This author has replaced the Maxwell equations with the corresponding mutual energy principle, self-energy principle as the axioms in electromagnetic eld theory. The advantage of doing this is that it can overcome the diculty of the Maxwell equations, which conicts to the energy conservation law. The same conict also exists in the Schrödinger equation in the quantum mechanics. This author would like to intruded the mutual energy principle to quantum mechanics, but has met the diculty that there is no advanced solution for the Schrödinger equation. This diculty is overcome by introducing a negative radius. After this, all the theory about the mutual energy can be extend from the eld satisfying Maxwell equations to the eld satisfying Schrödinger equation. The Schrödinger equation can also be derived from the corresponding mutual energy principle. However, this doesn't mean both sides are equivalent. The mutual energy principle cannot derive a single solution of Schrödinger equation. The mutual energy principle can only derive a pair solutions of the Schrödinger equations. One is for retarded waves and another is for advanced waves. The retarded wave and the advanced wave must be synchronized. The solutions of the mutual energy principle is in accordance with the theory of the action-at-a-distance and the absorber theory. A action is done always between two objects, for example a source (emitter) and a sink (absorber). The mutual energy principle tell us that a particle is an action and a reaction between the source and the sink. In other hand the wave satisfying Schrödinger equation only need one source or one sink. From the mutual energy principle, it is easy to derive the mutual energy theorem, the mutual energy ow theorem, corresponding HuygensFresnel principle. All these will solve the wave-particle duality paradox.
This author has replaced the Maxwell equations with the corresponding mutual energy principle, self-energy principle as the axioms in electromagnetic eld theory. The advantage of doing this is that it can overcome the diculty of the Maxwell equations, which conicts to the energy conservation law. The same conict also exists in the Schrödinger equation in the quantum mechanics. This author would like to intruded the mutual energy principle to quantum mechanics, but has met the diculty that there is no advanced solution for the Schrödinger equation. This diculty is overcome by introducing a negative radius. After this, all the theory about the mutual energy can be extend from the eld satisfying Maxwell equations to the eld satisfying Schrödinger equation. The Schrödinger equation can also be derived from the corresponding mutual energy principle. However, this doesn't mean both sides are equivalent. The mutual energy principle cannot derive a single solution of Schrödinger equation. The mutual energy principle can only derive a pair solutions of the Schrödinger equations. One is for retarded waves and another is for advanced waves. The retarded wave and the advanced wave must be synchronized. The solutions of the mutual energy principle is in accordance with the theory of the action-at-a-distance and the absorber theory. A action is done always between two objects, for example a source (emitter) and a sink (absorber). The mutual energy principle tell us that a particle is an action and a reaction between the source and the sink. In other hand the wave satisfying Schrödinger equation only need one source or one sink. From the mutual energy principle, it is easy to derive the mutual energy theorem, the mutual energy ow theorem, corresponding HuygensFresnel principle. All these will solve the wave-particle duality paradox.
Quantum mechanics has the quantization. The quantization offer us a method from the mechanic equation to build the quantum wave equation. For example the Canonical quantization offers a method to build the Schrodinger equation from Hamilton in classical mechanics. This is also referred as first quantization. In general, Maxwell equation itself is wave equation, hence, it doesn't need the first quantization. There is second quantization, for electromagnetic field. The second quantization discuss how many photons can be created when the energy of electromagnetic field is known. This is not interesting to this author. This author is interested how to build a particle from the wave equations (Maxwell equations or Schrodinger equation). Here the particle should confined in space locally. It should has the properties of wave. Our traditional quantization is to find the wave equation. This author try to build a particle from this wave equation, this process can be called as particleization. This author has introduced the mutual energy principle, the mutual energy principle successfully solved the problem of conflict between the Maxwell equations and the law of the energy conservation. The mutual energy flow theorem is derived from the mutual energy principle. The mutual energy flow is consist of the retarded wave and the advanced wave. The mutual energy flow theorem tell us the total energy of the energy flow passes through any surfaces between the emitter to the absorber are all exactly same. This property is required by the photon and any particle in quantum mechanics. Hence, this author has linked the mutual energy flow to the photon and also other particle. The mutual energy flow has the property of waves and also confined in space locally. However there is still a problem, the field of an emitter or the field of an absorber decreases according to the distance from the field point to the source point. If the current (or charge) of a source or sink for a photon is constant. The energy of the photon which equals the inner product of the current and the field will depended on the distance between the the source and the sink of the photon. If the distance increases, the amount of photon energy will decrease to infinite small. This is not correct. The energy of a photon should be a constant E=hv. The energy of the photon cannot decrease with the distance between the emitter and the absorber. In order overcome this difficulty, this author suggests a normalization for the mutual energy flow. It is assume that the retarded wave sent from the emitter has collapse back in all direction. But the mutual energy flow build a energy channel between the source and sink. Since the energy can only go through this channel, the total energy of one photon must go through this channel. Hence, the total energy of the mutual energy flow has to be normalized to the energy of one photon. The mutual energy flow will increase to the energy of one photon. This leads that the amplitude of the wave does not decrease on the direction along the energy channel. The amplitude of the advanced wave also does not decrease on the direction of the energy channel. The electromagnetic wave in the space between an emitter (source) and an absorber (sink) looks like a wave inside a wave guide. The wave in a wave guide, the amplitude does not decrease alone the wave guide if the loss of energy can be omitted. This wave guide can be referred as the “nature wave guide”. In the nature wave guide the advanced wave leads the the retarded wave, hence, the retarded wave can only goes at the direction where has advanced wave. The retarded wave also leads the advanced wave. The advanced wave can only goes in the direction of the retarded wave. This normalization process successfully particularized the the mutual energy flow. This author believe this theory about the normalization/particleization of the mutual energy flow is also correct for other particle for example electron.
The particle and wave duality is solved through the self-energy and the mutual energy principles. Welch has introduced the time-domain reciprocity theorem in 1960. This author have introduced the mutual energy theorem in 1987. It has been proved that the above two theorems are same theorem in time-domain or in Fourier domain. This author believe there is an energy flow from transmitting antenna to the receiving antenna. Hence this theorem is a energy theorem instead of a mathematical theorem i.e. the reciprocity theorem. This author found that the mutual energy is the additional energy when the two waves are superposed comparing to the situation when the two waves alone stayed in the space. It is often asked that if the two waves are identical what is the energy after the two waves are superposed, 4 or 2 times? this author's answer are 2 or 4 depending whether the sources of the waves are involved or not. However this author noticed that a more important situation, which is the superposition of two waves: one is retarded wave sent from the emitter, another is the advanced wave sent from the absorber. This situation actually described the photon. This author have found that, instead there are two photons the retarded photon and the advanced photon like some author believed, there is only one photon. The reason is that the two waves the retarded wave and the advanced wave they both bring one photon energy, which are sent to the space, but these energy are returned with the time-reversed waves. The additional energy because of the superpose process of the two waves is just with 1 photon's energy instead of 2 photon's energy. This energy is sent from the emitter to the absorber. These build this author's photon model. This photon model is proved by this author through the notice of the conflict between the energy conservation and both the superposition principle and the Maxwell equations for single charge. This conflict force this author introduced the mutual energy principle and the self-energy principle. Self-energy principle tell us the self-energy (the wave's energy before superposed) time-reversal return to its source and hence do not transfer any energy from emitter to the absorber. The mutual energy principle tell us that transferring the energy from the emitter to the absorber is only done by the mutual energy flow. This author also proved that the mutual energy flow theorem, which says that the energy transferred by mutual energy flow is equal in any surface between the emitter to the absorber. The wave function collapse process is explained by the two processes together the first is the self-energy time-reversal return to their sources (instead of the targets), the second is that the mutual energy flow brings a photon's energy package from emitter to the absorber. The wave's probability property is also explained that because only when a retarded wave synchronized with an advanced wave the energy can be transferred. The photon energy is transferred only when the retarded wave (one of solution of Maxwell equations) and the advanced wave (another solution of the Maxwell equations) are synchronized, otherwise the two waves are returned by two time-reversal waves. Time-reversal wave are not satisfy Maxwell equations but satisfy the time-reversal Maxwell equations. Hence, 4 time-reversal Maxwell equations which describe the two additional time-reversal waves are added to Maxwell equations. Hence, the photon's package wave is consist of 4 waves which are corresponding to 4 self-energy flows. There are two additional energy flows, which are the mutual energy flows that is responsible for transferring the energy from emitter to the absorber. The time-reversal mutual energy flow which is responsible to bring the energy back from the absorber to the emitter if the absorber only obtained a half photon or a part of photon.
According to classical electromagnetic theory, quantum mechanics, and quantum field theory, particles are often associated with self-energy flow. In electromagnetic theory, the so-called self-energy flow corresponds to the Poynting vector of energy flow, and in quantum mechanics, it represents the probability current density. The author believes this is likely incorrect. The author finds that self-energy flow can be replaced by mutual energy flow. Mutual energy flow consists of retarded waves emitted by the source and advanced waves emitted by the sink. The source includes the primary coil of a transformer, the transmitting antenna, radiation atoms, and the cathode emitting electrons. The sink includes the secondary coil of the transformer, the receiving antenna, absorbing atoms, and the anode receiving electrons. Originally, all energy flows are generated by the source. If we assume energy flow is generated jointly by the source and the sink, we first need to compress the field produced by the source to half its original size. In this way, the source produces half of the field, and the sink produces the other half. The mutual energy flow generated by the source and sink together will exactly match the original self-energy flow produced by the source. However, there is still a problem. If the energy flow is mutual energy flow rather than self-energy flow, then self-energy flow should not transfer energy. However, if we do not alter the original electromagnetic theory or quantum theory, self-energy flow still transmits energy. This creates an absurd situation where mutual energy flow particles and self-energy flow particles are two distinct particles, both transferring energy. The author finds that both self-energy flows in electromagnetic theory and quantum theory should be reactive power and therefore do not transfer energy. The author adds a magnetic field, source, and sink to the Schrödinger equation and extends the mutual energy flow law from electromagnetic theory to quantum theory. The author then corrects the phase of the magnetic field in Maxwell’s electromagnetic theory and quantum theory by 90 degrees. After the correction, both self-energy flows in the two theories become reactive power. Thus, particles, including photons, electrons, and all other particles, are mutual energy flows, not self-energy flows. For electromagnetic theory, the author also proves that the composite energy flow of many photon mutual energy flows coincides with the self-energy flow corresponding to the Poynting vector calculated from Maxwell’s electromagnetic theory. Thus, electromagnetic waves have three different modes: (1) broadcasting mode, where the source and sink simultaneously emit spherical waves at random, which decay with propagation distance. (2) Photon mode, where the retarded wave emitted by the source and the advanced wave emitted by the sink are perfectly synchronized. At this point, the advanced wave forms the waveguide of the retarded wave, and the retarded wave forms the waveguide of the advanced wave. Due to the interference between the retarded and advanced waves, the two waves interfere constructively along the line connecting the source and the sink and weaken in other directions, ultimately turning into quasi-planar waves along the connection line between the source and sink. These retarded and advanced waves constitute the mutual energy flow, which is the photon. (3) The average energy flow consisting of countless photons is consistent with the energy flow calculated according to the Poynting vector. Therefore, Maxwell’s electromagnetic theory is still valid for calculating average radiation electromagnetic energy flow. However, calculating photons requires the author’s electromagnetic theory. The situation for electrons or other particles is roughly similar.
Combined with the principle of mutual energy of electromagnetic field, mutual energy theorem, mutual energy flow theorem and Wheeler - Feynman absorption theory, we propose a mutual energy flow interpretation for quantum mechanics. The Wheeler-Feynman absorber theory suggests that the light wave is composed of both the advanced and the retarded waves. John Cramer gave a simple model of photon with plane waves. The author put forward the mutual energy theorem and the principle of mutual energy. The principle of mutual energy is a formula which removes the remainder of the self-energy terms from the Poynting theorem. The principle of mutual energy as the electromagnetic field axiom can ensure that the classic electromagnetic theory building does not fall to overcome the many difficulties if the Maxwell equations have been applied as axioms. From the principle of mutual energy, radio waves, light waves are all consists of the retarded wave of the emitters and the advanced wave of the absorber. The retarded wave and the advanced wave are not simply superimposed, but combined very close together into one. The retarded wave and the advanced wave constitute the mutual energy flow. The energy flows from the emitter to the absorber in the form of energy flow. The integral of the flux of the energy flow to time is constant on any surface between the emitter and the absorber, which is the energy of the photon. Photon is nothing else, it is just the mutual energy flow. In this paper, we should further clarify that the mutual energy flow theorem tells us that the waves in the space are much like the waves in the waveguide as a quasi-plane wave. Although this quasi-plane wave is composed of the retarded wave of the emitter and the advanced wave of the absorber, we can also think of it as a purely retarded wave. The vacuum space naturally constitutes the waveguide of the mutual energy flow. Since the field between the charge and the other charge can be regarded as a plane wave, it naturally avoids the problem of self-energy divergence. Recently, Afshar experiments supported the John Cramer transactional interpretation. We found that, this experiment can also be explained by using the principle of mutual energy flow. Hence, Afshar experiment also supports the author's mutual energy flow interpretation. To make our theory self-consistent, we also introduced the principle of the return of the self-energy flow.
Since proposing the electromagnetic mutual energy theorem and its inner product concept in 1987, and subsequently improving Huygens' principle (1989), the author formally established the mutual energy flow theorem in 2017, aiming to completely replace the traditional Poynting vector (self-energy flow) with mutual energy flow. He also proposed a process analogous to wave function collapse in quantum mechanics, namely the “reverse collapse” of self-energy flow back to its origin. In 2022, the author further clarified that since self-energy flow must be reactive power and does not transfer energy, the mutual energy flow theorem should be updated to the mutual energy flow law. Following the extension of this idea to the Schrödinger equation (by defining a “magnetic field” and correcting the phase to establish a particle-like mutual energy flow), this paper demonstrates the universality of this framework for the Klein-Gordon (K-G) equation. By constructing a magnetic field for the K-G equation and correcting its phase, the corresponding mutual energy flow law is derived. The research confirms that this mutual energy flow, formed by the coupling of retarded and advanced waves, is generated at the source and annihilated at the sink, perfectly unifying the wave and particle properties and providing a solid theoretical foundation for understanding wave-particle duality.
Einstein guessed that the macroscopic electromagnetic wave is built by thousands of photons, however, no one has offered a theory about how the macroscopic electromagnetic wave is built from photons. A concrete theory about photons is needed to answer this question. Current theory for photons is Maxwell’s equation which has the solution of waves, but it is difficult to describe the photon as a particle. There is the paradox problem of wave-particle duality. This article offers one solution to solve this problem by introducing the normalized mutual energy flow. The interaction of the retarded wave and advanced wave produce the mutual energy flow. The mutual energy flow satisfies the mutual energy flow theorem. The mutual energy flow theorem tells us that the energy that goes through each surface between the emitter and the absorber is all same. That means the mutual energy flow is different in comparison to the waves. The wave, for example, the retarded wave, its amplitude is decreased with the distance from the source to the point of the field. The mutual energy flow does not decrease. The author noticed this and claimed that the photon is the mutual energy flow. In this article the author updated this claim that the photon is the normalized mutual energy flow. Here the normalization of mutual energy flow will normalize the mutual energy flow to the energy of a photon, which is E = hf. E is the energy of the photon; h is Planck constant; f is the frequency of the light. This normalization is similar to the normalization in quantum mechanics. After this normalization the relation between an electromagnetic wave and photon as a particle becomes clear. This article will prove that the macroscopic wave of an electromagnetic field can be built by thousands of normalized mutual energy flows, which describes the photons. The mutual energy flow is an interaction of the retarded wave and the advanced wave. The retarded wave and the advanced wave satisfy the Maxwell equations. There are two additional waves which are the time-reversal waves which satisfy time-reversal Maxwell equations. The advanced wave and the two time-reversal waves are all real and physical electromagnetic fields. The time-reversal waves cancel all self-energy flows of the retarded wave and advanced wave. Hence, the waves do not carry any energy, the energy is only transferred by the normalized mutual energy flows which are the photons. Hence, all energy is transferred by the photon instead of waves. This offers a solution to paradox of the duality of wave-particle.