Towards a Picture of the Natural World Derived from Relativity and Quantum Theory

Abstract

I contend that physics should provide a coherent account of reality, in addition to being an efficient algorithm for the prediction of empirical results. This article offers pictures of reality derived from theories of modern physics. In particular, it is shown that Bose quantum fields may be interpreted as pure wave fields via the Weyl–Wigner representation, the most relevant result being the existence of a stochastic vacuum field corresponding to the quantum vacuum fluctuations of the standard, canonical, formulation of field theory. That field provides explanations for the particle (photons) behavior of the electromagnetic field. Also, a realistic interpretation is offered for interference experiments with actual particles, like atoms. The interpretation of classical general relativity is standard but emphasis is given to the principle of equivalence. Problems like spacetime singularities (black holes) and the empirical violation of Bell inequalities are touched on but slightly. Asides from these problems, the main incompleteness of the article is the absence of a realistic interpretation of Fermi fields.

1. Introduction

1.1. The Picture of Reality

The aim of this article is to offer a consistent view of physical reality derived from relativity and quantum theory. I shall offer a personal opinion that may disagree with common wisdom in several respects. Also, the paper has a limited scope because dark points still remain. Nevertheless, I hope that the article may stimulate further advances in our view of nature.

The starting point is an epistemology of physics that I support, which was well summarized in the initial paragraph of the celebrated EPR article: “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” [1].

The distinction between reality and concepts means that different theoretical formalisms may exist for the description of a given domain of reality. Therefore, a specific theory, or a particular formalism, may be very efficient in order to derive predictions for the results of experiments, but other theories, or other formulations of the same theory, may be more suitable to get a picture of reality. This is specially true with respect to quantum theory, as I shall discuss in Section 3, Section 4, Section 5, Section 6 and Section 7. However, I claim that getting a picture of reality is an essential aspect of physics, or at least a basic ingredient for a philosophical approach to the natural world.

1.2. The Difficulties of a Realistic Interpretation of Quantum Theory

For the sake of clarity about the purpose of this article, I shall start with comments on the change in the views of the physical world with the advent of relativity and quantum theories. Before the beginning of the 20th century, people believed in absolute space as the framework for all physical processes, and also in absolute time that flows irreversibly; that is, with a future dramatically different from the past. In space, there is matter that consists of atoms; that is, small particles that may remain bound, giving rise to molecules, liquids or solids. The atoms have a random motion that we call heat and it may explain, statistically, the laws of thermodynamics. The motion of bodies under forces is governed by Newton’s laws of dynamics. Two fundamental forces were known: gravitational and electromagnetic, formulated by Newton and Maxwell, respectively. The existence of electromagnetic radiation was known; in particular, it explained the nature of light.

This world view, that defines “classical physics”, was incomplete and, in several respects, wrong. However, it provided a picture of the physical world that was clear; that is, free from internal contradictions. This classical view changed dramatically in the first few years of the 20th century. The theory of relativity modified our view of space and time. Firstly, they were unified in a spacetime that was later declared to have intrinsic curvature and, finally, the curvature was tied to energy. The relativistic world view, although strange or contrary to cherished prejudices, still provided a clear picture of physical reality. Indeed, Einstein’s relativity theory is considered a part of classical physics.

In sharp contrast, quantum theory soon appeared to have dark aspects. In particular, from the beginning it involved a contradiction between the well established wave theory of light and the corpuscle theory proposed by Einstein in 1905. In fact, this assumption, rather than Planck’s quantum of action, was the starting point for the problems of interpretation of quantum theory. These problems did not diminish, rather they increased during the early years of quantum theory and culminated when Heisenberg proposed his “quantum mechanics”, ignoring any attempt to provide a picture of reality, even rejecting it as misleading. In the rest of this subsection, I will further comment on the difficulties for the interpretation of quantum theory.

Quantum mechanics is routinely used in laboratories with great success, but no consensus on its interpretation has emerged [2]. Indeed, many interpretations have been proposed, as may be seen in a recent Oxford Handbook about quantum interpretations [3], which contains articles from more than fifty authors, supporting several different views. In my opinion, the absence of a consensus is a most important problem in fundamental physics. Also, I believe that having an interpretation that enjoyed broad acceptance would be of great practical relevance because its absence slows down progress in several areas of research, e.g., quantum information.

In this paper, I seek an interpretation of modern physics that continues the tradition of classical physics. Rather than attempting a definition of that interpretation, I shall clarify the subject with a paradigmatic example: Newton’s theory of the solar system. The theory began with an intuitive model where the Sun is at the center and there are rotating planets moving around it; that is the old heliocentric hypothesis, which became hegemonic after the observations and arguments of Copernicus and Galileo. This simple model offers a qualitative understanding of many observed facts: the turning of day and night, or summer and winter, the eclipses, and the sea tides. The model was complemented with the ascription of numerical quantities to the various elements of the model, after the careful observations of J. Kepler. The heliocentric model was a description, or picture, of reality that became a physical theory when Newton achieved a wonderful synthesis of the observed facts using his celebrated laws of dynamics and gravity, respectively. They allow an accurate quantitative account of the observations and measurements, with the possibility of predicting future events, like the dates of eclipses. In summary, the solar system theory consists of (1) a picture of reality and (2) Newton physical laws.

The solar system theory may be compared with the quantum-mechanical atom, where the theory provides rules for the prediction of empirical results but not a clear picture of reality. In fact, our knowledge about the atom is plagued with dark points. For the hydrogen atom, the quantum formalism of Schrödinger provides a wavefunction, $\psi \left(\mathbf{r},t\right),$ but the actual meaning of $\psi$ is unknown. Max Born proposed that ${\left|\psi \right|}^2$ gives the probability of finding the electron at $\left(\mathbf{r},t\right)$ in a position measurement. But Born’s proposal is just a rule that allows predictions about the result of eventual human observations. Quantum mechanics is treated as an algorithm for the prediction of empirical results, but it does not provide a clear picture of reality. For instance, it does not inform whether the electron is in a position at every time, or whether it becomes localized by the act of measuring the position.

Einstein was aware of the problem, as shown by a dialogue with Heisenberg that took place in 1926 [4]. Einstein opened the conversation with a question: “What you have told us sounds extremely strange. You assume the existence of electrons inside the atom...But you refuse to consider their orbits”. The conversation continued for a while and, after Heisenberg’s arguments, 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”. Heisenberg arguments were the seed of the Copenhagen interpretation of quantum mechanics, mainly supported by Niels Bohr. This considers quantum mechanics as just an algorithm for the prediction of the results of observations or measurements (an extremely efficient one, in fact), but it does not provides a picture of the world. In this article I support the criticism of Einstein, and seek a world picture resembling the classical one, but resting on modern physics. In order to give a short name to the sought interpretation, I will use the word “realistic”, although I am aware that the choice of name may be controversial. However, the word realism has become popular in the context of the Bell inequalities, which may justify the election.

In my view, there are two main difficulties in terms of reaching a satisfactory (realistic) interpretation of quantum theory: (1) the abstract character of the standard (“canonical”) formulation in terms of vectors and operators in Hilbert space, and (2) the wave-particle duality. The solution to the former problem is the use of an alternative formalism, which I propose to be the Wigner (or Weyl–Wigner) representation for Bose quantum fields; see Section 3.2 below. However, I have not yet a similar proposal for Fermi fields. Indeed, I believe that relativistic quantum fields are the fundamental inhabitants of our world, while non-relativistic quantum mechanics is an approximation of field theory. Therefore, the interpretations of elementary quantum mechanics should come later than that of fields, which will be the subject of Section 3.6 below. The wave-particle duality will be discussed in more detail in Section 6.

In summary, the mainstream of the physicist community believe that quantum mechanics does not admit an interpretation resembling that of classical physics. Nevertheless, I am convinced that it is, in fact, possible. This article discusses different cases where a realistic (classical-like) interpretation is feasible. Unfortunately, my proposal is incomplete, the lack of a realistic interpretation for Fermi quantum fields being the most relevant deficiency.

1.3. Plan of the Article

This paper includes a review of my work on quantum interpretation from about the year 2020 [5]. I shall start with the interpretation of (classical) relativity because it provides the theory of space and time, which are the framework for all phenomena in the material world. Quantum theory is currently perceived as more fundamental, hence, people have attempted to treat (general) relativity in the framework of quantum theory; for instance, attempting to “quantize gravity”, rather than trying to develop a quantum field theory in the framework of curved spacetime. For example, assuming that spacetime should be treated as stochastic, the randomness gives rise to a stochastic behavior of the fields via the Einstein equation (indeed, my belief is that quantum fields are actually stochastic fields). There are historical reasons for the perception of a quantum supremacy. Firstly, the great predictive power of quantum theory, which has been tested more times and with higher precision than general relativity. In particular, quantum electrodynamics is the most accurately tested theory of physics, leading to spectacular quantitative agreement with empirical results. Secondly, quantum theory is the basis for recent technological developments, while the applications of general relativity are almost always restricted to astrophysics and cosmology. However, I think that quantum theory must be studied in the framework of general relativity and not the opposite way around.

In the following Section 2, I will offer the picture of space and time that, in my view, emerges from relativity, and I shall postpone the interpretation of quantum theory for later sections. Section 3.1 provides arguments supporting my opinion that it is convenient to start with quantum fields, rather than non-relativistic quantum mechanics, in order to obtain a picture of reality. Thus, after a brief review of the Wigner representation in Section 3.2, and a digression on the canonical quantization, Section 3.4 deals with a realistic interpretation of the quantized electromagnetic field. In Section 3.5, I shall discuss several consequences. In Section 3.6, I revisit the quantization of other Bose quantum fields, but it is pointed out that a similar (realistic) approach does not yet exist for Fermi fields. After that, the quantization of general relativity is discussed in Section 4, and quantum particles in non-relativistic motion are studied in Section 5. Section 6 deals with the realistic interpretation of the wave-particle duality, in particular the phenomena that apparently prove the existence of particles of light (“photons”). Finally, in Section 7 I briefly discuss the phenomenon of quantum “entanglement” and comment on the Bell inequalities. General conclusions of the article are presented in Section 8.

Asides from the lack of interpretation for Fermi quantum fields and the absence of a clear interpretation of quantum gravity, there are two criticisms to this article that I foresee. Firstly, in relation to the matching of general relativity with quantum theory, the arguments offered in the present paper would fail in some cases; namely, when there are spacetime singularities (black holes). Secondly, my view about non-local phenomena (like the alleged empirical violation of the Bell inequalities) is contrary to common opinion in the physicist community. These criticisms are sound and I will comment on them in Section 4.3 and Section 7.2, respectively.

2. Space and Time in Relativity Theory

2.1. From Newton Absolute Space to the Spacetime of Special Relativity

The view of space and time has been dramatically modified with the advent of relativity theory. Before the beginning of the XX Century, the prevailing opinion was that space and time were absolute realities, independent of matter and of the means of observation. Contrary to that belief, relativity theory began with the assertion that both space and time should be treated as means to describe properties of matter that depend on, or are relative to, the observer. Hence, the name “relativity” for the theory. It is paradoxical that, in the final form of the theory, i.e., general relativity, spacetime became the most fundamental reality of the material world; although its geometry is closely tied to matter, it is in the form of quantum fields.

The question about absolute space became relevant when Newton formulated the laws of mechanics, proposing that the acceleration is proportional to the applied force. This led to the question, acceleration with respect to what? Newton attempted to give an answer with the celebrated experiment involving a bucket filled with water. When the water rotates, it is observed that the center of the bucket is depleted with respect to the periphery. It was observed that this centrifugal effect depends on the rotation of water, but it is independent of the rotation, or not, of the bucket. This fact led Newton to assume the existence of an “absolute space” with respect to which the motion should be defined. (For a historical account with quotes by Newton himself and other authors, including Einstein, see [6]). However, the need for an absolute space may be avoided if we assume that “the relevant acceleration is relative” to the average mass of distant matter (e.g., galaxies), as proposed by Ernst Mach [7]. The “Mach principle” had a important influence on Einstein’s thinking [8].

Absolute time had been supported even more strongly as a result of the dramatic experience of human beings with the irreversible course of our (human) life, from youth to old age. In Einstein’s special theory of relativity (of 1905), absolute time was the most dramatic rejection. Indeed, the theory includes the possibility that the time lapse between two events, say A and B, depends on the observer. Furthermore, it may be that event A precedes event B for one observer, but B precedes A for another observer in relative motion with the former. However, in 1908 Minkowski showed that the mathematical structure of special relativity may be seen as the substitution of a unified spacetime for separated space and time. Events, i.e., points in spacetime, may be represented by four real numbers $\left\{x,y,z,t\right\}$ with the property that for two different events, A and B, both their distance $d_{AB}$ and time lapse $t_{AB},$ that is (assuming Cartesian coordinates)

$$d_{AB}=\sqrt{{\left(x_A-x_B\right)}^2+{\left(y_A-y_B\right)}^2+{\left(z_A-z_B\right)}^2},t_{AB}=t_B-t_A,$$

(1)

may be “relative”, but there is a quantity, the interval, which is “absolute”, and independent of the observer. It is defined by

$$I=\sqrt{\left|d_{AB}^2-t_{AB}^2{c}^2\right|},$$

(2)

where c is the speed of light. Space and time were unified as spacetime; mathematically, a pseudo-Euclidean variety with four dimensions. In the words of Einstein “physics became the study of a 4-dimensional object, rather than the evolution of a 3-dimensional object” [8]. The spacetime, defined so far (that is, according to special relativity), is named “Minkowski space”. Relevant properties of that space are summarized as follows.

The free (inertial) motion corresponds with the position, r, of a particle changing in proportion to time t, that is

$$\mathbf{r}={\mathbf{r}}_0+\mathbf{v}t,\left|\mathbf{v}\right|\le c,$$

(3)

where c is the speed of light in vacuum. (Note that in this equation and in the rest of the paper we use boldface for vector quantities and italics for scalars). The existence of an upper limit to the possible velocities is one of the main consequences of relativity theory. Equation (2) implies that for a body in inertial motion, in particular at rest, the following quantity defined between two given events (points in Minkowski space) A and B, is invariant

$${\tau }_{AB}\equiv \sqrt{t_{AB}^2-d_{AB}^2{c}^{-2}}=t_{AB}\sqrt{1-{\displaystyle \frac{v^2}{c^2}}},$$

(4)

that is, it is the same for all inertial observers. It is named “proper time”, and it corresponds with the time measured by an observer at rest with respect to the body. The time $t_{AB}$, seen by another inertial observer, is never smaller than the proper time ${\tau }_{AB}$.

Proper time may be defined more generally, for any body with arbitrary motion between events A and B placed at a distance $d_{AB},$ via the integral

$${\tau }_{AB}={\int }_{t_A}^{t_B}\sqrt{1-{\displaystyle \frac{{\left|\mathbf{v}\left(t\right)\right|}^2}{c^2}}}dt,d_{AB}=\left|{\int }_{t_A}^{t_B}\mathbf{v}\left(t\right)dt\right|.$$

(5)

A consequence of this definition is that the proper time ${\tau }_{AB}$ between two given spacetime points is a maximum for an observer moving with constant velocity, $\mathbf{v}$; that is, when the motion is inertial.

For a correct understanding of general relativity, a historical digression is convenient, which is made in the following.

2.2. In Search of a Relativistic Field Theory of Gravity

The origin of general relativity was the search for a relativistic theory of gravity. In fact, Newton’s law

$$F_{grav}=G{\displaystyle \frac{Mm}{r^2}},$$

(6)

does not fit in special relativity; indeed, it predicts instantaneous actions at a distance. Then, searching for a relativistic theory of gravity was one of the main scientific goals after Einstein’s special relativity of 1905. In order to see the difficulties with regard to that aim, let us consider the formulation of Newton’s gravity as a field theory.

The concept of field of force had been introduced by M. Faraday, in around 1840, in relation to electromagnetism, in order to avoid the problem of actions at a distance that had troubled Newton. Applying the concept of field to gravitation, from Equation (6) we may get the gravitational potential $\varphi \left(\mathbf{r}\right)$ and the field intensity $\mathbf{g}\left(\mathbf{r}\right)$ created by a mass distribution $\rho \left(\mathbf{r}\right),$ as follows

$$\varphi \left(\mathbf{r}\right)=-G\int d^3{\mathbf{r}}^{\prime }\rho \left({\mathbf{r}}^{\prime }\right){\left|{\mathbf{r}}^{\prime }-\mathbf{r}\right|}^{-1},\mathbf{g}\left(\mathbf{r}\right)=-▽\varphi \left(\mathbf{r}\right),$$

(7)

where $▽\equiv \left(\partial /\partial x,\partial /\partial y,\partial /\partial z\right).$ The gravitational force $\mathbf{f}$ on a (pointlike) test particle with mass m placed at $\mathbf{r}$ would be

$${\mathbf{f}}_{grav}=m\mathbf{g}\left(\mathbf{r}\right).$$

(8)

The acceleration of the particle by the gravity force changes its kinetic energy; hence, conservation of total energy implies that the field itself should possess energy, which may be transferred to the particle. In fact, we must ascribe a negative energy density ${\rho }_{grav}\left(\mathbf{r}\right)$ to the gravitational field as follows

$${\rho }_{grav}\left(\mathbf{r}\right)=-{\displaystyle \frac{1}{2G}}{\left|\mathbf{g}\left(\mathbf{r}\right)\right|}^2.$$

(9)

We might compare the passage, from Newton’s non-relativistic theory of gravity, to a relativistic theory, with the route from the non-relativistic Coulomb law of electrostatic to the relativistic Maxwell theory of electromagnetism. Of course, that process required a lot of experimental and theoretical work, which lapsed for most the 19th century. Also, Maxwell’s theory preceded Einstein’s special relativity, but, conceptually, the latter may be considered the framework for the former.

The relevant point is that Newton’s theory of gravity, as well as both Coulomb’s and Maxwell’s theories, are linear in the sense that the effects (attraction or repulsion forces) are proportional to the causes (masses or electric charges and currents, respectively). However, any relativistic generalization of Newton’s gravitational theory could not be linear. In fact, special relativity established that energy E is related to mass m via

$$E=m{c}^2,$$

(10)

c being the speed of light. Therefore, a relativistic theory of gravity should take into account the (negative) mass-energy of the field Equation (9). Hence, the gravitational field $\mathbf{g}\left(\mathbf{r}\right),$ Equation (7), should be replaced by the following one

$$\mathbf{g}\left(\mathbf{r}\right)=-\int d^3{\mathbf{r}}^{\prime }{\left|{\mathbf{r}}^{\prime }-\mathbf{r}\right|}^{-1}\left[G\rho \left({\mathbf{r}}^{\prime }\right)-c^{-2}{\left|\mathbf{g}\left({\mathbf{r}}^{\prime }\right)\right|}^2\right],$$

(11)

where the latter term comes from the energy of the field itself. Actually, that term is usually negligible, hence, Equation (7) may be a good approximation for Equation (11). However, from a fundamental point of view, that term also gravitates; hence, the equation should be modified again, giving rise to the appearance of another term of order $c^{-4}$, and this would give rise to a term of order $c^{-6},$ and so on. This shows the non-linear character of the relativistic gravity theory that is wanted, which gives rise to big difficulties.

We may compare Equation (8) with the electrostatic force from another point of view. We can write the electric force in the form

$${\mathbf{f}}_{electric}=q\mathbf{E}\left(\mathbf{r}\right),$$

(12)

where q is the electric charge of a particle and $\mathbf{E}\left(\mathbf{r}\right)$ the electric field. In both cases, electrostatic and gravitational, the equation of motion for a particle should be Newton’s second law of mechanics, that is

$$\mathbf{f}=m\mathbf{a},$$

(13)

where $\mathbf{a}$ is the acceleration. The point is that in the electric case the force depends on the electric charge $q,$ but the acceleration in terms of the force involves a different parameter, the mass m. In sharp contrast, in the gravitational case, the mass appears in both the force Equation (8) and the law of motion Equation (13), which looks somewhat strange. This peculiar fact may be stated by saying that all bodies experience the same acceleration in a gravitational field because, from Equations (8) and (13), we may get

$$\mathbf{a}=-\mathbf{g}.$$

(14)

In particular, all bodies fall with the same acceleration near the Earth’s surface (neglecting perturbations by the air), as had been discovered by Galileo, and which was essential for Newton’s proposal of a universal gravitational law, Equation (6). It is universal because it governs both the motion of celestial bodies, like that of the moon around the earth, and the motion of bodies near the earth’s surface.

In 1907, Einstein realized that Equation (14) might be a fundamental law of nature, that is “the gravity field is equivalent to an acceleration”, which is known as the Principle of equivalence. Einstein interpreted the equivalence principle as stating that (1) the motion under gravity is just the “natural free motion”, i.e., purely inertial, which explains Galileo’s discovery, and (2) the inertia at a point is determined by the distribution of masses around it. Einstein considered the equivalence principle, and his interpretation, to be the “glücklichste Gedanke meines Lebens” (the happiest though of my life) [8]. The principle requires, or strongly suggests, that spacetime is curved; hence, inertial motion ceases to be a straight line with constant velocity, which might explain the motion of bodies under “gravity”. In the years from 1907 to 1915, Einstein studied Riemann’s theory of manifolds with curvature; he worked hard and finally arrived at general relativity, “the most beautiful theory of physics”, in the words of Lev Landau.

2.3. Spacetime and Matter in General Relativity

General relativity (GR) introduced three innovations: (1) it reinforced the relevance of spacetime as a fundamental framework, (2) it led to the need for spacetime curvature, and (3) it strongly tied spacetime with matter. These innovations gave rise to the picture that follows.

In contrast to the (flat) Minkowski spacetime of special relativity, in GR, the spacetime is a variety with intrinsic curvature [9], as described in the following. Events, i.e., points in spacetime, may be represented by four coordinates, the real numbers $\left\{x^1,x^2,x^3,x^4\right\}.$ Then, curvature may be derived from the expression of the infinitesimal interval $ds$ amongst two close events in terms of the coordinates, that is

$$d{s}^2={\displaystyle \sum _{\mu =1}^4}{\displaystyle \sum _{\nu =1}^4}{g}_{\mu \nu }\left(x^1,x^2,x^3,x^4\right)d{x}^{\mu }d{x}^{\nu },$$

(15)

which is known as the metric. Actually, $ds$ corresponds with an infinitesimal proper time, as defined in Equation (4).

The method to determining curvature via the metric was introduced by the mathematician Carl F. Gauss for surfaces. The points of a surface may be determined by two coordinates (x,y), and the metric $dl,$ which gives the infinitésimal distance between two points, and may be written in terms of these coordinates, that is

$$d{l}^2=A\left(x,y\right)d{x}^2+B\left(x,y\right)d{y}^2+C\left(x,y\right)dxdy,$$

which can be compared with Equation (15) for four coordinates of spacetime. From the functions A, B, C, Gauss derived a single number, named curvature of the surface at the point (x,y), which corresponds with $1/R^2,R$ being the radius of the sphere most close to the surface near the said point. For instance, if A and B are constant (independent of x,y) and C = 0, Gauss curvature is nil (i.e., $R\to \infty )$ and the surface is said to be flat. Bernhard Riemann generalized Gauss’ theory for varieties with N dimensions, and Einstein used Riemann’s theory for four-dimensional spacetime.

In Riemann’s theory, combining the elements of the metric tensor $g_{\mu \nu },$ Equation (15), with their first and second derivatives with respect to the coordinates, it is possible to get the Riemann tensor $R_{\mu \nu \lambda \sigma }$, whose nil value is a necessary and sufficient condition for zero curvature, the spacetime being flat (Minkowski) in that case. On the other hand, matter (here the word matter includes radiation) is characterized by quantities like energy, momentum and angular momentum, that may be summed up via an energy–momentum tensor $T_{\mu \nu }^{matt}$. The fundamental equation of general relativity, Einstein’s equation, relates curvature with the energy–momentum tensor of matter and radiation $T_{\mu \nu }^{matt}$, that is

$$R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R\equiv G_{\mu \nu }=-8\pi G{T}_{\mu \nu }^{matt},$$

(16)

where G is Newton’s constant of gravity. The left side $G_{\mu \nu },$ named Einstein tensor, involves the Ricci tensor $R_{\mu \nu }$ and the scalar R, which may be derived from the Riemann tensor, that is

$$R_{\mu \nu }=\sum _{\lambda \sigma }{g}^{\lambda \sigma }{R}_{\lambda \mu \sigma \nu },R=\sum _{\mu \nu }{g}^{\mu \nu }{R}_{\mu \nu }.$$

(17)

A popular understanding of Equation (16) may be summarized by the Wheeler slogan: “Spacetime tells matter how to move, matter tells spacetime how to curve”. Indeed, spacetime curvature is characterized by Einstein’s tensor $G_{\mu \nu }$ and matter (including radiation) by the energy–momentum tensor $T_{\mu \nu }^{matt}.$ In the next section, I shall propose a conceptual simplification of this view of general relativity.

I stress, again, that gravity is not a force according to GR, but that the acceleration of the body’s motion is a consequence of spacetime curvature. In fact, the inertial motion corresponds with a path in spacetime that may be parametriced by a parameter s, via four functions $x_j\left(s\right),$ j = 1, 2, 3, 4, with the condition that the proper time is a maximum between an initial event, $s=s_a,$ and a final one, $s=s_b,$ that is

$$s_{ab}\equiv {\int }_{s_a}^{s_b}ds=\mathrm{maximum}.$$

This is a generalization to curved space of the condition stated after Equation (5) in flat (Minkowski) space. The motion that maximizes proper time may be labeled a geodesic in spacetime. Thus, free motion follows spacetime geodesics.

Let us comment on the fact that Einstein’s tensor is nil in regions without matter, but Riemann’s tensor may be finite (not nil) in those regions. In fact, the nil value of the Riemann tensor $R_{\lambda \mu \sigma \nu }$ is a necessary and sufficient condition for the absence of curvature, but, in contrast, the nil value of Einstein tensor $G_{\mu \nu }$ is necessary but not sufficient. The reason for that is a peculiar property of the Riemann curvature in four dimensions, namely the Riemann tensor may be written as a sum of two tensors, as follows (see e.g., [9] ),

$$\begin{array}{ccc}\hfill R_{\lambda \mu \sigma \nu }& =& M_{\lambda \mu \sigma \nu }+C_{\lambda \mu \sigma \nu },\hfill \\ \hfill M_{\lambda \mu \sigma \nu }& =& {\displaystyle \frac{1}{2}}\left(g_{\lambda \sigma }{R}_{\mu \nu }-g_{\lambda \nu }{R}_{\mu \sigma }-g_{\mu \sigma }{R}_{\lambda \nu }+g_{\mu \nu }{R}_{\lambda \sigma }\right)\hfill \\ & & -{\displaystyle \frac{R}{6}}\left(g_{\lambda \sigma }{g}_{\mu \nu }-g_{\lambda \nu }{g}_{\mu \sigma }\right).\hfill \end{array}$$

(18)

$C_{\lambda \mu \sigma \nu }$ is named the Weyl tensor and it does not contribute to the Einstein tensor, that is, $G_{\mu \nu }$ may be got by putting $M_{\lambda \mu \sigma \nu }$ in place of $R_{\lambda \mu \sigma \nu }$ in Equation (17), as may be easily checked. This fact may be stated by saying that the curvature (Riemann) tensor is in part energy–momentum (that is, a linear function of the Ricci tensor) and in part “gravitation” in itself (Weyl tensor). This may be seen by putting Equation (16) in Equation (18), which gives

$$\begin{array}{ccc}\hfill M_{\lambda \mu \sigma \nu }& =& -4\pi G\left(g_{\lambda \sigma }{T}_{\mu \nu }-g_{\lambda \nu }{T}_{\mu \sigma }-g_{\mu \sigma }{T}_{\lambda \nu }+g_{\mu \nu }{T}_{\lambda \sigma }\right)\hfill \\ & & +{\displaystyle \frac{4GT}{3}}\left(g_{\lambda \sigma }{g}_{\mu \nu }-g_{\lambda \nu }{g}_{\mu \sigma }\right),\hfill \end{array}$$

(19)

where G is Newton’s constant, $T_{\mu \nu }$ the momentum–energy tensor, and $T=\sum g^{\rho \tau }{T}_{\rho \tau }.$

This allows a nice interpretation for the strange “action at a distance” of gravity. The pre-relativistic solution to the problem was the introduction of the concept of field of force, Equation (9), but it gave rise to difficulties, as commented on in Section 2.2. In sharp contrast, in GR the Weyl tensor appears naturally when Einstein’s Equation (16) is solved (integrated). Therefore, the curvature, and, consequently, the motion of matter, is determined by the Weyl tensor in regions without matter (nor radiation) where the tensor $T_{\mu \nu }$ is nil and, consequently, it is also the case that $M_{\lambda \mu \sigma \nu }=0$. This is the case for the instance in the exterior of spherical bodies like the Earth where the inertial motion consists of elipses that are geodesics in spacetime, but the Einstein tensor $G_{\mu \nu }$ is nil there. We may say that the Weyl tensor plays, in GR, the role of the gravitational field in Newtonian theory, but in GR it appears naturally while in Newtonian gravity it is an ad hoc supplement.

Integrating Equation (16) means getting the metric elements as functions of the coordinates; that is, determining the functions $g_{\mu \nu }\left(x_1,x_2,x_3,x_4\right)$ for some region defined by appropriate boundary conditions (e.g., the whole of checked meaning retained space, assuming that curvature goes to zero at infinity). The integration requires that the metric tensor $g_{\mu \nu }$ should be twice derivable with respect to the coordinates $\left\{x^1,x^2,x^3,x^4\right\}.$ (In view of this requirement for the derivability of the metric tensor, it is not strange that Einstein was reluctant to believe in the existence of actual singularities [8], which, nevertheless, at present are assumed to exist in black holes, see Section 4.3.

2.4. Is General Relativity a Field Theory of Gravity?

In Maxwell’s electromagnetic theory, the field appears as something real. The typical example is light, which is just an electromagnetic field. It travels from the source to the detector (say, from the Sun to the eyes of people on Earth). From Maxwell time, the relevance of fields has increased and the current belief is that the inhabitants of the universe are Relativistic Quantum Fields. Therefore, Quantum Field Theory (QFT) has become the fundamental theory of nature. It has led to predictions that have achieved a truly spectacular agreement with empirical data, specially in the domain of Quantum Electrodynamics. This fact has given rise to the widespread opinion that general relativity should be quantized like all other known fields of the standard theory of fundamental particles, hence the attempt to treat GR as the relativistic field theory of gravity. I do not agree: I believe that GR is not a field theory of gravity, but the denial that gravity is a field. It is a theory of spacetime and its relation with matter.

In my view, GR might be a field theory of gravity if the quantities $g_{\mu \nu }$ were interpreted as potentials of the field, this property being unrelated in principle to spacetime curvature. If this were the case, it would be plausible to have a different tensor determining the metric. This possibility has been studied, for instance, by A. A. Logunov in his “Relativistic theory of gravity” [10]. I believe that the interpretation of Einstein’s Equation (16), with the double role of determining the spacetime curvature and being a gravitational field, is unnecessary and destroys the conceptual simplicity of GR. My point of view is that GR is quite different from the relativistic fields of high energy physics.

It is true that the study of systems with strong energy density requires a joint treatment of spacetime curvature and quantum features, as is the case in spacetime singularities (in black holes and the very early universe). This has given rise to a research program known as “quantum gravity”. However, I believe that what is needed is an appropriate quantization of spacetime, as I will discuss in Section 4, but not a quantization of gravity because “gravity” is just the name given to the fact that spacetime has intrinsic curvature, a fact that modifies the evolution of fields.

2.5. A Geometrical View of Mechanical Quantities

General relativity allows us to connect the field equations with the spacetime curvature via three steps. Firstly, there are equations that provide the mass-energy contents of the fields in the form of an energy–momentum tensor $T_{\mu \nu }.$ Then, Equation (16) relates that tensor to Einstein’s $G_{\mu \nu }$. Finally, the mathematical theory of Riemann relates Einstein’s tensor to the metric tensor. I argue that we can make a conceptual simplification, reducing the steps from three to two. We cannot remove the former and the latter steps, but we may remove the second step because Einstein’s equation is just an equality of two tensors modulo the Newton constant G. Thus, it is enough to take Einstein equation as an identity, rather than an equality relating different concepts in order to reduce the mentioned steps from three to two. With that view, Equation (16) might be seen as a kind of “dictionary” that translates from mechanical language (the energy–momentum tensor) to geometrical language (Einstein’s tensor). The dimensional (Newton) constant is needed because, for historical reasons, we use units of length (and time, when the speed of light is $c=1$) in the metric tensor, and, therefore, in Einstein’s tensor too, which are different from the units used in the energy–momentum tensor (mass, energy, momentum or pressure).

Of course, I do not mean that Einstein equation is a trivial discovery. It involves the highly non-trivial assumption that spacetime is curved, which strongly changes our view of the world. What I mean is that GR allows a beautiful geometrical interpretation of mechanical concepts like energy, momentum and angular momentum. Alongside the historical development of physics, people introduced these concepts, which were fundamental for the development of physics, but now we have a charming interpretation for them thanks to Einstein’s equation. I argue that they are forms of spacetime curvature.

The proposal to change the interpretation of Equation (16) from equality to identity may seem a mere semantic issue, but it allows a better understanding of our world. Firstly, it makes more compelling the opinion that there are not four fundamental forces in nature, just two, because gravity is not a force and electromagnetism is unified with the weak interactions; or only one, if electroweak forces were unified with the strong interaction. Secondly, it simplifies the picture of the world because it reduces the number of concepts needed for the description, the dynamical variables not being primitive but geometrical concepts.

The picture that emerges is that the world consists of fields in a curved spacetime, possibly also particles, but I shall exclude particles at this stage; see below. The numerical values of the fields (possibly with several components each) at every point in a region are constrained by spacetime curvature, the field equations and boundary conditions. In turn, the fields determine the Einstein tensor via the field equations in a region of spacetime. Every Einstein tensor is associated with a class of curvature, the class consisting of all Riemann tensors giving the same Ricci tensor or, equivalently, the same energy–momentum tensor.

This picture corresponds with what is named “classical general relativity”. In order to discuss the “quantization of general relativity” (QGR) it is necessary to study the interpretation of quantum theory, which will be the subject of Section 3. Quantization of GR will be discussed in Section 4, but before this I shall comment on the irreversibility of time, in Section 2.6, which follows.

2.6. Irreversibility in the Universe

Although marginal for this paper, I will comment briefly on the so-called arrow of time. From ancient times, people believed that space does not possess any special direction, it is the presence of Earth that causes the difference between vertical and horizontal directions. More common was the belief in an essential direction of time distinguishing the past from the future. Physicists, however, were reluctant to admit that this “arrow of time” is fundamental. As Einstein stated, “For us convinced physicist the distinction between past and future is an illusion, although a persistent one” [11]. This is supported by the fact that no violation has been found of the product of the three fundamental discrete symmetries; that is, CPT, which could provide a truly fundamental physical arrow of time. Of course, it is true that the fulfillment of CPT in the fundamental laws of physics is compatible with the violation of C, P and T symmetries.

Thus, the observed arrow of time required an explanation. Boltzmann’s statistical proof that a closed system evolves spontaneously toward equilibrium was an important achievement that elucidated irreversibility in many cases, but it cannot provide an explanation as to why “we grow old”; something that was recognized in Boltzmann’s time. The true reason for the existence of a general arrow of time on Earth, and, hence, the irreversibility of our lives, derives from the fact that the Earth receives energy from the Sun at high temperatures (about 6000 K) and re-emits it at low temperatures (about 300 K), thus producing an increase in entropy. In turn, this is a consequence of the expansion of the universe, which leads to irreversible star evolution. Thus, the appropriate explanation for the irreversibility on Earth has come after the discovery that the universe is expanding.

3. Quantum Theory: The Weyl–Wigner Representation

3.1. Lack of Consensus on the Interpretation: The Realistic Approach

At the end of the 19th century, there was an apparently well established picture of matter, resting on the physics known at that time. This state of affairs was dramatically altered by quantum theory. In fact, quantization was not conceived as the substitution of a new picture of reality for the classical image, but the withdrawal of the old picture without the advent of a new one. Indeed, quantum mechanics was proposed by Heisenberg with an explicit resignation regarding pictures of reality.

The problem remains until today. In fact, no consensus on its interpretation has emerged [3]. The result is that, after one century of quantum mechanics, we find ourselves in a strange situation. Everybody who has learned quantum mechanics agrees on how to use it but we do not understand the meaning of this strange conceptual apparatus that each of us uses so effectively to deal with our world.

The initial formulation of the theory was a non-relativistic quantum mechanics of particles (QM in the rest of this section). Popular approaches to interpreting quantum theory usually begin with QM, with the hope of extending, later, the interpretation to relativistic quantum theory. I think that this is an error because the quantum particles studied in QM cannot be treated as similar to classical particles. In fact, from a fundamental point of view, it is not obvious why the non-relativistic approximation of interacting relativistic quantum fields may be treated as a set of particles under the action of a electromagnetic field, as usual: for instance, in the case of electrons and nuclei in the quantum theory of atoms, molecules and solids. However, the particles are under the action of fields (including their vacua), with these fields being hidden in the non-relativistic approximation. In my opinion, the action of the fields is essential for the quantum behavior of the particles. Indeed, in Section 5 below, I will present a model where the introduction of a non-local potential may explain the wave behavior of particles (e.g., interference of electrons, neutrons or atoms). In my view, that potential simulates the action of the hidden fields in the non-relativistic approximation. In summary, the quantum particles in the non-relativistic approximation are actually rather complex objects consisting of interacting fields dressing bare particles. Thus, I believe that, in order to find a realistic interpretation of quantum theory, it is suitable to try to understand, firstly, the (relativistic quantum) fields rather than the quantum mechanics of non-relativistic particles.

The purpose of Heisenberg was to formulate QM with ingredients as close as possible to measurable quantities. Thus, his quantum mechanics substituted arrays of numbers, e.g., frequencies and intensities of atomic spectra, for the dynamical variables of classical mechanics like position, momentum or energy. The said arrays of numbers (matrices in mathematical language) may be multiplied with each other, but the product is not commutative. Dirac replaced the matrices with abstract vectors and operators in a linear space, which, after mathematical elaboration by J. von Neumann, is defined as a Hilbert space. The result is an elegant formalism that has become the canonical formulation of quantum theory. However, it does not offer an intuitive picture of reality in contrast to what happens in classical physics. Thus, QM looks like an (efficient) algorithm for the prediction of empirical results, whose physical interpretation is dark.

The alternative Schrödinger’s wave mechanics appeared, after the work of L. de Broglie, as an attempt to unify two images popular in classical physics, but incompatible with each other: particles (localized) and waves (extended). Schrödinger’s initial proposal for understanding his “wavefunction” as a continuous distribution of mass or electric charge failed because the localized (particle) behavior of electrons was proved necessary in order to understand many-electron atoms. Max Born introduced a practical interpretation of the wavefunction assuming that its modulus square is a probability density for the position of a particle. From that time on, the probability amplitude (e.g., for the positions of particles) has been the cornerstone of the whole quantum theory. However, in my view, “probability amplitude” is a strange union of unrelated words that many people admits as an explanation, e.g., of wave-particle duality, but which I do not.

In summary, none of the two initial formulations of QM leads naturally to a picture of physical reality. The consequence has been the early supremacy of Heisenberg–Bohr’s (Copenhagen) pragmatic approach, which values the predictive power of the theory but rejects physical pictures as misleading. Later on, many other interpretations have been proposed [3], but none fully satisfactory most people.

I am convinced that (1) a (realistic) interpretation providing a picture of reality is possible, (2) it cannot be achieved from the canonical (Hilbert space) formalism, but from the Wigner representation, maybe with some modifications, (3) we should start with the interpretation of quantum fields, and (4) after that, we might get a picture of the mechanics of quantum particles.

As a consequence, in the following I shall, firstly, revisit the formalism initiated by Weyl and Wigner, which leads to a realistic interpretation of quantum Bose fields, in particular electromagnetism (in Section 3.2, Section 3.3, Section 3.4 and Section 3.5). There is not yet a similar formulation for Fermi fields; hence, our interpretation will be incomplete. For non-relativistic quantum motion, a realistic interpretation is provided in Section 5 below, but it is convenient to first deal with spacetime quantization, which will be covered in Section 4.

3.2. Weyl–Wigner Formalism for Quantum Particles

In 1932, Wigner introduced a new formalism for quantum mechanics that has a classical flavor [12]. He proposed the following representation, $W_{\psi }(x,p)$, for the state of a particle with wavefunction $\psi \left(x\right)$ (in one dimension for simplicity):

$$W_{\psi }(x,p)\equiv {\displaystyle \frac{1}{\pi \mathit{ħ}}}\int \psi (x+y)\psi (x-y)exp\left(2ipy/\mathit{ħ}\right)dy,$$

(20)

which is named the Wigner function of the quantum state. The generalization to N particles with 3N coordinates and 3N momenta is straightforward. It is also possible to define a Wigner representation for observables; hence, both states and observables become functions in phase space, that is, the space of the coordinates $\left\{x_j\right\}$ and momenta $\left\{p_j\right\}$ of the particles. The expectation values ${〈M〉}_f$ are obtained via integrals like

$${〈M〉}_f=\int M\left(\left\{x_j,p_j\right\}\right)f\left(\left\{x_j,p_j\right\}\right){\Pi }_{j}d{x}_{j}d{p}_j,$$

(21)

where f and M represent the state and the observable, respectively. The numerical values of these expectations agree with those got from the previous formulations, in particular the canonical, Hilbert space, formalism [13,14]. As a result, the Wigner representation is a different formalism for the same physical theory; that is, quantum mechanics.

The proof of equivalence may be most easily seen via the Weyl transform, which leads to the Wigner representation, which starts from the canonical (Hilbert space) formalism rather than from (Schrödinger’s) wave mechanics, as in Equation (20). Weyl introduced his transform in 1927 as a method of quantization [15]. We may suppose the following naive argument at the origin of the Weyl transform. For a relation in classical mechanics, e.g.,

$$F\left(x,p\right)=0,$$

we might find a similar relation involving operators, $F^{\prime }(\widehat{x},\widehat{p}),$ via the transform

$$F^{\prime }\left(\widehat{x},\widehat{p}\right)=\int F\left(x,p\right)\delta \left(x-\widehat{x}\right)\delta \left(p-\widehat{p}\right)dxdp,$$

(22)

where $\delta \left(\right)$ are Dirac deltas. (From now on I will label operators by a “hat”, e.g., $\widehat{x},\widehat{p}).$ Equation (22) is a symbolic expression, not a sensible mathematical equation, because Dirac delta is defined for numerical arguments, but not for operators. In order to give a meaning to Equation (22), we may substitute integral representations for the deltas. That is, (modulo appropriate regularization)

$$\delta (x-y)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }exp\left[i\lambda \left(x-y\right)\right]d\lambda .$$

(23)

This equality is valid for numerical x and y, but the integrand on the right side is meaningful even if x and/or y are operators. Hence, we might substitute integral representations for the deltas in Equation (22). However, a difficulty remains because the operators $\widehat{x}$ and $\widehat{p}$ do not commute; hence, the integral representation associated with the symbolic expression $\delta \left(x-\widehat{x}\right)\delta \left(p-\widehat{p}\right)$ is different from that associated with $\delta \left(p-\widehat{p}\right)\delta \left(x-\widehat{x}\right).$

Weyl proposed a transform that leads from classical functions like $F(x,p)$ to (quantum) functions $F_{sym}\left(\widehat{x},\widehat{p}\right).$ The subindex $sym$ means symmetrical order; that is, writing any product involving operators in all possible orderings and dividing by the number of terms, for instance

$${\left({\widehat{x}}^2\widehat{p}\right)}_{sym}={\displaystyle \frac{1}{3}}\left({\widehat{x}}^2\widehat{p}+\widehat{x}\widehat{p}\widehat{x}+\widehat{p}{\widehat{x}}^2\right).$$

(24)

The Weyl transform may be written (for a single particle in one dimension) as

$$F_W\left(\widehat{x},\widehat{p}\right)={\displaystyle \frac{1}{4{\pi }^2}}\int F(x,p)exp\left[i\lambda \left(x-\widehat{x}\right)+i\mu \left(p-\widehat{p}\right)\right]d\lambda d\mu ,$$

(25)

whose generalization to many particles in 3D is straightforward.

Most interesting in the following is the inverse Weyl transform, which may be written as follows:

$$F(x,p)={\displaystyle \frac{1}{4{\pi }^2}}\int d\lambda d\mu Tr\left\{{\widehat{F}}^{\prime }exp\left[i\lambda \left(\widehat{x}-x\right)+i\mu \left(\widehat{p}-p\right)\right]\right\},$$

(26)

where ${\widehat{F}}^{\prime }$ is an operator and $Tr$ means the Trace operation. It is not difficult to prove that, if we write ${\widehat{F}}^{\prime }$ in the form $\left|\psi \rangle \langle \psi \right|$, Equation (26) leads to Equation (20), which proves the equivalence with the Wigner representation (that in the following I will name Weyl–Wigner, WW for short). The inverse Weyl transform allows getting, from states and observables in the canonical (Hilbert space) formalism, the corresponding states and observables in WW. The expectation values, calculated in WW, Equation (21), agree with those calculated in the canonical formalism; hence, WW represents the same physical theory as the canonical formulation in terms of Hilbert spaces because both predict the same measurable quantities (expectation values).

The Wigner representation has a classical flavor, but the classical appearance is misleading. In fact, the Wigner functions $f\left(\left\{x_j,p_j\right\}\right)$ are not positive definite, in general; hence, the states cannot be interpreted as probability distributions in phase space. Therefore, the Wigner representation is currently seen as just a useful calculational tool for some specific problems (of quantum statistical mechanics, in particular), but it does not offer a realistic interpretation of the quantum mechanics of particles.

In summary, the formalism of non-relativistic QM, plus the rules named “measurement theory”, provide a good tool for the prediction of the results of experiments, but they do not give clues for a picture of reality. Therefore, QM is not a good starting point for achieving a realistic interpretation of quantum theory.

3.3. Difficulties with a Picture of Reality of the Canonical Quantized Fields

I shall deal with the electromagnetic field or, more generally, Bose fields. The standard method by which to describe a classical field is to expand it in plane waves or, more generally, normal modes. For instance, in the simple case of a scalar (Bose) field, the expansion in plane waves may read

$$\varphi (\mathbf{r},t)=\sum _{\mathbf{k}}{\varphi }_{\mathbf{k}}\equiv \sum _{\mathbf{k}}\sqrt{{\displaystyle \frac{2\pi \mathit{ħ}}{\omega V}}}\left[a_{\mathbf{k}}exp\left(i\mathbf{k}·\mathbf{r}-i{\omega }_{l}t\right)+a_{\mathbf{k}}^{*}exp\left(-i\mathbf{k}·\mathbf{r}+i\omega t\right)\right].$$

(27)

The amplitudes of the modes are conveniently written using two complex conjugate quantities (c-numbers) $\left\{a_j,a_j^{*}\right\},$ where j labels a mode. The standard (canonical) quantization method consists of promoting the amplitudes to be operators $\left\{{\widehat{a}}_j,{\widehat{a}}_j^{†}\right\}$ with appropriate commutation rules, that is

$${\widehat{a}}_j{\widehat{a}}_k^{†}-{\widehat{a}}_k^{†}{\widehat{a}}_j={\delta }_{jk},{\widehat{a}}_j{\widehat{a}}_k-{\widehat{a}}_k{\widehat{a}}_j={\widehat{a}}_j^{†}{\widehat{a}}_k^{†}-{\widehat{a}}_k^{†}{\widehat{a}}_j^{†}=0,$$

(28)

where ${\delta }_{jj}=1,{\delta }_{jk\ne j}=0$. The evolution of the field operators $\left\{{\widehat{a}}_j\left(t\right),{\widehat{a}}_j^{†}\left(t\right)\right\}$ has formal similarity with the motion of mechanical harmonic oscillators, which may be shown performing the change in variables

$${\widehat{x}}_j\equiv {\displaystyle \frac{c}{\sqrt{2}{\omega }_j}}\left({\widehat{a}}_j+{\widehat{a}}_j^{†}\right),{\widehat{p}}_j\equiv {\displaystyle \frac{i\mathit{ħ}{\omega }_j}{\sqrt{2}c}}\left({\widehat{a}}_j-{\widehat{a}}_j^{†}\right),$$

(29)

where $\mathit{ħ}$ is Planck’s constant, c the velocity of light and ${\omega }_j$ the frequency of the normal mode.

The vacuum state is represented either by the state vector $\mid vac\rangle$ or by the density operator

$$\widehat{\rho }=\mid vac\rangle \langle vac\mid ,$$

(30)

fulfilling

$${\widehat{a}}_j\mid vac\rangle =0=\langle vac\mid {\widehat{a}}_j^{†},$$

for any annihilation operator ${\widehat{a}}_j,$ 0 being here the nul vector in the Hilbert space.

Excited pure states of the radiation field in the canonical formalism are associated with vectors in the Hilbert space that are obtained by repeated application of the creation operators to the vacuum state. Thus, a generic pure state may be represented as follows

$${\widehat{f}}^{†}\mid 0\rangle ,$$

(31)

where ${\widehat{f}}^{†}$ means any polynomial of the creation operators $\left\{{\widehat{a}}_j^{†}\right\}.$ Mixed states are probability distributions of pure states, and are represented by density operators. The observables, the expectation values and the evolution are well known and I shall skip them.

Up to here, I have been discussing the formalism, now I will pass to the interpretation. We might try to interpret the formalism as representing a wave field with a somewhat strange representation by vectors and operators in a Hilbert space. However, the standard opinion is not this, but that the formalism is assumed to represent both waves and particles. Indeed, relativistic quantum fields are supposed to represents a kind of particle each, every particle corresponding to a definite spin s and mass m. For instance, for the electromagnetic (EM) field $s=1,m=0$. In fact, the operators $\left\{{\widehat{a}}_j^{†}\right\}$ are believed to create particle states: photons in the case of the EM field. For instance, the vector (of the Hilbert space) ${\widehat{a}}_j^{†}\mid 0\rangle$ is a one-photon state. However, this assumption gives rise to a difficult problem of interpretation. In fact, the state represented by ${\widehat{a}}_j^{†}\mid 0\rangle$ will be associated with a plane wave in the expansion of the field, corresponding to one of the terms in Equation (27). But it may be a spherical wave for another choice of expansion. Thus, the picture of the boson offered by ${\widehat{a}}_j^{†}\mid 0\rangle$ depends on our choice of expansion. Furthermore, bosons (in particular photons) are intuitively associated with (small) particles rather than with extended plane (or spherical) waves.

In summary, any clear interpretation of a quantum Bose field as consisting of waves and particles is not possible. Hence, most people support the belief that quantum field theory cannot provide a picture of reality, and even that it should not. I strongly disagree. I am convinced that the picture is possible, but as wave fields only. The particle behavior would be an effect of the vacuum field. In my view, the picture is crystal clear using the WW representation rather than the canonical, Hilbert space, formalism.

3.4. Weyl–Wigner Formulation of the Quantum Electromagnetic Field

3.4.1. The Formalism

The WW formalism, developed for particles, may be extended to fields, in particular, the electromagnetic (EM) field. In the following, I provide a short review of the formalism; details may be seen elsewhere [16]. The Weyl transform Equation (26) leads from the operators Equation (29) to classical-like amplitudes. This fact allows for a straightforward formulation of the electromagnetic field in the WW representation [16], which may be extended to other Bose fields [17].

As in the WW formalism for particles, the Weyl transform Equation (26) allows us to derive the product of amplitudes in WW for any product of creation or annihilation operators in the canonical (Hilbert space, HS) formalism. In fact, if we have a symmetric product of operators like ${\left({\widehat{a}}_j^m{\widehat{a}}_j^{†n}\right)}_{sym}$ in the canonical (HS) formalism, the WW counterpart is

$${\left({\widehat{a}}_j^m{\widehat{a}}_j^{†n}\right)}_{sym}\to a_j^m{a}_j^{*n}$$

(32)

where sym stands for symmetric and it means writing a sum of the $m+n$ operators in all possible orderings and then dividing by the number of terms; that is, $\left(m+n\right)!/(m!n!)$. If the product of operators in the canonical (Hilbert space, HS) formalism is not symmetric, it is possible to get a symmetric expression using the commutation rules in Equation (28). Simple examples of the transform from HS to WW are

$${\widehat{a}}_j^{†}{\widehat{a}}_j\to {\left|a_j\right|}^2-{\displaystyle \frac{1}{2}},{\widehat{a}}_j^{†}{\widehat{a}}_j{\widehat{a}}_j^{†}{\widehat{a}}_j\to {\left|a_j\right|}^4-{\left|a_j\right|}^2.$$

(33)

The vacuum state in WW may be got by inserting the density operator $\widehat{\rho },$ Equation (30), in place of $\widehat{M}$ in Equation (26). We get, after some algebra,

$$W_0=\prod _j{\displaystyle \frac{2}{\pi }}exp\left(-2{\left|a_j\right|}^2\right),$$

(34)

which is normalized for the integration with respect to ${\prod }_j$ d Re a_jd Im $a_j$. Taking the Lorentz invariance of the vacuum field into account, the mean energy $E_j$ associated with a field component of the vacuum field in the expansion Equation (27) is proportional to ${\omega }_j$ and it is necessary to choose $〈E_j〉$$={\displaystyle \frac{1}{2}}\mathit{ħ}{\omega }_j$ [18]. This fact was already taken into account for the choice of coefficients in Equation (27). Hence, Equation (34) leads to

$$W_0=\prod _j{\displaystyle \frac{2}{\mathit{ħ}{\omega }_j}}exp\left(-{\displaystyle \frac{2E_j}{\mathit{ħ}{\omega }_j}}\right),$$

(35)

where $E_j$ is the mean energy of mode j, and the normalization is appropriate for integration with respect to ${\prod }_{j}d{E}_j$.

The WW counterparts of the states Equation (31) may be obtained by taking the Weyl transform Equation (26) into account, which gives, for every one of these states, a function of the amplitudes $\left\{a_j\right\}$, named the Wigner function, of the state. Thus, the Wigner functions of the canonical (HS) pure states are as follows

$$W_{\psi }\left(\left\{a_j\right\}\right)=T_W\left({\widehat{f}}^{†}\mid 0\rangle \langle 0\mid \widehat{f}\right),$$

(36)

where ${\widehat{f}}^{†}\mid 0\rangle \langle 0\mid \widehat{f}$ is the density operator corresponding to the state vector Equation (31) and $T_W$ means the (inverse) Weyl transform Equation (26).

Observables are represented by functions of the electric and magnetic field, which are related to the observables in the canonical (HS) formalism via the expansion in normal modes followed by the Weyl transform Equation (26). The expectation value of an observable in a state is obtained via the integral of the product of the corresponding functions of the amplitudes.

Equation (26) allows us to get the WW counterparts of the observables in the HS formalism. In particular, the free field Hamiltonians are, respectively,

$${\widehat{H}}_{HS}=\mathit{ħ}\sum _j{\omega }_j({\widehat{a}}_j^{†}{\widehat{a}}_j+{\displaystyle \frac{1}{2}})={\displaystyle \frac{1}{2}}\mathit{ħ}\sum _j{\omega }_j({\widehat{a}}_j^{†}{\widehat{a}}_j+{\widehat{a}}_j{\widehat{a}}_j^{†}),H_{WW}=\mathit{ħ}\sum _j{\omega }_j{\left|a_j\right|}^2.$$

(37)

However, in the canonical formalism it is common to change the order of the operators by putting the annihilation to the right, which is known as the “normal ordering rule”. Using that rule, the canonical and WW Hamiltonians become, respectively,

$${\widehat{H}}_{HS}^{normal}=\mathit{ħ}\sum _j{\omega }_j{\widehat{a}}_j^{†}{\widehat{a}}_j,H_{WW}^{normal}=\mathit{ħ}\sum _j{\omega }_j\left({\left|a_j\right|}^2-{\displaystyle \frac{1}{2}}\right),$$

(38)

where I have taken Equation (33) into account. Hence, the vacuum energy is defined as zero in HS, but this choice requires a reinterpretation in WW, as shown below; see comments on Equation (68).

Expectation values in the canonical formalism read $Tr(\widehat{\rho }\widehat{M})$, or, in particular, $\langle \psi \mid \widehat{M}\mid \psi \rangle ,$ and the translation to the WW formalism leads to the integral of the product of two functions of the amplitudes, that is,

$$Tr(\widehat{\rho }\widehat{M})=\int W_{\widehat{\rho }}\left\{a_j,a_j^{*}\right\}{W}_{\widehat{M}}\left\{a_j,a_j^{*}\right\}\prod _{j}d\mathrm{Re}{a}_{j}d\mathrm{Im}{a}_j,$$

(39)

where $W_{\widehat{\rho }}$ and $W_{\widehat{M}}$ are the counterparts of a density operator, $\widehat{\rho },$ and a quantum observable $\widehat{M}.$ A particular case of Equation (39) is the vacuum expectation value where $W_{\widehat{\rho }}$ becomes $W_0.$

The evolution of the states in the WW formalism is given by the Moyal equation,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial W}{\partial t}}& =& {\displaystyle \frac{2}{\mathit{ħ}}}{\displaystyle \sum _{n=0}^{3N}}{\displaystyle \frac{{\left(-1\right)}^n}{\left(2n+1\right)!}}{\left[{\displaystyle \frac{\mathit{ħ}}{2}}\left({\displaystyle \frac{\partial }{\partial x_j}}{\displaystyle \frac{\partial }{\partial p_j^{\prime }}}-{\displaystyle \frac{\partial }{\partial p_j}}{\displaystyle \frac{\partial }{\partial x_j^{\prime }}}\right)\right]}^{2n+1}\hfill \\ & & \times \left[W\left(\left\{x_j,p_j\right\}\right)H_{part}\left(\left\{x_j^{\prime },p_j^{\prime }\right\}\right)\right]\equiv {\left\{W,H_{part}\right\}}_M,\hfill \end{array}$$

(40)

where we should identify $\left\{x_j^{\prime },p_j^{\prime }\right\}=\left\{x_j,p_j\right\}$ after performing the derivatives [5,14]. ${\left\{W,H_{part}\right\}}_M$ is named the Moyal bracket. For simplicity, I have written it in terms of canonical variables of mechanics, but the evolution in Equation (40) is also valid for the radiation field if we perform a c-number change in variables similar to Equation (29).

In the case of the free EM field, the Hamiltonian $H_{WW}$ Equation (37) is quadratic in the amplitudes; hence, terms with $n\ne 0$ do not contribute to Equation (40). Only terms without the Planck constant remain. Then, Moyal’s bracket becomes Poisson’s, which proves that the evolution of the quantized free EM field in the WW formalism is just the classical (Maxwell) evolution [19].

Furthermore, the interaction of the Hamiltonian $H_{int}$ of the field with a system of charged particles is given in terms of the potential vector, which is also linear in terms of the field amplitudes $\left\{a_j,a_j^{*}\right\}$. However, $H_{int}$ is not quadratic in terms of the coordinates and momenta of the particles in general; hence, the evolution of the particles is not classical. In particular, the Moyal Equation (40) for the particles depend on the Planck constant $\mathit{ħ}$, which, however, does not appear in the field evolution. For details, see [19].

3.4.2. Realistic Interpretation

Up to here, we have the mere WW formalism. In order to make a realistic interpretation possible, we must introduce several assumptions that do not follow on from the Weyl transform. The most relevant refers to the states. In fact, the Weyl transform of the states, as defined in the standard formalism, cannot be states in WW if we want a realistic interpretation. For instance, as is well known, the Weyl transform of a single-photon state (the Wigner function of the state) is not positive definite, therefore, it could not be considered a physical state in a realistic interpretation of the WW formalism.

In general, Equation (36) suggests an interpretation of the Wigner function $W_{\psi }\left(\left\{a_j\right\}\right)$ as a probability distribution of amplitudes. However, a necessary condition for this interpretation would be that $W_{\psi }$ is a non-negative definite and normalized. The latter constraint holds, provided that the state vector Equation (31) is normalized, which I assume. However, the positivity causes a problem because there are many canonical states in Equation (31), whose Wigner function is not positive. Thus, the set of states in WW does not fit in with the set of states of the canonical formalism. Indeed, we shall assume that the physical states of the field in WW should correspond to radiation with (positive) a probability distribution of field amplitudes $\left\{a_j,a_j^{*}\right\}$ superposed to the vacuum field ZPF. I believe that this difficulty does not prevent a realistic interpretation of the WW formalism for the quantized EM field.

Equations (34) and (35) strongly suggest an interpretation of the vacuum state $W_0,$ and of the quantized EM field in the WW formalism as a probability distribution of amplitudes. Thus, it suggests a picture of the quantum vacuum as a real random radiation, a stochastic field, filling space with the distribution Equation (35). That random radiation has a mean energy${\displaystyle \frac{1}{2}}\mathit{ħ}{\omega }_j$ per normal mode, and it is currently named the zeropoint field (ZPF).

In spite of the very different definition of states in either the canonical formalism or WW, I conjecture that all experiments, in the domain of validity, that may be interpreted with the canonical formalism might also be interpreted within WW. Indeed, my belief about the general interpretation of quantum theory may be put as follows: The “measurement theory” is an addition that, although useful for some calculations, should not be taken as an essential part of the theory. Here, the measurement theory includes the definitions of states and observables. In my view, choosing the appropriate “quantum state”, representing a preparation, and the adequate “quantum observable”, representing an observation or measurement, is a difficult task that should be carefully studied in every actual experiment.

I shall finish the section by elucidating why the WW formalism is appropriate for the EM field but not for the non-relativistic quantum mechanics of particles (QM). Indeed, in both cases, the set of states in WW does not fit in with the set of canonical (Hilbert space, HS) formalisms due to a requirement of positivity in the former that is not demanded in HS. The response is that the situation is quite different in both cases, for the following reasons:

1. In a quantized EM field (QEM), the ground state has a Wigner function, Equation (34), which is positive definite. In QM, the ground state of a system of particles frequently has a Wigner function that is not positive.

2. The evolution of the free QEM field is governed by the classical Maxwell–Lorentz laws, which preserve the positivity of probability distributions. This is not the case in QM, which is governed by the Moyal Equation (40).

3. The states of QEM are most often produced by the action of macroscopic devices on the vacuum, which would give rise to states with a positive Wigner function. This happens, for instance, in spontaneous parametric down conversion, leading to entangled photon pairs; see Section 7.1 below.

4. In the canonical (HS) interpretation of experiments, n-photon states usually appear but at intermediate stages of the calculation, which may not be positive when translated to the WW formalism via Equation (26). However, there is no need to ascribe physical reality to these intermediate (mathematical) elements of the calculation.

5. In typical QED calculations the “photon propagator” involves the vacuum expectation value and that state has a positive Wigner function.

3.5. Weyl–Wigner Formalism for Quantum Electrodynamics

In Section 3.4.1 and Section 3.4.2 I have reviewed the properties of the electromagnetic field alone, but quantum electrodynamics would also involve the charges. The combination of the quantized EM field with Fermi fields, that is relativistic QED, cannot be treated within the WW formalism because we do not have an appropriate (realistic) interpretation of Fermi fields. The combination of the EM field with non-relativistic particles also cannot be treated within WW because the Wigner representation for those particles does not admit a realistic interpretation in general, as commented on above.

3.5.1. Quantum Optics

Quantum optics is a domain where the field interacts with matter and both may be treated within the WW formalism when the EM field interacts with macroscopic bodies. Indeed, macroscopic bodies may be treated within classical electrodynamics, and the combination with the quantized EM field gives, precisely, a study within the WW formalism. In this case, the WW treatment corresponds just with classical Maxwell–Lorentz electrodynamics with the addition a zeropoint Gaussian random field in the vacuum [5,20]. In some cases, that treatment has given rise to things, like “negative probability distributions”, that cannot admit a realistic interpretation [21]. However, I am convinced that all these problems might be eliminated with a careful treatment within WW.

Actually, the phenomena within quantum optics that present the greatest difficulties for a realistic interpretation are photon entanglement and the optical tests of Bell inequalities, which are discussed in Section 7.1 and Section 7.2. In the following, I recall three simple but interesting examples that have been studied in more detail elsewhere [22]. Further examples will be provided in Section 6, which is devoted to the wave-particle duality.

3.5.2. Casimir Effect

This is the attraction between two parallel perfect conduction plates placed in a vacuum. The reason for the force is that the plates restrict the possible modes of the radiation field because, in equilibrium, the component of the electric field parallel to the plate surface should be nil. Thus, the vacuum energy of the ZPF with the plates in place is different from the energy with the plates removed, and the dependence of the energy with the distance between plates gives rise to a force. The calculation in the WW formalism is closely related to the standard one in HS [18]. The picture that we get in WW is that the pressure of the ZPF is different on the two sides of each plate, which gives rise to the force.

3.5.3. Atoms in Cavities

As shown in the theory of the Casimir effect, the ZPF radiation modes in confined space are different from the modes in free space. Then, assuming that spontaneous emission is partially stimulated by the ZPF, the lack of some radiation modes would prevent excited atoms’ decay emitting radiation in the said modes. In fact, the experiments have shown the inhibition of atomic decay in cavities giving rise to increased lifetimes of excited atoms. For a semiclassical model, see [23].

3.5.4. Stability of Matter: Ground State of the Hydrogen Atom

The atom cannot be studied within the WW formalism because it involves charged particles, not just the field. This means that he prediction obtained may not allow a realistic interpretationprovides a realistic interpretation. However, it is the case that it for some properties of the atom. The interpretation is interesting because it suggests an explanation for the stability of matter.

In a simplified model, the hydrogen atom consists of two particles, proton and electron, each characterized 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. In classical mechanics the electron may move around the nucleus, say in a circle, having energy E and we might write the following equalities

$$\left|E\right|={\displaystyle \frac{1}{2}}m{v}^2={\displaystyle \frac{1}{2}}{\displaystyle \frac{e^2}{r}},v=r\omega ,$$

(41)

According to classical electrodynamics, the electron would radiate, leading to a collapse of the atom but, taking the ZPF into account as a real random radiation, the atom may also absorb energy from the field. The combination of emission and (random) absorption perturbs the motion, which would be irregular, not circular. However, it is plausible that the equations in (41) are roughly fulfilled, on average. A dynamical equilibrium may arise when the atomic kinetic energy becomes equal to the energy of the radiation mode, having the same frequency, that is, $\left|E\right|\sim {\displaystyle \frac{1}{2}}\mathit{ħ}\omega$. Indeed, this equality is plausible because the electron will interact most strongly with such modes. Hence, the energy and the size of the atom may be got by removing the quantities v and $\omega$ from the equations in (41), which leads to

$$E\sim -{\displaystyle \frac{m{e}^4}{2{\mathit{ħ}}^2}},r\sim {\displaystyle \frac{{\mathit{ħ}}^2}{m{e}^2}},$$

(42)

which is in agreement with the quantum prediction and experiments.

3.6. Realistic Interpretation of Bose Fields via the WW Formalism

I believe that Bose fields are continuous fields similar to the electromagnetic one, the particle behavior being an effective property, see Section 5 below. In contrast, I have not a clear picture of Fermi fields, but I conjecture that they might consist of a sea of particles and antiparticles similar to the original picture of Dirac.

Thus, we might assume that in nature there are bare particles with a definite mass and charge, but that physical particles have quite different values for those quantities, as shown by the need for renormalization techniques in quantum electrodynamics. The reason for the difference is the “dressing” due to many quantum fields including their vacua. For instance, a physical electron should be seen as an extended object with a size of order equal to the Compton wavelength, having observable mass and charge far from the bare quantities. The electron position might be defined by the center of the charge distribution, but the momentum and angular momentum would involve substantial contributions from fields. Hence, in contrast to classical mechanics, the electron state cannot be represented by a point in phase space. In particular, its future evolution is not determined by just initial position and momentum. For this reason, I do not propose to get a picture of QM using the Weyl–Wigner formalism, which, nevertheless, does yield a fairly realistic interpretation for the EM field, as studied in Section 4. For Bose fields, other than electromagnetism, the WW treatment is similar. Thus, WW strongly suggests a realistic interpretation for the quantized Bose fields [17].

In summary, there are several formalisms for the study of quantum systems and in this article we deal with two of them: canonical (Hilbert space, HS) and Weyl–Wigner (WW). Both are valid as calculational tools with different efficiency, the canonical one being most useful in general. Only one, WW, provides a physically realistic interpretation for Bose fields and none of them for Fermi fields or non-relativistic quantum mechanics, as was commented on Section 3.

In fact, we do not have a transform for Fermi fields that would play the role of Weyl transform for Bose fields. Thus, our realistic interpretation of quantum fields is not yet complete. In my view, getting an appropriate formalism for Fermi fields would be a dramatic improvement for the interpretation of quantum theory. It would permit a realistic interpretation of the whole quantum field theory. For the evolution of quantum particles in non-relativistic motion, there is a formalism allowing a realistic interpretation, and this is presented in Section 5 below.

4. Quantum Gravity

4.1. The Difficulties with Matching General Relativity with Quantum Theory

Einstein’s Equation (16) is meaningful in a classical framework, but it does not fit into quantum theory. In fact, the Einstein tensor on the left side of Equation (16) is a classical (c-number) quantity, but in quantum theory the stress-energy tensor on the right side should be an observable that in the canonical (HS) formalism is represented in terms of operators on Hilbert space. Solutions to the problem might be either substituting a c-number tensor for the stress-energy operator on the right side or substituting operators for the components of the Einstein tensor on the left side. However, none of these solutions is good. An approximation to the former solution might be achieved by substituting the expectation number of the stress-energy operator ${\widehat{T}}_{\mu \nu },$ in the appropriate state $\mid \psi \rangle ,$ for the operator itself. That is, the following would be substituted for Equation (16)

$$G_{\mu \nu }=-8\pi G〈\psi \left|{\widehat{T}}_{\mu \nu }^{matt}\right|\psi 〉.$$

(43)

This equation has been used with success in some cases, but it is just a semiclassical approximation that ignores quantum fluctuations.

The alternative solution would be to promote the metric tensor elements, $g_{\mu \nu },$ to be operators, say ${\widehat{g}}_{\mu \nu },$ and, hence, to get a quantum operator form of the Einstein tensor ${\widehat{G}}_{\mu \nu }.$ However, there is an ambiguity in passing from ${\widehat{g}}_{\mu \nu }$ to ${\widehat{G}}_{\mu \nu }$ because the operators ${\widehat{g}}_{\mu \nu }$ would not commute with each other and with their derivatives, in general. For these reasons, people have attempted to find a theory that unifies quantum theory with general relativity (GR) via other approaches, a program known as “quantum gravity” [24].

Actually, the main motivation for quantum gravity is the attempt to deal with the problem of singularities in spacetime. Singularities are predicted at the center of collapsed astrophysical bodies (black holes) and also in the very early universe. Near a singularity, both quantum and “gravitational” (i.e., general relativistic) effects are equally relevant. This is in contrast with what happens far from singularities. Indeed, quantum effects are relevant in laboratories where gravity may be neglected, and gravity is relevant in astrophysics where quantum effects are negligible (maybe with some exceptions, see [25]). In both these regimes, quantization of gravity is not needed. I shall comment on black holes in Section 4.3 below. For the universe, there are strong arguments for the assumption that the universe started about 1.4 × ${10}^{10}$ years ago and there was a period with very strong mass density where both quantum and gravitational effects were important. The study of the early universe is out of the scope of this article.

4.2. The Search for Quantum Gravity

A usual approach to quantum gravity has been to reinterpret general relativity as a “field theory of gravity”, which might be quantized like other relativistic fields. A justification for treating GR as a field theory has been that there is an alternative road to Einstein’s Equation (16), not starting from the equivalence principle (i.e., the equivalence between gravity and acceleration), but deriving GR as a gauge theory whose associated quantum particle, the graviton, is massless with spin 2. The meaning of Einstein’s Equation (16) as a relation between matter and spacetime is usually maintained; hence, spacetime itself is taken both as a quantum field and the ground for all fields, which to me looks bizarre.

A canonical quantization procedure in analogy with other fields, e.g., electromagnetic, is not possible due to the nonlinear character of GR. Other quantization methods have led to theories that are not renormalizable. Consequently, several different routes have been devised without complete success until now. For instance, string theory, loop quantum gravity, noncommutative geometry and others [24].

In Section 2.4 I have criticized the opinion that general relativity may be seen as a field theory of gravity. Therefore, I propose a different approach for the quantization of GR. I support the view that Equation (16) is just the relation between the curvature of spacetime and the energy and momentum of the true fields, say those of elementary particle physics. Thus, the unification of quantum theory with general relativity should consist of the study of quantum fields in curved spacetime, and attributing to spacetime properties induced by the fact that actual fields are quantized. In the approach via WW, these properties consist, essentially, of the existence of vacuum fields, ZPF. Then the stochastic character of the ZPF would lead to the necessity of assuming a stochastic character for the curvature of spacetime. That is, to study spacetime via a probability distribution of (classical) metrics $g_{\mu \nu }$, each one giving rise to a different Einstein tensor $G_{\mu \nu }.$ The probability distribution of metrics should fit in the distribution of energy- momentum tensors via Equation (16). Thus, I conclude that spacetime has fluctuations at all scales. In any case, we should neither treat spacetime as a field nor gravity as a force (see Section 2.3), and maintain Einstein’s Equation (16) as a valid relation between matter and spacetime, both being treated as random.

4.3. The Problem of Singularities: Black Holes

The current opinion is that many stars may collapse after some period of cooling and/or contraction. The collapse gives rise to a Schwarzshild singularity in collapsed stars (black holes) [26]. It is remarkable that the possible existence of actual, physical, singularities was rejected by several celebrated authors, including Einstein [8,27]. However, a number of theoretical studies have led to acceptance regarding the collapse of spherical compact objects with a high ratio mass by radius; that is, the theory that M/R > 2G, is unavoidable. After an early work by Oppenheimer et al. [28,29], this opinion was strongly advocated by J. A. Wheeler et al. [30] in around 1960, and it has, allegedly, been supported by observations in subsequent years. The amount of work on black holes carried out over the last 70 years has been enormous. Hence, it is now current wisdom, although there is still some controversy about the subject; see [31]. I hope that the subject will be clarified in the future. The possible singularity associated with the big bang will not be discussed in this article.

5. The Motion of Non-Relativistic Quantum Particles

The concept of a particle in relativistic motion is inappropriate in quantum theory. (Here, relativity refers to the special theory). The reason is that in the relativistic domain it is necessary to deal with the creation and annihilation of particles, which is studied by relativistic quantum field theory. According to the interpretation of this article, Bose fields are wavelike, the particle behavior being an effect of the ZPF, as will be discussed in detail in Section 6. For Fermi fields, there is no formulation similar to WW able to provide a clear realistic picture. So, the following will be devoted to a realistic interpretation of the mechanics of particles in non-relativistic motion; that is, with velocity much smaller than the speed of light.

As discussed in Section 3.4.2, WW does not provide a realistic picture for particles because they are always dressed with quantum fields, which strongly modifies their motion. In the following, I propose an interpretation resting on the formulation of quantum mechanics via path-integrals, introduced by Feynman in 1948. The application to non-relativistic quantum mechanics is studied in a book by Feynman and Hibbs [32]. In the following sections, I review, briefly, the realistic interpretation presented elsewhere [33].

5.1. Path-Integral Formulation of Quantum Mechanics

Feynman proposed calculating the amplitude that a particle placed at position ${\mathrm{x}}_0$ at time $t=0$ reaches at position x at time t, as follows (in one dimension)

$$A(x_0,0\mid x,t)=\int d{x}_1\dots \int d{x}_{n-1}A(x_0,0\mid x_1,t_1)\dots A(x_{n-1},t_{n-1}\mid x,t).$$

(44)

The set of positions $\left\{x_0,x_1,\dots x\right\}$ defines a (discrete) path; hence, Equation (44) is an integral of discrete paths. From Equation (44), we may obtain the corresponding probability; that is,

$$P\left(x\right)\propto {\left|A(x_0,0\mid x,t)\right|}^2.$$

(45)

The time intervals may be chosen to be identical; that is, $t_{j+1}-t_j=\epsilon ,$ with $\epsilon$ as small as desired. At the limit $\epsilon \to 0,A(x_0,0\mid x,t)$ becomes an integral of continuous path amplitudes.

In the case of one-dimensional motion in a potential $V\left(x\right)$, the partial amplitudes are defined as follows:

$$\begin{array}{ccc}\hfill A\left(x_{j-1},t_{j-1}\mid x_j,t_j\right)& =& \sqrt{{\displaystyle \frac{m}{2\pi i\mathit{ħ}\epsilon }}}exp\left({\displaystyle \frac{i\epsilon }{\mathit{ħ}}}{L}_j\right),\hfill \\ \hfill L_j& \equiv & {\displaystyle \frac{1}{2}}m{\left({\displaystyle \frac{x_j-x_{j-1}}{\epsilon }}\right)}^2-{\displaystyle \frac{1}{2}}\left[V\left(x_{j-1}\right)+V\left(x_j\right)\right]\hfill \end{array}$$

(46)

where m is the mass of the particle. (This expression differs from the original one of Feynman [32] because I have substituted ${\displaystyle \frac{1}{2}}\left[V\left(x_{j-1}\right)+V\left(x_j\right)\right]$ for $V\left[\left(x_{j-1}+x_j\right)/2\right]$ for later convenience. Both formulations agree in the limit $\epsilon \to 0).$

The amplitude $A(x_0,0\mid x,t)$ is named the “propagator” of the wavefunction $\psi \left(x,t\right)$; it allows us to get the wavefunction at time t from the wavefunction at time 0; that is,

$$\psi \left(x,t\right)=\int d{x}_0\psi \left(x_0,t\right)A(x_0,0\mid x,t).$$

(47)

Hence, the propagator $A(x_0,0\mid x,t)$ fulfills the Schrödinger equation with the initial condition

$$A(x_0,0\mid x,0)=\delta \left(x-x_0\right),$$

where $\delta \left(x\right)$ is Dirac’s delta. Thus, the propagator is the Green’s function of the Schrödinger equation.

The path integrals formulation may be generalized to three dimensions, to many-particles- and also to relativistic-field theory. It has an extremely important role in modern theoretical physics, both because it is well adapted to derive general properties, e.g., symmetries and, due to the relevance for actual calculations, it is the seed of Feynman graphs in covariant perturbation theory [34]. Dealing with formal and calculational aspects lies outside the scope of this section, which is devoted to the physical interpretation of the Feynman formalism in (non-relativistic) quantum mechanics.

5.2. Transition Probability as a Sum of Paths Probabilities

In the following, I present a formulation for the motion of a quantum particle in terms of probabilities (rather than amplitudes!) of paths, with the condition that the transition probability agrees with the square modulus of the Feynman amplitude Equation (45), that is,

$$P(x_0,0\to x,t)={\left|A(x_0,0\mid x,t)\right|}^2.$$

If we take the (continuous) set of paths as discrete for the sake of clarity, and we generalize to three dimensions, our aim is to get the transition probability as a sum of probabilities of paths, that is

$$P({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b)=\sum _k{W}_k({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b).$$

(48)

Here, every value of the index k corresponds with a possible path of the particle with end points $({\mathbf{r}}_a,t_a)$ and $({\mathbf{r}}_b,t_b)$. The problem is to find “weights” $W_k$ that could be interpreted as probabilities, in order to provide an intuitive picture of the quantum evolution as a random motion of particles. In the following, I propose a method to get the said weights. In some cases to be studied below, the weights are non-negative definite and, therefore, may be interpreted as probabilities, in other cases the formalism should be slightly modified in order to get positivity.

I shall start from the 3D generalization of the amplitude Equation (46), that is,

$$\begin{array}{ccc}\hfill A\left({\mathbf{x}}_a,t_a\to {\mathbf{x}}_b,t_b\right)& =& \underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi i\mathit{ħ}\epsilon }}\right)}^{3n/2}\int d{\mathbf{x}}_{n-1}\dots \int d{\mathbf{x}}_1\hfill \\ & & {\displaystyle \prod _{j=1}^n}exp\left\{{\displaystyle \frac{im}{2\mathit{ħ}\epsilon }}{\left|{\mathbf{x}}_j-{\mathbf{x}}_{j-1}\right|}^2-{\displaystyle \frac{i\epsilon }{2\mathit{ħ}}}\left[V\left({\mathbf{x}}_{j-1}\right)+V\left({\mathbf{x}}_j\right)\right]\right\},\hfill \end{array}$$

(49)

where $\epsilon \equiv t_j-t_{j-1},$ is independent of j and ${\mathbf{x}}_0\equiv {\mathbf{x}}_a,{\mathbf{x}}_n\equiv {\mathbf{x}}_b.$ The limit $\epsilon \to 0$ should be understood with $n\to \infty$ fulfilling

$$\underset{\epsilon \to 0}{lim}\left(n\epsilon \right)=t_b-t_a.$$

(50)

Actually, the integrals involved are not convergent, therefore, an appropriate regularization is implicit.

The transition probability is the square modulus of the transition amplitude, which becomes [33]

$$\begin{array}{ccc}\hfill P({\mathbf{r}}_a,t_a& \to & {\mathbf{r}}_b,t_b)=A\left({\mathbf{x}}_a,t_a\to {\mathbf{x}}_b,t_b\right)A^{*}\left({\mathbf{y}}_a,t_a\to {\mathbf{y}}_b,t_b\right)\hfill \\ & =& \underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi \mathit{ħ}\epsilon }}\right)}^{3n}\int d{\mathbf{x}}_{n-1}\dots \int d{\mathbf{x}}_1\int d{\mathbf{y}}_{n-1}\dots \int d{\mathbf{y}}_1\hfill \\ & & \times {\displaystyle \prod _{j=1}^n}exp\left\{{\displaystyle \frac{im}{2\epsilon \mathit{ħ}}}\left[{\left|{\mathbf{y}}_j-{\mathbf{y}}_{j-1}\right|}^2-{\left|{\mathbf{x}}_j-{\mathbf{x}}_{j-1}\right|}^2\right]\right\}\hfill \end{array}$$

(51)

$$\begin{array}{ccc}& & \times {\displaystyle \prod _{j=1}^n}exp\left\{-{\displaystyle \frac{i\epsilon }{2\mathit{ħ}}}\left[V\left({\mathbf{x}}_j\right)-V\left({\mathbf{y}}_j\right)\right]\right\},\hfill \end{array}$$

(52)

where I identified ${\mathbf{x}}_a={\mathbf{y}}_a={\mathbf{r}}_a,{\mathbf{x}}_b={\mathbf{y}}_b={\mathbf{r}}_b$ and reordered the integrals. I have represented vectors with bold face letters, so that $\int d{\mathbf{x}}_j$, $\int d{\mathbf{y}}_j$ are triple integrals over the whole 3D space and I have included the parameters m and $\mathit{ħ}$, following Feynman [32]. With that choice, the quantity $P({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b)$ has dimensions of probability per square volume. Then, the probability that a particle is in some finite volume B at time $t_b$ conditional to be in another finite volume A at an earlier time $t_a$ will be

$$P\left(A\to B\right)={\int }_{{\mathbf{r}}_a\in A}d{\mathbf{r}}_a{\int }_{{\mathbf{r}}_b\in B}d{\mathbf{r}}_{b}P({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b).$$

In order to proceed, I shall make a change in variables, that is,

$${\mathbf{r}}_j={\displaystyle \frac{1}{2}}\left({\mathbf{x}}_j+{\mathbf{y}}_j\right),{\mathbf{u}}_j={\mathbf{x}}_j-{\mathbf{y}}_j,0\le j\le n.$$

(53)

Hence, Equation (52) becomes, reordering the exponentials,

$$\begin{array}{ccc}\hfill P({\mathbf{r}}_a,t_a& \to & {\mathbf{r}}_b,t_b)=\underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi \epsilon \mathit{ħ}}}\right)}^{3n}\int d{\mathbf{r}}_{n-1}\dots \int d{\mathbf{r}}_1\int d{\mathbf{u}}_{n-1}\dots \int d{\mathbf{u}}_1\hfill \\ & & \times {\displaystyle \prod _{j=1}^{n-1}}exp\left[-{\displaystyle \frac{im}{\epsilon \mathit{ħ}}}{\mathbf{u}}_j·\left({\mathbf{r}}_{j-1}-2{\mathbf{r}}_j+{\mathbf{r}}_{j+1}\right)\right]\hfill \\ & & \times {\displaystyle \prod _{j=1}^{n-1}}exp\left\{{\displaystyle \frac{i\epsilon }{2\mathit{ħ}}}\left[V({\mathbf{r}}_j-{\displaystyle \frac{1}{2}}{\mathbf{u}}_j)-V({\mathbf{r}}_j+{\displaystyle \frac{1}{2}}{\mathbf{u}}_j)\right]\right\},\hfill \end{array}$$

(54)

where ${\mathbf{r}}_0={\mathbf{r}}_a$, ${\mathbf{r}}_n={\mathbf{r}}_b$ and ${\mathbf{u}}_0={\mathbf{u}}_n=0$; hence, the probability may be written

$$P({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b)=\underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi \mathit{ħ}}}\right)}^3{\epsilon }^{-3n}\int d{\mathbf{r}}_{n-1}\dots \int d{\mathbf{r}}_1{\displaystyle \prod _{j=1}^{n-1}}{Q}_j,$$

(55)

where

$$\begin{array}{ccc}\hfill Q_j& \equiv & {\left({\displaystyle \frac{m}{2\pi \mathit{ħ}}}\right)}^3\int d\mathbf{u}exp\left(-i{\displaystyle \frac{m}{\mathit{ħ}}}\mathbf{u}·{\mathbf{s}}_j\right)\hfill \\ & & \times exp\left\{{\displaystyle \frac{i\epsilon }{2\mathit{ħ}}}\left[V({\mathbf{r}}_j-{\displaystyle \frac{1}{2}}\mathbf{u})-V({\mathbf{r}}_j+{\displaystyle \frac{1}{2}}\mathbf{u})\right]\right\},\hfill \end{array}$$

(56)

with

$${\mathbf{s}}_j\equiv {\displaystyle \frac{{\mathbf{r}}_{j-1}-2{\mathbf{r}}_j+{\mathbf{r}}_{j+1}}{\epsilon }}={\displaystyle \frac{{\mathbf{r}}_{j+1}-{\mathbf{r}}_j}{\epsilon }}-{\displaystyle \frac{{\mathbf{r}}_j-{\mathbf{r}}_{j-1}}{\epsilon }}\equiv {\mathbf{v}}_j-{\mathbf{v}}_{j-1},j=1,2,\dots n-1.$$

(57)

The quantity ${\mathbf{s}}_j$ has the physical meaning of velocity change at time t and the ratio ${\mathbf{s}}_j/\epsilon$ might be interpreted as an acceleration. However, the limit $\epsilon \to 0$ may not exist; that is, the instantaneous velocity and acceleration are not well defined in general.

Performing the integrals in ${\mathbf{u}}_j$ is not possible without a knowledge of the potential, $V\left(\mathbf{r}\right),$ but to the lowest order in $\mathit{ħ}$ the integrals are simple. In fact, approximating $V\left({\mathbf{r}}_j\pm {\mathbf{u}}_j/2\right)$ to the first order in ${\mathbf{u}}_j$ in the second exponent of Equation (56) and then integrating with respect to $\mathbf{u},$ I get

$$\begin{array}{ccc}\hfill P({\mathbf{r}}_a,t_a& \to & {\mathbf{r}}_b,t_b)=\underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi \mathit{ħ}\epsilon }}\right)}^{3\left(n+1\right)/2}\int d{\mathbf{r}}_1\dots \int d{\mathbf{r}}_{n-1}\hfill \\ & & \times {\displaystyle \prod _{j=1}^{n-1}}{\delta }^3\left({\mathbf{s}}_j+{\displaystyle \frac{2\epsilon }{m}}\nabla V\left({\mathbf{r}}_j\right)\right)+O\left({\mathit{ħ}}^3\right),\hfill \end{array}$$

(58)

This corresponds with a motion fulfilling at every time

$$m{\displaystyle \frac{{\mathbf{r}}_{j-1}-2{\mathbf{r}}_j+{\mathbf{r}}_{j+1}}{{\epsilon }^2}}=-\nabla V\left({\mathbf{r}}_j\right),$$

(59)

that is, the (discretized) classical equation of motion. Thus, Equation (58) provides the classical limit of quantum mechanics when $\mathit{ħ}\to 0.$

It is interesting that the classical motion, Equation (59), is obtained without any approximation when the potential, $V\left({\mathbf{r}}_j\right),$ is at most quadratic in the coordinates because, in this case, Equation (59) is exact (no term $O\left({\mathit{ħ}}^3\right)$ appears). This might be interpreted by saying that in “linear problems the quantum particle follows the classical path”. The typical example is the harmonic oscillator. This is the reason why quantum mechanics of linear systems looks semiclassical. In this case, all quantum effects derive from the fact that the initial wave function cannot be localized in a too small region due to the Heisenberg uncertainty principle, a constraint which does not appear in Feynman’s path integrals formalism, and should be put as an additional constraint. In contrast, it does appear in the canonical Hilbert space formalism, where Heisenberg uncertainty relations are a consequence of the commutation rules.

5.3. Path Weights in Terms of the Fourier Transform of the Potential

It is convenient to perform a change leading to a more simple description of the transition probability, but equivalent to Equation (54) in the limit $\epsilon \to 0,n\to \infty .$ After some algebra, we get the following transition probability [33],

$$\begin{array}{ccc}\hfill P({\mathbf{r}}_a,t_a& \to & {\mathbf{r}}_b,t_b)=\underset{\epsilon \to 0}{lim}{\left({\displaystyle \frac{m}{2\pi \mathit{ħ}}}\right)}^3{\epsilon }^{-3n}\int d{\mathbf{r}}_{n-1}\dots \int d{\mathbf{r}}_1{\displaystyle \prod _{j=1}^{n-1}}{Q}_j,Q_j=D_j+\epsilon F_j\hfill \\ \hfill D_j& \equiv & {\delta }^3\left({\mathbf{s}}_j\right),F_j\equiv -{\displaystyle \frac{m^3}{2{\mathit{ħ}}^4}}\mathrm{Im}\left[\tilde{V}\left({\displaystyle \frac{2m{\mathbf{s}}_j}{\mathit{ħ}}}\right)exp\left(-{\displaystyle \frac{2im{\mathbf{s}}_j·{\mathbf{r}}_j}{\mathit{ħ}}}\right)\right],\hfill \end{array}$$

(60)

where ${\mathbf{r}}_o={\mathbf{r}}_a,{\mathbf{r}}_n={\mathbf{r}}_b,$ and ${\mathbf{s}}_j$ are the change in velocity at time $t_j;$ see Equation (57). The Fourier transform is here defined as follows:

$$\tilde{V}\left(\mathbf{w}\right)\equiv \int d\mathbf{x}exp\left(i\mathbf{w}.\mathbf{x}\right)V\left(\mathbf{x}\right),$$

(61)

$\mathbf{w}$ and $\mathbf{x}$ being 3D vectors. Calculating the transition probability $P({\mathbf{r}}_a,t_a\to {\mathbf{r}}_b,t_b)$ is involved because the changes in the velocity and the positions are related via Equation (57).

5.4. Realistic Interpretation

Two interesting questions are whether the paths involved in Equation (60) are continuous and whether the quantities $Q_j$ are positive (or zero). The answers to both questions are affirmative [33]. However, the quantities ${\mathbf{s}}_j$, which would represent the instantaneous acceleration in the limit $\epsilon \to 0$, are not well defined. That is, the functions $\mathbf{r}\left(t\right)$ are continuous but only once derivable.

Equation (60) affords a formulation of the quantum motion of particles in the form of a probability distribution of possible paths from the initial position ${\mathbf{r}}_a$ at time $t_a$ to a final position ${\mathbf{r}}_b$ at time $t_b.$ The motion has a random character, although quite different from the most popular Brownian motion. It is interesting that the probability of every path depends on the Fourier transform $\tilde{V}\left(2s_j\right)$ of the potential, $V\left(\mathbf{r}\right),$ hence, the effect of $V\left(\mathbf{r}\right)$ on the motion of the particle is non-local. A plausible explanation for this fact is that the particle motion is influenced by the fluctuating spacetime curvature discussed in Section 3.6, and also by the interaction with the vacuum fields discussed in Section 3.4.

It is remarkable that these influences, which are rather involved, give rise to a relatively simple action via the Fourier transform of the external force (here treated as deriving from a potential $V\left(\mathbf{r}\right)$). This may be put in a different form, namely the question of how the quantum effects may be taken into account via the rather simple mathematical formalism of Hilbert spaces? Indeed, we have shown that Feynman path integral formalism for quantum particles is equivalent to Schrödinger formulation, and this is equivalent to the canonical HS formalism.

5.5. Scattering Experiments: Born Approximation

An application of the formalism here proposed is the study of the scattering of a particle by a potential. In those experiments, a source emits particles, all of them with velocity ${\mathbf{v}}_a$. A fraction of the particles cross a target region where they experience the force due to the potential $V\left(\mathbf{r}\right).$ Then, the particles emerge from the target with velocities $\left\{{\mathbf{v}}_b\right\}$ different from the initial one ${\mathbf{v}}_a$ and eventually arrive at a detector. The target is, in practice, small (microscopic), while the distances from the target to either the source or the detector are both large (macroscopic). In the formalism of this article, I assume that the particles are small (or pointlike) corpuscles. No waves appear.

The quantity of interest in scattering experiments is the differential cross section, $\sigma \left(\theta ,\varphi \right)$. It is proportional to the number of particles per unit solid angle that leave the target with a velocity ${\mathbf{v}}_b$ in the direction determined by the angles $\left(\theta ,\varphi \right)$. In our formalism, we may write

$$\sigma \left(\theta ,\varphi \right)\propto \int d{\mathbf{r}}_a\rho \left({\mathbf{r}}_a\right)\int v_b^{2}d{v}_b{P}_v\left({\mathbf{r}}_a\to {\mathbf{v}}_b\right),$$

(62)

where $P_v\left({\mathbf{v}}_b\right)$ is the probability that a particle emerging from the point ${\mathbf{r}}_a$ of the source with velocity ${\mathbf{v}}_a$ reaches the velocity ${\mathbf{v}}_b$ after crossing the target. The integral, with respect to the modulus of ${\mathbf{v}}_b$, takes into account that only the direction of ${\mathbf{v}}_a$ matters, not the modulus. The triple integral with respect to ${\mathbf{r}}_a$ is necessary in order to sum over all possible initial positions of the particle in the source. However, the cross section should be independent of the density $\rho \left({\mathbf{r}}_a\right)$ of particles. Then, I shall assume that the velocity is in the direction of the Z axis and the density $\rho$ is homogeneous in a slab with limits $\alpha \le z\le \beta .$ Then, the following should be substituted for Equation (62)

$$\sigma \left(\theta ,\varphi \right)\propto {\int }_{-\infty }^{\infty }d{x}_a{\int }_{-\infty }^{\infty }d{y}_a\int v_b^{2}d{v}_b{P}_v\left({\mathbf{r}}_a\to {\mathbf{v}}_b\right).$$

(63)

Calculating exactly $P_v\left({\mathbf{v}}_b\right)$ is involved, but it is relatively simple in the Born approximation.

In our approach, the Born approximation consists of writing the product ${\prod }_{j=1}^{n-1}{Q}_j$ as an expansion in powers of $\epsilon F$, taking Equation (60) into account and truncating the expansion to second order; that is,

$$\begin{array}{ccc}\hfill {\displaystyle \prod _{j=1}^{n-1}}{Q}_j& =& {\displaystyle \prod _{j=1}^{n-1}}(D_j+\epsilon F_j)\simeq {\displaystyle \prod _{j=1}^{n-1}}{D}_j+\sum _k\left\{{\displaystyle \prod _{j=1}^{k-1}}{D}_j\right\}\epsilon F_k\left\{{\displaystyle \prod _{l=k+1}^{n-1}}{D}_l\right\}\hfill \\ & & +\sum _{ki}\left\{{\displaystyle \prod _{j=1}^{k-1}}{D}_j\right\}\epsilon F_k\left\{{\displaystyle \prod _{l=k+1}^{i-1}}{D}_l\right\}\epsilon F_i\left\{{\displaystyle \prod _{r=l+1}^{n-1}}{D}_r\right\}.\hfill \end{array}$$

(64)

It can be shown that the sum ${\sum }_k\epsilon F_k$ consists of n terms; hence, it remains finite in the limit $\epsilon \to 0;$ see Equation (50). Therefore, the small parameter in the expansion is actually the potential V, see Equation (60), as is typical in the Born approximation. Products like ${\prod }_{j=1}^{k-1}{D}_j$ correspond to motion in straight lines and constant velocity from the time $t_j$ to the time $t_{k-1}$. Terms like $F_k$ give the probabilities of the possible changes in velocity $s_k$ at time $t_k;$ see (60).

The first two terms in Equation (64) do not contribute to the cross section; hence, the third term is dominant in the expansion. That term leads to the well known result of Born cross section [33]. That is, in terms of the initial and final wavevectors associated with the particle, ${\mathbf{k}}_a$ and ${\mathbf{k}}_b$, respectively, we get

$$\sigma \left(\theta ,\varphi \right)={\displaystyle \frac{1}{16{\pi }^2}}{\left|\int d\mathbf{x}exp\left[-i\mathbf{x}·\left({\mathbf{k}}_b-{\mathbf{k}}_a\right)\right]V\left(\mathbf{x}\right)\right|}^2,\mathbf{k}={\displaystyle \frac{m\mathbf{v}}{\mathit{ħ}}}.$$

5.6. Interference Experiments with Particles

The wave behavior of electrons, proposed by L. de Broglie in 1923, soon led to experiments proving their interference. Later on, similar experiments have been performed with neutrons, atoms and even molecules. The results may be explained with the formalism presented in this section.

Indeed, Born’s approximation allows us to calculate the result of a simple interference experiment. In fact, let us consider a particle with initial velocity ${\mathbf{v}}_0=\left(0,0,v_0\right),$ which is moving in the Z direction and eventually arrives at a region with the potential

$$V\left(\mathbf{r}\right)=C\left\{exp\left[-\lambda {\left|\mathbf{r}+\mathbf{a}\right|}^2\right]+exp\left[-\lambda {\left|\mathbf{r}-\mathbf{a}\right|}^2\right]\right\},\mathbf{a}\equiv \left(a,0,0\right),$$

(65)

which is a model for a screen with two holes, chosen for an easy calculation. Obtaining the cross section via Born’s approximation is not difficult using Equation (64). We get

$$\sigma \propto exp\left[-{\displaystyle \frac{{\mathbf{v}}^2+{\mathbf{v}}_0^2-2v_0{v}_z}{2\lambda }}\right]{cos}^2\left(a{v}_x\right),$$

where $\mathbf{v}=\left(v_x,v_y,v_z\right)$ is the final velocity. Hence, the cross section becomes, taking the conservation of energy into account,

$$\sigma =C^2{\displaystyle \frac{\pi }{{\lambda }^3}}exp\left[-{\displaystyle \frac{2v_0^2{sin}^2\theta }{\lambda }}\right]{cos}^2\left(a{v}_{0}sin\theta cos\varphi \right).$$

(66)

Assuming that particle detections are observed as spots produced on a screen placed parallel to the $XY$ plane, we would observe typical interference fringes with a decreasing intensity in both directions X and Y and a maximum at $x=y=0$.

The point of this calculation is that a wave behavior of particles is absent; particles remained during the interference experiments. The wave behavior in the interference is an effect of the non-local action of the potential. However, it is plausible to assume that the said action is mediated by some “hidden” waves, which fits in with the assumption that quantum particles are not simple objects, but they are ”dressed” with waves at a difference with the classical ones. In fact, I propose that the mentioned waves may be just some combination of the fluctuating spacetime and the ZPF of all quantum fields, in particular the electromagnetic one, but not only it.

5.7. Discussion

I have shown that in non-relativistic quantum mechanics (without spin) it is possible to picture the transition probability in terms of particle paths. A path may be defined by the positions $\left\{{\mathbf{r}}_a\equiv {\mathbf{r}}_0,\dots {\mathbf{r}}_j,\dots {\mathbf{r}}_b\equiv {\mathbf{r}}_n\right\}$ at times $t_{a,}\dots t_j\equiv t_a+j\epsilon ,\dots t_b$ or, what is equivalent, the initial and final positions plus the velocity changes $\left\{{\mathbf{s}}_j/\epsilon \right\}$ at times $t_j.$ Eventually, we should consider the limit $n\to \infty$ with $n\epsilon =t_b-t_a.$

In summary, the formalism suggests an intuitive picture of non-relativistic quantum mechanics in terms of probabilities of the possible paths of particles. The particle’s motion is represented by a stochastic process such that there is a random change in velocity at every time $t_j$, with a probability depending on the potential over a large region around the position of the particle (indeed, it derives from the Fourier transform of the potential; see Equation (60)). The wave behavior, e.g., in experiments on atom interference, may be interpreted assuming that the motion of the particles is governed by a law (different from Newton’s) where the “acceleration” depends on the potential of a whole spatial region, at a difference with the local action of classical dynamics. I have dealt with a single particle, but the generalization to N interacting particles is straightforward, except for the possible effects of quantum statistics.

With this interpretation, the interference experiments with particles (e.g., atoms) might be explained without assuming that those particles possess a wave nature or that they may cross two distant slits at the same time. But I stress again that we remain at the level of non-relativistic quantum mechanics. I do not claim that a similar interpretation may be extended to relativistic quantum field theory when spin plays a role, or even to atoms or molecules when (Bose or Fermi) statistics are relevant.

6. Wave-Particle Duality

6.1. Realistic Interpretation of Photon Effects via the ZPF

The difficulty with reconciling the wave and particle behavior of light is a big obstacle for the classical-like interpretation of quantum phenomena. In particular, the celebrated experiment on the anticorrelation and recombination of light after a beam splitter [35] (see below) has been used in popular books in order to argue against the possibility of getting pictures of reality for quantum phenomena, e.g., [36].

The origin of the wave-particle duality goes back to Einstein’s proposal that light consists of particles, later called photons. Hence, he derived successfully the laws of the photoelectric effect (an achievement which was mentioned as merit for his Nobel Prize [8]). Einstein’s proposal of photons was actually unnecessary because the photoelectric laws may be derived from the weaker assumption that light is absorbed in discrete amounts of energy $h\nu ,$ which had been proposed (or suggested) five years earlier by Planck. It is true that Einstein seemed not too happy with the dual nature of light and attempted a fusion of waves and particles in later articles [37]. Indeed, in 1916, he introduced the concept of radiation needless, as a kind of alternative to “particles of light” (photons).

In the canonical (HS) formalism, some reality is ascribed to a zeropoint field (ZPF), assuming the existence of vacuum fluctuations. These fluctuations are supposed to consist of “short lived virtual particles”, a sentence that is senseless in our realistic interpretation due to the ambiguous meaning of the word “virtual”. However, the ZPF has a great relevance in the quantitative prediction of measurement results, both in quantum electrodynamics (QED) and in quantum optics (see e.g., [18]). In the WW formalism, the predicted EM radiation-filling space is interpreted as a real stochastic field, as discussed in Section 3.4.

In the following section, I present several phenomena that in principle might be quantitative interpreted by standard quantum (field) theory, but the eventual calculation does not provide a picture of reality. Therefore, I shall propose only heuristic qualitative or semiquantitative models in the following section.

6.2. Photon Detection: Photocounts

In order to take account of “photon detection”, 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 behavior of radiation. In fact, the former are caused by the granular (atomic or molecular) nature of the plate, and the photocounts in a detector derive from the fact that photon counters are manufactured so that they click whenever the radiation arriving at the detector transfers to it enough energy, which is compatible with light being waves [38,39]. It is true that in this case discriminating (weak) light signals, coming to the detector from a source, from the assumed (strong) vacuum radiation (ZPF), looks like searching for a needle in a haystack. However, there are models for photodetection that avoid the problem [38,39]. The models take advantage of the fact that the ZPF flux comes from all directions; it has rotational invariance on average. Hence, we may assume that the clean effect is nil because the pressure from different directions cancels out. In contrast, the radiation coming from a source is directional.

Let us now compare the interpretation of the radiation energy measurement in either the canonical or WW formalisms. In HS, the probability distribution of values got in the measurement of an observable $\widehat{M}$ in the state with density matrix $\widehat{\rho },$ may be obtained via the moments; that is, the expectation values of powers of the observable, $〈M^n〉,$ as follows:

$$〈M^n〉=Tr\left({\widehat{M}}^n\widehat{\rho }\right).$$

(67)

In WW, the expectation becomes an integral of the observable, written in terms of the amplitudes, weighted by the probability distribution in the (mixed) state; see Equation (39). In the case of the EM field, the most relevant observable is the energy, where $\widehat{M}$ becomes the Hamiltonian operator.

As an illustration, let us calculate the expectation value of the energy within WW for the general state $W_{\psi }^{total}$ given in Equation (36). In order to agree with HS predictions, we shall use the normally ordered Hamiltonian Equation (38). For a state $\mid \varphi \rangle$ of the field, the calculation, firstly in HS, then in WW, may be written as

$$\begin{array}{ccc}\hfill 〈E〉& =& 〈\varphi \left|\sum _l\mathit{ħ}{\omega }_l{\widehat{a}}_l^{†}{\widehat{a}}_l\right|\varphi 〉\hfill \\ & =& \sum _l\mathit{ħ}{\omega }_l\int W_{\varphi }\left(\left\{a_l\right\}\right)\left({\left|a_l\right|}^2-{\displaystyle \frac{1}{2}}\right)d\mathrm{Re}{a}_{l}d\mathrm{Im}{a}_l\hfill \\ & =& \sum _l\mathit{ħ}{\omega }_l\int {\left|a_l\right|}^2\left(W_{\varphi }\left(\left\{a_l\right\}\right)-W_0\left(\left\{a_l\right\}\right)\right)d\mathrm{Re}{a}_{l}d\mathrm{Im}{a}_l,\hfill \end{array}$$

(68)

where $W_{\varphi }\left(\left\{a_l\right\}\right)$ is the state (Wigner function) that in WW represents the HS state $\mid \varphi \rangle$ and $W_0$ the vacuum Wigner function Equation (34). The result is that the ZPF does not contribute to detection, which is in agreement with the HS result. The property may be stated by saying that photodetectors are sensitive only to radiation that excludes the ZPF. For a physically realistic interpretation of Equation (68) in WW, and a more extended discussion about photodetectors, see [38].

6.3. Absorption of Light in Discrete Amounts and the Photoelectric Effect

As is well known, in order to derive his radiation law, in 1900 Planck introduced the hypothesis that absorption and emission of radiation takes place in “quanta” of energy $E=\mathit{ħ}\omega .$ In the following, I will show that absorption in discrete amounts may be explained by the action of the ZPF.

It is plausible that absorption of radiation takes place via resonance with material oscillators. Thus, we may consider a light signal with wavevector ${\mathbf{k}}_0$ that arrives at a material having weakly bound electrons whose motion possesses a relevant component with frequency $c\left|{\mathbf{k}}_0\right|$ and it is parallel to ${\mathbf{k}}_0.$ The ZPF may be described in terms of plane waves and we are interested in those having wavevectors $\mathbf{k}$ near ${\mathbf{k}}_0$. From time to time it may happen that several of these waves have phases close to the incoming signal, say in a frequency range $\Delta \omega ,$ so that they may interfere constructively, giving rise to an unusually large intensity during some coherence time T of order $1/\Delta \omega$. In this case, a transfer of energy to the detection material will be most probable and an electron may be ejected. It may be shown that the absorbed energy would be of an order of twice the mean energy per mode; that is, $E\sim \mathit{ħ}\omega ,$ see [22].

6.4. Linear Momentum of the “Photon”: Radiation Needles

In his 1916 work on the emission of light by atoms, Einstein predicted that it should be directional and random. The latter feature bothered Einstein by the apparent violation of causality. The former was taken as a reinforcement of the concept of photon, which acquired definite linear momentum in addition to energy. Actually, both features are straightforward consequences of the ZPF. It is plausible that emission is stimulated by radiation either from or belonging to the ZPF. In the standard quantum language, the former is named stimulated and the latter spontaneous. The former would be in the same direction as the incident beam, but the latter in a random direction due to the stochasticity of the ZPF.

A more detailed, but semiquantitative, description of the emission is as follows. Let us assume that a strong fluctuation of the ZPF with frequency $\omega$ arrives at an atom and it happens that $\omega$ 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 that of the incoming wave. 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 would be constructive in the forward direction and mainly destructive in all other directions. The outgoing energy would be concentrated within the region where the phase difference is small, with the boundary defined by the following relation with the distance, d, and the half angle, $\theta$, as seen from the atom. Then, we have

$${\displaystyle \frac{d}{cos\theta }}-d\sim {\displaystyle \frac{\lambda }{2}}\Rightarrow \theta \sim \sqrt{{\displaystyle \frac{\lambda }{d}}},$$

(69)

where $\lambda$ is the wavelength. If we take d to be the coherence length of the emitted light wavepacket (the alleged “photon”), for typical atomic emissions we have $d\sim 1$ m, $\lambda \sim 1\mu ,$ so that $\theta \sim {10}^{-3}$. This fits with Einstein’s proposal of “needles of radiation” in the atomic emission.

6.5. 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 an X-ray, with frequency ${\omega }_1,$ and one electron, giving rise to another photon with smaller frequency, ${\omega }_2,$ at an angle $\theta$ with the incident radiation and a recoil electron. Indeed, the (relativistic) kinematics may be explained assuming that there are incident and outgoing radiation needles having energies $\mathit{ħ}{\omega }_1$ and $\mathit{ħ}{\omega }_2,$ respectively, and the electron is initially at rest. In summary, quantum electrodynamics (in the canonical formalism) gives a quantitative account of the phenomenon, including the cross section of the process, but it does not offer a clear intuitive picture. The WW formalism for the field, in particular the random stochastic vacuum field, provides a stochastic picture if we substitute radiation needles for photons [22]. However, the derivation of the cross section in WW seems involved and will not been attempted here.

6.6. Anticorrelation and Recombination Experiment

A remarkable experiment showing the particle behavior of light is the anticorrelation after a beam splitter [35]. In the experiment, a light beam is sent to a balanced non-polarizing beam splitter BS1. Two detectors, say A and B, placed in front of the two outgoing channels, may measure the single, ${\mathrm{P}}_A$ and ${\mathrm{P}}_B$, and coincidence, ${\mathrm{P}}_{AB}$, detection probabilities within a small time window. We expect that the probabilities fulfill

$$r\equiv {\displaystyle \frac{P_{AB}}{P_A{P}_B}}\simeq {\displaystyle \frac{〈I^2〉}{{〈I〉}^2}},$$

(70)

assuming that the probabilities are proportional to the intensities arriving at both detectors, these supposed identical. The measured rates are given by the products of detection probability times and the number of windows in a unit time interval.

If the radiation has a sure (nonfluctuating) intensity, like in a laser beam, then we have

$$〈I^2〉={〈I〉}^2\Rightarrow r={\displaystyle \frac{〈I^2〉}{{〈I〉}^2}}=1$$

meaning that the detections are uncorrelated. On the other hand for chaotic (e.g., thermal) light we would have

$$〈I^2〉=2{〈I〉}^2\Rightarrow r={\displaystyle \frac{〈I^2〉}{{〈I〉}^2}}=2.$$

The change from $r=1$ to $r=2,$ a phenomenon known as “photon bunching”, has been interpreted as a quantum effect due to the Bose character of photons. But a simple classical explanation is that it derives from correlated Gaussian fluctuations of the chaotic light.

The particle behavior of light appears if the radiation incoming BS1 is weak, and also the set up is appropriate in order to prepare the beam as a series of "single photon states" in quantum language. In this case, quantum theory predicts, and the experiment confirms [35], that the value of r, Equation (70), is much smaller than unity. The current quantum explanation is that photons are not divided, but go to one of the channels each. However, in the WW formalism the particle behavior is caused by the ZPF [20,22]. In fact, in BS1, in addition to the signal entering one incoming channel, there is another incoming channel where ZPF may enter, which interferes with the signal beam. The interference should be destructive in one of the outgoing beams if it is constructive in the other one, by conservation of energy. If we assume that radiation is detected only when it is more intense than the average ZPF beam, then there may only be detection in one of the detectors, so explaining the anticorrelation.

Grangier et al. [35] also showed a wave behavior of light in the recombination experiment, where the detectors in front of BS1 are removed and, via appropriate mirrors, the two beams emerging from BS1 are sent via two incoming channels of another beam splitter BS2. The intensity emerging from one of the outgoing channels of BS2 corresponds with the superposition of the beams arriving at the incoming channels. The result of the experiment is that detection is observed only in one the detectors placed in front of the outgoing channels of BS2, but which detector clicks depends on the difference between the two path lengths in the travel of light between BS1 and BS2. The standard quantum explanation is that light behaves as a wave and the recombination reproduces the initial beam sent to BS1.

The recombination may also be easily interpreted within the WW formalism, but I omit the details [20,22].

However, the current interpretation is that the experiments show both the particle and the wave behavior or light; the former in the anticorrelation and the later in the recombination. Thus, the experiment is mentioned in popular books as a proof of the impossibility of a realistic interpretation of quantum mechanics.

7. Entanglement and Bell Inequalities

7.1. Entangled Photon Pairs from Parametric Down Conversion

Entanglement is a quantum property that may be easily defined mathematically within the canonical (Hilbert space) formalism, but the definition does not provide an intuitive picture of the phenomenon. It is currently seen as a specific quantum form of correlation, which is claimed to be dramatically different from the correlations that appear in classical physics. However, the WW formalism provides a picture for entangled “photon pairs”, at least those produced via “spontaneous parametric down conversion” (SPDC), which is the most widely used method to get entangled photon pairs. The production of “photon pairs” via SPDC is as follows [40].

A laser beam with frequency ${\omega }_0$ is sent to an appropriate crystal possessing nonlinear electric susceptibility. Then, as a rainbow appears in the opposite side of the crystal, typically, two (“conjugated”) beams A and B are selected amongst those in the rainbow, with frequencies ${\omega }_a$ and ${\omega }_b$ fulfilling

$${\omega }_a+{\omega }_b={\omega }_0,$$

We may assume that a light beam with (complex) amplitude $a,$ and frequency ${\omega }_a,$ from the ZPF, enters the crystal on the same side as the laser. Then, the interaction of the laser, the beam a and the electrons of the crystal give rise to radiation of light with amplitude $a^{*}$ in a different direction. In the same direction as $a^{*},$ a beam with amplitude $b,$ also from the ZPF, enters the crystal and its interaction with the laser and the electrons produces radiation with amplitude $b^{*}$ and frequency ${\omega }_b.$ It is the case that this beam travels in the same direction as $a.$ The result is that two rays emerge from the crystal and they may be represented as functions of time t by

$$A\left(t\right)=a\left(t\right)+Db{\left(t\right)}^{*},B\left(t\right)=b\left(t\right)+Da{\left(t\right)}^{*},$$

(71)

where D is a complex parameter, $\left|D\right|<<1.$ The direction of the outgoing beam A is the same as that of the incoming beam a and the produced beam $b^{*}$, and, similarly, B, b and $a^{*}$ are colinear. Of course, other radiation from the ZPF may enter the crystal, giving rise to other beams that are not collected in the experimental set up. The beams actually selected in the experiment are the outgoing A and B. For a derivation within classical electrodynamics (but taking the ZPF as real), see Ref. [40].

The calculation within the Weyl–Wigner formalism of the correlation between $A\left(t\right)$ and $B\left(t\right)$ agrees with the predictions of the canonical formalism for the correlation experiment. However, the common interpretation of the canonical calculation is that the beams A and B consist of pairs of photons, with one photon in each beam, which are entangled. Our analysis within WW provides an intuitive picture where the strong correlation amongst the beams A and B comes from the fact that the beams have fluctuating intensity and the positive fluctuations of $a^{*}$ coincide in time with those of a. This is also the case for b and $b^{*}.$ As a consequence, there will also be correlated fluctuations amongst A$\left(t\right)$ and B$\left(t\right)$. Assuming that detection events happen at those times when the beams arriving at the detectors have high intensity, then coincidence detections may happen most frequently when fluctuations coincide. The relevant result is the prediction that the coincidence detection rate $R_{AB}$ may be close to the single detection rates. That is,

$$R_A\simeq R_B\simeq R_{AB},$$

(72)

which is to be compared with the typical classical situation (without ZPF) where $R_{AB}<<R_A\simeq R_B$. This analysis within WW provides an intuitive picture of entanglement [22,38,39].

In the canonical (Hilbert space) derivation, the result involves the commutation relations amongst creation and annihilation operators, but it does not provide any intuitive picture. In the WW the “photon entanglement” appears as a correlation between the fluctuations of the field intensities of two beams. The canonical (Hilbert space) counterpart of Equation (71) is

$$\widehat{A}=\widehat{a}+D{\widehat{b}}^{*},\widehat{B}=\widehat{b}+D{\widehat{b}}^{*}.$$

(73)

7.2. Local Realism and Bell Inequalities

As is well known, in 1964 Bell derived some inequalities that should be fulfilled by any local hidden variables (LHV) model. Later, the concept of LHV was generalized and the Bell inequalities are currently assumed to be valid for any realistic local model. Thus, the inequalities are used in order to discriminate between quantum theory and local realistic theories. It is assumed that any empirical violation of a Bell inequality would imply “the death of local realism”.

Many experimental tests of the inequalities have been performed, those involving “entangled photon pairs” being the most relevant. The current opinion is that, in fact, local realism has been empirically refuted [41]. However, I believe that the subject is not yet closed, but this belief would require careful and long arguments in order to be convincing. Thus, the matter will not be studied further in this article. My arguments may be seen in the references [42,43].

8. Conclusions

I contend that physics should provide a coherent account of reality, in addition to offering an algorithm for the prediction of empirical results. However, the mainstream of the physicist community believes that quantum mechanics does not allow a realistic interpretation. In particular, quantum fields, allegedly, show a simultaneous behavior as particles (localized) and waves (extended), in spite of these being contradictory concepts. This fact has given rise to a variety of interpretations of quantum mechanics, none of which have reached a consensus. The Copenhagen interpretation is still the most popular but, in fact, it is rather the statement that no interpretation is needed, just that a good predictive power is required for physical theories. I do not agree, and I have made efforts to get a “realistic” view of nature that reaches consensus as it happens in classical physics.

My efforts have had only partial success, as is reviewed in this article. The most relevant achievement is the proof that the Wigner (or Weyl–Wigner, WW) representation predicts, for Bose quantum fields, the same (correct) results as the most common canonical formalism resting on Hilbert spaces. The WW formalism leads to a straightforward realistic interpretation of the Bose fields as continuous, that is, wavelike. The most important result is the existence of vacuum fields in the form of Gaussian random radiation (sometimes named the zeropoint field, ZPF). The study of these results is made in Section 3.

The existence of the ZPF allows a qualitative interpretation of many empirical results attributed to the “corpuscular behavior (or nature)” of the fields. This is the subject of Section 6.

The wave behavior of particles like electrons, neutrons, atoms or molecules in interference experiments may be explained by non-local forces. That is, we may assume that the force may be mediated by some (hidden, not well known) fields spread out over long distances. A simple model is presented in Section 5, where the motion of non-relativistic particles is formulated as a probability distribution of paths, which depend non-locally on the Fourier transform of the potential. This allows for a simple interpretation of particle interference, which is given in Section 5. In particular, in the two-slit experiment, one slit may influence the motion of a particle placed near another slit and separated from the former by a macroscopic distance.

Finally, I offer an intuitive picture of entanglement, in particular entangled photon pairs, in Section 7, where I also briefly comment on the Bell inequalities.

The treatment of classical relativity is standard, except that I include a philosophical proposal for the interpretation of general relativity in Section 2.5, which is independent from the rest of the article. Quantum gravity is treated rather superficially in Section 4 because I do not have a clear opinion about it (and lack sufficient familiarity with the subject). But, I suggest that spacetime might be quantized as a consequence of the matter quantization via Einstein’s equation; hence, quantized spacetime would mean assuming the existence of a probability distribution of metrics.

My article has an important incompleteness because I cannot give any hint of a realistic interpretation of Fermi fields. Also, I touch but slightly on the important problems of spacetime singularities (on black holes and the very early universe) and the alleged loophole-free empirical violation of Bell inequalities. However, I hope that the article provides an advance in our understanding of physical reality.