Is the Photon a Self-Energy Flow or a Mutual-Energy Flow?

Introduction

The Problem of Wave-Particle Duality

Whether electromagnetic waves are generated solely by the source or jointly by the source and the sink has long puzzled scientists. The term "source" refers to the transmitting antenna, the light source of a laser, or the primary coil of a transformer. The term "sink" refers to the receiving antenna, the screen that absorbs the laser, or the secondary coil of a transformer.

Traditional electromagnetic field theory has always considered the energy flow to be a self-energy flow. The so-called self-energy flow refers to the energy flow produced solely by the source itself. In electromagnetic field theory, it corresponds to the Poynting vector; in quantum mechanics, the self-energy flow corresponds to the energy flow associated with the probability current. Traditional electromagnetic field theory and quantum theory assume that the sink does not generate any field. The sink is merely a passive receiver of the field. However, a number of scientists have argued that the sink emits advanced waves, as opposed to the retarded waves emitted by the source. Notable among these are the action-at-a-distance theories by Schwarzschild and others [1-3], Wheeler and Feynman’s absorber theory [4,5], Cramer’s transactional interpretation of quantum mechanics, and Stephenson’s advanced wave theory [6]. The author also supports this view [7].

Even if one accepts the role of the sink, there are still two possibilities: one is that the source generates its own self-energy flow and the sink generates its own self-energy flow – Cramer’s transactional interpretation falls into this category. Only the author advocates that energy is transmitted by mutual energy flow [7], which is jointly generated by the retarded wave from the source and the advanced wave from the sink.

This paper does not approach mutual or self-energy flow from the perspective of electromagnetic theory itself, but rather investigates the possibility of using mutual energy flow to replace self-energy flow, focusing on the interchangeability of their respective energy flow densities.

Reciprocity Theorem, Mutual-Energy Theorem, Mutual-Energy Flow Theorem, and Energy Conservation Law

Consider the Poynting vector S = E × H, which satisfies the time-integrated Poynting theorem[8],

Equation (12) ensures that the above conditions are satisfied. These energy conservation conditions cannot be derived from classical Maxwell theory [7].

Focus of This Paper

This paper no longer investigates mutual energy flow from the perspective of electromagnetic fields. Instead, it analyzes the problem from the perspective of the Poynting vector and the mixed Poynting vector. We assume the author does not know whether energy is transmitted by mutual energy flow or self-energy flow but attempts to replace self-energy flow with mutual energy flow to explore what consequences may result.

Poynting Vector and Mixed Poynting Vector

Traditional Electromagnetic Field Theory

Traditional theory asserts that only self-energy flows carry energy. Thus, S₁ = E₁ ×H₁ * does the work, while S₂ from the sink is negligible  or ignored. In quantum theory, the EM wave (E₁, H₁) is emitted as a spherical wave, gradually collapsing into a plane wave (photon) directed toward the sink.

Advanced Waves from the Sink

By symmetry, energy might also be carried by the sink’s advanced wave SS₂ E₂ H^∗2 . Initially spherical, it collapses toward the source.

Coexistence of Retarded and Advanced Waves

According to Cramer’s transactional interpretation, the source emits a retarded wave and the sink emits an advanced wave. Their handshake creates a photon. Between source and sink, the superposition yields destructive interference outside the region, forming:

Concurrent Self- and Mutual-Energy Transfer

Mutual-Energy Transfer Only

Comparison

In the table “Transfer” means the energy transfer by which type of energy flow; and are the phase difference between electric field and magnetic field; “Collapse” tells which energy flow shoud collapse; “Self-Energy” means self energy flow; “MEF” means the mutual energy flow; “factor” is the compress factor of the energy flow; “Gerneration” tells whether this energy flow generates from a source; “Annihilation” tells whether this energy flow annihilates at a sink.

Method 1 corresponds to classical electromagnetic theory and quantum mechanics, where energy transfer is achieved by SS₁ only. SS₁ results in generation but no annihilation. Methods 2 and 4 have not been proposed by others; the author includes them here to illustrate alternative possibilities. Method 3 is the transactional interpretation of quantum mechanics proposed by Cramer in 1986 [14,15]. This interpretation allows for sources and sinks, as well as retarded and advanced waves. However, it does not compute mutual energy flow.

Method 5 was proposed by the author in 2017 [13]. It does not modify the magnetic field obtained from Maxwell’s equations but instead adds a time-reversed electromagnetic wave to classical theory. The main drawback of this method is that one must introduce a time- reversed Maxwell’s equations.

The first four cases all require wave collapse. The fifth involves reverse wave collapse. Only the sixth does not require any wave collapse! Wave collapse is a physical process, yet there is no established equation describing it. On the other hand, reverse collapse can be described by an equation. If wave collapse could also be described by an equation, it would imply a modification of Maxwell’s electromagnetic theory.

Hence, all these methods inherently involve some correction to Maxwell’s theory. However, only the sixth case explicitly modifies the magnetic field. Moreover, mutual energy flow requires a normalization factor of 1/2 to transfer energy. This normalization factor is not derivable from Maxwell’s equations; it originates from the idea of half-retarded and half-advanced waves [4,5], which again constitutes a correction to Maxwell’s theory.

Examples

Transformer

The energy flow in a transformer cannot be due to self-energy flow. This example relates to the so-called Maxwell-Lodge effect. In fact, the Aharonov-Bohm (AB) effect in quantum mechanics is also connected to this issue [16,17].

Assume a primary coil that is infinitely long, with its magnetic field completely confined inside the coil. The secondary coil is wound outside the primary coil at radius r=R₂, as shown in Figure 1

Dipole Antenna Pair

We assume that the source initially emits retarded spherical waves randomly in all directions, and the sink also emits advanced waves in all directions. Suppose the retarded wave from the source and the advanced wave from the sink are synchronized. This means when the electromagnetic wave from the source reaches the sink, the sink emits an advanced wave. Once synchronized, the advanced wave becomes a guiding wave for the retarded wave and vice versa. Due to the interference effect, both waves become quasi-plane waves directed from the source to the sink. This is equivalent to the collapse of both into plane waves.

Since the sink is to the right of the source, we assume the right side of each current emits retarded waves, and the left side emits advanced waves. At the current elements, the wave types flip. Thus, to the right of both currents, we have retarded waves, and to the left, advanced waves. Therefore:

The sign changes of H₁ and H₂ can be explained either by Ampère’s law or by the wave-type inversion across the current – that is, the switch between retarded and advanced waves, as described by Eq. (12). For E₁, H₁, the wave is advanced for x < 0 and becomes retarded for x > 0. For E₂, H₂, it is advanced for x < L and retarded for x > L. This way, the mutual energy flow density Sm is generated at the source (x = 0) and annihilated at the sink (x = L), accurately describing photon creation and annihilation. This example corresponds exactly to Case 6 in the comparison table.

It is evident that when calculating the mutual energy flow, the electric and magnetic fields can be derived under magneto-quasistatic conditions, then modified using the appropriate retarded or advanced phase factors. Of course, Maxwell’s equations can also be used directly, but afterward, one must correct the magnetic field using Eq. (12). The electromagnetic field of a photon is a plane (or quasi- plane) wave and does not contain the static component E_s = -∇, so there is no need to apply correction to E_s.

Figure 3: The transmitting and receiving antennas are both dipoles. The retarded and advanced waves are reactive powers and appear to collapse back in all directions.

Only along the source-sink line does mutual energy flow exist. In this example, the source initially emits retarded spherical waves, and the sink emits advanced spherical waves. After synchronization due to mutual interaction, both waves interfere and become quasi-plane waves directed from source to sink. This naturally explains the wave collapse. Alternatively, one can view the retarded and advanced waves as reactive power that is emitted into space and simultaneously collapses back to the source or sink. However, along the source- sink line, due to the existence of mutual energy flow, the energy of one photon is exactly transferred – see Figure 3. This figure also corresponds to Case 5 in the comparison table.

Case 5 is indeed a promising approach. However, it requires manually introducing time-reversed electromagnetic waves. Time-reversed EM waves may also generate time-reversed mutual energy flows, which might cancel the original mutual energy flow. Hence, it is not yet a perfect solution.

In Case 6, the self-energy flow formed by electric and magnetic fields is reactive power. During one oscillation cycle, the wave propagates forward for half the time and backward for the other half. On average, there is no net energy transfer. Since these waves do not transfer energy, they do not need to be interpreted probabilistically. Reactive waves and probabilistic interpretations serve similar roles. In this case, energy is fully transferred by the mutual energy flow.

Conclusion

If mutual energy flow is to replace self-energy flow, it must be normalized by a factor of 1/2 because it consists of two terms, which otherwise would double the energy flow. If energy is indeed transferred via mutual energy, then self-energy flow should not carry energy – otherwise, two types of photons would exist: self-energy photons and mutual-energy photons, contradicting observations. We observe only one type of photon. To prevent self-energy from carrying energy, it must either undergo reverse collapse or be constructed with a 90-degree phase difference between electric and magnetic fields. The latter requires a correction to classical Maxwell’s theory: namely, a 90-degree phase adjustment to the far-field magnetic component. This adjustment allows the magnetic field’s initial phase to be derived under magneto-quasistatic conditions, with the radiation field acquired by appending retarded or advanced phase factors.

The fields between a transmitting and receiving antenna, due to interference, become quasi-plane waves. The mutual energy flow calculated from them is created at the source and annihilated at the sink – precisely matching photon behavior. Thus, the mutual energy flow theory proposed herein can accurately describe photons. These ideas are also applicable to quantum mechanics.

References

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