SEDE BOGOTA DNSAV

NEGATIVE ENERGY ELECTROMAGNETIC RADIATION AND THE INTERPRETATION OF QUANTUM MECHANICS

Barbosa E. and González F.
Department of Physics, National University of Colombia

Transversal 33 No. 106-45, Bogotá, Colombia

A slight generalization of the classical Maxwell equations obtained by taking complex solutions for the electromagnetic fields emitted by the moving electronic charge in atoms is discussed. Because of this generalization, we are led to the introduction of the idea of negative energy radiation. If this generalization is correct, we have a possibility for a reconsideration of the current probabilistic interpretation of quantum mechanics.

PACS number(s): 03.65.Bz, 32.80.-t

I. INTRODUCTION

Just after the outcome of quantum mechanics, physicists started to look for a quantum theory of radiation. An approach is that of taking the complex vector as a complex solution of the Maxwell equations. This complex vector is a mathematical representation of the normal electromagnetic waves [1]. It is, however, possible to take the complex conjugate solution of the time-reversed Maxwell equations as a mathematical representation of negative energy radiation.

In concordance with the Stückelberg-Feynman [2-3] interpretation of time reversal, we state that this complex conjugate vector represents positive energy electromagnetic radiation going backward in time. We also assume that negative energy radiation going backward in time is physically equivalent to positive energy radiation going forward in time.

In this paper, we propose to include into the physical theory the idea of negative energy radiation. This idea might apparently seem strange; it provides, nonetheless, a theoretical tool for a reconsideration of the probabilistic interpretation of quantum mechanics.

II. SCHRÖDINGER-DIRAC EQUATION FOR ELECTROMAGNETIC RADIATION

When a radiative transition takes place, we have a changing configuration of electric and magnetic fields. The Maxwell equations, which govern the behavior of this electromagnetic field pattern, can be written in a form that resembles that of the Schrödinger or the Dirac equations. This is essentially what Oppenheimer [1], Landau and Peierls [4], Kursunoglu [5], and some others did. They have introduced the complex electromagnetic vector

In terms of this complex vector, the Maxwell equations

Can be written as a single equation

Introducing the three rotation matrices

and the momentum operator, Maxwell’s equations take the form

This equation is clearly of the same form as the Dirac equation for the electron. Although these three by three matrices are different from the four by four matrices of the Dirac equation, the similarity is still very striking. We may call this the Schrödinger-Dirac equation for the electromagnetic radiation. The physics described by this equation is of course the same as that described by Maxwell’s equations; they are mathematically equivalent.

A recent approach, in which, the electromagnetic radiation is treated as a standard quantum system characterized by the Hamiltonian, was published by W. J. Deal [6]. His paper has the significant title "Quantum Mechanics of Photons." Instead of the vector, Deal introduces the normalized wave function

where is the energy difference between the energies of the two atomic states involved in the transition originating the photon. We prefer to write our equations in terms of h, because we are convinced that the electromagnetic field equations of Maxwell are satisfied at the atomic scale. We also do this in agreement to Lamb [7], who clearly has in mind that it is not convenient to talk about photons. For the understanding of the physical phenomena involving interactions between matter and radiation, it is enough to talk about radiation. By the way, the photons of Deal’s paper are not identical with the quanta of energy we obtain, when we use quantum operators for the electromagnetic field. Deal’s photons are constituted by the electromagnetic field pattern generated by the changing distribution of electronic charges in an atomic radiative transition.

On the other hand, from a purely mathematical point of view, it is also possible to write the Maxwell equations in terms of the complex conjugate vector.

(7)

The Schrödinger-Dirac equation for electromagnetic radiation can then be written as

(8)

This equation differs from the original one (5) in a minus sign. It can also be written as

(9)

which is evidently the time-reversed equation of the complex Maxwell equation (5).

We contend that this time-reversed equation also governs the behavior of a tangible electromagnetic field as real as that described by equation (5). Definitely, in a radiative atomic transition from an upper to a lower state, represents electromagnetic radiation going out from the emitting atom. We claim now that represents electromagnetic radiation going into the atom. The physical effects electromagnetic radiation going into the atom cannot be distinguished from those of a configuration of electromagnetic fields described by the solution of the equation (9). This equation also governs the behavior of electromagnetic fields E and B satisfying normal Maxwell’s equations, but as we shall see later, this configuration of electromagnetic fields has a negative energy density.

We propose, summarizing, that both equations

(10)

represent electromagnetic radiation, but in the interaction between the radiation represented by the solution with the – sign and an atom, energy is transferred from the electromagnetic fields to the atom. We can define Hamiltonian operators for the electromagnetic radiation similar to the Hamiltonians corresponding to material particles

(11)

These quantum Hamiltonians are similar to the classical Hamiltonian. However, in the general quantum case, we have an extra matrix operator , and both signs .

The Hamiltonian of the electromagnetic radiation is similar to that of the relativistic electron

(12)

where, are the four by four matrices introduced by Dirac to linearize the nonrelativistic Schrödinger Hamiltonian. Electromagnetic radiation, contrary to electrons, has no mass, a fact readily shown explicitly in the above equations.

III. MECHANISM OF EMISSION AND ABSORPTION OF RADIATION BY ATOMS

The behavior of an electron making a transition from an upper to a lower stationary energy state is described by the time dependent Schrödinger equation of the system, composed by the atom and the radiation. The Weisskopf-Wigner approximation [8] can be used to show that the charge distribution behaves at any time like a rotating electric dipole. The fields radiated by electronic charge distributions rotating within the atom can be calculated using Maxwell’s equations.

There is no doubt that an outgoing electromagnetic wave radiated by an atom is represented by the complex vector. We think further that an electromagnetic wave with positive energy density going backward in time is equivalent to an electromagnetic wave of negative energy going forward in time. Therefore, we assert that the complex vector represents an electromagnetic wave of negative energy propagating in time from the present to a future time t.

If our interpretation is correct, it can be said that the absorption of ingoing electromagnetic waves by an atom is equivalent to the emission of outgoing electromagnetic waves of negative energy. Former workers have abandoned negative energy solutions on different grounds. We believe nevertheless, that their arguments are not strong enough. Mathematical completeness and a deeper physical insight into the problem of emission and absorption of radiation deserve consideration of these negative energy solutions. These negative energy electromagnetic waves are characterized by the fact that when they interact with electric charges, energy is transferred from the fields to the charges. There is no a priori reason, why this should not be so. Why does nature should have a preferred direction in the transference of energy from matter to radiation? Why it is not possible a process in which energy flows from the electromagnetic radiation to the matter? Our mathematical model definitely allows a description of this reversed process.

IV. MOMENTUM, ENERGY, AND SPIN OF ELECTROMAGNETIC WAVES

The momentum densities for the positive and negative electromagnetic waves are given by

which can also be written as

The Poynting vector of negative energy electromagnetic waves has a direction opposite to that of the propagation vector. As a result, in an interaction between a negative energy electromagnetic wave and an electron, energy is transferred from the radiation field to the atom. Until now, the possibility of occurrence of this phenomenon has been totally excluded from electromagnetism. Based on our next discussions, we believe however that we can improve our present models to explain the interaction between matter and radiation.

The energy densities for positive and negative electromagnetic waves are given by

which up to the , is similar to the classical _expression

We can also find that the spin momentum density of both types of electromagnetic radiation is given by the _expression

where is the total energy of the electromagnetic field. Integrating over all space, we get for the total spin.

As in the Dirac theory of the electron, it can be shown that the Hamiltonians do not commute with the angular momentum. They do commute with the sum of the angular momentum and the total spin.

Our image of the electromagnetic waves spin is essentially the same as that of the electron spin introduced by Ohanian [9]. He has shown that the spin of the electron, consequently also that of the electromagnetic waves, can be regarded as a circulating flow of energy in the wave fields.

V. PHYSICAL MEANING OF THE NEGATIVE ENERGY ELECTROMAGNETIC RADIATION

The fields E and B, which constitute the electromagnetic waves are the fields radiated by the changing distributions, in a radiative transitions, of the electronic charges within the atoms. Their values at different points in space and time can be calculated using the classical electrodynamics.

We have already remarked that, without any doubt, represents outgoing electromagnetic waves radiated by an atom making a transition from an upper to a lower state of energy. What is then the physical interpretation of the complex conjugate solution of the Maxwell equations?

Following the Stückelberg-Feynman interpretation of time reversal, we can say that represents ingoing electromagnetic waves being absorbed by such an atom. These ingoing waves are formally equivalent to outgoing waves of negative energy. We assert then that these negative energy waves are also made up by tangible real electromagnetic fields.

When electromagnetic radiation is emitted by an atom, energy is transferred from the atom to the field. Conversely, when electromagnetic radiation is absorbed by an atom, energy is transferred from the fields to the atom. If this absorption process can correspondingly be viewed as the process of emission of a negative energy waves, then we can easily understand that there are no instantaneous absorption processes. On the other hand, when an atom emits negative energy electromagnetic waves, it gains energy making a transition from a lower to an upper energy state.

Mostly the only effect of negative energy electromagnetic waves is to induce resonant transitions from an upper to a lower state. Analogously the only effect of positive energy electromagnetic waves is to induce resonant transitions from a lower to an upper state. This means that spontaneous emission is not as spontaneous as we are used to believe. We now assert that “spontaneous” radiative transitions are induced by negative energy electromagnetic fields. All positive electromagnetic radiation, emitted by atoms, is emitted by stimulated emission caused by negative energy electromagnetic fields. Positive energy radiation, on the contrary, induces resonant transitions from a lower to an upper state. This also means that absorption, or emission of negative energy radiation, is induced by positive energy radiation.

Negative energy emission means just that positive energy is transferred from all other corpuscles in the universe to the atom. Another possible interpretation is to consider that in this process positive energy is absorbed. This view should be more familiar and completely describable in terms of the normal Maxwell equations (not reversed in time). In this way, the physical interpretation of negative energy states of the electromagnetic field is easier to understand than the negative energy states of electrons. In other words, when an atom starts emitting negative energy radiation, all other bodies in the universe would accelerate coherently. The reason for the coherent response of the rest of the universe, to the stimulated absorption of positive energy electromagnetic waves, is yet to be investigated.

These ideas might seem far-fetched. However, our interpretation has an important advantage. We can disregard the wave-particle dual nature of matter as well as the probabilistic interpretation of quantum mechanics. We allege, as Schrödinger did, that only waves have real existence: particles are simply wave-packets.

VI. DERIVATION OF PLANCK'S RADIATION LAW

The calculation of the spectrum of the blackbody radiation was the first problem of physics in which the need for the quantum hypothesis arose. Our point of view based on negative energy radiation must be consistent with Planck's radiation law.

In our model, matter and radiation interact permanently with each other by emitting positive and negative energy electromagnetic waves. We can say that in thermal equilibrium, atoms are constantly emitting positive as well as negative energy radiation at the same rate. We now emphasize again that the absorption of positive energy electromagnetic waves is equivalent to the emission of negative energy electromagnetic waves. Similarly, the emission of positive energy electromagnetic waves can be considered as the absorption of negative energy electromagnetic waves.

We are convinced that there are only induced processes. Spontaneous emission of positive energy electromagnetic waves can be understood as an emission stimulated by negative energy electromagnetic waves. There is no need to distinguish at all between stimulated and spontaneous emission. Einstein’s [10] hypothesis of stimulated emission of radiation is rather peculiar. It is very difficult to understand how an atom upon which positive energy is impinging tends to decrease its internal energy.

Let us consider a blackbody cavity with superconducting walls. For a system of atoms placed inside it, the Boltzmann distribution defines a value for the temperature of a system of atoms with energy levels and,

(17)

where and are the number of atoms in the upper and lower states, and.

In the blackbody problem, physicists have considered so far that the main observable process is the emission of positive energy electromagnetic waves, which in our view is equivalent to the absorption of negative energy electromagnetic waves. We believe that there are indeed radiation and absorption of negative energy electromagnetic waves. Inside a blackbody cavity, we have not only positive electromagnetic energy but also a certain amount of negative one. If this system is to be in thermal equilibrium, the ratio of the density of positive energy to the density of negative energy should also be equal to the Boltzmann factor. This means that if the spectral distribution of positive energy radiation inside a black-body is given by, then the spectral distribution of negative energy radiation should be given by. Otherwise, we emphasize again; there cannot be thermal equilibrium between matter and radiation. The energy density of radiation of a given frequency is the sum of the positive energy density and the negative energy density. We can say just that positive and negative energy electromagnetic fields cancel each other.

Thus, in thermal equilibrium, the constant number of transitions from the upper to the lower state per unit time is proportional to the difference between the positive and negative energy multiplied by the number of atoms in the upper state. This constant is usually designated by A.

(18)

Hence, we have for the energy density of the positive energy electromagnetic radiation,

(19)

which is the Planck's radiation formula.

VII. CONCLUSION

Contrary to the current belief, we are of the opinion that radiation of positive energy in fact inhibits the emission of radiation, instead of stimulating it. There are some experiments, which can be used to support our view. Lange and Walther [11], using a one-atom maser, have recently observed the suppression of spontaneous emission by atoms in a superconducting cavity. According to our model, waves reflected back into the cavity produces an increase in the energy of the radiating atom, decreasing thereby the rate of radiation, which means a suppression of “spontaneous” emission. Our negative energy radiation model can also be used to interpret other experiments in which the modification is not necessarily a suppression of spontaneous emission, but an enhancement of the atomic radiation.

It is remarkable that in our positive-negative energy model of the radiation-matter interaction, it is not necessary to introduce any point-particle hypothesis of light. The wave fields E and B combined in the complex vector, and its conjugate, are the only physical entities we need for the description of electromagnetic waves.

If our point of view is correct, it is easy to understand how the wave and corpuscular models coexist together. The words, corpuscle and particle, do not mean definitely point-particle. They are words to describe a wave packet. The electromagnetic wave-packets correspond to the point-particles many physicists are used to think of. We can say that corpuscles of light, or photons (with the permission of Lamb !) have finite dimensions, namely the dimensions of these wave-packets.

The only thing we have added is the notion that there exists some anti-electromagnetic waves. We can say that electromagnetic radiation is some kind of matter without mass, to which the principle of matter-antimatter conjugation including negative energy states should also be applied.

The current probabilistic interpretation of quantum mechanics leads to very confusing pictures of the physical processes involved in the emission and absorption of electromagnetic radiation by atoms. It leads to a point-particle model of photons and electrons. Maybe Born [12] himself was not completely satisfied with such an interpretation. His first arguments for the probabilistic interpretation were rather weak, as we see from his words:

“As a trial, I want to pursue the following interpretation: the driving field, represented by a scalar function y of the coordinates of all the involved particles and the time, propagates according to the Schrödinger equation. Energy and momentum are transferred as if corpuscles (electrons) were really flying around. The orbits of these corpuscles are defined only as far as the energy and momentum conservation theorems allow. Besides, for the occurrence of a given orbit only a probability, determined through the distribution of the function y, can be given. It would be possible to summarize somewhat paradoxically: the motion of particles follows probabilistic rules, but the probability itself propagates in concordance with the causality principle.”

After 69 years with the foregoing ‘trial’ interpretation of Quantum Mechanics, the associated duality principle in which the wave and corpuscular models paradoxically coexist is still alive. Born introduced the probabilistic interpretation, because at that moment it was apparently impossible to explain how an electron or a photon wave could be absorbed.

We need neither the idea of a point-particle moving in a definite trajectory nor the instantaneous reduction of the wave packets. The picture of extended photons and electrons is not new. There have been several proposals for extended electrons, and consequently for extended photons, including an outstanding one of Dirac [13].

With our wave field interpretation, based upon a matter-antimatter conjugation, it is easy to come to an interpretation of absorption processes. According to our model, all the interaction processes between matter and radiation are to be described with the electromagnetic waves of positive and negative energy.

Most people have believed that phenomena as the photoelectric effect and the Compton effect require a point-particle model of the photon. Scully and Sargent [14] pointed out however, among other things, that the photon classical electromagnetic wave theory of light, together with a wave theory of electrons, is enough to explain the photoelectric effect. Wentzel [15] was the first physicist to give such a derivation. On the other hand, many other workers, as reviewed in a paper of Kidd, Ardini, and Anton [16], have made alternate derivations of the Compton equation using a pure wave model of light. The cross section for the scattering of radiation by free electrons, given by the Klein-Nishina formula [17], can also be obtained with pure waves.

It remains to clarify how an electron is absorbed in the measurements of scattering cross-sections. In electron absorption processes, it was necessary to introduce a point-particle model, and consequently a probabilistic interpretation, as Born did. We have no doubt, after all, that the detection of a scattered electron is nothing more than a resonant transfer of energy from the scattered electron wave to the bounded electron waves in the detection system. Calculations in this direction are still to be carried out.

Anyhow, if the introduction of the negative energy radiation idea is correct, it would allow us to understand radiation-matter interaction phenomena as pure wave phenomena. Therefore, the probabilistic interpretation proposed by Born to account for particle-like interactions between matter and radiation would not be necessary anymore.
ACKNOWLEDGMENTS

The authors are grateful to D. Buriticá, J. Charris, G. Valencia, M. Balaguera and M. Yoshida for valuable criticisms and helpful discussions.

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