Stochastic Interpretation of Quantum Mechanics Assuming That Vacuum Fields Are Real

Stochastic Interpretation of Quantum Mechanics Assuming That Vacuum Fields Are Real

Emilio Santos. Departamento de Física, Universidad de Cantabria, 39005 Santander, Spain.

Received 27 December 2021 / Accepted 24 April 2022 / Published 4 May 2022. Foundations 2022, 2, 409–442.

Keywords: interpretations of quantum mechanics · realism · stochastic · vacuum fields.

Abstract

We characterize the electromagnetic vacuum as a stochastic field. Some consequences, like the particle behaviour of light, are studied. The stochastic approach is connected with the standard Hilbert space formalism via the Weyl transform. Several experiments involving spontaneous parametric down conversion are studied comparing Hilbert space and Weyl–Wigner formalisms. This allows an intuitive picture of entanglement to be obtained as a correlation between field fluctuations in distant places, involving the vacuum fields. The analysis shows that the Bell definition of local realism is not general enough, whence the reported violation of Bell inequalities does not refute local realism.

1 Introduction

Almost one century after the discovery of quantum mechanics we still lack any consensus about what one is actually talking about as one uses it. “There is a gap between the abstract terms in which the theory is couched and the phenomena the theory enables each of us to account for so well. Because it has no practical consequences for how we each use quantum mechanics to deal with physical problems, this cognitive dissonance has managed to coexist with the quantum theory from the very beginning.”

The discrepancy about the correct approach to the theory appeared very early, two extremes corresponding to the creators of ‘wave mechanics’ (de Broglie, Schrödinger) and those of ‘quantum mechanics’ (Heisenberg, Bohr, Pauli). People in the former group attempted to get a picture of the microworld, without real success. Those in the latter group supported the view that a picture of reality is not needed. The absence of a satisfactory picture, in spite of a big effort by some people, combined with the mathematical elegance of the (Hilbert space) formalism of quantum mechanics plus its spectacular success in the quantitative predictions of empirical evidence, led the mainstream of the community to support the Heisenberg–Bohr view (the Copenhagen interpretation). In recent times, alternative interpretations have been proposed, like ‘many worlds’, bizarre in my opinion, or explanations for the lack of consensus, like QBism. The latter rests on the belief that “the absence of conceptual clarity for almost a century suggests that the problem might lie in some implicit misconceptions about the nature of scientific explanation”. In Section 2 of this article I will provide arguments for both the possibility and the usefulness of a realistic interpretation of quantum mechanics.

I believe that in order to achieve a realistic interpretation, we must assume that quantum vacuum fields are real. On the other hand, a plausible explanation for the stability of atoms, without departing from Maxwell theory, is the existence of a background radiation filling space. The conjunction of the two facts suggests the identification of the background radiation with the quantum vacuum electromagnetic field, taken as a stochastic field. Indeed, relativistic invariance leads to the spectrum of the possible radiation, modulo a unique parameter fixing the scale. If we identify that parameter with the Planck constant, there is agreement between the properties of the assumed background radiation and the quantum vacuum field. Section 3 of the article provides a more detailed exposition of this idea.

In Section 4, we study the characterization of the vacuum electromagnetic radiation as a stochastic field. Section 5 deals with the particle properties of light explained as due to the action of the stochastic vacuum fields. The quantitative connection of the stochastic approach with the standard quantum Hilbert space formalism is made via the Weyl transform, studied in Section 6. Section 7 shows that entanglement and the violation of Bell inequalities may be understood as effects of the vacuum fields. Finally, in Section 8, I offer several ideas for the search of a more complete realistic interpretation of quantum theory.

2 Understanding vs. using quantum mechanics

2.1 Pragmatic approach

None of the interpretations of quantum mechanics proposed till now offer a clear intuitive picture of the quantum world. Nevertheless, most physicists do not worry for the lack of a picture and embrace a pragmatic approach close to the early proposal of Bohr and Heisenberg, usually known as the Copenhagen interpretation.

Behind the pragmatic approach there is usually a philosophical position about physics (or science in general) that may be summarized as follows. It is taken for granted that a physical theory has at least two components: (1) the formalism, or mathematical apparatus, of the theory, and (2) the correspondence rules that establish a link between the formalism and the results of observations or measurements. As an example, let us consider the formalism of quantum mechanics based on the mathematical theory of Hilbert spaces. The formalism involves two kinds of operators, density operators that represent states, and self-adjoint operators that represent observables. The link with the measurement results is given by the postulate that the expectation value, the trace of the density operator times the observable, corresponds to the statistical mean of the values obtained when one realizes several measurements on identically prepared systems by means of an appropriate apparatus.

If we assume that the formalism and the correspondence rules are the only objects required to define a physical theory, in the sense that the statistical regularities need not be further explained, then we get what has been called a minimal instrumentalistic interpretation of the theory. It may be identified with the purely pragmatic approach mentioned above. Most people claiming to support that approach accept the following positions:

1. The notion of an individual physical system ‘having’ or ‘possessing’ values for all its physical quantities is inappropriate in the context of quantum theory.
2. The concept of ‘measurement’ is fundamental in the sense that the scope of quantum theory is intrinsically restricted to predicting the results of measurements.
3. The spread in the results of measurements on identically prepared systems must not be interpreted as reflecting a ‘lack of knowledge’ of some objectively existing state of affairs.

The instrumentalistic approach is quite different from, or even opposite to, the realistic view traditional in classical physics. Between these two extremes there are a variety of approaches.

2.2 Realistic interpretation

The main opponent to the purely pragmatic approach to quantum mechanics was Albert Einstein. Indeed, his discussions with Niels Bohr are the paradigm of a scientific debate, hard in the scientific arguments but hearty from the personal point of view. One of the most celebrated moments of the debate was a 1935 article by Einstein, Podolsky and Rosen. It begins as follows: “Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves” (my emphasis).

I strongly support Einstein’s view, that is, I believe that a realistic interpretation is possible. The main point is the claim that any physical theory should offer a physical model in addition to the formalism and rules for the connection with the experiments. The latter are obviously essential because they are required for the comparison of the theory with empirical evidence, which is the test for the validity of the theory. In my opinion, physical models are also necessary in order to reach a coherent picture of the world. Many quantum physicists apparently support the uselessness of pictures, but it is the case that when they attempt popular explanations of quantum phenomena they frequently propose actual pictures, many of them rather bizarre. For instance, it has been claimed that quantum mechanics compels us to believe that there are a multiplicity of ‘me’ in parallel universes (the many worlds interpretation) or that an atom may be present in two distant places at the same time. This is an indication that the need of “picturing the reality to ourselves” cannot be easily dismissed. Furthermore, the existence of physical models might open the possibility for new developments and applications of quantum theory and, therefore, it is not a purely academic question.

An illuminating confrontation between pragmatic and realistic epistemologies is the conversation of Heisenberg with Einstein that took place in Berlin in 1926, as remembered by Heisenberg himself. The most relevant part is reproduced in the following:

“Einstein opened the conversation with a question that bore on the philosophical background of my recent work. ‘What you have told us sounds extremely strange. You assume the existence of electrons inside the atom, and you are probably quite right to do so. But you refuse to consider their orbits, even though we can observe electron tracks in a cloud chamber. I should very much like to hear more about your reasons for making such strange assumptions’. ‘We cannot observe electron orbits inside the atom’, I must have replied, ‘but the radiation which an atom emits during discharges enables us to deduce the frequencies and corresponding amplitudes of its electrons. After all, even in the older physics wave numbers and amplitudes could be considered substitutes for electron orbits. Now, since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits.’ ‘But you don’t seriously believe’, Einstein protested, ‘that none but observable magnitudes must go into a physical theory?’.”

The conversation continued for a while and at the end Einstein warned: “You are moving on very thin ice. For you are suddenly speaking of what we know about nature and no longer about what nature really does. In science we ought to be concerned solely with what nature does” (my emphasis). Einstein’s arguments are a clear support to a realistic epistemology, and I fully agree with his views about the foundations of quantum physics.

I propose that the difficulties for a realistic interpretation of quantum phenomena do not derive from the empirical facts, or not only. Nevertheless, most textbooks of quantum mechanics emphasize the difficulty, or impossibility, in interpreting typical quantum phenomena with a realistic view. The purpose of this article is to show that, in fact, those phenomena are compatible with a picture of the microworld. Of course, the picture is somewhat different from the one offered by classical physics, but not dramatically different.

3 Vacuum fields, the clue for a realistic interpretation

The belief that the vacuum is not empty has been supported by many people from long ago. It goes back, at least, to the idea of the ether in the 19th century, that apparently was excluded by relativity theory. However, it reappeared with the development of quantum theory. Thus, for instance, de Broglie’s theory or the hydrodynamical interpretation of the Schrödinger equation by Madelung suggest a vacuum that is not empty. This has led to many attempts to derive quantum theory, or at least the Schrödinger equation, from the existence of a subquantum fluid. For instance, in a recent attempt by Sbitnev, the Schrödinger equation is deduced from two equations: continuity and Navier–Stokes. In the latter, the gradient pressure is slightly modified, the extra term describing a change of the pressure induced by a change of entropy.

In this article, I also assume that the vacuum contains several stochastic fields, which precisely correspond to the quantum vacuum fields. Here, I will study only one of them, namely the vacuum electromagnetic radiation. In the following, I show that the stability of matter compels us of, or strongly suggests, the existence of stochastic fields which may be identified with the quantum vacuum. We start discussing two relevant predictions of the vacuum electromagnetic field: the energy and size of atoms and the Casimir effect.

3.1 The stability of atoms rests on vacuum radiation

Soon after the Rutherford experiment of 1911 that led to the nuclear atom, Bohr proposed in 1913 a model that involved postulates contradicting classical electrodynamics. The common wisdom was, and still is, that the contradiction cannot be avoided. That it appears even for the most basic empirical fact, the stability of the atoms. However, this claim is flawed.

Indeed, if studied within classical electrodynamics, a hydrogen atom, consisting of one proton and one electron, cannot be stable if isolated. The reason is that an electron moving around the proton would radiate, and therefore the atom will lose energy until it collapses. However, the argument is not valid if there are many atoms in the universe, because if all atoms radiate, the hypothesis of isolation is not appropriate. It is more plausible to assume that there is some amount of radiation filling space. Then, every atom would sometimes radiate but other times it would absorb energy from the radiation, eventually arriving at a dynamical equilibrium. This may explain, at least qualitatively, the stability of the atom. The moral is that the matter and radiation of the universe cannot be treated independently, and the complexity of the universe compels us to treat the radiation as a background stochastic field. The electron of a hydrogen atom would then move in a random way around the nucleus. I propose that the probability distribution of electron positions is what the Schrödinger wavefunction provides via Born’s rule.

3.2 Spectrum of the vacuum radiation

It is plausible that the statistical properties of the background radiation that we have assumed are homogeneous, isotropic and Lorentz invariant. The most relevant statistical property is the spectrum, defined as the radiation energy per unit volume and unit frequency interval. It is the case that a spectrum compatible with said constraints must be proportional to the cube of the frequency.

Equation (1). The spectrum equals h-bar times the cube of the angular frequency, divided by two pi squared times the cube of the speed of light, where the constant is to be determined in order to fit empirical results. Thus, we will identify it with the Planck constant.

A standard method to study the radiation field in free space is to expand it in plane waves (or in normal modes if it is enclosed in a cavity). Equation (2). In free space, the number of modes per unit volume and unit frequency interval is the square of the angular frequency divided by pi squared times the cube of the speed of light.

Equation (3). Taking Equation (1) into account, the vacuum radiation field is equivalent to an energy of one half h-bar omega per normal mode of the radiation. Equation (3) is just one half the “quantum” of energy introduced by Planck in his pioneer derivation of the radiation law that gave birth to quantum theory. In the following, I will derive some consequences of the existence of vacuum radiation with spectrum Equation (1).

3.3 The energy and size of the hydrogen atom

Via a heuristic approach, it is possible to derive the typical sizes and energies of quantum systems governed by electromagnetic interactions. Let us consider the example of a hydrogen atom consisting of two particles, proton and electron, characterized each by the mass and the electric charge. The proton mass being much larger than the electron mass, we may study the atom assuming that the proton is at rest and the motion of the electron is such that the atom is in a dynamical equilibrium with radiation. In our study of the electron motion, it is plausible that the main interaction with the vacuum radiation takes place via those normal modes of the field that have frequencies close to those of the electron motion. Additionally, the mean kinetic energy of the electron should be close to half the average energy of those normal modes which have the greatest interaction with the atom. As the potential energy is twice the kinetic energy with the sign changed, in view of the virial theorem, the total energy should be the negative of the kinetic energy.

Equation (4). Then, if the electron moved around the nucleus in a circle having energy E (that is, with balanced emission and absorption of radiation), we might write three equalities: the magnitude of the energy equals one half the electron’s kinetic energy and also one half the electrostatic energy at radius r; the speed equals r times the angular frequency; and the magnitude of the energy is of order one half h-bar omega, the latter corresponding to the condition of dynamical equilibrium with radiation.

Of course, the motion is perturbed by the action of the vacuum fields, whence the electron motion would be very irregular, not circular, but it is plausible that Equation (4) might be roughly fulfilled on the average. Equation (5). Hence the energy and the size of the atom may be obtained by removing the speed and the frequency from Equation (4), which gives the energy as minus the electron mass times the fourth power of the charge divided by twice h-bar squared, and the radius as h-bar squared divided by the electron mass times the charge squared — in rough agreement with the quantum prediction and with experiments.

In this example, we have used a heuristic approach; a rigorous stochastic treatment would be more lengthy because it should involve also the vacuum electron–positron field and possibly other fields, something that will not be studied in the present article (however, it is remarkable that the standard quantum formalism allows a relatively simple treatment). It is the case that the study of the vacuum electromagnetic radiation field interacting, via Maxwell–Lorentz laws, with electric charges or macroscopic bodies reproduces several quantum predictions. Indeed, extensive research on this line has been made, which is known as stochastic, or random, electrodynamics (SED). In fact, there are also cases where the SED predictions disagree with quantum mechanics (and experiments), a fact that we may attribute to the neglect of: (1) other vacuum fields, like electron–positron, and (2) the back action of the charges that would modify the vacuum radiation. A case where the SED treatment fully agrees with the quantum one is the Casimir effect that we briefly revisit in the following.

3.4 Stationary states of charged particles: Bohr atomic model

The existence of vacuum radiation suggests a physical picture for the energy spectra of quantum systems, not only for their ground states. In particular for the hydrogen atom, as is shown in the following.

The motion of a charged particle, say an electron, under the action of the vacuum radiation would be very irregular due to the action of the strong high frequency components of the field, see Equation (1). That is, the position may change dramatically in a short time interval, something similar to Brownian motion. In contrast, some memory may be conserved for long times. The latter prediction may be illustrated by the fact that the mean velocity of a free particle does not change with time if only the force due to the vacuum radiation acts on it. Equation (6) shows this: the time average of the velocity equals the initial velocity, because the average force due to the vacuum radiation is nil, that radiation being assumed isotropic, whence all directions of the force would be equally probable. In contrast, when the particle is not free, the evolution of the mean velocity is involved because the actions of a given force and of the vacuum force are not independent.

We conclude that the existence of a vacuum field implies that the instantaneous velocity is not well defined and cannot be measured. However, we might measure the mean velocity during a not too small time interval. Hence, a simultaneous measurement of position and velocity is not possible, which in quantum mechanics is quantitatively stated via the Heisenberg uncertainty relations. In spite of this, we may assume that when a particle follows a path close to classical, although suffering strong shaking, large deviations from the classical orbit might be scarce. Thus, in the hydrogen atom, some orbits of the electron that would be periodic according to classical mechanics would be relatively stable. These orbits have constant angular momentum (they are ellipses or in particular circles). It is not strange that these were the orbits quantized in the Bohr–Sommerfeld model of the atom. The idea was the basis of the “old quantum theory”, where quantization was made in terms of action-angle variables. Of course, the old quantum theory is known to be a semi-classical approximation to modern quantum mechanics.

In the following, I will give arguments that may provide a physical picture for the atomic Bohr model, which rests on two celebrated postulates. The second one is just the assumption, proposed earlier by Planck, that the absorption of radiation takes place in the form of “quanta” with energy equal to h-bar times the angular frequency, Equation (7). A heuristic derivation of this relation will be seen in Section 5.2 below, as due to the interference between any given radiation and the vacuum field.

In order to get a physical picture of the first Bohr postulate, we will study just circular orbits. We may assume that relatively stable orbits have discrete energies, labelled by an integer n, with n = 1 corresponding to the ground state of Equation (5). In fact, the existence of a discrete set of (almost) stable orbits cannot be easily derived from our assumption of a real vacuum radiation, but if we accept this assumption, the full spectrum of energies of the atom may be derived as follows. We shall study transitions between two close orbits taking Equation (7) into account: Equation (8) states that the energy difference between neighbouring levels equals h-bar times the frequency of the absorbed or emitted radiation.

Now it is plausible to assume that the radiation frequency is related to the rotation frequency of the electron, whence we may write Equation (9): that energy difference is approximately h-bar times the mean of the rotation frequencies in the two states. The rotation frequencies may be related to the energies according to classical electrodynamics, using the first three relations of Equation (4), so that the square of the rotation frequency equals eight times the cube of the energy magnitude divided by the electron mass times the fourth power of the charge. Equation (10) then gives a difference equation for the level energies, which provides the set of energies for the stable states in the Bohr model.

An approximate solution of Equation (10) may be easily obtained when n is much greater than one. In fact, we may take the variable n as continuous and substitute a differential equation, Equation (11), for Equation (10). Equation (12). The solution of the differential equation gives the energy of level n as the electron mass times the fourth power of the charge, divided by twice h-bar squared times n squared, where the integration constant is fixed so that the ground state energy agrees with Equation (5). It may be seen that Equation (12) is also an approximate solution of Equation (10) and it is equivalent to Bohr’s first postulate, Equation (13), which states that circular orbits with angular momentum equal to n times h-bar are stable.

Actually, Equations (9) to (11) are not new, but similar equations have been used in the past. Indeed, the fact that the frequency of emitted or absorbed radiation agrees with the rotational frequency of the electron for orbits with large n is well known as a typical example of Bohr’s correspondence principle. More recently, equations similar to our Equation (10) have been used in order to derive the spectrum of the hydrogen atom, for instance in an article by Oks and Uzer which we comment on below.

Whether our approach is classical may be controversial; certainly the assumption of a vacuum radiation fulfilling Equation (1) is alien to classical physics, but other similar derivations cannot be labeled classical either. For instance, in the paper quoted above, the authors used Dirac’s generalized dynamics formalism, where constraints may be included in the Hamiltonian. That is a classical mechanical approach, but it is not classical electrodynamical because the coupling of charged particles with the radiation field is not included. In the application to the hydrogen atom, the quoted authors take into account the electrostatic force between electron and nucleus, but not the full interaction that should include the radiation emitted by any charge according to Maxwell theory. Then, after a number of assumptions without any clear justification, amongst them the obviously non-classical Planck hypothesis, Equation (7), they arrive at a set of static solutions, Equation (14), in which the radius does not change with time. Static solutions are not only counterintuitive, but they violate the Earnshaw theorem, which states that according to Maxwell electrodynamics, no stable state exists for any system of charged particles at rest. In any case, the main purpose of our approach in this paper is to get physical pictures of quantum phenomena, not to make classical-like derivations.

The conclusion of this section is that classical electrodynamics combined with the assumption of a (vacuum) radiation field filling space suggests an intuitive picture for the quantization of the hydrogen atom that might be extended to other quantum systems.

3.5 The Casimir effect

The Casimir effect consists of the attraction between two parallel perfectly conducting plates in vacuum. Equation (15). The force per unit area depends on the distance between the plates: it is minus pi squared h-bar c divided by 240 times the fourth power of the separation, a force confirmed empirically.

The reason for the attraction may be understood qualitatively as follows. In equilibrium, the electric field of the vacuum radiation (that we will label zeropoint field, ZPF) should be nil on any plate surface, otherwise an electric current would be produced. This fact constrains the possible normal modes of the radiation, mainly those having wavelengths comparable to or longer than the separation, but the distribution of high frequency (short wavelength) modes would be barely modified by the presence of the plates. If we assume that an effective cut-off exists at a wavelength of order the separation times a constant K, then the decrease in energy of the ZPF in the space between plates follows from integrating the spectrum of Equation (1) up to the cut-off frequency; the derivative of that energy per unit area with respect to the separation agrees with Equation (15) if K is about 6.

The rigorous SED derivation is similar to the quantum-mechanical calculation, just substituting stochastic averages for quantum vacuum expectations. It consists of determining the normal modes of the radiation when the plates are at a given distance and then attributing a mean energy of one half h-bar omega to every mode. The energy diverges if we sum over all radiation modes, but the force per unit area is finite and it reproduces Equation (15). A regularization procedure is required in order to get the result. The physical picture of the phenomenon is that the radiation pressures on both faces of each plate are different and this is the reason for a net force on the plate. The Casimir effect is currently considered the strongest argument for the reality of the quantum vacuum fields. For us, it is especially relevant because it provides an example of the fact that what matters is the difference between the radiation arriving at the two faces of a plate, rather than the total radiation acting on one side. A similar behaviour will be assumed for photocounters in Section 7.4.

(Section 4, The vacuum radiation as a stochastic field, is omitted for length; it establishes the Gaussian probability distribution of the field amplitudes in each normal mode, and the corresponding distributions for other states of the field. The complete text is at the source.)

5 The particle behaviour of light

In this section, we shall show that the vacuum radiation, taken as a stochastic field, provides hints for a realistic interpretation of the particle behaviour of light.

5.1 What is a photon?

Maxwell theory establishes that light consists of electromagnetic waves. However, this view was allegedly superseded by the proposal that light consists also of particles, later named photons. The wave-particle behaviour is the main mystery of quantum mechanics, in the words of Feynman, and it prevents a clear understanding of the theory. In the following, I provide qualitative explanations for some examples of the particle behaviour of light within Maxwell theory. That behaviour may be understood as being due to the existence of the vacuum stochastic radiation studied in the previous section.

In the year 1900, Planck assumed that energy exchanges between matter and light take place in discrete amounts (“quanta”) of energy related to the frequency by Equation (25), the energy equal to h-bar times the angular frequency. Five years later, Einstein went further, postulating that light itself consists of particles with that energy, whence he derived the law of the photoelectric effect. In fact, the Planck assumption is sufficient to derive that law, without the stronger Einstein postulate. Assuming that when monochromatic light arrives at an appropriate material only one quantum may be absorbed at a time, a part of it used to extract an electron and the rest to supply it kinetic energy, we get the law of the photoelectric effect: the electron’s kinetic energy is the quantum minus the work function, provided the quantum exceeds the work function, and there is no effect otherwise. The constraint that radiation may be absorbed only in amounts fulfilling Equation (25) is the first example of particle-like behaviour, that we will explain qualitatively in Section 5.2.

In the celebrated 1916 article about the absorption and emission of radiation, Einstein arrived at the conclusion that the radiation emitted by an atom possesses well defined momentum, in his words it appears in the form of radiation needles. The two commented claims by Einstein led to the popular belief that light consists of particles (photons) with definite energy and momentum each. Compton experiments in 1923–1924 are commonly viewed as a confirmation of that belief. A semi-quantitative explanation of the radiation needles will be provided in Section 5.3. More recently, experiments have been performed in optics that dramatically exhibit wave-particle behaviour of light. I will comment on them in Section 5.4.

5.2 Understanding quanta: discrete energy exchanges

Firstly, I point out that the absorption of light in the form of localized spots in a photographic plate or clicks in a photodetector are not valid arguments for the particle behaviour of radiation. In fact, the former is caused by the granular (atomic or molecular) nature of photographic plates. The latter derive from the fact that photocounters are manufactured so that they click when the radiation arriving during a detection time surpasses some threshold, which is compatible with light being continuous (waves).

The absorption of light in discrete amounts, fulfilling Equation (25), may be understood as follows. Let us consider a light signal with a given wavevector that arrives at a material having weakly bound electrons. The vacuum radiation may be described in terms of plane waves, and we are interested in those waves with wavevectors near the signal’s. From time to time, it may happen that several of these waves have phases close to the incoming signal, whence they will interfere constructively giving rise to an unusually large intensity during some time T. In this case, a transfer of energy to the material will be more probable, for example an electron may be ejected. We may identify T with the coherence time of radiation consisting of the signal plus the vacuum radiation able to interfere constructively with it. The question is how much energy may be transferred.

The wavevectors of the vacuum field effective for the transfer of energy should be close to the signal’s in order that interference takes place, differing only slightly in modulus or direction or both. It is plausible to identify the angular spread with the ratio of the frequency spread to the frequency, and the coherence time with pi divided by the frequency spread. The effective area for absorption is then of order one half the square of the coherence length times the square of the angular spread, which reduces to one half the square of pi c over omega. The effective intensity — energy per unit area per unit time — is the speed of light times the spectrum times the frequency interval. Equation (26) multiplies these together to give the absorbed energy. Equation (27). Taking the spectrum to be that of the vacuum field, Equation (1), the absorbed energy comes out as pi over four times h-bar omega, in rough agreement with Equation (25).

5.3 Radiation needles and the Compton effect

An interpretation of the needle radiation that appears in the emission of light by atoms is as follows. In our stochastic interpretation, the emission is not spontaneous but induced by the vacuum field (or zeropoint field, ZPF). Then let us assume that in a fluctuation a strong plane wave of the ZPF with some frequency arrives at an atom and it happens that this frequency is also one of the possible frequencies for emission from the excited atom. Then the arriving plane wave component of the ZPF may induce the emission of radiation with the same frequency and phase as the incoming wave. Thus, the emitted radiation should correspond to the addition of the amplitudes (not the intensities) of the incoming plane wave plus the emitted spherical wave. The frequencies being equal there would be interference and it is not difficult to show that it will be constructive in the forward direction and mainly destructive in all other directions.

More quantitatively, the outgoing energy will be concentrated within the region where the phase difference is small. Equation (28) puts the boundary of that region at a half angle of order the square root of the wavelength divided by the distance. If we take that distance to be the coherence length of the emitted “photon”, for typical atomic emissions of about one metre and a wavelength of about one micron, the half angle is about one part in a thousand. This fits with Einstein’s proposal of “needles of radiation” and, in addition, it explains the random character of the direction of emission. In our interpretation, the stochastic character of the ZPF is the cause of the randomness.

This provides the picture of a localized photon as a concentration of radiation energy that nevertheless has a frequency relatively well defined. Furthermore, that frequency is plausibly related to the emitted energy by the Planck equation, taking the arguments of the previous section into account. In any case, the coherence time of the radiation needle cannot be larger than the lifetime of the atomic state.

We may apply that photon model to the case of an atomic cascade where two photons are emitted within a short time interval. Then the picture that emerges is the existence of two “needles of radiation” moving in different directions. In particular, if both the initial and the final state of the atom have zero spin and the photons are emitted in opposite directions, then the angular momenta of the two photons should be opposite by angular momentum conservation, whence they will be strongly correlated in polarization. The quantum formalism predicts that they will be maximally entangled, but I will not provide an interpretation of photon entanglement at this moment, see below Section 7.5. The polarization correlation will diminish if the photons are emitted at an angle smaller than 180 degrees, and this causes that no Bell inequality may be violated in experiments using photon pairs from atomic cascades. Several atomic cascade tests of the Bell inequalities were performed in the decade 1975–1985.

As another example, I propose a semi-quantitative model for the Compton effect. As is well known, Compton’s was the experiment that the scientific community accepted as the final proof of the existence of photons. The experiment is usually understood as a collision between one photon of X-ray and one electron, giving rise to another photon with smaller frequency at an angle with the incident one and a recoil electron. Indeed, the (relativistic) kinematics may be explained assuming that the incident and outgoing photons have energies given by the Planck relation, and the electron is initially at rest. Quantum electrodynamics gives a quantitative account of the phenomenon, including the cross section of the process, but it does not offer an intuitive picture. On the other hand, there have been several attempts at a semiclassical explanation that I will not revisit here.

A stochastic interpretation might be achieved if we substitute radiation needles for photons. A rough model is as follows. Let us consider an incoming monochromatic X-ray beam. By the arguments leading to Equation (27), we may assume that the beam contains radiation wavepackets with energy h-bar omega and momentum h-bar omega over c. From time to time, a large fluctuation of the ZPF may cross the incoming X-ray beam at an angle, in a region where there are weakly bound electrons. If the ZPF fluctuation has an appropriate frequency, it could interfere with the radiation of the X-ray beam producing a concentration of energy in a direction at a smaller angle, which may accelerate one electron in that direction. The electron will radiate with energy and momentum determined by the conservation laws.

5.4 The wave-particle behaviour of light in optics

In quantum optics, the experiments may be usually interpreted in terms of light waves, the particle behaviour being apparent only in photodetection. Detectors will not be studied here in detail, but we may plausibly assume that the particle behaviour of light in detection is usually related to the corpuscular nature of atoms, or electrons in detectors. However, there are cases when this explanation is not sufficient or not appropriate. We will study two examples: anticorrelation after a beam-splitter in the following and entangled photon pairs later.

A simple beam-splitter (BS) may just consist of a slab of transparent material. If a light beam impinges at a point of the slab, a part of the beam intensity is transmitted and another part reflected. The relative intensities of the outgoing fields depend on the refraction index of the material and the angle of incidence. In this way we have an elementary beam-splitter with one incoming channel and two outgoing channels. Actually, we have another incoming channel via a light beam arriving on the opposite side of the slab that gives rise to two new outgoing channels. In practice, the plate is used so that the transmitted light from the first incoming channel is superposed to the reflected light of the second incoming channel, and the light reflected from the former is superposed to the light transmitted from the latter. In this way, we would have two incoming channels and two outgoing ones. In practice, beam-splitters may be more sophisticated, for example involving piles of plates (used in many tests of Bell inequalities). Sometimes, the BS polarizes the light, thus acting as a polarizer or a polarization analyzer.

In the following, I study in more detail a balanced non-polarizing BS. Equation (29). If the field amplitudes of the incoming beams are E1 and E2, then the amplitudes in the outgoing channels are one over the square root of two times (E1 plus i E2), and one over the square root of two times (E2 plus i E1). The imaginary unit is appropriate if we treat the electromagnetic fields in the complex representation, as we will do throughout this article. From Equation (29), it is obvious that the energy is conserved in the BS: the sum of intensities in the incoming channels equals the similar sum in the outgoing ones. In the experiment studied in the following, the field arriving at one of the incoming channels will be a signal and a vacuum field at the other one.

5.5 Anticorrelation-recombination experiment

A dramatic exhibition of the wave-particle behaviour of light is the anticorrelation-recombination experiment. A weak radiation signal, allegedly consisting of well separated photons, is sent to one of the incoming channels of a balanced beam splitter BS1, and two photodetectors, A and B, are placed in front of the outgoing channels. No coincidences are observed, which shows the corpuscle behaviour of light: a photon is not divided, but goes to one of the detectors. If the detectors are removed and the two outgoing radiation beams are recombined via the two incoming channels of another beam splitter BS2, then the detection in one of the outgoing channels depends on the length difference between the two paths from BS1 to BS2, this being a typical wave behaviour.

Our stochastic interpretation is as follows. If we assumed that the vacuum quantum fields were not real fields, then only the signal field entering BS1 should produce outgoing fields, in every one of the two outgoing channels. However, if the vacuum fields are real, there is another (vacuum) field with a frequency similar to the signal entering BS1 via the second incoming channel, and interference is produced. Equation (30) gives the two outgoing fields as the signal and the vacuum field combined with a relative factor of i and divided by the square root of two.

Equation (31). Depending on the relative phases, one of the intensities may be large and the other one small: each outgoing intensity is one half the sum of the signal and vacuum intensities, plus or minus the product of their amplitudes times the cosine of their relative phase. On the other hand, the vacuum intensities on the two channels would ideally be equal.

If we assume that detection is roughly proportional to the part of the arriving intensity that surpasses the ZPF level, then the single rate is one half the average of the signal intensity minus the vacuum intensity, and the coincidence rate is one quarter the average of the square of that difference, minus one half the product of the mean signal and mean vacuum intensities. This result shows that for weak signals, that is when the signal intensity is not much greater than the vacuum intensity, the coincidence detection rate is inhibited — the coincidence rate is much smaller than the product of the single rates, as observed in the commented experiment. (We define the rate as a dimensionless probability of detection per time window).

In contrast, for macroscopic (classical) light the signal intensity is much greater than the vacuum intensity, and the ratio of the coincidence rate to the product of the single rates is the mean of the squared intensity divided by the square of the mean intensity. Hence, if the radiation has fixed (nonfluctuating) intensity, like laser light, that ratio is one, meaning that the detections are uncorrelated. On the other hand, for chaotic light, where the field fluctuations are Gaussian, the ratio is two, meaning that the detections by Alice and Bob are positively correlated. The change from one to two, a phenomenon known as “photon bunching”, has been interpreted as a quantum effect attributed to the Bose character of photons. In our stochastic interpretation, it is the consequence of the correlated fluctuations derived from the Gaussian character of chaotic light.

Equation (32). In the recombination process, the fields of Equation (30) will enter BS2, giving rise in one of the outgoing channels to an intensity equal to one half the sum of the signal and vacuum intensities, plus one half their difference times the cosine of the relative phase due to the different path lengths. The device used in the experiment, consisting of two beam splitters and two mirrors in between, is called a Mach–Zehnder interferometer. The detection rate is proportional to one half the average of the difference of intensities times one plus that cosine, meaning that a 100% visibility may be achieved.

Thus, we have a wave explanation for one of the most dramatic particle behaviours of light, the anticorrelation after a beam splitter. The anticorrelation is usually named “photon antibunching” and it is considered a typically quantum phenomenon, that cannot be explained by classical theories. Of course, it can be explained if we do assume that the vacuum fields are real stochastic fields. The evolution of these fields is classical (Maxwellian), but the assumption of real vacuum fields is alien to classical physics. I stress that the Planck constant appears fixing the scale of the vacuum fields.

(Section 6, Connection with the quantum formalism, is omitted for length; it develops the Weyl transform and the Wigner function, treats the divergence of the vacuum energy, and gives the Weyl–Wigner formalism in quantum field theory together with the correspondence between states of radiation and expectation values in the two formalisms. Sections 7.1 to 7.4 are likewise omitted; they define entanglement and the Bell inequalities, and work through spontaneous parametric down conversion and a stochastic model of the correlation experiment including a model of the photodetector. The complete text is at the source.)

7 Entanglement and Bell inequalities

7.5 Understanding entanglement

The strong correlation exhibited by the comparison of the two coincidence predictions is a consequence of the phenomenon of entanglement and it is labeled strange from a classical point of view. In our stochastic interpretation, it is due to the fact that the signal field produced in the crystal is correlated with the ZPF field that had entered the crystal; similarly for the correlation involving the other signal. That is, the strong correlation appears because the same normal modes of the radiation appear in both fields, the one that goes to Alice and the one that goes to Bob.

Now I shall stress the relevance of the vacuum fluctuations in order to understand the difference between the “classical correlation” and “entanglement”. In the evaluation of the averages we have taken the Gaussian distribution of field amplitudes into account, which gives the fourth moment of each amplitude equal to twice the square of its second moment, Equation (78). Now let us assume that we had used instead a sure (that is, not fluctuating) distribution, Equation (79), a Dirac delta fixing each amplitude. Equation (80). In that case the fourth moment would equal the square of the second moment, and Equation (81) the coincidence probability would factorize into the product of the two single probabilities, meaning that there was no correlation between Alice’s and Bob’s detections. In contrast, a strong positive correlation is obtained if we take the fluctuations into account. This happens when the field is assumed Gaussian, which leads to a stronger correlation.

We conclude that the strong positive correlation associated with entanglement requires that the fluctuations are correlated. That is, the high probability of coincidence detection requires a strong positive correlation between fluctuations of the fields arriving at Alice and Bob, respectively. This leads to a physical (realistic) interpretation as follows: entanglement is a correlation between fluctuations of fields in distant places. In our example, the correlation of fluctuations involves the vacuum fields and might be labeled entanglement between a signal and the vacuum.

7.6 The violation of Bell inequalities

The interpretation of parametric down conversion experiments in terms of stochastic processes, including the vacuum fields, allows local models violating a Bell inequality. This contradicts the wide consensus that Bell’s is the unique local realistic formalism appropriate for experiments measuring correlations between distant parties. Our proof consists of exhibiting a local model for an experiment leading to predictions that violate a Bell inequality. In the construction of the model, we are free to fix the fields produced in the source, but then we should obtain the predictions using classical laws and determining the correlations as in the earlier equations.

Equation (82) proposes a model in which the fields produced in the source are two vectors, each built from the signal and idler amplitudes combined with the vacuum amplitudes, written in horizontal and vertical unit vectors. Now I assume that the first field goes to Alice, who possesses a polarization analyzer at an angle theta with the horizontal in front of her detector; Equation (83) gives the field arriving at her detector, plus some amount of ZPF. I also assume that Bob has a polarization analyzer at an angle phi with the horizontal in front of his detector, so that Equation (84) gives the field arriving at his detector.

The single detection probability for either party comes out as one half the squared coupling, Equations (85)–(89). The calculation of the coincidence probability is more involved, although still straightforward: it proceeds by splitting each intensity into partial intensities, Equations (90)–(91), discarding the terms that do not contribute and the term of higher order in the coupling, and evaluating the surviving expectations of the field amplitudes, Equations (92)–(93). Equation (94). The result is that the coincidence probability equals one half the squared coupling times the squared cosine of the difference of the two analyzer angles.

The predictions of our model may violate the Bell inequality. In fact, we may consider an experiment where Alice measures with her detector when the polarizer is put at either of two angles and, similarly, Bob at either of two angles. The predicted probability of a single count by either is one half the squared coupling. The coincidence probability is given by Equation (94) and the violation of the Bell inequality is produced if the four angles are chosen as zero, pi over four, pi over eight and three pi over eight. Inserting these in the inequality gives, on the left side, the squared coupling, and on the right side the squared coupling times one plus the square root of two, all over two — which is larger, so the Bell inequality is violated. That is, our model agrees with both the standard quantum predictions and a local realistic view of nature. We conclude that Bell inequalities are not necessary conditions for local realism, contrary to the current wisdom.

8 Conclusions

8.1 Quantum states

A big difficulty for a realistic interpretation of quantum theory is that the concepts of state and measurement have been highly idealized. This has led to the attempt of achieving a picture of the quantum world, with the (implicit) assumption that preparations and measurements are simple processes because their mathematical representations in the Hilbert space formalism are simple, that is vectors and self-adjoint operators. However, the physical processes involved are not simple.

Let us study the question of what is a quantum state. In classical physics, the concept of state is certainly simple, it rests on the concept of isolation for either a particle or a wave or any combination of them. However, it is a common view that neither the concept of a particle nor the concept of a wave may be transferred to quantum physics. Thus, the standard answer to the question whether the electron is a particle or a wave is neither. The answer involves a contradiction: anything is either localized (particle) or extended (wave), of course with respect to some reference size, say for an electron compared with an atom. I believe that a more correct answer is that the electron is both. In fact, an electron cannot be seen as an isolated point particle. The physical electron corresponds to a cloud of interacting electrons and positrons, electromagnetic radiation and other fields with a mass m and net charge e. The cloud may have a size possibly as large as the Compton wavelength.

This statement may be put in a different form as follows. The vacuum consists of a set of real fluctuating fields that are modified by the presence of an electron. In this paper, we claim that the fields are real, in contrast with the common opinion that they are virtual (I believe that virtual is a word without any clear meaning that is used in order to avoid commitment with either the assertion that the fields are real or they are not). In summary, in contrast with the classical domain, particles like electrons cannot be seen as having states defined in a manner as simple as in classical mechanics.

We may ask what is the physical interpretation of the state in a more complex system like an atom. It is not just a system of Z + 1 point (or small) particles, that is the nucleus plus Z electrons. In the study of the atom, the nucleus might perhaps be treated as a particle localized in a region far smaller than the atom, but this is not the case for the electrons. What exists is a large number of electrons and positrons that are created (maybe with emission of radiation) or annihilated (with absorption of radiation) in pairs, with a conservation of the total electric charge. Many other quantum fields are likely involved that may correspond to modifications of the vacuum. In summary, I believe that the quantum state of any physical system is a quite complex structure consisting of many interacting fields evolving in time.

Sometimes it is argued that in a nonrelativistic treatment, the possible creation or annihilation of electron–positron pairs should not be taken into account because the energies required are far larger than typical atomic energies. However, the argument is flawed. In classical electrodynamics, the total mass-energy of, say, two electrons plus a positron at extremely small distances may not be greater than the mass of a single electron due to a possibly strong electrostatic negative energy of interaction. In summary, the internal structure of quantum systems like atoms should always be treated taking many (relativistic) quantum fields of the vacuum into account. Of course, this is actually accepted by most people when it is recognized that in renormalization calculations, the bare mass or charge are quite different from the physical ones. The simple change from bare to physical quantities abridges a complicated phenomenon, but the quantum formalism has the virtue that quite complex structures like atoms may be treated using simple equations like Schrödinger’s. That equation is just a (fairly good) approximation.

In this respect, my view is quite different from the common one. I do not believe that quantum equations are exact when we ignore the interaction with the vacuum fields and corrections appear when the interaction is switched on. Interactions with vacuum fields are not small corrections, they are precisely the cause of the difference between classical and quantum physics. Indeed, classical physics is obtained from quantum physics when the Planck constant goes to zero, but that constant is just the parameter that fixes the scale of the vacuum fields (see Section 3), so that setting it to zero means ignoring the vacuum fields.

It is remarkable that quantum theory may be formulated using simple mathematical objects (that is, vectors and operators in a Hilbert space) and relations between them in order to describe very complex phenomena.

8.2 Measurements

Measurements have been still more idealized than states in standard books or papers on quantum mechanics. My opinions on this subject fully agree with Einstein’s. I quote him:

“You must appreciate that observation is a very complicated process. The phenomenon under observation produces certain events in our measuring apparatus. As a result, further processes take place in the apparatus, which eventually and by complicated paths produce sense impressions and help us to fix the effects in our consciousness. Along this whole path — from the phenomenon to its fixation in our consciousness — we must be able to tell how nature functions, must know the natural laws at least in practical terms, before we can claim to have observed anything at all. Only theory, that is, knowledge of natural laws, enables us to deduce the underlying phenomena from our sense impressions. When we claim that we can observe something new, we ought really to be saying that, although we are about to formulate new natural laws that do not agree with the old ones, we nevertheless assume that the existing laws — covering the whole path from the phenomenon to our consciousness — function in such a way that we can rely upon them and hence speak of observations.”

In summary, a realistic interpretation of quantum theory cannot be achieved by attempting to interpret directly the (Hilbert space) formalism. That formalism is a simple, although extremely efficient, algorithm in order to calculate relevant predictions for the results of experiments. In some cases, alternative formalisms may be better in order to get a physical picture of phenomena, even if they are less efficient for calculations. In particular, for the radiation field, the Weyl–Wigner formalism is superior to Hilbert space in this respect.

Funding. This research received no external funding.

Conflicts of interest. The author declares no conflict of interest.

Open Access. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license, https://creativecommons.org/licenses/by/4.0/. Copyright © 2022 by the author. Licensee MDPI, Basel, Switzerland.

(Sections 4, 6 and 7.1 to 7.4, and the reference list of 46 items, are omitted for length; the complete text is at the source.)