User:Imrecons/sandbox

=mutual energy= The concept of the mutual energy is consist of mutual energy formula, mutual energy theorem, mutual energy flow.

Introduction of mutual energy
Poynting vector can be write as following,

where $$\mathbf{E}$$ is electric field. $$\mathbf{H}$$ is magnetic H-field. Point vector are energy intensity of the energy flow of the electromagnetic field. Assume that,


 * $$\mathbf{E}=\mathbf{E}_1+\mathbf{E}_2$$


 * $$\mathbf{H}=\mathbf{H}_1+\mathbf{H}_2$$

Hence we have,



\mathbf{\mathbf{E}\times\mathbf{H}}=(\mathbf{E}_1+\mathbf{E}_2)\times (\mathbf{H}_1+\mathbf{H}_2) $$



=\mathbf{E}_1 \times \mathbf{H}_1+ (\mathbf{E}_1 \times \mathbf{H}_2+\mathbf{E}_2 \times \mathbf{H}_1) +\mathbf{E}_2 \times \mathbf{H}_2 $$

Hence we can define



\mathbf{E}_1 \times \mathbf{H}_1 $$



\mathbf{E}_2 \times \mathbf{H}_2 $$

as self-energy items of the Poynting vector. And define



(\mathbf{E}_1 \times \mathbf{H}_2+\mathbf{E}_2 \times \mathbf{H}_1) $$

as the mutual energy items of the Poynting vector.

Poynting theorem
Where

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

is the Poynting vector.

$$ u=\frac{1}{2}(\mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}) $$ is the energy of electric field.

Superposition principle


\mathbf{E}=\sum_{i=1}^{N}\mathbf{E}_{i} $$



\mathbf{H}=\sum_{i=1}^{N}\mathbf{H}_{i} $$

Assume $$\mathbf{E}_{i}$$ and $$\mathbf{H}_{i}$$ are the electromagnetic field of $$i$$-th charge. Assume that $$i$$-th charge accelerates or decelerates.

Poynting theorem of $$N$$ charges
Substitute superposition principle to Poynting theorem

mutual energy items of the Poynting theorem
Subtract the self-energy items from the Poynting theorem of $$N$$ charges, the mutual energy items of the Poynting theorem can be obtain

This can be referred as mutual energy formula.

Lorentz reciprocity theorem
Where $$\omega$$ is frequency.

Welch's reciprocity theorem
where $$t$$ is time. This theorem is introduced by W. J. Welch in 1960 （W. J. Welch）.

Conjugate transform
It is not clear who first introduced the concept of the conjugate transform, but the details theory of the conjugate transform can be found in （Jin Au Kong）. It is important that if a field satisfies the Maxwell equations, after the conjugate transform, it still satisfies the Maxwell equations. If the original field is retarded wave, after the transform it becomes advanced wave. Vice Versa, if the original field is advanced wave, after the transform it becomes the retarded wave.

$$ \mathbf{\mathbb{C}}(\mathbf{E}(t),\mathbf{H}(t),\mathbf{J}(t),\mathbf{K}(t),\mathbf{\boldsymbol{\epsilon}}(t),\mathbf{\boldsymbol{\mu}}(t))=(\mathbf{E}(-t),-\mathbf{H}(-t),-\mathbf{J}(-t),\mathbf{K}(-t),\mathbf{\boldsymbol{\epsilon}}(-t),\boldsymbol{\mathbf{\mu}}(-t)) $$

or

$$ \mathbf{\mathbb{C}}(\mathbf{E}(\omega),\mathbf{H}(\omega),\mathbf{J}(\omega),\mathbf{K}(\omega),\mathbf{\boldsymbol{\epsilon}}(\omega),\mathbf{\boldsymbol{\mu}}(\omega))=(\mathbf{E}^{*}(\omega),-\mathbf{H^{*}}(\omega),-\mathbf{J^{*}}(\omega),\mathbf{K}^{*}(\omega),\mathbf{\boldsymbol{\epsilon^{*}}}(\omega),\boldsymbol{\mathbf{\mu^{*}}}(\omega)) $$

Where $$\mathbf{\mathbb{C}}$$ is the conjugate transform. $$\mathbf{E}$$ is electric field. $$\mathbf{H}$$ Magnetic field. $$\mathbf{J}$$ current intensity. $$\mathbf{K}$$ magnetic current intensity. $$\mathbf{\boldsymbol{\epsilon}}$$ is permittivity, $$\mathbf{\boldsymbol{\mu}}$$ is permeability, $$t$$ is time, $$\omega$$ is frequency.

Rumsey's reciprocity theorem
V.H. Rumsey has introduced his summarize the Lorentz reciprocity theorem as "action and reaction". He has apply the complex conjugate transform to the his "action and reaction" theorem and obtained a new reciprocity theorem （V.H. Rumsey）,

Rumsey's reciprocity theorem was derived by apply the conjugate transform to the Lorentz reciprocity theorem.

Inner product space for the electromagnetic fields on a closed Surface
Shuang-ren Zhao has defined the inner product for any two electromagnetic fields which are (Shuang-ren Zhao) ,

$$(\xi_{1},\xi_{2})_{\Gamma}=\oint_{\Gamma}(\mathbf{E}_{1}(\omega)\times\mathbf{H}_{2}^{*}(\omega)+\mathbf{E}_{2}^{*}(\omega)\times\mathbf{H}_{1}(\omega))\cdot\hat{n}d\Gamma $$

where $$\xi=[\boldsymbol{E},\boldsymbol{H}]$$, $$\tau=[\boldsymbol{J},\boldsymbol{K}]$$, Shuang-ren Zhao has proved that the above inner products, satisfy the Inner product space 3 definitions. If $$\tau_2$$ is taken as a unit vector of ether current $$\boldsymbol{J}_2$$ or $$\boldsymbol{K}_2$$, the field $$\xi_1$$ can be calculated ether on the original source $$\boldsymbol{J}_1$$ or on the surface $$\Gamma$$. $$\Gamma$$ is any surface outside the two volumes $$V_1$$ and $$V_2$$.

$$\hat{n} $$ is a unit surface normal vector. Shuang-ren Zhao has proved that this kind of inner product satisfy inner product space 3 conditions,


 * Conjugate symmetry:
 * $$( \xi_{1},\xi_{2}) =( \xi_{2},\xi_{1})^*$$


 * Linearity in the first argument:
 * $$\begin{align}

( a \xi_{1},\xi_{2}) &= a ( \xi_{1},\xi_{2}) \\ ( \xi_{1}+\xi_{2},\xi_{3}) &= ( \xi_{1},\xi_{3}) + ( \xi_{2},\xi_{3}) \end{align}$$


 * Positive-definiteness:
 * $$\begin{align}

( \xi,\xi) &\geq 0 \\ ( \xi,\xi) &= 0 \Leftrightarrow x = \mathbf{0} \,. \end{align}$$

According to this theory that the inner product of a retarded wave $$\xi_{1}$$ and an advanced wave $$\xi_{2}$$ vanish, if the sources of the two wave are inside the surface $$\Gamma$$, i.e.,

where $$\tau_1,\tau_2 \in V$$ are the source of $$\xi_1,\xi_2$$. $$\Gamma$$ is the boundary surface of the volume $$V$$

The inner product can also be defined in a completed surface or infinite surface $$\Gamma$$ which is between the volume $$V_1$$ and $$V_2$$. In this case the inner product of a retarded wave send from $$\tau_1$$ and the the advanced wave $$\xi_2$$ send from $$\tau_2$$ are not zero.

Where $$\Gamma1$$ and $$\Gamma2$$ are similar suface like $$\Gamma2$$

The mutual energy theorem
Shuang-ren Zhao has introduced the mutual energy theorem (Shuang-ren Zhao) in early of 1987.

$$ (\tau_{1},\xi_{2})_{V_{1}}=\int_{V1}(\boldsymbol{J}_{1}(\omega)\cdot\boldsymbol{E}_{2}^{*}(\omega)+\boldsymbol{K}_{1}(\omega)\cdot\boldsymbol{H}_{2}^{*}(\omega))dV$$

$$ (\xi_{1},\tau_{2})_{V_{2}}=\int_{V2}(\boldsymbol{E}_{1}(\omega)\cdot\boldsymbol{J}_{2}^{*}(\omega)+\boldsymbol{H}_{1}(\omega)\cdot\boldsymbol{K}_{2}^{*}(\omega))dV $$

The mutual energy theorem is similar to Rumasy's reciprocity theorem, but Shuang-ren Zhao thought this is a energy theorem instead some kind of reciprocity theorem. Since this theorem can be applied to a system with a transmitting antenna and receiving antenna. Shuang-ren Zhao believe this theorem tell us that the energy received by the receiving antenna is equal to the part of energy send from the transmitting antenna to the receiving antenna.

The time-domain cross-correlation reciprocity theorem
Adrianus T. de Hoop published the time-domain cross-correlation reciprocity theorem in the end of 1987 （Adrianus T. de Hoop） which can be seen as following,

Huygens–Fresnel principle
Shuang-ren Zhao emphases that the mutual energy theorem is an energy theorem instead of some kind of reciprocity theorem. The theorem described an energy in the space. This theorem can be seen as Huygens–Fresnel principle (Shuang-ren Zhao), which can be written as,

where

$$(\xi_{1},\xi_{2})_{\Gamma}=\oint_{\Gamma}(\mathbf{E}_{1}(\omega)\times\mathbf{H}_{2}^{*}(\omega)+\mathbf{E}_{2}^{*}(\omega)\times\mathbf{H}_{1}(\omega))\cdot\hat{n}d\Gamma $$

$$\Gamma$$ is any close surface or infinite big surface separating $$V_1$$ and $$V_2$$. We take the direction of $$\hat{n}$$ is from $$V_1$$ to $$V_2$$.

Assume $$\mathbf{J}_{2}=\delta(x-x')\mathbf{\hat{m}}$$

Assume $$\mathbf{K}_{2}=\delta(x-x')\mathbf{\hat{m}}$$

The forgot second Lorentz reciprocity theorem
I. V. Petrusenko introduced the forgot second Lorentz reciprocity theorem in 2009 （I. V. Petrusenko）. . These theorems have been rediscovered for many times later, this shows they are very important.

The relation of the these theorems
It is not difficult to prove that the mutual energy theorem （Shuang-ren Zhao） and the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop） are same theorem connected by Fourier transform, one is in the Fourier frequency-domain, another is in time-domain. The method of this mutual energy theorem is similar to （V.H. Rumsey） The reciprocity theory in arbitrary time-domain （W. J. Welch） is a special case where $$\tau=0$$ of the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop） The forgot second Lorentz reciprocity theorem （I. V. Petrusenko） is also same to （V.H. Rumsey） Hence, All the above theorems can be see as one theorem. The same mathematical formula has two major applications (1) is used as reciprocity for example to find the directivity diagram of the receiving antenna from the directivity diagram of the transmitting antenna, in this case this formula can be referred as a reciprocity theorem; (2) to find the energy transfer between the transmitting antenna and the receiving antenna, then the same formula can be referred as the mutual energy theorem.

If this theorem is applied as the reciprocity theorem, doesn't mater there is advanced wave, since in the reciprocity theorem, it is allowed among the two fields one is a real field another is a virtual filed. It will be no problem that a virtual field to be an advanced field. However when this theorem is applied as an energy theorem that need the two fields are all real in physics. The advanced field must be allowed in physics.

mutual energy flow
The mutual energy flow theorem is same as the Huygens–Fresnel principle (Shuang-ren Zhao) in mathematics. However Shuang-ren Zhao realized there is an energy flow transferred from the transmitting antenna to the receiving antenna. Hence give this formula a new name, mutual energy flow theorem,

The inner product of the mutual energy flow theorem can also be defined in the time domain, normally there is $$\tau=0$$. Hence there, is,

$$(\xi_{1},\xi_{2})_{\Gamma}=\int_{t=-\infty}^{\infty}\oint_{\Gamma}(\mathbf{E}_{1}(t)\times\mathbf{H}_{2}(t+\tau)+\mathbf{E}_{2}(t)\times\mathbf{H}_{1}(t+\tau))\cdot\hat{n}d\Gamma dt$$

Which is the energy go through the surface $$\Gamma$$.

$$ (\tau_{1},\xi_{2})_{V_{1}}=\int_{t=-\infty}^{\infty}\int_{V1}(\boldsymbol{J}_{1}(t)\cdot\boldsymbol{E}_{2}(t+\tau)+\boldsymbol{K}_{1}(t)\cdot\boldsymbol{H}_{2}(t+\tau))dVdt $$

$$ (\xi_{1},\tau_{2})_{V_{2}}=\int_{t=-\infty}^{\infty}\int_{V2}(\boldsymbol{E}_{1}(t)\cdot\boldsymbol{J}_{2}(t+\tau)+\boldsymbol{H}(t)\cdot\boldsymbol{K}_{2}(t+\tau))dVdt $$

can be defined as the mutual energy flow going through the surface $$\Gamma$$.

The wave expansion
Assume there are a complete set of the electromagnetic fields $$\xi_0$$,$$\xi_1,\xi_2,\xi_3,...$$. Assume these electromagnetic fields are normalized, i.e.,

$$(\xi_i,\xi_j)=\delta_{ij}$$

Then any electromagnetic field can be expanded as,

$$ \xi=\sum_{i=0}^{N}a_i \xi_i $$

where

$$ (\xi,\xi_j) = (\sum_{i=0}^{N}a_i \xi_i,\xi_j)=a_j $$

Hence, we have

$$ \xi=\sum_{i=0}^{N}(\xi,\xi_i) \xi_i $$

Spherical waves expansion
Shuang-ren Zhao proposed the spherical wave expansion method . The spherical wave expansion can be written as, $$ \xi_{nm}=[M_{nm},\frac{j}{\eta} N_{nm}] $$ $$ \eta_{nm}=[N_{nm},\frac{j}{\eta} M_{nm}] $$

Where the factor, $$ \frac{j}{\eta} $$ make the second term is just a magnetic field, when the first term in the square brackets of the above formula is an electric field. $$ j=\sqrt{-1} $$. where

$$ M_{nm}= \nabla \times [g_{n}rh_n(kr)Y_n^m(\theta,\phi)] $$

$$ N_{nm}=\frac{1}{k} \nabla \times M_{nm} $$

$$ h_n $$ 是$$n$$ 阶第二类Hankel函数， $$ k=\omega \sqrt{\epsilon \mu} $$

$$ \eta=\sqrt{\frac{\mu}{\epsilon} } $$

$$ r,\theta,\phi $$ are coordinates of sphere.The corresponding unit vector are $$ r,\theta,\phi $$

$$ Y_n^m(\theta,\phi)=[(2n+1)\frac{(n-m)!}{(n+m)!}]^{\frac{1}{2}}P_n^m(cos\theta)\exp(jm\phi) $$

$$ P_n^m $$ is Legendre function， $$ n=0,1,... $$ $$ m=0,\plusmn 1,......\plusmn n $$

The formula of the spherical wave expansion can be written as,

$$ \xi=\sum_{nm}(a_{nm}\xi_{nm}+b_{nm}\eta_{nm}) $$

$$ a_{nm}=(\xi,\xi_{nm}),\,\,\,\,b_{nm}=(\xi,\eta_{nm}) $$

The mutual energy theorem is also a reciprocity theorem
The same theorem Zhao calls it the mutual energy theorem, but Welch and de Hoop does call it the time domain reciprocity theorem. In fact, this formula can be used as a reciprocity theorem. That is, the conjugate transformation is performed on all fields in the mutual energy theorem.

$$ -(\mathbb{C}\tau_{1},\mathbb{C}\xi_{2})=(\mathbb{C}\xi_{1},\mathbb{C}\xi_{2})=(\mathbb{C}\tau_{1},\mathbb{C}\xi_{2}) $$

得，

$$ (\xi_{2},\tau_{1})=(\xi_{1},\xi_{2})=-(\xi_{2},\tau_{1}) $$

Hence, in the original mutual energy theorem $$\tau_1$$ is emitter, $$\xi_1$$ is the retarded wave, $$\tau_2$$ is the absorber, $$\xi_2$$ is advanced wave. After the transform, $$\tau_2$$ becomes the emitter, $$\xi_2$$ becomes the retarded wave, $$\tau_1$$ becomes the absorber, $$\xi_1$$ becomes the advanced wave.

According to this reciprocity theorem, the directivity diagram of the receiving antenna is the same as that of the directivity diagram of the transmitting antenna. The pattern of the same electric charge where it is absorbed is the same as the pattern when it is radiated.

The calculation of the radiation and the absorption of the antenna
Assume the current element of the antenna 1 is $$\mathbf{J}_1$$, Which produce the retarded wave, $$[\mathbf{E}_1,\mathbf{H}_1]$$，

Assume the current element of the antenna 2 is $$ \mathbf{J}_2 $$ produce the advanced wave, $$[\mathbf{E}_2,\mathbf{H}_2]$$，The mutual energy theorem,

$$ -\int_{V2}\mathbf{E}^*_2 \cdot \mathbf{J}_1 dV dt = \int_{V1} \mathbf{E}_1 \cdot  \mathbf{J}^*_2  dV dt$$

tell us, the advanced wave of the receiving antenna $$ \mathbf{E}_2 $$ sucks the power form $$ \mathbf{J}_1 $$. And this power is same as the retarded wave $$ \mathbf{E}_1 $$ offers the power to the receiving antenna $$ \mathbf{J}_2 $$

We know that the directivity of the transmitter antenna is easier to be calculated but is more difficult to be measured. In contrast, the directivity diagram of the receiving antenna is not easier to be calculated, but is easier to be measured. Apply this theorem, we can apply the calculated directivity diagram of the radiation as the directivity diagram of the receiving antenna. And also can apply the measured directivity diagram of the receiving antenna as the directivity diagram of the radiation.

接收天线方向图的计算的争议
关于接收天线的方向图的计算， 应用互能定理同应用互易定理在求接收天线的方向图这点上是一致的. 但是如果用两者计算接收天线上的电流就不同了. 两者关键不同是在对接收天线的认识上. 应用互能定理计算，必须承认接收天线辐射超前波. 用互易定理计算其实认为接收天线辐射滞后波. 究竟接收天线辐射超前波还是滞后波这个问题是有争议的. 承认超前波的学者可以用互能定理计算接收天线的方向图. 不承认超前波的学者使用洛伦兹互易定理计算接收天线的方向图. 尽管在计算方向图上两者同样有效，但从物理上讲，两者中只有一个是真正的物理定理. 另一个这个物理定理的一个变换，可以看成数学定理. 互能定理的提出者赵双任认为互能定理是一个能量定理，当然也是一个物理定理. 互易定理仅仅是互能定理的共轭变换，不是一个物理定理.

mutual energy principle
=Time-reversal waves=

We know that Maxwell equations has two kind solution, retarded solution and advanced solution. Corresponding to the retarded solution there is the retarded wave. Corresponding to the advanced solution there is advanced wave. Advanced wave do not satisfy our traditional causality. Many people think the advanced wave are not real wave. But there are a few very famous physics believe the advanced wave exist. Assume the retarded wave and advanced wave all exist. The question is that is the advanced wave the time-reversal wave of the retarded wave? The answer is negative. Maxwell equations are not time-reversible. We can define the time-reversal waves for the retarded wave and define the time-reversal wave for the advanced wave. The question is the time-reversal waves exist or not?

Conjugate transform
It is not clear who first introduced the concept of the conjugate transform, but the details theory of the conjugate transform can be found in （Jin Au Kong）. The conjugate transform can be seen in following,

$$ \mathbf{\mathbb{C}}(\mathbf{E}(t),\mathbf{H}(t),\mathbf{J}(t),\mathbf{K}(t),\mathbf{\boldsymbol{\epsilon}}(t),\mathbf{\boldsymbol{\mu}}(t))=(\mathbf{E}(-t),-\mathbf{H}(-t),-\mathbf{J}(-t),\mathbf{K}(-t),\mathbf{\boldsymbol{\epsilon}}(-t),\boldsymbol{\mathbf{\mu}}(-t)) $$

or

$$ \mathbf{\mathbb{C}}(\mathbf{E}(\omega),\mathbf{H}(\omega),\mathbf{J}(\omega),\mathbf{K}(\omega),\mathbf{\boldsymbol{\epsilon}}(\omega),\mathbf{\boldsymbol{\mu}}(\omega))=(\mathbf{E}^{*}(\omega),-\mathbf{H^{*}}(\omega),-\mathbf{J^{*}}(\omega),\mathbf{K}^{*}(\omega),\mathbf{\boldsymbol{\epsilon^{*}}}(\omega),\boldsymbol{\mathbf{\mu^{*}}}(\omega)) $$

Where $$\mathbf{\mathbb{C}}$$ is the conjugate transform. $$\mathbf{E}$$ is electric field. $$\mathbf{H}$$ Magnetic field. $$\mathbf{J}$$ current intensity. $$\mathbf{K}$$ magnetic current intensity. $$\mathbf{\boldsymbol{\epsilon}}$$ is permittivity, $$\mathbf{\boldsymbol{\mu}}$$ is permeability, $$t$$ is time, $$\omega$$ is frequency.

It is important that if a field satisfies the Maxwell equations, after the conjugate transform, it still satisfies the Maxwell equations. If the original field is retarded wave, after the transform it becomes advanced wave. Vice Versa, if the original field is advanced wave, after the transform it becomes the retarded wave.

Time reversal transform
Time reversal transform is different with conjugate transform. It can be defined as following $$ \mathbf{\mathbb{R}}(\mathbf{E}(t),\mathbf{H}(t),\mathbf{J}(t),\mathbf{K}(t),\mathbf{\boldsymbol{\epsilon}}(t),\mathbf{\boldsymbol{\mu}}(t))=(\mathbf{e}(-t),\mathbf{h}(-t),-\mathbf{j}(-t),-\mathbf{k}(-t),\mathbf{\boldsymbol{\epsilon}}(-t),\boldsymbol{\mathbf{\mu}}(-t),) $$

or

$$ \mathbf{\mathbb{R}}(\mathbf{E}(\omega),\mathbf{H}(\omega),\mathbf{J}(\omega),\mathbf{K}(\omega),\mathbf{\boldsymbol{\epsilon}}(\omega),\mathbf{\boldsymbol{\mu}}(\omega))=(\mathbf{e}^{*}(\omega),\mathbf{h^{*}}(\omega),-\mathbf{j^{*}}(\omega),-\mathbf{k}^{*}(\omega),\mathbf{\boldsymbol{\epsilon^{*}}}(\omega),\boldsymbol{\mathbf{\mu^{*}}}(\omega)) $$

Where $$\mathbf{\mathbb{R}}$$ is time reversal transform. $$[\mathbf{e},\mathbf{h}]$$ is the time-reversal electromagnetic field. $$[\mathbf{j},\mathbf{k}]$$ is time-reversal electric current intensity and time-reversal magnetic current intensity. After the time-reversal transform $$\mathbf{\mathbb{R}}$$, a normal electromagnetic field become time-reversal magnetic fields. The time-reversal electromagnetic fields do not satisfy Maxwell equations, but they satisfy time-reversal Maxwell equations.

Assume there is a retarded wave, the time-reversal wave corresponding to this retarded wave is not the advanced wave. The advanced wave is from current time go to the the past time. The time reversal wave is from a future time go to the the current time. Advanced wave still satisfy Maxwell equations, it is a normal electric field. Time-reversal wave does not satisfy Maxwell equations, it is not a normal electromagnetic field.

Time reversal Maxwell equations

 * {| class="wikitable" style="text-align: center;"

! scope="col" style="width: 15em;" | Name ! scope="col" | Integral equations ! scope="col" | Differential equations ! scope="col" style="width: 21em;" | Meaning


 * time-reversal Gauss's law
 * $$\nabla \cdot \mathbf{e} = \frac {\rho} {\varepsilon_0}$$
 * The electric flux through a closed surface is proportional to the charge inside an enclosed volume.
 * time-reversal Gauss's law for magnetism
 * $$\nabla \cdot \mathbf{b} = 0$$
 * The magnetic flux through a closed surface is zero (i.e. there are no magnetic monopoles)
 * time-reversal Maxwell–Faraday equation (Faraday's law of induction)
 * $$\oint_{\partial \Sigma} \mathbf{e} \cdot \mathrm{d}\boldsymbol{l} =  \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{b} \cdot \mathrm{d}\mathbf{S} $$
 * $$\nabla \times \mathbf{e} = \frac{\partial \mathbf{b}} {\partial t}$$
 * The work per unit charge required to move a charge around a closed loop equals the rate of decrease of the magnetic flux through an enclosed surface.
 * time-reversal Ampère's circuital law (with Maxwell's addition)
 * $$\oint_{\partial \Sigma} \mathbf{b} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 \left(\iint_{\Sigma} -\mathbf{j} \cdot \mathrm{d}\mathbf{S} - \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{e} \cdot \mathrm{d}\mathbf{S} \right)$$
 * $$\nabla \times \mathbf{b} = \mu_0\left(-\mathbf{j} - \varepsilon_0 \frac{\partial \mathbf{e}} {\partial t} \right) $$
 * The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through an enclosed surface.
 * }
 * time-reversal Ampère's circuital law (with Maxwell's addition)
 * $$\oint_{\partial \Sigma} \mathbf{b} \cdot \mathrm{d}\boldsymbol{l} = \mu_0 \left(\iint_{\Sigma} -\mathbf{j} \cdot \mathrm{d}\mathbf{S} - \varepsilon_0 \frac{\mathrm{d}}{\mathrm{d}t} \iint_{\Sigma} \mathbf{e} \cdot \mathrm{d}\mathbf{S} \right)$$
 * $$\nabla \times \mathbf{b} = \mu_0\left(-\mathbf{j} - \varepsilon_0 \frac{\partial \mathbf{e}} {\partial t} \right) $$
 * The magnetic field induced around a closed loop is proportional to the electric current plus displacement current (proportional to the rate of change of electric flux) through an enclosed surface.
 * }
 * }

Time reversal Waves
The waves satisfy time-reversal Maxwell equations are time reversal waves. There are two kind of time-reversal waves. One is the time-reversal wave corresponding to the retarded wave, another is the time-reversal wave corresponding to the advanced wave.

Here, the time-reversal wave is a possible physical wave exist in the nature. It satisfy time-reversal Maxwell equations.

It is worth to say that there is also a kind technology is referred as time-reversal focus method, which is applied normal electromagnetic field to simulate some property of time-reversal waves,  please see Time reversal signal processing for details. Here the time-reversal wave doesn't mean that kind of technology.

Background of time-reversal wave
Here some thing related the time-reversal wave will be discussed. Perhaps the time-reversal wave alberta doesn't involve.

Action at a distance
The theory of action-at-a-distance are introduced by （K. Schwarzschild） （H. Tetrode） （A.D. Fokker）. According to this theory, a electric current will produce two electromagnetic potentials or two electromagnetic waves: one is the retarded wave, another is advanced wave. The emitter can send the retarded wave, but in the same time it also sends an advanced wave. The absorber can send the advanced wave, but in the same time it also sends a retarded wave. According to this theory, the sun cannot send the radiation wave out, if it stayed alone in the empty space. Infinite absorbers are the reason that the sun can radiate its light. The action can be written as following,

where

$$s_{ij}^{2}=(x_{i\mu}-x_{j\mu})(x_{i}^{\mu}-x_{j}^{\mu})$$

$$ds=c^{2}dt^{2}-dx_1^{2}-dx_2^{2}-dx_3^{2}$$

Absorber theory
The absorber theory is introduced by Wheeler and Feynman （J. A. Wheeler） （J. A. Wheeler）. The absorber theory is build on the top of the above theory of the action-at-a-distance （A.D. Fokker） （K. Schwarzschild） （H. Tetrode） . A cording to this theory, electromagnetic field has no its own freedom. The electromagnetic field is adjective field. It is only a bookkeeper for the action or reaction between at least two charges. That means without a test charge or absorber, only the emitter alone can not produce the radiation. Absorber theory try to offer a better explanation to the recoil force of a accelerated or decelerated charge in empty space. The recoil force has been introduced by Dirac （P. A. M. Dirac）. But Wheeler and Feynman do not satisfy that Dirac did not offer a reasonable reason of that formula. Wheeler and Feynman try to use the absorbers stayed on the infinite big sphere to explain the formula given by Dirac. The absorber theory also emphases the importance of the absorber in the radiation process.

Transactional interpretation for quantum mechanics
The transactional interpretation of quantum mechanics introduced by John Cramer (John Cramer). The transactional interpretation is build on the top of Wheeler–Feynman absorber theory. In this theory, the emitter can send a offer wave to the absorber, when the absorber receive the offering wave, it can send a confirmation wave to the emitter. These two waves can have a handshake. This handshake process is the transactional process. In this process the photon or other particle is produced. The confirmation wave is advanced wave.

Welch's reciprocity theorem
This theory is introduced by W. J. Welch proposed in 1960 （W. J. Welch）. The theorem can be seen in the following,

In order to prove the above formula, it is required to prove a surface integral vanishes. The surface is on the infinite big sphere. The proof of the vanish of the surface integral on infinite big sphere need the two waves one is retarded wave and another is advanced wave.

Rumsey's reciprocity theorem
V.H. Rumsey has introduced his summarize the Lorentz reciprocity theorem as "action and reaction". He has apply the complex conjugate transform to the his "action and reaction" theorem and obtained a new reciprocity theorem （V.H. Rumsey）,

Inner product space for the electromagnetic fields on a closed Surface
Shuang-ren Zhao has defined the inner product for any two electromagnetic fields which are (Shuang-ren Zhao) ,

$$(\xi_{1},\xi_{2})_{\Gamma}=\oint_{\Gamma}(\mathbf{E}_{1}(\omega)\times\mathbf{H}_{2}^{*}(\omega)+\mathbf{E}_{2}^{*}(\omega)\times\mathbf{H}_{1}(\omega))\cdot\hat{n}d\Gamma $$

where $$\xi=[\boldsymbol{E},\boldsymbol{H}]$$, $$\tau=[\boldsymbol{J},\boldsymbol{K}]$$, Shuang-ren Zhao has proved that the above inner products, satisfy the Inner product space 3 definitions. If $$\tau_2$$ is taken as a unit vector of ether current $$\boldsymbol{J}_2$$ or $$\boldsymbol{K}_2$$, the field $$\tau_1=\boldsymbol{J}_1,\boldsymbol{K}_1$$. $$\xi_1$$ can be calculated ether on the original source $$\boldsymbol{J}_1$$ or on the surface $$\Gamma$$. $$\Gamma$$ is any surface outside the two volumes $$V_1$$ and $$V_2$$.

$$\hat{n} $$ is a unit surface normal vector. Shuang-ren Zhao has proved that this kind of inner product satisfy inner product space 3 conditions.

Zhao's mutual energy theorem
Shuang-ren Zhao has introduced the mutual energy theorem (Shuang-ren Zhao) in early of 1987. Shuang-ren Zhao emphases that the mutual energy theorem is an energy theorem instead of some kind of reciprocity theorem. The theorem described an energy in the space.

where

$$\Gamma$$ is any close surface or infinite big surface separating $$V_1$$ and $$V_2$$. We take the direction of $$\hat{n}$$ is from $$V_1$$ to $$V_2$$.

$$ (\tau_{1},\xi_{2})_{V_{1}}=\int_{V1}(\boldsymbol{J}_{1}(\omega)\cdot\boldsymbol{E}_{2}^{*}(\omega)+\boldsymbol{K}_{1}(\omega)\cdot\boldsymbol{H}_{2}^{*}(\omega))dV$$

$$ (\xi_{1},\tau_{2})_{V_{2}}=\int_{V2}(\boldsymbol{E}_{1}(\omega)\cdot\boldsymbol{J}_{2}^{*}(\omega)+\boldsymbol{H}_{1}(\omega)\cdot\boldsymbol{K}_{2}^{*}(\omega))dV $$

de Hoop's reciprocity theorem
Adrianus T. de Hoop published the time-domain cross-correlation reciprocity theorem in the end of 1987 （Adrianus T. de Hoop） which can be seen as following,

Zhao's mutual energy flow theorem
This theorem can be seen as Huygens–Fresnel principle (Shuang-ren Zhao), which can be written as,

Huygens–Fresnel principle tell us that the field not only can be calculated from the current it is also can be calculated on the surface $$\Gamma$$.

Recently Shuang-ren Zhao define $$(\xi_{1},\xi_{2})_{\Gamma}$$ as mutual energy flow. Hence the above theorem can be referred as the mutual energy flow theorem （Shuang-ren Zhao） .

The mutual energy flow theorem tell us the mutual energy flow doesn't attenuate like the waves. The wave attenuate when it travels. For the mutual energy flow, the energy go through any surface between the emitter and the absorber are all equal. The mutual energy flow is a energy flow starts from emitter ends at the absorber. It transfer the energy from a point to another point. These characters are just the photon need. Hence, Shuang-ren zhao believe that the photon actually is the mutual energy flow. The mutual energy flow is thin in the two ends and think in the middle between the two ends and hence, can easily explain the double-slice experiment. The mutual energy flow is consist of a retarded wave and a advanced wave this can also easily explain the delayed choice experiment and delayed quantum erase experiment.

The forgot second Lorentz reciprocity theorem
I. V. Petrusenko introduced the forgot second Lorentz reciprocity theorem in 2009 （I. V. Petrusenko）.

It is similar to the Rumsey's reciprocity theorem or Zhao's mutual energy theorem.

Poynting theorem of the time-reversal wave
Apply time-reversal transform to the Poynting theorem,

where


 * $$U = \frac{1}{2}\left(\epsilon_0 \mathbf{E}\cdot\mathbf{E} + \frac{1}{\mu_0}\mathbf{B}\cdot\mathbf{B}\right)$$

we can obtained the Poynting theorem for time-reversal wave, which is in the following,


 * $$u = \frac{1}{2}\left(\epsilon_0 \mathbf{e}\cdot\mathbf{e} + \frac{1}{\mu_0}\mathbf{b}\cdot\mathbf{b}\right)$$

Time-reversal wave can balance out or cancel the normal wave
Here the normal wave are the retarded wave and the advanced wave which satisfy Maxwell equations. The normal wave for example the retarded wave can be canceled or balanced out by the time-reversal wave corresponding to retarded wave. Similarly, the advanced wave can be canceled or balanced out by the time-reversal wave corresponding to the advanced wave.

If the time-reversal wave exist it can balance out the normal electromagnetic fields which means it is possible that,

Hence if time-reversal wave exist, and if they exist together with the normal waves. That will make the normal wave not carry energy.

In quantum physics, the waves are probability wave. A probability wave cannot carry energy. In order to make the wave not carry energy the time-reversal wave is required.

The self-energy principle
The self-energy principle which says that for any radiation wave, retarded wave or advanced wave, there exist the corresponding time-reversal wave cancel it or balanced out it. The retarded wave is canceled by the time-reversal wave corresponding to the retarded wave. The advanced wave is cancel or balanced by the time-reversal wave corresponding to the advanced wave. Hence, the radiation waves do not carry and do not transfer any energy.

Waves do not carry energy sound strange, However, waves in quantum physics are probability wave which is pure mathematic wave. A mathematic wave should not carry energy. Energy should be carried by particles for example photons, but not carried by waves. In the following section the mutual energy principle will solve the energy transfer problem.

It should be notice that for wave there are difference between macroscopic wave and microscopic wave. Here, it is assumed that in the microscopic situation, the waved do not carry energy, which do not deny it is still possible that in the macroscopic situation, wave still can carries the energy. But the the macroscopic wave consists of infinite particles for example photons. Hence energy is still carried by particles instead of microscopic wave.

The mutual energy principle of $$N$$ charges
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 :

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:

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

The mutual energy principle is used as axioms to replace the Maxwell equations. The reasons Maxwell equations should be replaced are,

(1) With Maxwell equations, the wave-particle duality paradox still cannot solved.

(2) With Maxwell equations, we do not clear what is the relationship between the advanced wave and the retarded wave.

(3) Maxwell equations together with superposition principle conflict with energy conservation law this will be show in the flow section.

What is the reason the mutual energy formula can be applied as axiom,

(1) From Maxwell equations, the action-at-a-distance principle cannot be derived. Maxwell equations is not equivalent with the action-at-a-distance principle. The theory of action-at-a-distance are introduced by （K. Schwarzschild） （H. Tetrode） （A.D. Fokker）. But the mutual energy formula can be seen as the equivalent formula of the principle action at a distance.

(2) From this mutual energy formula the mutual energy theorem, mutual energy flow theorem  can be derived directly without any additional prerequisite.

Deriving the mutual energy principle by the conflict
Here we not really derive or prove the mutual energy principle. It is referred as principle or axiom that should not be derived. Here we just offers the reason why the self energy principle and the mutual energy principle should be suggested.

Add the energy conservation to the top of Maxwell electromagnetic field theory
the conflict among Maxwell equation, superposition principle and energy conservation condition is a reason that the self-energy principle and the mutual energy principle was proposed. The total energy of the interaction of the $$N$$ charge is,

$$w_{ij}=\mathbf{J}_i \cdot E_j $$, there is no the items $$w_{ii}=\mathbf{J}_i \cdot E_i $$. In the above equation each charge is subjected to the force of other $$N-1$$ charges. It is assumed that the charge can not offer itself any force.

Assume the wave of each charge satisfies the Maxwell equations. From Maxwell equation the Poynting theorem can be derived,

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


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


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


 * $$ \mathbf{D}=\epsilon_0 \mathbf{E} $$,
 * $$ \mathbf{B}=\mu_0 \mathbf{H} $$

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.

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

The thought of proof
Substitute Eq.(3) to Eq.(2), the Poynting theorem for $$N$$ charges can be obtained as follows,

If we can prove

we have,

This formula is referred as the mutual energy formula. It is very important and later will be applied as principle. If we can have,

and

we can obtained the energy conservation law,

This wave we can prove the energy conservation law from Poynting theorem (or Maxwell equations) and superposition principle.

There is conflict
Eq.(7) and Eq.(8) can be proved in the following section. How ever Eq.(5) can not be proved. Let us just accept Eq.(5) or accept the following,

This is referred as self-energy conditions. It is not correct fully and will be adjust latter. But if without it, we have conflict.

If the energy conservation of $$N$$ charges Eq.(1) is correct, then the power term of the $$N$$ charge in the Poynting theorem, without Eq.(10) is $$\sum^{N}_{i=1} \sum^{N}_{j=1}w_{ij}$$ which is too high to be estimated! If this power item is overestimated, in fact, means Eq.(10) should be correct some how. And hence, we obtained Eq.(6). In Eq.(6) the total power is $$\sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i}w_{ij}$$ that is correct. Eq.(6) is referred as the mutual energy formula, since the above conflict (the Maxwell equation plus superposition principle conflict with energy conservation). Shuang-ren Zhao would like to apply Eq.(6) as axioms to replace the Maxwell equation and superposition principle. Form Eq.(6) the energy conservation law Eq.(3) can be proved (see following section) that means Eq.(6) doesn't conflict with energy conservation law. Eq.(6) is referred as Mutual energy principle.

The Maxwell equations can be derived from the mutual energy principle
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 $$

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 the 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 physical 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.

Second time conflict
When we obtained the Maxwell equations, even there is some difference, the Maxwell equations mast exist as a pair, if Maxwell equation is established. the Poynting theorem for single charge should also be established,

This conflict to the self-condition Eq.(10). In order to solve this conflict, we have to introduce the self-energy principle. This means the wave of single charge, the retarded wave or the advanced wave all send out, but there is the time-reversal wave which can balance out all the energy of the retarded wave and the advanced wave, i.e.,

This way, the second conflict will be solved. Eq.(12-14) can be seen as the formula of the self-energy principle. This principle tell that the energy flow of the waves are canceled and hence they together do not carry any energy.

The proof of Eq.(8)
Eq.(8) can be rewritten as,


 * $$ \sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i} \int_{t=-\infty}^{\infty} \int_V \frac{\partial u_{ij}}{\partial t} dV dt=0 $$

We only need to prove,


 * $$ \int_{t=-\infty}^{\infty}\frac{\partial u_{ij}}{\partial t}dt=0 $$

or


 * $$ [u_{ij}]_{-\infty}^\infty=u_{ij}(\infty)-u_{ij}(-\infty)=0 $$

We can assume, that $$u_{ij}(-\infty)=u_{ij}(+\infty) = constant $$ and hence the formula is proved. Proof finish.

The proof of Eq.(7)
E.(7) can be rewritten as,
 * $$ -\int_{t=-\infty}^{\infty} \sum^{N}_{i=1} \sum^{N}_{j=1,j < i} \int_V (\nabla\cdot\mathbf{S}_{ij}+\nabla\cdot\mathbf{S}_{ji})dV dt=0$$

Assume $$N=2$$, we will first prove it in the situation where $$N=2$$ and the widen it the more general situation, i.e.,


 * $$ -\int_{t=-\infty}^{\infty} \sum^{2}_{i=1} \sum^{2}_{j=1,j < i} \int_V (\nabla\cdot\mathbf{S}_{ij}+\nabla\cdot\mathbf{S}_{ji}) dV dt=0$$

we only need to prove,


 * $$ \int_v (\nabla\cdot\mathbf{S}_{12}+\nabla\cdot\mathbf{S}_{21}) dV=0$$

or


 * $$ \int_V \nabla\cdot(\mathbf{E}_1 \times \mathbf{H}_2+\mathbf{E}_2 \times \mathbf{H}_1) dV=0$$

or

Assume $$\Gamma$$ is a infinite big sphere. The retarded field $$(E_1,H_1)$$ and $$(E_2,H_2)$$ one is retarded Wave, one is advanced wave, if in time $$t=0$$, the emitter charge send a retarded wave, and in some time later for example t=T the emitter send the advanced wave. The two waves reach the infinite $$\Gamma$$, one in the future, another is in the past. Hence do not  nonzero in the same time. Hence this integral vanishes. This result is also true to the case where $$N$$ is arbitrary. The proof finished.

The proof of the mutual energy theorem
We have proved Eq.(7) and Eq.(8), afterwards we obtained Eq.(9) which is the extended mutual energy theorem for $$N$$ charges. Eq.(9) can be rewritten as,

Assume $$N=2$$


 * $$\int_{t=-\infty}^{\infty}\sum^{2}_{i=1} \sum^{j < i}_{j=1}\int_V (w_{ij}+w_{ji}) dv dt=0\,\,\,\,\,\,\,\,\,\,\,(17)$$

or


 * $$\int_{t=-\infty}^{\infty} \int_V (w_{12}+w_{21}) dV dt=0$$

or


 * $$-\int_{t=-\infty}^{\infty} \int_{V1} w_{12} dV dt=\int_{t=-\infty}^{\infty}\int_{V2} w_{21} dV dt$$

or

This is the mutual energy of two charges. Considering there is the conditions for example $$\mathbf{J}_1$$ and $$\mathbf{J}_2$$ must inside of volume $$V$$, and there is no any current at outside of the surface, Eq.(15) is established. If the above condition does not satisfy. In general we still have,

This is also referred as the Mutual energy theorem.

The Mutual energy flow theorem
The mutual energy flow theorem originally is used as Huygens–Fresnel principle (Shuang-ren Zhao). Recently Shuang-ren Zhao realized that the inner product of two fields $$(\xi_1,\xi_2)_\Gamma$$ is a energy flow. Hence the following formula,

can be seen as the mutual energy flow theorem. Where

$$ Q=(\xi_1,\xi_2)_\Gamma=\int_{t=-\infty}^{\infty} \iint\limits_{\Gamma} (\mathbf{E}_1 \times \mathbf{H}_2 + \mathbf{E}_2 \times \mathbf{H}_1)\cdot \mathbf{n} d\Gamma dt $$

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

We can define,

$$ q= \iint\limits_{\Gamma} (\mathbf{E}_1 \times \mathbf{H}_2 + \mathbf{E}_2 \times \mathbf{H}_1)\cdot \mathbf{n} d\Gamma  $$

For each other. 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 $$\Gamma$$.

Let $$\Gamma_1$$ be a surface surrounds $$V1$$. $$Q1$$ is the energy flowing through the surface $$\Gamma_1$$. $$\Gamma_2$$ is a surface that surrounds $$V2$$. $$Q2$$ is the energy flowing through the surface $$\Gamma_2$$. 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.

The proof of the mutual energy flow theorem
Started from Eq.(19), assume that the volume $$V$$ is just take on the $$V1$$, we have,

similar we can choose the volume $$V$$ as $$V2$$

In the above Eq.(22) the surface normal $$\mathbf{n}$$ is direct to the outside. Adjust it to direct inside, we have,

Now the normal surface vector $$\mathbf{n}$$ in Eq.(21) and Eq.(23) are all direct from $$V1$$ to $$V2$$. Substitute Eq.(21) and Eq.(23) to the mutual energy theorem Eq.(18), we obtain Eq(20). Proof finish.

Macroscopic wave can be consist of infinite photons which are the mutual energy flows
=dddddddddddd =

The relation of the these theorems
It is not difficult to prove that the mutual energy theorem （Shuang-ren Zhao） and the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop） are same theorem connected by Fourier transform, one is in the Fourier frequency-domain, another is in time-domain. The method of this mutual energy theorem is similar to （V.H. Rumsey） The reciprocity theory in arbitrary time-domain （W. J. Welch） is a special case where $$\tau=0$$ of the time-domain cross-correlation reciprocity theorem（Adrianus T. de Hoop）. The forgot second Lorentz reciprocity theorem （I. V. Petrusenko） is also same to （V.H. Rumsey）. Hence, All the above theorem can be see as one theorem.

The Lorentz reciprocity theorem
In the Lorentz reciprocity theorem,

$$ \int_{V_{2}}\boldsymbol{E}_{2}(\omega)\cdot\boldsymbol{J}_{1}(\omega)dVdt=\int_{V_{2}}\boldsymbol{E}_{1}(\omega)\cdot\boldsymbol{J}_{2}(\omega)dV $$

all fields are retarded potential. However conjugate transform can be applied to one of the two fields inside the reciprocity theorem. In this case the two fields one become retarded field another become advanced field. Hence, Lorentz reciprocity theorem together with conjugate transform is equal to the mutual energy theorem Shuang-ren Zhao） or time-domain cross-correlation reciprocity theorem （Adrianus T. de Hoop）.

The difference between the Lorentz reciprocity theorem and the mutual energy theorem/time-domain cross-correlation reciprocity theorem
In Lorentz reciprocity theorem the two fields are all retarded field. In the mutual energy theorem or cross-correlation time-domain reciprocity theorem, the two fields are one is retarded wave send from the transmitting antenna, another is the advanced wave send from the receiving antenna.

The mutual energy theorem or time-domain cross-correlation reciprocity theorem are energy theorem. It describe the energy relation of the two antenna. The Lorentz reciprocity theorem reciprocity theorem is a mathematic theorem which is conjugate transform of the mutual energy theorem/time-domain cross-correlation reciprocity theorem.

Other applications
This theorem can be applied to calculate the directivity diagram receiving antenna similar to the Lorentz reciprocity theorem. The inner product of electromagnetic fields are defined（Shuang-ren Zhao） （Shuang-ren Zhao） can be applied to waves expansions, for example with spherical waves （Shuang-ren Zhao） and plane waves （Shuang-ren Zhao）.

Useful
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:

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:

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,


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


 * $$ w_{ij} = \mathbf{E}_i \cdot  \mathbf{J}_j  $$

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,


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

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

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

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, $$\sum^{N}_{i=1} \sum^{N}_{j=1}w_{ij}$$ 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,

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,

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

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 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 as following,

In the above


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

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.

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,

$$  \int_{t=-\infty}^{\infty} \int_V (\mathbf{E}_1 \cdot \frac{\partial \mathbf{D}_2}{\partial t} +\frac{\partial \mathbf{D}_1 }{\partial t} \cdot \mathbf{E}_2 +\mathbf{H}_1 \cdot \frac{\partial \mathbf{B}_2}{\partial t} +\frac{\partial \mathbf{B}_1 }{\partial t} \cdot \mathbf{H}_2) dV dt=0$$

From this the mutal energy theorem can be obtained,

If the two electromagnetic fields $$\mathbf{E}_1$$, $$\mathbf{H}_1$$ and $$\mathbf{E}_2$$ One of the advanced waves, one is the retarded 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 nonzere at the same time a the large Sphere. So there are,

and hence,

$$ \int_{t=-\infty}^{\infty}\int_V (\mathbf{E}_1 \cdot \mathbf{J}_2 +\mathbf{E}_2  \cdot \mathbf{J}_1) dV dt = 0 $$

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,

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 it 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 a energy send 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 meaning of the mutual energy theorem is only same in the formula. There is big difference in the meaning of the theorems. The new mutual energy theorem tell us that all energy send from the emitter are received by the absorber. There is no any energy send to the whole space. Corresponding to the retarded wave, the energy send 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 the mutual energy flow. The mutual energy flow is responsible for the energy transfer from the emitter to the absorber.

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.

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.

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 $$\mathbf{E}_{i,j}$$

$$\mathbf{E}_{i,j}=\frac{\mathbf{F}_{i,j}}{q_j},\,\,\,\,\,\,\,\,\,\, \mathbf{F}_{i,j}=k q_i q_j \frac{\mathbf{r}_{i,j}}{r_{i,j}^3} $$

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.

$$\mathbf{E}_j=\sum_{i=1,i \ne j}^{N} E_{i,j}$$

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 $$\sum_{i=1,i \ne j}^{N}$$ 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.

Photon is the mutual energy flow
We know that photon is an energy pack. But because it is called the particle, it is natural to imagine that the photon should have a nucleus like an atom, and that it 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 from the principle of mutual energy tells us that photons are reciprocal flow. There is nothing like a photon's nuclear thing. Before we say that the photon is a wave we imagine it is a lagging wave, is a wave that diffuses into the whole space. The principle of reciprocity tells us that photons are neither lagged waves nor advanced waves. Photon is composed of hysteresis and super-wave common energy flow. The reciprocal flow is not a diffuse wave, and the energy flow is similar to the plane wave in the waveguide. Mutual flow supports point-to-point energy propagation. Here we still say that light 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 that the kind of diffuse to the entire space of the retarded wave, it looks like a  quasi-plane wave in a waveguide which is rough 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 radiator and the absorber, the waveguide can become very thick. If the partition is placed between the emitter absorbers and the double slits are engraved on the baffle, the mutual energy flow will interfere. So like a wave. So the mutual energy flow can explain the wave-like duality of 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 that is between $$V1$$ and $$V2$$. This amount is actually the energy of the photon. Because the $$A1$$ is small at $$A1$$ and $$A2$$, the energy is concentrated at a point so that it can be seen as a particle. In the middle of $$V1$$ and $$V2$$, the interdiffit stream becomes very thick, so it is like a wave. This is reason why the light has the wave-particle duality.

The experiment of the double slits
The mutual energy flow is very thick at the middle and very thik at the two heads, in the middle part can place a partition, in the partition carved two gaps. Let the light pass through the gaps. It is conceivable that interference will occur when the mutual energy flow passes through the double slits. All 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 mutal energy flow, it goes at the same time from the two slits to transfer energy. The photon itself is the mutual energy flow, part of the energy flow from the first gap through the other part hord the other gap. Photons are interdependent streams consisting of retarded waves and advanced waves, which can transfer energy. This explanation, double-slits experiment, and it 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 the absorber can only receive right-handed photons, the emitter must also emit dextral photons. The photon is left or right, not just by the emitter, and the left or right 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 is knows that at the moment when the photon is emitted, A right-handed photon.

In the above we have said that the photon is generally described as a two-charge system, the system has only one emitter, 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 mutal 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 charge of a non-linear material and are absorbed by both absorbs in the system, so the interacting system has a total of three charges, and the three charge systems 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-frequancy 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 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 principle of mutual energy can explain the quantum entanglement phenomenon.

Why is the Maxwell equations just mathematical equations, a probabilistic equations, rather than a physical equations?
We found that the Maxwell equations must have two groups to pass 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 this 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 optical 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, the mutual energy flow is photon. Photon transfers energy from the emitter to the absorber. This energy transfer process is irreversible.

Since the self-energy flow can be returned, that is to say that this process is reversible. A self-energy flow process is completely offset by the same time reversal process. So the self-energy flow does not deliver any energy. So the overall effect can be seen as a mathematical process rather than a physical process. Whether it has issued a self-energy flow, and then have self-energy flow back, or did not issue a self-energy flow, the effect is the same. This question has yet to be studied further.

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, but what is behind this nature 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 mutal energy flows removes the mystery of the Copenhagen school of probability interpretation.

The explanation using the mutual energy principle to the electromagnetic field in the radio frequency band
In the band of light, due to the radiation emitted by the emitter, the same wave with the absorber is a very short signal, the two signals are synchronized, or at the same time is a small probability event. In the radio frequency band, the radio signal is a continuous signal, which can be synchronized with the infinite number of charges in the environment. Let us assume that there are $$N$$ charges in the world, $$N$$ close to infinity. Assuming that one of them is retarded, every charge in the world is involved in the system of this $$N $$ charge. If it is an absorber, it sends a super-wave to receive the retared wave, and if it is an emitter charge that is close to the emitter charge just now, it emits more energy under the original emitter charge.

In the radio band, the Poynting theorem, the Maxwell equations are still established. But at this time, the Poynting theorem and Maxwell equation is only similar to the establishment. Although this approximation is approximation with only an infinitely small differenc.

The Poynting theorem in the radio band
When $$N$$ is close to infinity,

The difference between

$$\sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i}w_{ij}$$

and

$$\sum^{N}_{i=1} \sum^{N}_{j=1}w_{ij}$$

is

math>\sum^{N}_{i=1} w_{ii}

compare to itself is an infinite small, that is,

$$\lim_{N \to \infty} \frac{\sum^{N}_{i=1} w_{ii}}{\sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i}w_{ij}}=0$$

Hence we can use $$\sum^{N}_{i=1} \sum^{N}_{j=1}w_{ij}$$ to replace $$\sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i}w_{ij}$$. Similarly, the other terms in the mutual energy principle can have similarly replacement, Hence we can use $$\sum^{N}_{i=1} \sum^{N}_{j=1}$$ to replace $$\sum^{N}_{i=1} \sum^{N}_{j=1,j \ne i}$$. Hence We can obtain $$N$$ charge Poynting theorem, which is written as following,

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

It is worth mentioning that, from the mathematical point of view, the above formula is a very accurate formula, with the principle of mutual energy is only an infinitesimal, but physically speaking, the above formula with the principle of mutual nature is essential difference. The principle of mutual energy does not recognize the physical function of self-energy flow and other self-energy iterms. $$N$$ charge of the Poynting theorem emphasizes the role of self-energy and self-energy iterms. The principle of mutual energy is now regarded as the axiom of the electromagnetic field, so the Poynting theorem of the $$N$$ charge is still a mathematic approximation, not a physical formula or theorem. consider

The above formula is the superimposition principle,We will discuss it in next section? apply the above formula, we obtains,

Thus we have obtain the Poynting theorem. Although the above theorem is exactly the same as the traditional Poynting theorem, it is quite different. The field in the Poynting theorem is the field of all charge generation, including the advaned waves generated by the absorber. Here, the current density $$\mathbf{J}$$ also includes the current density of the absorber that produces the advanced wave.

in the radio band superposition principle
We assume that $$N$$ is near infinity. In this case when we complement the principle of mutual energy. These self-energy items must be physically zero. But the mathematical calculation of these phases is not zero, when these values which are not zero complement the principle of mutual energy, we can apply the principle of superposition of electromagnetic fields,

To simplify the settlement of electromagnetic fields. Further, we can get the Poynting theorem, Maxwell equations (see the next section).

Thus the superposition principle is an approximate mathematical formula, and when $$N$$ is near infinity, this approximation is a very good approximation. There is only an infinitely small amount between the approximation and the exact value. The superposition principle is not a physical rationale.

On the other hand, the Maxwell equations need to be supported by the principle of superimposition. For example, the Maxwell equations for the calculation of a single charge is established. If there is no superposition principle, we can not prove that the multi-charge Maxwell equations is also established. The principle of mutual energy is unlike the Maxwell equations, which does not require the support of the superposition principle. The principle of mutual energy itself is true for two or more charge systems. Using the principle of reciprocity as the axiom of the theory of electromagnetic field, eliminating the dependence on the principle of superimposition. Making the electromagnetic field theory axiom system can be reduced by one, so become more concise.

When the $$N$$ is relatively small, for example, in the case of photons, this superposition principle is completely ineffective.

The Maxwell equation in the radio frequency band
The Poynting theorem can rewritten as,

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

Or,

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

Considering $$\mathbf{E}$$ and $$\mathbf{H}$$ is not zero. They are not linearly related to the amount, so there are

The above derivation is not strict, in fact, from the Poynting theorem can not be strictly deduced Maxwell equations. The Poynting theorem can be strictly deduced from the Maxwell equations. Thus the Maxwell equation is the sufficit condition of the Poynting theorem. The Maxwell equations are not a necessary condition for the Poynting theorem. But the above derivation suggests that the Poynting Theorem is almost a necessary condition for the Maxwell equations. Thus the Poynting theorem is almost equivalent to the Maxwell equations.

When we need to solve the Poynting theorem, the solution of the Maxwell equations is, of course, the solution of the Poyntin's theorem. Thus we obtained the Maxwell equation from the Poynting theorem.

On the other hand, we know that the Poynting theorem can deduce the mutual energy theorem, the mutual energy theorem can deduced, the Lorentz reciprocity theorem, and the Lorentz reciprocity theorem can deduce the Green's function solution. Thus all the solutions of the Maxwell equation can be obtained. We can derive the Maxwell equations from all the solutions of the Maxwell equations. Thus we can also derive from the Poynting theorem to the Maxwell equations, although this derivation is not very strict. In general, it can be argued that the Poynting theorem is equivalent to the Maxwell equation.

Thus we got the Maxwell equation. So the radio, microwave aspects of the problem can still use the Maxwell equation to solve, our previous classic electromagnetic field theory is still applicable.

It should be noted, however, that the theory of the electromagnetic field derived from the principle of mutual energy can not leave the current generated by the absorber and can not leave the advanced wave. In order to minimize the effect of the absorber and the advanced wave, we can think that the absorber is evenly distributed on the infinite sphere. So that we can completely use the emitter to solve the Maxwell equations. The resulting solution should be very close to the solution of the sphere where the absorber is at infinity. Of course, we can also solve the Maxwell equations by assuming that the absorber is at the boundary of the infinite sphere.

In summary, the Maxwell equation derived from the principle of mutual energy is the same as the classical Maxwell theory. But the classic Maxwell theory does not emphasize the absorber, not emphasizing the advance wave. Everything can be done by the emitter with the retarded wave. The Maxwell equations derived from the principle of mutual energy still requires the existence of the absorber, which must exist. The mutual energy flow of the advanced wave and the retarded wave is completely retarded. So there is no contradiction with the causal relationship. The absence of an absorber only an emitter is a simplification of the situation of the absorber on an infinite sphere. From the physical point of view, the absorber, the advanced wave must exist.

Conclusion
The theory of the Maxwell equations is only very accurate in the radio microwave band theory, but it encounters a lot of difficulties in the light band. 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 be unified under a theoretical framework. Although the mutual energy flow in the principle of mutual energy is composed of the advanced wave and retarded wave, the mutual energy flow is completely retarded, so the principle of mutual energy does not violate the causal relationship. From the above analysis can clearly see that the principle of mutual energy is a self-consistent theory.