{2}\cdot \mathbf {J} _{1}dVdt=Q_{1}=Q=Q_{2}=\int _{t=-\infty }^{\infty }\int _{V1}\mathbf {E} _{1}\cdot \mathbf {J} _{2}dVdt$

where

$Q=\int _{t=-\infty }^{\infty }\iint \limits _{A}(\mathbf {E} _{1}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot d\mathbf {A} dt$

$A$ are any surface between $V1$ and $V2$. The unit normal vector of the surface $A$ is from $V1$ to $V2$.

We can define,

$q=\iint \limits _{A}(\mathbf {E} _{1}\times \mathbf {H} _{2}+\mathbf {E} _{2}\times \mathbf {H} _{1})\cdot d\mathbf {A}$

It is the flow of energy flow from $V1$ to $V2$.

$Q$ hence is the time integral of flux of the mutual energy flow, and hence is the energy go through the surface $A$.

Let $A1$ be a surface surrounds $V1$. $Q1$ is the energy flowing through the surface $A1$. $A2$ is a surface that surrounds $V2$. $Q2$ is the energy flowing through the surface $A2$. Here we assume that the normal of all surfaces is from $1$ to $2$. $Q1$ and $Q2$ are defined similarly to $Q$ and are no longer listed.

Mutual energy flow waveguide

The flow of mutual energy is much like the electromagnetic field in the waveguide. This field can be seen as a plane wave. But the waveguide is very special, two ends are very small. As small as an electric charge. The middle is very thick. This waveguide is a natural energy waveguide, where the wave can be seen as a quasi-plane wave.

This natural waveguide is not only present between the emitter and the absorber, but also between the transmitting antenna and the receiving antenna. Between the transmitting antenna and the receiving antenna, the energy transfer is also done by the mutual energy flow. The difference from the antenna to the emitter/absorber is the antenna send the wave to the whole space, only a small part energy is sent from the transmitting antenna to the receiving antenna. But for the system of the emitter and the absorber, the energy is only sent from the emitter to the absorber. There is no any energy sent from emitter to the whole space. The self-energy is actually sent from the emitter to the whole space, but this energy is canceled by the time reversal waves.

The mutual energy flow is retarded

Although the interoperable flow is composed of two electromagnetic fields, one is the field of the retarded wave, one is the field of the advanced wave. But the mutual energy flow is indeed retarded only. That is to say, mutual energy flow does not violate the causal relationship.

Mutual flow is retarded does not violate the causal relationship, but tells us that a physical quantity with energy must retardedly spread out, must not violate the causal relationship, must be local. However, for the information without energy, not the same, form example the quantum entanglement is nolocal. The photon can tell the emitter the state of the absorber, and the time it takes for the information to be passed can be negative. So the principle of mutual energy is not opposed to non-local information transmission.

Mutual energy flow interpretation of quantum mechanics

Wave function collapse

Considering that the transmission of photon energy is from a point to point, the waves in Maxwell's theory are diffused throughout the space, and in order to explain that photons naturally have the speculation that the light waves collapsed onto their absorbers. This guess is just a very rough assumption. No one gives a concrete equation about the collapse of the wave function. In fact, the wave function collapse process can be explained with the principle of mutual energy and self-energy principle. We know that the energy transfer between the emitter to the absorber is done by the mutual energy flow. Completed the energy transfer from the emitters to the absorber, the self-energy flow comeback. These two processes together constitute the same effect with the wave function collapse. The mutual energy flow, self-energy flow flow back, have an accurate mathematical description. Not just is a qualitative theory but is also a quantitative theory, so it is much more accurate than just talking about the wave function collapse.

Electromagnetic radiation process of light waves

Assuming that the charge absorbs energy when releasing the supercurrent, the charge releases energy when releasing the hysteresis wave. The charge that emits energy is called a radiator. The charge that absorbs energy is called the absorber. In fact there is a scatterer, which absorbs the energy and radiates the absorbed part of the energy. For the sake of simplicity, we break the scatterer into radiators and absorbers. The charge jumps from the high energy level to the low energy level in the atom, and the energy is reduced, so the radiation energy produces a hysteresis wave. The charge jumps from the low energy level to the high energy level to absorb the energy and thus generate the precursor wave. Assume that these waves are randomly generated spontaneously.

It is assumed that the high level charge can spontaneously randomly transition to the low potential. Produce a hysteresis wave. If the hysteresis wave does not run into the front wave. Can not produce reciprocal flow, no photon generation. In front of us when it comes from the flow back, the radiation charge back to the original position. From the low point transition back to high potential. Waiting for the next chance.

After the next chance, the charge jumps again from the high energy level to the low energy level, issuing a hysteresis wave. If the hysteresis wave happens to encounter a leading wave, that is, there is a leading wave synchronized with this hysteresis wave. Mutual energy flow, which also means that there are photons emitted from the radiator to meet the super-wave absorber. After the energy flow is completed from the radiator to the absorber, the energy flow returns automatically. The energy transfer of energy flow is an irreversible process. Since the flow of energy from the entire space, there is no material to absorb it, so it has to return to its source.

The absorber spontaneously jumps from low energy level to high energy level. Generate the advanced wave, if not run into any retarded wave, this super wave is only self-flowing wave, the wave automatically returns its source. Absorber to restore the principle of low energy level position. If the advanced wave coincides with the retarded wave emitted by the emitter, the absorber absorbs the energy of a photon from the retarded wave. The self-energy flow of the advanced wave radiate by the absorber is come back to the absorber.

So the self-energy flow does not pass the energy, but it acts as a complement to the mutual energy flow, regardless that appeare the mutual energy flow or not, since the self-energy flow has to return to its source. If the mutual energy flow is generated, the energy of a photon is transferred from the emitter to the absorber.

self-energy flow is issued, and then returned, or did not issue, that is, the space was only the mutual energy flow, the effect is the same. It is easier to accept the return from the flow of energy and return. Because of this, the charge can spontaneously radiate the retarded wave and the advanced wave. Retarded and advanced waves encounter is a random event, two very short waves just synchronize, which of course is a small probability event, the key it is a probability event. This probability should be proportional to the square of the amplitude of the retarded wave and the advanced wave. This is a good explanation of the real reason why photons are absorbed only by probability.

Defect of electromagnetic field superposition principle

Electromagnetic field of charge

For a $N$ charge system, how should the electromagnetic field in space be defined? If the electric field of a charge $i$ at charge $j$ is $\mathbf {E} _{i,j}$

$\mathbf {E} _{i,j}={\frac {\mathbf {F} _{i,j}}{q_{j}}},\,\,\,\,\,\,\,\,\,\,\mathbf {F} _{i,j}=kq_{i}q_{j}{\frac {\mathbf {r} _{i,j}}{r_{i,j}^{3}}}$

where $q_{i}$, $q_{j}$,is amount of charge at charge $i$ and charge $j$. $k$ is a constant. $r_{i,j}$ is the distance from $q_{i}$ to $q_{j}$. $\mathbf {r} _{i,j}$ is corresponding vector of distance. We can difine the electric field at any place with a charge.

$\mathbf {E} _{j}=\sum _{i=1,i\neq j}^{N}E_{i,j}$

The classical electromagnetic theory holds that the electric field of $x$ in any space can be expressed as (note that we assume that there is no charge at $x$

$\mathbf {E} _{x}=\sum _{i=1}^{N}E_{i,x}$

This is called the superimposition principle of the field. Shuang-ren Zhao believes this principle has problem. Which is correct $\sum _{i=1}^{N}$ or $\sum _{i=1,i\neq j}^{N}$ to define the field? None is suitable. The above equation overestimate the value of the electric field. The superposition principle of the above equation is also the main reason for the overestimation of the energy of the Poynting theorem with $N$ charge. So the electric field in the charge position is not defined correctly. The field where there is no charge in the space can still be superimposed. So for any position $x$ electric field can be write as following,

$\mathbf {E} _{x}=[E_{1,x},E_{2,x},E_{i,x}...E_{N,x}]$

It is known that the Maxwell equations need the help of superposition principle, without the principle of the superposition, even if the Maxwell equation for a single charge is established, it still can not prove the Maxwell equations for many charges. The principle of mutual energy does not need to be based on the superposition principle of fields. The principle of mutual energy is based on the interaction between all fields, so the above definition of the field is sufficient for the mutual energy principle.

Also if some of the charge movement, its total magnetic field can be listed as follows,

$\mathbf {H} _{x}=[H_{1,x},H_{2,x},H_{i,x}...H_{N,x}]$

In short, the principle of superposition is problematic, but fortunately the theory of mutual energy principle does not depend on the superposition principle. In this theoretical system, the electromagnetic field does not have to be added up. It is enough for a simple list of the fields.

The mutual energy principle have replace all the 3 conditions, (1) Maxwell equation for single charge, (2) superposition principle, (3) energy conservation condition. This is also the reason shuang-ren Zhao would like to take the mutual energy principle as an axiom for electromagnetic theory.

The principle of mutual energy interpretation of light

Photon is the mutual energy flow

We know that photon is an energy package. But because it is called the particle, it is natural to imagine that the photon should have a nucleus like an atom, and there is a field outside the nucleus. This field constitutes a light wave. When we ask which slit the photon is a pass, in fact we are asking which slit the nucleus of the photon go through in the double slits situation. But the theory of the mutual energy principle tells us that photons are the mutual energy flow. There is nothing like a photon's nuclear. When we say that the photon is a wave, we imagine it is a retarded wave. This wave diffuses into the whole space. The mutual energy principle tells us that photons are neither retarded waves nor advanced waves. Photon is the mutual energy flow which is composed of retarded and advanced wave. The mutual energy flow is not a diffuse wave, and the mutual energy flow is similar to a plane wave in a wave-guide. The mutual energy flow supports point-to-point energy propagation. Here we still say that light wave is made up of many photons, but according to the mutual energy flow theorem photon is not a particle, the photon is the mutual energy flow. It can be said that the photon is a wave, but it is not the traditional wave which will diffuse to the entire space. The mutual energy flow looks like a quasi-plane wave in a wave-guide which is thick in the middle and peaky at the two ends. This wave is like a wave and also like a particle. This wave will converge to a point like a particle when approaching the emitter and the absorber. But between the emitter and the absorber, the wave-guide can become very thick. If a partition board is placed between the emitter and the absorber and the double slits are engraved on the partition board, the mutual energy flow will interfere. So like a wave. So the mutual energy flow can explain the wave and particle duality of the light.

Wave particle duality

The above-mentioned mutual energy flow theorem can be used to explain the wave-particle duality problem. Photons are the energy flow from the emitter to the absorber. The energy transfer of photons can be described by the mutual energy flow theorem. The mutual energy flow is always constant on any surface between $V1$ and $V2$. This amount is actually the energy of the photon. Because the mutual energy flow is thin at $A1$ and $A2$, the energy is concentrated at a point so that it can be seen as a particle. In the middle between $V1$ and $V2$, the mutual energy flow becomes very thick, so it looks like a wave. This is the reason why the light has the wave and particle duality.

The mutual energy theorem and the mutual energy flow theorem can guarantee the energy conservation, the energy sent by the emitter are received all by the absorber. There is no any energy lost. The moment conservation is also guaranteed by the energy flow. Hence the mutual energy flow can realize all mechanic properties of the photon.

The experiment of the double slits

The mutual energy flow is very thick at the middle between the emitter and the absorber and very thin at the two ends. If in the middle part can place a partition board, in the partition board can carve two slits. Let the light pass through the slits. It is conceivable that interference will occur when the mutual energy flow passes through the double slits. When we ask which slit in the double slits the photon go through, this question itself has some problems. Photons are in the form of the mutual energy flow, it goes at the same time from the two slits to transfer the energy. The photon itself is the mutual energy flow, part of the energy flow from the first slit, part of energy go through the other slit. Photons are mutual energy flow consisting of retarded waves and advanced waves, which can transfer energy. According to this explanation, the double-slits experiment is not a mysterious any more.

Quantum entanglement

According to the principle of mutual energy, photons in the time of launch, there have been identified absorber. If a absorber can only receive right-handed photons, the emitter must also emit a right hand photon. The photon is left or right, not decided just by the emitter, and also must be determined by the absorber at the same time. Since the absorber emits an advanced wave, if the absorber absorbs only the right-handed photons, this information can notify the emitter by the advance wave before the absorption is generated, so that the emitter knows that at the moment when the photon is emitted, a right-handed photon must be sent.

In the above we have said that the photon is generally described as a two-charge system, the system has only an emitter and an absorber. The emitter produces a retarded wave, and the absorber produces an advanced wave. These two random events are just synchronized to form an mutual energy flow. This mutual energy flow is the photon. Quantum entanglement occurs when the charges of the system is more than two.

For example, when the first photon encounters a non-linear optical material, two lower-frequency photons are produced, which are radiated from a single charge of a non-linear material and are absorbed by both absorbers in the system. So the interacting system has total of three charges. The three-charge system produce two photons, and this two lower-frequency photons are entangled. This is because if an absorber receives only the right-handed photons, the emitter already knows that it must radiate a right-handed photon to the right-handed absorber. Since the angular momentum of two low-frequency photons must be equal to the angular momentum of the incidence photon with higher frequency, we assume that the angular momentum of the incidence photon with higher frequency of is zero. So that the angular momentum of another lower-frequency photon must be left-handed. So the second lower-frequency photon corresponding to the absorber is only possible to obtain a left-handed photon. The time from which the absorber of the first lower frequency photon transmits the signal from the absorber to the emitter is negative (since this transfer is done by the advanced wave). The time that the second photon is transmitted from the emitter to the second absorber is positive. A positive time is offset by the same negative time, so the time between the two lower frequencies of photons is almost zero. This time is far less than the time of light walking. So it is superluminal. Therefore, the entanglement phenomenon is also superluminal and non-local. So the theory of the principle of mutual energy and the self-energy principle can explain the quantum entanglement phenomenon.

Why the Maxwell equations can be seen as mathematical equations, a probabilistic equations?

We found that the Maxwell equations must have two groups to produce the energy flow. If there is only one group equations now, how do we look at this group equations and their solutions? We can regard these equations as mathematical equations, and their solutions are also mathematical or probabilistic.

The theory of the mutual energy principle tell us that the emitter randomly radiate retarded wave, the absorber randomly radiate advanced wave. Because of the mechanism so-called the return of the self-energy flow, these waves are not carrying energy. Can be seen as a mathematical wave. Corresponding to a retarded wave, there are thousands of thousands of absorbers in the environment, and these absorbers are randomly irradiated in time. Which advanced wave can be synchronized with the retarded wave is obviously a probabilistic problem. This probability should also be proportional to the square of the absolute value of the complex amplitude of the retarded wave. This is why the solution of the Maxwell equation in the light band is related to the root cause of the probability. When there is an advanced wave just synchronize with this retarded wave, the mutual energy flow is generated. This mutual energy flow is the photon. Photon transfers energy from the emitter to the absorber through the mutual energy flow. This energy transfer process is irreversible.

Since the self-energy flow can be returned, that is this process is reversible. A self-energy flow process can be completely offset by the corresponding time reversal process. So the self-energy flow does not deliver any energy. So the overall effect of the solution of the Maxwell equation can be seen as a mathematical process rather than a physical process. Whether it has issued a self-energy flow (then there is time-reversal self-energy flow canceled it) or it did not issue any self-energy flow at all, the effect is same.

This probability explanation emphasizes the reason for the probability of occurrence, unlike the Copenhagen school that the probability is only the nature of the quantum mechanics, but what is behind this nature, it did not tell us anything. This is also the reason why Einstein doubts the probabilistic interpretation of the Copenhagen school. The interpretation of the probability of the mutual energy flows removes the mystery of the Copenhagen school of the probability interpretation.

It should be clear here we only say the Maxwell equations can be seen as a probability equations. Actually even the time reversal process canceled the self-energy process, the self-energy process and the corresponding time-reversal process are all physical process. There are two "real" energy flow, but they canceled.

Conclusion

The theory of the retarded wave of the Maxwell equations is only accurate in the radio microwave band theory, but it encounters a lot of difficulties in the light band. It cannot explain the mechanics character of photon for example, energy conservation, momentum conservation. The principle of mutual energy can take the place of the Maxwell equations, becomes the new axiom of electromagnetic field theory. Starting from the principle of mutual energy, all bands include radio, microwave, infrared, light, ultraviolet, X-ray, gamma light can all be unified under a same theoretical framework. Although the mutual energy flow in the principle of mutual energy is composed of the advanced wave and retarded wave, it is completely retarded. Hence, mutual energy flow does not violate the causal relationship.

The Maxwell equation is only partially correct. The superposition principle has problem. These two are conflict with the energy conservation condition. Hence Maxwell equation and the superposition principle can not be used as axioms for electromagnetic theory. The axiom of electromagnetic field theory have to be replaced with the mutual energy principle and the self-energy principle.

This theory tell us the electromagnetic field is not just the retarded wave, there are also the advanced wave, and 2 time reversal waves corresponding to the retarded wave and the advanced wave. The retarded wave is canceled by its corresponding time reversal wave. The advanced wave is also canceled by its corresponding time reversal wave. These 4 wave corresponding to 4 energy flow. These 4 energy flow cancel completely. However the retarded wave and the advanced wave can produce a mutual energy flow which is responsible the transfer of the energy of the photon.

From the above analysis can clearly see that the theory of the principle of mutual energy is a self-consistent theory.

References

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