Advanced wave

Advanced wave is also referred as advanced potential, advanced field and advanced solution. The word advanced potential is often applied in the electromagnetic field theory, where Maxwell equations are involved. Advanced wave and advanced field are often applied in quantum physics, hence, not only the Maxwell equations, but also Schrödinger equation, Dirac equation and Klein–Gordon equation are involved. Advanced solution is often used when we solve an equation, for example Maxwell equation. An advanced wave is a wave sent from the current time to the past time. Advanced wave is different with retarded wave or retarded potential which sends from the current time to the future time. Advanced waves violate our traditional understanding of causality. However, many physicists including Albert Einstein, John Archibald Wheeler, Richard Feynman and John G. Cramer believe the advanced wave or advanced field are a real phenomenon in physics. There are two kinds of theoretical support for the advanced wave: one kind came from the publications in quantum physics, another kind came from classical electromagnetic field theory.

Advanced wave

Advanced wave in 1D empty space for a point source

In 1D space for example wave-guide, if the source intensity is

$f({\boldsymbol {r}}_{0},t)$

where ${\boldsymbol {r}}_{0}$ is the source position. $t$ is the time.

The advanced wave has the following form,

$f({\boldsymbol {r}}_{0},t_{a})$

where ${\boldsymbol {r}}$ is a remote point and

$t_{a}=t+{\frac {||{\boldsymbol {r}}-{\boldsymbol {r}}_{0}||}{c}}$

where $c$ is speed for example light speed. It allowed the wave differ with the above form in the nearby region close to the source, but at least at the far-field region, the wave should be like above.

Advanced wave in 3D empty space for a point source

In 3D empty space, assume there is a point source at $r_{0}$, the source function is

$f({\boldsymbol {r}}_{0},t)$

where ${\boldsymbol {r}}_{0}$ is the source position. $t$ is the time.

the advanced wave can be described as following form,

${\frac {1}{||{\boldsymbol {r}}-{\boldsymbol {r}}_{0}||}}f({\boldsymbol {r}}_{0},t_{a})$

where ${\boldsymbol {r}}$ is a remote point and

$t_{a}=t+{\frac {||{\boldsymbol {r}}-{\boldsymbol {r}}_{0}||}{c}}$

where $c$ is speed for example light speed. It allowed the wave differ with the above form in the nearby region close to the source, but at least at the far-field region, the wave should be like above.

The advanced wave is a wave runs from current time to the past, in contrast the retarded wave, which runs from current time to the future.

If the source is not point source, the advanced wave can be obtained by applying the superposition principle to all point sources.

The fundamental theories which support the advanced wave

Maxwell equations

The Maxwell equations have two solutions one is the retarded potential, another is the advanced potential. Hence, Maxwell equations support the advanced wave.

Weber electrodynamics

Weber electrodynamics is introduced earlier than Maxwell equations. Weber electrodynamics is belong to an action principle which does not reject the advanced wave.

Dirac equation

Dirac equation also support the retarded wave and the advanced wave.

Klein–Gordon equation

Klein–Gordon equation has two solutions, one is retarded solution, another is advanced solution.

Schrödinger equation

Schrödinger equation has only one retarded solution. But it is easy to produce an advanced solution to them, which is the complex conjugate of the retarded solution.

The theories coming from quantum physics, which support the advanced wave

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,

$S=-\sum _{i}m_{i}c\int ({\frac {dx_{i\mu }}{d\tau _{i}}}{\frac {dx_{i}^{\mu }}{d\tau _{i}}})^{\frac {1}{2}}d\tau _{i}-\sum _{i}\sum _{j<i}{\frac {e_{i}e_{j}}{c}}\int \int \delta (s_{ij}^{2}){\frac {dx_{i\mu }}{d\tau _{i}}}{\frac {dx_{j}^{\mu }}{d\tau _{j}}}d\tau _{i}d\tau _{j}=\mathrm {extremum}$

where

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

$ds=c^{2}dt^{2}-dx_{1}^{2}-dx_{2}^{2}-dx_{3}^{2}$

Ritz-Einstein debate

Walter Ritz and Albert Einstein has a debate in 1909 about the time-asymmetric macroscopic phenomena. Ritz believe that this time-asymmetric macroscopic phenomena is because of the fundamental physical law. Hence, only the retarded wave is allowed in the physics, the advanced wave is forbidden in the physics. In contrast, Einstein believe that this time-asymmetric macroscopic phenomena is because of statistics. Hence, for the microscopic phenomena is still possible to be time-symmetric, that means the retarded wave and advanced wave are all allowed in physics. （Walter Ritz and Albert Einstein）.

Dirac theory

Dirac introduce a classical electromagnetic field method to calculate the self-force of a accelerative or accelerative electron. In this method the difference between retarded potential and advanced potential is involved （P. A. M. Dirac）.

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.

Wheeler's delayed choice experiment

Wheeler's delayed choice experiment is introduced in 1978-1984. 30 years before, Wheeler introduced the absorber theory in which he support the existence of the advanced wave. Wheeler's delayed choice experiment can be seen as a result of advanced wave. Hence, Wheeler's delayed choice experiment support (or at least not reject) the concept of advanced wave. This experiment is proved to be true quickly after it is suggested.

Delayed choice quantum eraser

Delayed choice quantum eraser is an experiment following the Wheeler's delayed choice experiment. It can be seen as another testimony (or at least not reject) for advanced wave.

The tutorial about advanced wave

Lawrence M. Stephenson has a publication about the advanced potential (Lawrence M. Stephenson), which is easy to read and can be used as a tutorial for the knowledge of the advanced wave.

The theories in classical electromagnetic field, in which the advanced wave is involved

The reciprocity theory in arbitrary time-domain

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

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

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.

The new 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）,

$-\int _{V1}\mathbf {E} _{2}^{*}(\omega )\cdot \mathbf {J} _{1}(\omega )dVdt=\int _{V2}\mathbf {E} _{1}(\omega )\cdot \mathbf {J} _{2}^{*}(\omega )dV$

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),

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

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 $\Gamma _{s}$. $\Gamma _{s}$ 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:

${\begin{aligned}(a\xi _{1},\xi _{2})&=a(\xi _{1},\xi _{2})\\(\xi _{1}+\xi _{2},\xi _{3})&=(\xi _{1},\xi _{3})+(\xi _{2},\xi _{3})\end{aligned}}$

• Positive-definiteness:

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

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 waves are inside the surface $\Gamma _{s}$, i.e.,

$(\xi _{1},\xi _{2})_{\Gamma _{s}}=0$

where $\tau _{1},\tau _{2}\in V$ are the source of $\xi _{1},\xi _{2}$. $\Gamma _{s}$ is the boundary surface of the volume $V$. $\Gamma _{s}$ can be chosen on the sphere with infinite radius.

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 $\xi _{1}$ and $\xi _{2}$ is not zero, if $\xi _{1}$ is the retarded wave sends from $\tau _{1}$. $\xi _{2}$ is the advanced wave sends from $\tau _{2}$.

$(\xi _{1},\xi _{2})_{\Gamma _{1}}=(\xi _{1},\xi _{2})_{\Gamma }=(\xi _{1},\xi _{2})_{\Gamma _{2}}\neq 0$

Where $\Gamma _{1}$ is the boundary surface of volume $V_{1}$ and $\Gamma _{2}$ is the boundary surface of volume $V_{2}$. $\Gamma$ is any surface between $\Gamma _{1}$ and $\Gamma _{2}$. Assume all normal vector of the surface are at the direction from $V_{1}$ to $V_{2}$.

The 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. This theorem can be seen as Huygens–Fresnel principle (Shuang-ren Zhao), which can be written as,

$-(\tau _{1},\xi _{2})_{V_{1}}=(\xi _{1},\xi _{2})_{\Gamma }=(\xi _{1},\tau _{2})_{V_{2}}$

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 $V_{2}$.

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

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

The 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,

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

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.

The 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.

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）. Hence, Lorentz reciprocity can be written as following,

$-(\tau _{1},\mathbf {\mathbb {C} } \xi _{2})_{V_{1}}=(\xi _{1},\mathbf {\mathbb {C} } \tau _{2})_{V_{2}}$

Actually the mutual energy theorem Shuang-ren Zhao）, the new reciprocity theorem of Rumsey（V.H. Rumsey） is derived by applying the conjugate transform to the Lorentz reciprocity theorem. It is remarked to say that even the mutual energy theorem can be derived from Lorentz reciprocity theorem by using the conjugate transform, it is a independent theorem with the Lorentz reciprocity theorem. The reason is that the conjugate transform is not a mathematic transform like Fourier transform, it is physical transform according to Maxwell equations. Another reason is the field after the conjugate transform is changed. The advanced wave becomes to retarded wave and the retarded wave becomes to the advanced wave.

The mutual energy theorem is used as reciprocity theorem

The mutual energy theorem (include the Welch's reciprocity theorem and de Hoop's reciprocity theorem) can be applied as a reciprocity theorem.

$-(\tau _{1},\xi _{2})_{V_{1}}=(\xi _{1},\xi _{2})_{\Gamma }=(\xi _{1},\tau _{2})_{V_{2}}$

Apply the conjugate transform to all field with index 1 and index 2. We obtained

$-(\mathbb {C} \tau _{1},\mathbb {C} \xi _{2})_{V_{1}}=(\mathbb {C} \xi _{1},\mathbb {C} \xi _{2})_{\Gamma }=(\mathbb {C} \xi _{1},\mathbb {C} \tau _{2})_{V_{2}}$

It can be simplified as

$(\xi _{2},\tau _{1})_{V_{1}}=(\xi _{2},\xi _{1})_{\Gamma }=-(\tau _{2},\xi _{1})_{V_{2}}$

It can be seen that in this case the emitter and the absorber are just exchanged. In this meaning that the mutual energy theorem is also a reciprocity theorem.

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 fields. In the mutual energy theorem or cross-correlation time-domain reciprocity theorem, the two fields are one is retarded wave sending from the transmitting antenna, another is the advanced wave sending 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. It is also can be applied as some kind of reciprocity theroem

The Lorentz reciprocity theorem reciprocity theorem is a mathematical theorem which transform of one filed in the mutual energy theorem/time-domain cross-correlation reciprocity theorem. Lorentz reciprocity theorem can be applied to find the directivity diagram actually is also because there is the mutual energy theorem stand on the back of it. However most electronic engineer used Lorentz reciprocity theorem more because there the advanced wave is exclude.

The mutual energy flow and self energy flow

Assume there are two charges and the two fields are send from these two charges. The two fields can be superposed. The superposed field satisfy also the Poynting theorem. The Poynting vector can be written as

${\boldsymbol {E}}\times {\boldsymbol {H=}}({\boldsymbol {E}}_{1}+{\boldsymbol {E}}_{2})\times ({\boldsymbol {H_{1}+H_{2})}}$

$={\boldsymbol {E}}_{1}\times {\boldsymbol {H_{1}}}+{\boldsymbol {E}}_{2}\times {\boldsymbol {H}}_{2}+{\boldsymbol {(E}}_{1}\times {\boldsymbol {H}}_{2}+{\boldsymbol {E}}_{2}\times {\boldsymbol {H}}_{1})$

In this formula

${\boldsymbol {E}}_{1}\times {\boldsymbol {H}}_{1}$

and

${\boldsymbol {E}}_{2}\times {\boldsymbol {H}}_{2}$

can be defined as defined as energy intensity of the self-energy flows

${\boldsymbol {(E}}_{1}\times {\boldsymbol {H}}_{2}+{\boldsymbol {E}}_{2}\times {\boldsymbol {H}}_{1})$

can be defined as the energy intensity of the mutual energy flow （Shuang-ren Zhao）

When there is the definition of mutual energy flow, assume $\tau _{1}$ is an emitter. $\tau _{2}$ is an absorber and hence, $\xi _{1}$ is a retarded wave and $\xi _{2}$ is a advanced wave, then the original mutual energy formula can be also called mutual energy flow theorem:

$-(\tau _{1},\xi _{2})_{V_{1}}=(\xi _{1},\xi _{2})_{\Gamma _{1}}=(\xi _{1},\xi _{2})_{\Gamma _{2}}=(\xi _{1},\tau _{2})_{V_{2}}$

where $\Gamma _{1}$ and $\Gamma _{2}$ are any two surfaces between $V_{1}$ and $V_{2}$. And,

$(\xi _{1},\xi _{2})_{\Gamma }=\int _{t=-\infty }^{\infty }\iint _{\Gamma }({\boldsymbol {E}}_{1}\times {\boldsymbol {H}}_{2}+{\boldsymbol {E}}_{2}\times {\boldsymbol {H}}_{1})\cdot {\hat {n}}d\Gamma dt$

is the mutual energy go through the surface $\Gamma$. The surface can be any surface between the two sources $\tau _{1}$ and $\tau _{2}$.

$(\tau _{1},\xi _{2})_{V_{1}}=\int _{t=-\infty }^{\infty }\int _{V_{1}}({\boldsymbol {J}}_{1}(t)\cdot {\boldsymbol {E}}_{2}(t))dVdt$

$(\xi _{1},\tau _{2})_{V_{2}}=\int _{t=-\infty }^{\infty }\int _{V_{2}}({\boldsymbol {E}}_{1}(t)\cdot {\boldsymbol {J}}_{2}(t))dVdt$

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 .

Sphere wave expansion and plane wave expansion

The inner product space of electromagnetic fields with the mutual energy theorem（Shuang-ren Zhao） (Shuang-ren Zhao) has been applied wave expansions. for example with spherical waves （Shuang-ren Zhao） and plane waves (Shuang-ren Zhao).. The wave expansions can be applied to retarded waves and also advanced waves.

Citations

This article "Advanced wave" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Advanced wave. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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