# Toward an a-temporal interpretation of quantum potential.

Abstract

Bohm's pilot wave theory, introducing the concept of quantum potential, provided a causal, and at the same time nonlocal, description of quantum phenomena. Starting from some results of a-temporal gravitation and loop quantum gravity, we show here that subatomic particles can be interpreted, at the deepest level of reality, as the result of the interaction of energy (in the "entropy state") with one or more elementary grains of a-temporal physical space. A new interpretation of quantum potential is thus introduced and illustrated: The quantum potential intended as "special state of a-temporal physical space in presence of microscopic processes." When we take into consideration an atomic or subatomic process, an a-temporal physical space assumes the special "state" represented by quantum potential in consequence of the energy shifting between certain quanta of space; shifting that materializes the subatomic particle in exam in the different points of physical space (and which is determined by the vibrations at appropriate frequencies of the quanta of space occupied by the particle). We underscore that this new interpretation of quantum potential appears lawful also in virtue of the fact that both a-temporal physical space and quantum potential explain nonlocality, the instantaneous link between two subatomic particles independently from their distance. Finally, we show that this new interpretation of quantum potential provides a convincing reading of some experiments characteristic of atomic physics (double-slit interference, tunnelling).

Introduction

In Bohm's pilot wave theory, the basic idea is that a wave and a corpuscle contemporarily constitute each elementary particle of physics; here, the wave has the role to guide the corpuscle during its movement. The wave is described mathematically by Schrodinger's wavefunction [psi]. Under the guide of the wave, the movement of the corpuscle happens in agreement with a law of motion, which assumes the following form

[partial derivative]S / [partial derivative]t + [|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = 0

(where R is the amplitude and S the phase of the wavefunction, h is Planck's reduced constant, m is the mass of the particle and V is the classic potential) and thus is equal to the classical equation of Hamilton-Jacobi, except for the appearance of the additional term

Q = [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R

having the dimension of an energy and containing Planck's constant and (therefore) appropriately defined quantum potential. (1-3) The equation of motion of the particle can also be expressed in the form,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus equal to Newton's second law of classical mechanics, always with the additional term Q of quantum potential.

The movement of an elementary particle, according to Bohm's pilot wave theory, is thus tied to a total force that is given by the sum of two terms: A classical force (derived from a classic potential) and a quantum force (derived just from quantum potential). Quantum potential must not be considered a term that is introduced ad hoc, contrary to the opinion of the supporters of the Copenhagen interpretation. Quantum potential plays an essential role in quantum formalism: In the plant of Bohm's theory, it emerges directly from Schrodinger's equation and without it, energy shouldn't be conserved. In fact, equation

[partial derivative]S / [partial derivative]t + [|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = 0

taking into account that the quantity is the total energy of the particle and that

- [partial derivative]S / [partial derivative]t

[|[nabla]S|.sup.2] / 2m

is its kinetic energy, can be also written in the equivalent form

[|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = - [partial derivative]S / [partial derivative]t

which can be seen as a real energy conservation law in quantum mechanics: Here one can easily see that without the quantum potential

Q = - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2] R / R

energy couldn't be conserved. It must be observed, as it was shown recently by Hiley, that quantum potential can also be derived by Heisenberg's formalism by choosing a particular representation for operators, and here such terms must be present to assure the conservation of the total energy of the system. (4)

By introducing quantum potential, Bohm demonstrated that it is possible to provide a causal, and at the same time nonlocal, interpretation of quantum mechanics. Bohm's theory clearly shows that nonlocal correlations between subatomic particles--and which constitute a fundamental feature of many-body systems--are caused by the action of quantum potential. For a many-body system, quantum potential acting on each particle is a function of the positions of all the other particles and thus, in general, doesn't decrease with distance. As a consequence, the contribution to the total force acting on the i-th particle coming from the quantum potential, i.e. [[nabla].sub.i]Q does not necessarily fall off with distance and, indeed, the forces between two particles of a many-body system may become stronger, even if |[psi]| may decrease in this limit. The equation of motion of the i-th in particle, in the limit of big separations, assumes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus depends on the coordinates of all the n particles of the system. This determines just nonlocal correlations in a many-body system.

Since 1979, the quantum potential approach (thank to the contributions, above all, of Philippidis, Dewdney, Hiley, and Vigier) has explained, in visualizable terms, many experimental results (for example, classic double-slit experiment, tunnelling, trajectories of two particles in a potential of harmonic oscillator, EPR-type experiments, experiments of neutron-interferometry). In 1984, Bohm proposed an idea to interpret quantum potential as a sort of "information potential"--the particles in their movement are guided by the quantum potential just as a ship on automatic pilot can be handled by radar waves of much less energy than that of the ship. On the basis of this interpretation, the results of double-slit experiment are explained by saying that the quantum potential contains an active information--for example the slits--and that this information manifests itself in the particles' motions.

Here a new interpretation of quantum potential is proposed: The idea that quantum potential can be seen as the "special state of a-temporal physical space in presence of microscopic processes," as the whole of the characteristics that a-temporal physical space assumes when we have quantum phenomena.

Some Considerations about Loop Quantum Gravity and A-temporal Physical Space

Some results of a-temporal gravitation theory and loop quantum gravity can be considered the starting points of an a-temporal interpretation of quantum potential.

A-temporal gravitation theory is a theory of gravitation that represents the natural consequence of the following idea. On the basis of our elementary perception, the passing of time cannot be perceived directly as matter and space; we can perceive only the irreversible changes of matter in space. It's beyond our power to establish if time actually exists as physical reality; we can say that time exists only as a stream of material changes happening in space. (5)

From this point of view (alternative in regards to those conventionally adopted in physics, but perhaps more correct and appropriate because it is more coherent with experimental facts), it derives that the stage in which physical phenomena have place isn't space-time, but really a four-dimensional a-temporal space. Phenomena happen in space-time only in the mathematical models of reality, which sometimes become more real than reality itself, which instead--in virtue of the elementary perception--turns out to be a-temporal. General relativity can, therefore, be interpreted in the following way: the density of this a-temporal space transmits gravity and its effect is to determine modifications in the geometrical properties of this a-temporal physical space.

The density D of physical space in a given volume depends on the amount of matter into it. In the center of a given material object, the density is D = m x G where m is the mass contained in that volume and G is the gravitational constant. In a-temporal gravitation theory, gravitational interaction is immediate, acting directly via the density of physical space. In other words, gravitational interaction is a-temporal (has no physical time), in the sense that no material change (travelling of wave or particle) through the a-temporal cosmic space is needed for its acting. (6)

On the other hand, loop quantum gravity predicts that physical space has a granular structure given by a net of intersecting loops, and it's these loops, these nodes that represent the quantum excitations of gravitational field, i.e., the elementary quanta of space. Area and volume of physical space are quantized. In particular, the elementary quanta of spatial volume reside precisely at the nodes of the net, while links (among the different nodes of the net) carry quantum numbers of area elements. These quantum numbers and their algebra look like the spin angular momentum numbers of elementary particles, and therefore the elementary grains of space, i.e., the loops of the net, can be appropriately defined spin elements or "spin networks." The picture of physical space provided by loop quantum gravity is practically the following: The elementary grains of space are represented by nodes of spin networks (they are called this because they satisfy an algebra analogous to that of spin), and their volume is given by a quantum number, which is associated with the node in units of the elementary Planck volume, V = [(<begin strikethrough>h<end strikethrough>G / [c.sup.3]).sup.3/2] where h is Planck's reduced-constant, G the universal gravitation constant, and c the speed of light. Two nodes are adjacent if there is a link between the two, in which case they are separated by an elementary surface the area of which is determined by the quantum number associated with that link. Link quantum numbers, j, are integers or half-integers and the area of the elementary surface is

A = 16[pi][V.sup.2/3] [square root of j (j+1)]

where V is the Planck volume. (7)

On the grounds of these results of loop quantum gravity and a-temporal gravitation theory, one can assume that the fundamental constituents of physical space (which can be called quanta of space) have the size of Planck's volume and are a-temporal entities. In virtue of the first law of thermodynamics, QS have not been created and cannot be destroyed. (7) Now, the interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes" arises just from the granular structure of a-temporal physical space. There is one other ingredient to add: How to interpret a material particle inside quantized a-temporal physical space.

The Interpretation of Subatomic Particles in Quantized A-temporal Physical Space

In the model proposed here, quanta of space (QS) are the fundamental constituents of a-temporal physical space (ATPS) and of all material objects. All physical reality we observe and experience is constituted by QS; both ATPS in which matter is present, and ATPS in which there isn't matter, are constituted by QS. The only difference is that ATPS, in absence of matter, has no entropy (there is no experimental evidence that such a space has entropy; its only property is density); instead, matter has entropy and can change its state. Therefore, one can say that QS, in absence of matter, are in the "non-entropy state" of energy (and it's for this reason that space in absence of matter appears to us "empty" or without changes), while QS constituting matter represent the "entropy-state" of energy (and it's for this reason that matter can change its state--for example, its position and its speed).

QS constituting ATPS change their electrical charge from positive to negative in a Planck time (5.39 * [10.sup.-44]s), vibrate at the "basic frequency" 0.19x[10.sup.-44][s.sup.-1], have a "basic energy" given by the relation Eqs = h*0.19x[10.sup.-44][s.sup.-1] where h is Planck's constant (6.626069x[10.sup.-34]Jxs), and thus Eqs = 1.26x[10.sup.10]J. QS of ATPS have a "bipolar nature" (namely, they change the electrical charge continuously) and are complete into themselves. Their existence doesn't depend on other physical entities. They have no radiation, no "dispersion of the energy"; their energy is always the same and precisely the basic energy Eqs = 1.26x[10.sup.10]J. This means that QS that build up ATPS have no entropy.

In ATPS endowed with a granular, quantized structure, every sub-atomic particle can be seen as the result of the interaction of energy (in the "entropy-state") with one or more QS. More precisely, one can suggest that particles that are devoid of internal structure such as quarks, leptons and intermediate bosons, can be seen as the result of the interaction of energy (in the "entropystate") with one quantum of space; instead, particles, which, turn out to be endowed with an internal structure such as baryons (constituted by three quarks) and mesons (constituted by a quark-antiquark pair) are given by the interaction of energy (in the "entropy-state") with more QS.

And what happens to a quantum of space when it changes its energy from the "non-entropy-state" to the "entropy-state", thus becoming material quantum? What allows a quantum of space to change its energy from the "non-entropy-state" to the "entropy-state"? What does it produce in this change in the state of energy? In this model based on ATPS, one can assume that a quantum of space changes its energy from the "non-entropy-state" to the "entropy-state," becoming a quantum of matter, as a consequence of the vibration at a certain appropriate frequency (lower than the basic one).

While QS vibrating at the basic frequency 0.19x[10.sup.-44][s.sup.-1] have energy in the "non entropy state" (and therefore constitute t "empty" ATPS, where matter is absent), QS vibrating at appropriate frequencies (lower than the basic one) assume energy in the "entropy state" and become so material quanta, endowed by mass and therefore perceivable by our senses.

QS become seat of a discrete quantity of energy (in the "entropystate") in virtue of a vibration at an appropriate frequency. All material particles are composed by QS that vibrate at certain frequencies and that, in virtue of this vibration, become seat of a discrete quantity of entropic energy. It's the vibration of QS of ATPS at appropriate frequencies, lower than the basic one, that make such QS material particles, endowed with an energy in the "entropy-state," able to modify their state.

For example, according to the view proposed here, the electron is just a particle derived from a quantum of space that vibrates at certain frequencies and that, because of this vibration, becomes seat of a discrete quantity of entropic energy. The discrete quantity of entropic energy that a quantum of space assumes when it gives origin to an electron depends on the characteristics of the region in exam, on the situation existing in that particular region of ATPS (namely on the type of the interaction to which the quantum of space is subjected). In fact, in this model, the fundamental interactions and physical fields can be interpreted as special states, as special ambient situations existing in ATPS in the presence of certain material particles, and produce modifications in the properties of ATPS (in particular, gravity has the effect to produce modifications in the geometrical properties of ATPS while the other three interactions--electromagnetic, weak and strong--determine modifications in the "vibrations" of QS of the region of ATPS in exam).

In general, according to our research, an a-temporal model of the universe could be based on the following postulates:

1. Energy is composed by basic packets of energy, which have the Planck size (here called "quanta of space"--QS). QS constituting physical space have no entropy.

2. QS form physical space and matter. The "arena" (stage) of the universe is "space-matter," physical reality is composed of space and matter. QS are each described by a wave function depending on its position in physical space, a quantum number, which indicates its orientation (as to an arbitrary axis) and a frequency of vibration. QS vibrating with the basic frequency constitute physical space, are devoid of mass, and are the "non-entropy state" of energy, and therefore, are not perceived by our senses. QS vibrating with the frequencies characteristic of material particles constitute matter and are the "entropy state" of energy (and therefore, are perceived by our senses). Each subatomic particle is the result of the interaction of energy in the "entropy state" with one or more QS, caused by the vibration of these QS at appropriate frequencies. In particular, particles devoid of internal structures, such as quarks, leptons and intermediate bosons, are the result of the interaction of energy in the entropy state with one quantum of space; particles endowed with an internal structure, such as baryons and mesons, are given by the interaction of entropic energy with more QS. QS vibrating at the frequencies of the electromagnetic spectrum generate electromagnetic waves (which propagate through physical space at the speed of light).

3. Time is an irreversible change of matter in physical space. Physical space itself is a-temporal (here called "a-temporal Physical Space"--ATPS). The universe is an a-temporal phenomenon where ATPS and matter are in a permanent dynamic equilibrium. There was no beginning of the universe and there will be no end.

4. The fundamental interactions and physical fields represent special states of ATPS and the different ambient situations existing in ATPS in presence of certain material particles. All interactions produce and determine modifications in the properties of ATPS. In particular, gravity has the effect to produce modifications in the geometrical properties (i.e., in the curvature) of ATPS. The other three interactions (electromagnetic, weak and strong) determine modifications in the vibrations of QS. They can change the frequencies of QS from the "basic frequency" (characterizing empty ATPS) to other appropriate frequencies less than the basic one (which can be the frequencies characteristic of material particles and/or of electromagnetic waves), or they can change the frequencies characteristic of some material particles into the frequencies characteristic of other material particles (and/or of electromagnetic waves). Each interaction is characterized by its own strength parameter, which indicates the intensity of the modifications induced in the region of ATPS under study, and its own particular range, i.e., the range in which the modifications of ATPS determined by the interaction happen. The density of ATPS transmits gravitational force; material objects move in the direction in which density of ATPS is increasing. The other forces are each mediated by the exchange of a particular boson: The photon for the electromagnetic interaction, the intermediate bosons [W.sup.[+ or -]] and [Z.sup.0] for the weak interaction, and the gluons for the strong interaction.

5. The description of a force is not altered by any modification of the length scales of rulers and of temporal scales of clocks utilized as measurement instruments (gauge invariance principle). Each interaction satisfies its peculiar gauge symmetry. (For example, in the case of the electromagnetic interaction, the quantum-mechanical description of experiments on charged particles is invariant under local phase transformations on the particle wavefunction, if one introduces a long-range field coupled to the charge-the electromagnetic field-and one makes simultaneously a suitable local gauge transformation on the electromagnetic potential. In the case of the strong interaction, the gauge symmetry is the isospin symmetry--the strong interactions are invariant under rotations in the isospin space). There are solid experimental results (for example, the fact that the Active Galactic Nucleus of the Milky Way Galaxy is "eating" the galaxy of Sagittarius, and at same time, continuously emits fresh gas) and theoretical results (a-temporal gravitation theory and loop quantum gravity), which support the ideas contained in some of these postulates. (6-8)

In order to illustrate the new interpretation of subatomic particles inside ATPS, consider an electron being in a stationary state of hydrogen atom. According to this model, the electron in a stationary state of hydrogen atom, i.e., the ambient situation existing in ATPS represented by the coulombian field created by a proton, can be seen as the result of the interaction of entropic energy, given by one of the eigenvalues of the energetic spectrum

[E.sub.n] = - 1/2 [m.sub.e] [e.sup.4] / [<begin strikethrough>h<end strikethrough>.sup.2][n.sup.2]

(where is [m.sub.e] the mass of electron and n is an integer positive number), with one quantum of space. This quantum of space becomes seat of this entropic energy as a consequence of the vibration at an appropriate frequency [v.sub.n], which is obtained by the relation:

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

and thus is given by

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

The particular value of the frequency of vibration determined by the coulombian potential in a quantum of space depends on the position of this quantum of space. One can say in fact that the coulombian potential created by a proton changes the frequency of a quantum of space being at a distance

[r.sub.n] = [n.sup.2][<begin strikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton, from the value given by the "basic frequency" to the value

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

(and therefore determines the appearance in it of the entropic energy).

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

For example, in the case in which the quantum of space is at a distance

[r.sub.1] = [<begin strikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton the coulombian potential produces the change of the frequency of this quantum of space from the value given by the "basic frequency" to the value

[v.sub.1] = - 2[pi] [sup.2][m.sub.e.][e.sup.4] / [h.sup.3]

(and therefore determines the appearance in this quantum of space of the entropic energy

[E.sub.1] = - [m.sub.e][e.sup.4] / 2[<begin srikethrough>h<end strikethrough>.sup.2]

equal to the first eigenvalue of the energetic spectrum of an hydrogen atom). A quantum of space at a distance

[r.sub.1] = [<begin srikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton, under the action of the coulombian potential,

V(r) = -[e.sup.2] / r

vibrates at the frequency

[v.sub.1] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3]

and becomes, therefore, seat of a quantity of entropic energy equal to the first eigenvalue of the energetic spectrum of an hydrogen atom, i.e., gives origin to an electron being in the first stationary state of the hydrogen atom.

Therefore, in this model, the electron being in a stationary state of the hydrogen atom derives from a quantum of space vibrating at appropriate frequencies less than the basic one. It's the vibration of a quantum of space at an appropriate frequency that causes the change of energy of this quantum of space from the "non-entropy state" to the "entropy state" and, therefore, creates the appearance of an electron in a stationary state of the hydrogen atom. Therefore, the following reading of the mathematical formalism, which concerns the electron of an hydrogen atom turns out to be lawful: In a given region of ATPS, the ambient situation represented by the coulombian field created by a proton determines a modification in the properties of that region--more precisely, it produces the change of the frequency of a quantum of space surrounding the proton from the value given by the "basic frequency" to one of the values

[v.sub.n] = - 2[pi] [sup.2][m.sub.e.][e.sup.4] / [h.sup.3][n.sup.2]

This quantum of space, vibrating at one of these frequencies [v.sub.n] becomes seat of a discrete quantity of entropic energy given by

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

and this means that it has given origin to an electron being in a stationary state of the hydrogen atom.

On the grounds of this interpretation of the electron being in a stationary state of the hydrogen atom, it also derives the following important consequence: One can say that it's the special "state" of ATPS (in this case represented by the coulombian potential) to "create" matter (in this case the electron being in a stationary state of hydrogen atom). In other words, one can also say that the presence, in a given point of ATPS, of a mass and a charge equal to the mass and the charge of electron is an effect of the vibration of a quantum of space at a frequency given by one of the values

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2],

caused by the ambient situation existing in the region of ATPS in examination.

Also, the quantum wave associated with the electron being in a stationary state of the hydrogen atom can be considered an effect of the vibration at the frequencies

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2];

in fact, since the appearance in a quantum of space of an entropic energy given by

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

is determined by its vibration at the frequency

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

and since an electron having energy equal to one of these values

[E.sub.n] [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

is described by an eigenfunction of the hydrogen atom, i.e.,

[[psi].sub.nlm] (r, v, [phi]) [<begin strikethrough>m<end strikethrough>.sub.e] [R.sub.nl](r)[Y.sup.m.sub.l](v, [phi]),

one can say that the quantum wave-function of the electron being in a stationary state of the hydrogen atom can itself be seen as a consequence of the vibration at the frequency

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

This particular example of the electron in a hydrogen atom shows the ordinary stationary states of energy predicted by quantum mechanics (for the particles devoid of internal structure) can be seen as the effect of the vibration of a quantum at appropriate frequencies characteristic of that particle; the fact that the frequency is quantized then implies the energy (in the entropy state) that a quantum of space acquires (as a consequence of that vibration) is quantized, and to each of these values there will correspond a different quantum wavefunction.

Therefore, on the grounds of this model, one can say that it's just the vibrations of QS at the frequencies characteristic of subatomic particles, which give place to the quantum waves associated to the material particles. The wave behavior of the subatomic particles arises from the vibrations of QS constituting them. In short, one can say that all derives from the vibrations at appropriate frequencies (characteristic of material particles). The vibrations of QS at appropriate frequencies create the appearance of material particles (in the sense that, because of them, these QS become seat of a discrete quantity of energy in the "entropy state") and, at the same time, create the wave behavior, the quantum waves associated to such particles. These quantum waves can be interpreted both in standard sense (as mathematical tools to compute certain probabilities) and in more "realistic" senses. We emphasize, however, that we prefer to interpret them in a realistic sense. We suggest, in particular, that these quantum waves can be interpreted like in Bohm's pilot wave theory.

Now, let's consider and compare between them the wave equation

[psi] = - [delta]Q[[psi], t] / [delta][psi]

of Bohm's quantum field theory concerning a massless quantum field and the classical massive Klein-Gordon equation [psi] = -[m.sup.2][psi]. In this regard, it has been shown that for particular solutions of the Klein-Gordon equation, the quantum potential associated with a one-quantum state acts so that the mass-less quantum field behaves as if it were a classical field with mass. As a consequence of this fact, in Bohm's quantum field theory, there is the possibility that the attribute of mass can be seen as an effect of quantum potential, i.e., that quantum potential can be viewed as the origin of the mass of a particle. (2)

If we take into account that in Bohm's quantum field theory, there is a possible link between the mass of a particle and the quantum potential, and that (in the model suggested here) the appearance of a mass means the appearance of an entropic energy, it seems lawful to think that the quantum waves actually guide the corresponding particles, through the action of quantum potential, during their motion, in the different points of ATPS. Taking into account the possible link between the mass of a particle and the quantum potential, on the grounds of the model proposed here, it appears lawful to suggest the idea that the role of quantum potential is just to transfer a discrete quantity of entropic energy from one quantum to another of ATPS (thus making a particle appear in the QS, which compose its trajectory). One can say that the quantum waves associated with material particles guide the corresponding particles in the regions where the wavefunction is more intense, through the action of quantum potential, which is responsible for the entropic energy shifting between the different QS composing the trajectory described by the particle in ATPS.

Quantum Potential as "Special State of A-temporal Physical Space in Presence of Microscopic Processes"

Studying the motion of a subatomic particle on the basis of the laws of Bohm's version of quantum mechanics and of our a-temporal model in quantized ATPS, we have the following results: On one side, in Bohm's version of quantum mechanics, the movement of the particle is determined by the sum of a classic potential and a quantum potential and, on the other side, in quantized ATPS, it's tied to the interaction of a discrete quantity of entropic energy with the various QS composing the trajectory described by the particle itself (interaction produced by the vibration of these QS at certain appropriate frequencies). In the model here, the vibration at appropriate frequencies of a quantum of space determines the interaction of energy in the "entropy-state" with this quantum of space and the appearance in it of a subatomic particle (or part of it); this vibration also produces a quantum wave that guides the particle during its motion, making it appear in the different points of ATPS. Therefore, the vibration of certain QS at appropriate frequencies always causes the motion of a subatomic particle.

The evolution of the wavefunction of a physical system happens in space-time only in the mathematical models of reality. According to this model, the wavefunction [psi] describing the state of a given physical system doesn't vary in time, but varies in a four-dimensional ATPS and the stream of changes that it has in space is itself time. This means that the coordinate t, which appears in Schrodinger's equation

i<begin strikethrough>h<end strikethrough> [partial derivative][psi] / [partial derivative]t = H[psi]

(H being the Hamiltonian of the system), doesn't represent time but rather the stream of changes that the physical system has in ATPS and, thus,

[partial derivative][psi] / [partial derivative]t

is the partial derivative of the wavefunction with respect to the stream of changes of the system in ATPS.

Also, the law of motion of the particle in Bohm's version of quantum mechanics

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can receive an analogous interpretation; here, t doesn't represent a "real" physical time, but rather the stream of changes of the particle in examination in ATPS. The acceleration

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

isn't the variation of the speed in time, but the stream of changes in which the speed is subjected in ATPS. Then as written above, the law of motion says that the total force (classic + quantum) acting on a physical system is tied to the stream of changes of the system's speed in ATPS (and thus, if the particle in examination is still, there isn't stream of changes in space and, therefore, no force acts on the particle).

Now, we can understand well the connection between the a-temporal interpretation of the law of motion in Bohm's pilot-wave theory and the interpretation of a subatomic particle in quantized ATPS. As we have said in the previous chapter, it's the vibration of the QS at appropriate frequencies that determines the appearance of a particle in ATPS and creates the wave that guides the particle during its motion. It then derives that the motion, the stream of changes to which a particle is subjected in ATPS, is itself tied to the vibration at appropriate frequencies of the QS composing its trajectory. Therefore, quantum potential itself is determined, in short, by the vibration at appropriate frequencies of some QS of ATPS.

The following idea is therefore proposed: It's completely lawful to think that there may be in some way a connection, a correspondence, between the entropic energy movement among the various QS, these elementary grains of ATPS (movement that, as said before, materializes a particle, making it appear in the different points of space), and the action of the total force, which produces the motion of that particle, as we know it from Bohm's pilot wave theory. The movement of a particle under the action of the total force (classic + quantum) can be seen as the effect of the energy transfer among the different QS in which the trajectory described by the particle can be decomposed. In other words, one can think that the total force (classic + quantum), which acts on a particle represents the immediate effect, i.e., the immediate manifestation of the entropic energy shifting between the various QS composing the trajectory described by this particle.

In this way, a new interpretation of quantum potential proposes itself: The idea that quantum potential has origin from this deeper level of reality, in which one assumes that physical space is a-temporal, composed by a net of QS having the size of Planck's length (which represent its elementary grains) and in which one assumes that subatomic particles appear in virtue of the interaction of a discrete quantity of entropic energy with one or more QS. In fact, this new interpretation implies a correspondence between Bohmian quantum potential and the energy shifting among the QS (responsible for the materialization of an elementary particle in different points of space). In regards to microscopic processes, ATPS assumes the special "state" represented by quantum potential in consequence of the energy shifting between certain QS, and, therefore, in consequence of the vibration of such QS at appropriate frequencies. In short, the movement of atomic and subatomic particles is determined by the energy shifting among the various QS (caused by the vibration at certain frequencies) and its in consequence of this energetic shifting that ATPS assumes the special "state" represented by quantum potential.

The interpretation of quantum potential as the "state" of ATPS in presence of microscopic processes can also be seen as a natural consequence that derives from quantum nonlocality. Nonlocality provides us with another element for which this new interpretation of quantum potential appears lawful. In Bohm's version of quantum mechanics, the instantaneous connection between two subatomic particles separated by big distances is explained by quantum potential. But at the same time, the nonlocality of quantum phenomena can also be seen as an effect of ATPS. One can think that it's ATPS transmitting the information between two particles--before being joined and then removed and carried at big distances one from the other--to let these two particles communicate instantaneously. Just as gravitational interaction is expected in a-temporal gravitation theory, the information between two quantum particles also is a-temporal (and has no speed). Hence, it immediately derives the possibility that there may be a sort of correspondence between quantum potential and ATPS, in particular that quantum potential can be interpreted as the special "state" of ATPS in presence of microscopic processes, and thus, of quantum particles. When one takes into consideration an atomic or subatomic process (such as, for example, the case of an EPR-type experiment, of two subatomic particles, before being joined and then separated and carried away at big distances one from the other), ATPS assumes the special "state" represented by quantum potential, and this allows an instantaneous communication between the two particles under study.

It's important to point out, finally, that this new interpretation of quantum potential as "special state of ATPS in presence of microscopic processes" is substantially compatible with the idea, originally proposed by Bohm and Hiley, to consider quantum potential as "information potential" and sheds new light on it. In fact, saying that quantum potential contains active information about the region of the experiment (and that this information manifests itself in the movement of the particles) is the equivalent of saying that it includes the whole of features that the region of experiment presents, and that such features determine the movement of the particles. Quantum potential contains active information just because it represents, at the deepest level, the state of ATPS in presence of microscopic processes; its way of acting derives from what happens at the deepest level of reality and it's from here that it arises its feature to contain an active information, in regards to the region of experiment.

After having illustrated and provided some important indications toward this new interpretation of quantum potential, in order to understand how it allows to throw new light on the behaviour of subatomic particles, let's see now in detail some fundamental experiments of atomic physics.

The Double-Slit Interference

The first important experiment that we confront is double-slit interference. It's the classic experiment in which a beam of electrons are sent, one at a time, to a double-slit, and then are revealed on a screen, where an interference figure is observed. As underlined by Feynman, the understanding of what happens in the double-slit experiment takes a part of the heart of the problems of quantum mechanics. In fact, standard quantum mechanics (Copenhagen interpretation) hardly enables us to understand what happens when electrons are sent toward a double-slit apparatus, or to understand why these electrons reproduce the interference figure. In the standard version of quantum mechanics, we can only say what the probability is--that the electron finishes in a certain point of the screen, but we cannot follow its motion, and we cannot speak of the electron's trajectory after it has passed through the slits. In short, we cannot provide a lecture in visualizable terms on this process.

Instead, Bohm's theory (on the grounds of calculations made by Philippidis, Dewdney and Hiley) explains what happens in the double-slit experiment in agreement with a principle of causality. (10) It recovers some form of visualizability; one can keep the concept of well-defined trajectories of the particles and the final position of the particle on the screen. This allows us to deduce through which slit it has passed. Bohm's theory shows how an accurate calculation of quantum potential for the double-slit experiment can generate the interference figure without leaving the concept of well-defined trajectories of the particles. By analyzing the double-slit experiment on the grounds of this theory, namely, one obtains the following results: Each single particle follows a precise and calculable trajectory, which comes from one slit or the other and this aggregate of trajectories produces the requested interference figure and, at the same time, shows that the final position of the particle on the screen permits us to deduce through which slit it has passed. Besides, if the distribution of the arrivals on the two slits is uniform, the action of quantum potential assures that the presence probability density of the particles remain equal to the modulus square of the wavefunction in all the following instances.

Now, utilizing the results displayed in the previous chapters about quantized ATPS, one can shed new light on double-slit experiment and provide a significant justification--at least in an interpretative way--of how the particles emitted by the source move in the region between the slits and the screen, and of their trajectories (which are considered "strange" by some physicists of Copenhagen interpretation).

The trajectory described by each subatomic particle can be interpreted as the result of the energy transfer between some QS; the QS occupied by a particle during its motion vibrate at appropriate frequencies and therefore assume a discrete quantity of entropic energy, perceivable to our senses. In consequence of the energy shifting between the different QS composing the trajectory described by each particle coming from the beam, ATPS of the double-slit experiment assumes the special state represented by quantum potential. If in Bohm's original theory, the trajectory described by each particle in the double-slit experiment is tied to quantum potential, we can now introduce our interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes." In fact, quantum potential, in our interpretation, draws its origin at the deepest level of reality in which physical space is a-temporal and endowed with a granular structure. The materialization of each subatomic particle in the different points is tied to an entropic energy transfer among the QS composing its trajectory. The region of double-slit assumes the special "state" represented by quantum potential--whose actions establish, according to Bohm's theory, one determined trajectory for each particle of the beam--by virtue of what happens at the deepest level of reality (i.e., by virtue of the interaction of entropic energy with certain QS, which are caused by the vibration at certain frequencies). It's the energy shifting between certain QS that produces the movement of each particle of the beam that is responsible for its following appearance in different points and thus, explains the particular way in which quantum potential acts. It's just in this deeper level of reality that quantum potential arises and emerges.

Therefore, if one utilizes this interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes," the aggregate of trajectories described by the electrons in double-slit experiment must not be considered devoid of sense, because they are produced by the events concerning the most fundamental level of reality. The following appearance of a particle in different points of ATPS is tied to the interaction of entropic energy with determined QS of the region of ATPS in exam (and which is determined by the vibration of these QS at appropriate frequencies). On the basis of this new interpretation of quantum potential, the following perspectives opens: When a single electron arrives on the slits, it's the particular "configuration" of QS composing that region of ATPS that determines what will happen to such electrons successively (i.e. after the passing through the slits). When the first electron is sent to the slits, the configuration of quantized ATPS is characterized by a net of QS with certain quantum numbers, and this net--which has quantum potential--acts in a certain way on that electron; it makes energy appear in certain points instead of others, which produces a determined trajectory for such electron. Thus, when the second electron arrives on the double-slit, we may conclude that the "configuration" of QS (composing that region of ATPS) is changed regarding the beginning; in the sense now that it will be characterized by a net of QS and characterized by different quantum numbers. This different configuration of ATPS determines a "different" action of quantum potential by allowing the electron to follow another trajectory (different from that of the first electron). This time, there is the vibration, and thus the creation of entropic energy, in different QS (in regards to those characterizing the state of space at the beginning of the experiment). These considerations can then be extended to all the particles, which are sent, one at a time, toward the double-slit. On the grounds of the interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes," it's possible to provide a significant justification--at least on interpretative levels--of the trajectory described by each particle of the beam.

Tunnelling

The second experiment we consider is the overcoming of a potential barrier by tunnelling. (11) If a beam of particles is sent, one at a time, toward a potential barrier, one observes experimentally that a given particle has a finite probability to overcome the potential barrier when its initial energy is less than the barrier. In the standard version of quantum mechanics, it's not possible to provide a reading in visualizable terms for this typical quantum phenomenon. Instead, according to Dewdney's results, Bohm's theory can provide a clear explanation, in terms of motions of single particles, of what happens when a single particle arrives on the potential barrier. This theory shows that some particles are reflected and some manage to overcome the barrier, according to their initial position. In particular, by applying Bohm's theory to this problem, one finds that in the barrier region, quantum potential is negative; this determines an effective lowering of the barrier, which, therefore, can be overcome by a particle (if it possesses sufficient energy) without violation of energy conservation law.

The interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes" also brings new light to this experiment about tunnelling. In virtue of the link between quantum potential and the energy shifting between the QS composing a region of ATPS, it's legitimate to think that the negative value of quantum potential in the barrier region is tied to the particular "configuration" of QS composing such region. Particles having bigger initial energy (and therefore speed) possess bigger probability to overcome the barrier. We can explain this by taking into account these particles pass more rapidly from one point to another of their trajectory. This means that for these particles endowed with bigger speed, the energy transfer between the different QS composing their trajectory happens more rapidly than for those particles having smaller initial speed; determining for particles endowed with bigger initial speed, a bigger probability to overcome the barrier.

In conclusion, according to the idea that at the most profound and fundamental level of reality, space is a-temporal and quantized, and that the motion of each subatomic particle in this space is tied to the entropic energy shifting between the various QS composing its trajectory, it's possible to provide a clear reading, a convincing explanation (at least at an interpretative level) of what happens in tunnelling experiment. It's possible to clearly understand why a particle endowed with sufficient energy can overcome the potential barrier.

Conclusions

The considerations addressed in this article about quantum potential introduce perspectives regarding the description of atomic processes. With the new interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes," it's possible here to obtain a more profound view, at a more fundamental and "microscopic" level, of atomic and subatomic processes. The trajectory described by a subatomic particle in a given quantum experiment can be seen as the consequence of the state of ATPS when that experiment is performed. ATPS assumes that state, which is represented by quantum potential, in virtue of the configuration of QS composing the region of ATPS in examination. It's the energy shifting between these particular QS composing a given region of ATPS--determined in turn by the vibration at appropriate frequencies--that materializes a subatomic particle in the different points of its trajectory. Therefore, we suggest that it is lawful to think that the way of acting of quantum potential emerge from here.

References

(1.) Fiscaletti, D. (2003). I fondamenti nella meccanica quantistica. Un'analisi critica dell'interpretazione ortodossa, della teoria di Bohm e della teoria GRW. Padova: CLEUP.

(2.) Holland, P. R. (1993). The quantum theory of motion. Cambridge, Massachusetts: Cambridge University Press.

(3.) Fiscaletti, D. (2004). "L'evoluzione dei concetti di onda pilota e potenziale quantistico dalle idee originali di de Broglie fino agli anni Ottanta." Quaderni di storia della Fisica, 12, 35-44.

(4.) Hiley, B. J. (2002). "From the Heisenberg Picture to Bohm: a New Perspective on Active Information and its relation to Shannon Information." In: Proc. Conf. Theory: reconsiderations of foundations. Vaxjo, Sweden: Vaxjo University Press, 141-162.

(5.) Sorli, A. and Sorli, I. (2004). "A-Temporal Gravitation." Electronic Journal of Theoretical Physics, 1 (2), 1-3. www.ejtp.com

(6.) Sorli, A. and Fiscaletti, D. (2005). "Active Galactic Nucleus as A Renewing System of the Universe." Electronic Journal of Theoretical Physics, 2 (6), 7-13. www.ejtp.com

(7.) Rovelli, C. (2003). "Loop quantum gravity." Physics World, 7 (11), 1-5.

(8.) Sorli, A. and Sorli, I. K. (2005). "From Space-Time to A-Temporal Physical Space." Frontier Perspectives, 14 (1), 38-40.

(9.) Fiscaletti, D. (2005). "A-Temporal Physical Space and Quantum Nonlocality." Electronic Journal of Theoretical Physics, 2 (6), 15-20. www.ejtp.com

(10.) Philippidis, C., Dewdney, C. and Hiley, B. (1979). Il Nuovo Cimento, 52B, 15.

(11.) Bergia, S. (2004). "La versione di Bohm della meccanica quantistica: variazioni sul tema." In: Quanti Copenaghen? Bohr, Heisenberg e le interpretazioni della meccanica quantistica, a cura di I. Tassani. Cesena: Il Ponte Vecchio, 179-199.

(12.) Dewdney, C. (1987). "Calculations in the causal interpretation of quantum mechanics." In: Quantum Uncertainties--Recent and Future Experiments and Interpretations. New York: Plenum Press, 19-40.

Davide Fiscaletti and Amrit Sorli SpaceLife Institute, Via Roncaglia, 35 61047 San Lorenzo in Campo (PU), Italy

FiscalettiDavide@libero.it; spacelife@libero.it

Bohm's pilot wave theory, introducing the concept of quantum potential, provided a causal, and at the same time nonlocal, description of quantum phenomena. Starting from some results of a-temporal gravitation and loop quantum gravity, we show here that subatomic particles can be interpreted, at the deepest level of reality, as the result of the interaction of energy (in the "entropy state") with one or more elementary grains of a-temporal physical space. A new interpretation of quantum potential is thus introduced and illustrated: The quantum potential intended as "special state of a-temporal physical space in presence of microscopic processes." When we take into consideration an atomic or subatomic process, an a-temporal physical space assumes the special "state" represented by quantum potential in consequence of the energy shifting between certain quanta of space; shifting that materializes the subatomic particle in exam in the different points of physical space (and which is determined by the vibrations at appropriate frequencies of the quanta of space occupied by the particle). We underscore that this new interpretation of quantum potential appears lawful also in virtue of the fact that both a-temporal physical space and quantum potential explain nonlocality, the instantaneous link between two subatomic particles independently from their distance. Finally, we show that this new interpretation of quantum potential provides a convincing reading of some experiments characteristic of atomic physics (double-slit interference, tunnelling).

Introduction

In Bohm's pilot wave theory, the basic idea is that a wave and a corpuscle contemporarily constitute each elementary particle of physics; here, the wave has the role to guide the corpuscle during its movement. The wave is described mathematically by Schrodinger's wavefunction [psi]. Under the guide of the wave, the movement of the corpuscle happens in agreement with a law of motion, which assumes the following form

[partial derivative]S / [partial derivative]t + [|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = 0

(where R is the amplitude and S the phase of the wavefunction, h is Planck's reduced constant, m is the mass of the particle and V is the classic potential) and thus is equal to the classical equation of Hamilton-Jacobi, except for the appearance of the additional term

Q = [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R

having the dimension of an energy and containing Planck's constant and (therefore) appropriately defined quantum potential. (1-3) The equation of motion of the particle can also be expressed in the form,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

thus equal to Newton's second law of classical mechanics, always with the additional term Q of quantum potential.

The movement of an elementary particle, according to Bohm's pilot wave theory, is thus tied to a total force that is given by the sum of two terms: A classical force (derived from a classic potential) and a quantum force (derived just from quantum potential). Quantum potential must not be considered a term that is introduced ad hoc, contrary to the opinion of the supporters of the Copenhagen interpretation. Quantum potential plays an essential role in quantum formalism: In the plant of Bohm's theory, it emerges directly from Schrodinger's equation and without it, energy shouldn't be conserved. In fact, equation

[partial derivative]S / [partial derivative]t + [|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = 0

taking into account that the quantity is the total energy of the particle and that

- [partial derivative]S / [partial derivative]t

[|[nabla]S|.sup.2] / 2m

is its kinetic energy, can be also written in the equivalent form

[|[nabla]S|.sup.2] / 2m - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2]R / R + V = - [partial derivative]S / [partial derivative]t

which can be seen as a real energy conservation law in quantum mechanics: Here one can easily see that without the quantum potential

Q = - [<begin strikethrough>h<end strikethrough>.sup.2] / 2m [[nabla].sup.2] R / R

energy couldn't be conserved. It must be observed, as it was shown recently by Hiley, that quantum potential can also be derived by Heisenberg's formalism by choosing a particular representation for operators, and here such terms must be present to assure the conservation of the total energy of the system. (4)

By introducing quantum potential, Bohm demonstrated that it is possible to provide a causal, and at the same time nonlocal, interpretation of quantum mechanics. Bohm's theory clearly shows that nonlocal correlations between subatomic particles--and which constitute a fundamental feature of many-body systems--are caused by the action of quantum potential. For a many-body system, quantum potential acting on each particle is a function of the positions of all the other particles and thus, in general, doesn't decrease with distance. As a consequence, the contribution to the total force acting on the i-th particle coming from the quantum potential, i.e. [[nabla].sub.i]Q does not necessarily fall off with distance and, indeed, the forces between two particles of a many-body system may become stronger, even if |[psi]| may decrease in this limit. The equation of motion of the i-th in particle, in the limit of big separations, assumes the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and thus depends on the coordinates of all the n particles of the system. This determines just nonlocal correlations in a many-body system.

Since 1979, the quantum potential approach (thank to the contributions, above all, of Philippidis, Dewdney, Hiley, and Vigier) has explained, in visualizable terms, many experimental results (for example, classic double-slit experiment, tunnelling, trajectories of two particles in a potential of harmonic oscillator, EPR-type experiments, experiments of neutron-interferometry). In 1984, Bohm proposed an idea to interpret quantum potential as a sort of "information potential"--the particles in their movement are guided by the quantum potential just as a ship on automatic pilot can be handled by radar waves of much less energy than that of the ship. On the basis of this interpretation, the results of double-slit experiment are explained by saying that the quantum potential contains an active information--for example the slits--and that this information manifests itself in the particles' motions.

Here a new interpretation of quantum potential is proposed: The idea that quantum potential can be seen as the "special state of a-temporal physical space in presence of microscopic processes," as the whole of the characteristics that a-temporal physical space assumes when we have quantum phenomena.

Some Considerations about Loop Quantum Gravity and A-temporal Physical Space

Some results of a-temporal gravitation theory and loop quantum gravity can be considered the starting points of an a-temporal interpretation of quantum potential.

A-temporal gravitation theory is a theory of gravitation that represents the natural consequence of the following idea. On the basis of our elementary perception, the passing of time cannot be perceived directly as matter and space; we can perceive only the irreversible changes of matter in space. It's beyond our power to establish if time actually exists as physical reality; we can say that time exists only as a stream of material changes happening in space. (5)

From this point of view (alternative in regards to those conventionally adopted in physics, but perhaps more correct and appropriate because it is more coherent with experimental facts), it derives that the stage in which physical phenomena have place isn't space-time, but really a four-dimensional a-temporal space. Phenomena happen in space-time only in the mathematical models of reality, which sometimes become more real than reality itself, which instead--in virtue of the elementary perception--turns out to be a-temporal. General relativity can, therefore, be interpreted in the following way: the density of this a-temporal space transmits gravity and its effect is to determine modifications in the geometrical properties of this a-temporal physical space.

The density D of physical space in a given volume depends on the amount of matter into it. In the center of a given material object, the density is D = m x G where m is the mass contained in that volume and G is the gravitational constant. In a-temporal gravitation theory, gravitational interaction is immediate, acting directly via the density of physical space. In other words, gravitational interaction is a-temporal (has no physical time), in the sense that no material change (travelling of wave or particle) through the a-temporal cosmic space is needed for its acting. (6)

On the other hand, loop quantum gravity predicts that physical space has a granular structure given by a net of intersecting loops, and it's these loops, these nodes that represent the quantum excitations of gravitational field, i.e., the elementary quanta of space. Area and volume of physical space are quantized. In particular, the elementary quanta of spatial volume reside precisely at the nodes of the net, while links (among the different nodes of the net) carry quantum numbers of area elements. These quantum numbers and their algebra look like the spin angular momentum numbers of elementary particles, and therefore the elementary grains of space, i.e., the loops of the net, can be appropriately defined spin elements or "spin networks." The picture of physical space provided by loop quantum gravity is practically the following: The elementary grains of space are represented by nodes of spin networks (they are called this because they satisfy an algebra analogous to that of spin), and their volume is given by a quantum number, which is associated with the node in units of the elementary Planck volume, V = [(<begin strikethrough>h<end strikethrough>G / [c.sup.3]).sup.3/2] where h is Planck's reduced-constant, G the universal gravitation constant, and c the speed of light. Two nodes are adjacent if there is a link between the two, in which case they are separated by an elementary surface the area of which is determined by the quantum number associated with that link. Link quantum numbers, j, are integers or half-integers and the area of the elementary surface is

A = 16[pi][V.sup.2/3] [square root of j (j+1)]

where V is the Planck volume. (7)

On the grounds of these results of loop quantum gravity and a-temporal gravitation theory, one can assume that the fundamental constituents of physical space (which can be called quanta of space) have the size of Planck's volume and are a-temporal entities. In virtue of the first law of thermodynamics, QS have not been created and cannot be destroyed. (7) Now, the interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes" arises just from the granular structure of a-temporal physical space. There is one other ingredient to add: How to interpret a material particle inside quantized a-temporal physical space.

The Interpretation of Subatomic Particles in Quantized A-temporal Physical Space

In the model proposed here, quanta of space (QS) are the fundamental constituents of a-temporal physical space (ATPS) and of all material objects. All physical reality we observe and experience is constituted by QS; both ATPS in which matter is present, and ATPS in which there isn't matter, are constituted by QS. The only difference is that ATPS, in absence of matter, has no entropy (there is no experimental evidence that such a space has entropy; its only property is density); instead, matter has entropy and can change its state. Therefore, one can say that QS, in absence of matter, are in the "non-entropy state" of energy (and it's for this reason that space in absence of matter appears to us "empty" or without changes), while QS constituting matter represent the "entropy-state" of energy (and it's for this reason that matter can change its state--for example, its position and its speed).

QS constituting ATPS change their electrical charge from positive to negative in a Planck time (5.39 * [10.sup.-44]s), vibrate at the "basic frequency" 0.19x[10.sup.-44][s.sup.-1], have a "basic energy" given by the relation Eqs = h*0.19x[10.sup.-44][s.sup.-1] where h is Planck's constant (6.626069x[10.sup.-34]Jxs), and thus Eqs = 1.26x[10.sup.10]J. QS of ATPS have a "bipolar nature" (namely, they change the electrical charge continuously) and are complete into themselves. Their existence doesn't depend on other physical entities. They have no radiation, no "dispersion of the energy"; their energy is always the same and precisely the basic energy Eqs = 1.26x[10.sup.10]J. This means that QS that build up ATPS have no entropy.

In ATPS endowed with a granular, quantized structure, every sub-atomic particle can be seen as the result of the interaction of energy (in the "entropy-state") with one or more QS. More precisely, one can suggest that particles that are devoid of internal structure such as quarks, leptons and intermediate bosons, can be seen as the result of the interaction of energy (in the "entropystate") with one quantum of space; instead, particles, which, turn out to be endowed with an internal structure such as baryons (constituted by three quarks) and mesons (constituted by a quark-antiquark pair) are given by the interaction of energy (in the "entropy-state") with more QS.

And what happens to a quantum of space when it changes its energy from the "non-entropy-state" to the "entropy-state", thus becoming material quantum? What allows a quantum of space to change its energy from the "non-entropy-state" to the "entropy-state"? What does it produce in this change in the state of energy? In this model based on ATPS, one can assume that a quantum of space changes its energy from the "non-entropy-state" to the "entropy-state," becoming a quantum of matter, as a consequence of the vibration at a certain appropriate frequency (lower than the basic one).

While QS vibrating at the basic frequency 0.19x[10.sup.-44][s.sup.-1] have energy in the "non entropy state" (and therefore constitute t "empty" ATPS, where matter is absent), QS vibrating at appropriate frequencies (lower than the basic one) assume energy in the "entropy state" and become so material quanta, endowed by mass and therefore perceivable by our senses.

QS become seat of a discrete quantity of energy (in the "entropystate") in virtue of a vibration at an appropriate frequency. All material particles are composed by QS that vibrate at certain frequencies and that, in virtue of this vibration, become seat of a discrete quantity of entropic energy. It's the vibration of QS of ATPS at appropriate frequencies, lower than the basic one, that make such QS material particles, endowed with an energy in the "entropy-state," able to modify their state.

For example, according to the view proposed here, the electron is just a particle derived from a quantum of space that vibrates at certain frequencies and that, because of this vibration, becomes seat of a discrete quantity of entropic energy. The discrete quantity of entropic energy that a quantum of space assumes when it gives origin to an electron depends on the characteristics of the region in exam, on the situation existing in that particular region of ATPS (namely on the type of the interaction to which the quantum of space is subjected). In fact, in this model, the fundamental interactions and physical fields can be interpreted as special states, as special ambient situations existing in ATPS in the presence of certain material particles, and produce modifications in the properties of ATPS (in particular, gravity has the effect to produce modifications in the geometrical properties of ATPS while the other three interactions--electromagnetic, weak and strong--determine modifications in the "vibrations" of QS of the region of ATPS in exam).

In general, according to our research, an a-temporal model of the universe could be based on the following postulates:

1. Energy is composed by basic packets of energy, which have the Planck size (here called "quanta of space"--QS). QS constituting physical space have no entropy.

2. QS form physical space and matter. The "arena" (stage) of the universe is "space-matter," physical reality is composed of space and matter. QS are each described by a wave function depending on its position in physical space, a quantum number, which indicates its orientation (as to an arbitrary axis) and a frequency of vibration. QS vibrating with the basic frequency constitute physical space, are devoid of mass, and are the "non-entropy state" of energy, and therefore, are not perceived by our senses. QS vibrating with the frequencies characteristic of material particles constitute matter and are the "entropy state" of energy (and therefore, are perceived by our senses). Each subatomic particle is the result of the interaction of energy in the "entropy state" with one or more QS, caused by the vibration of these QS at appropriate frequencies. In particular, particles devoid of internal structures, such as quarks, leptons and intermediate bosons, are the result of the interaction of energy in the entropy state with one quantum of space; particles endowed with an internal structure, such as baryons and mesons, are given by the interaction of entropic energy with more QS. QS vibrating at the frequencies of the electromagnetic spectrum generate electromagnetic waves (which propagate through physical space at the speed of light).

3. Time is an irreversible change of matter in physical space. Physical space itself is a-temporal (here called "a-temporal Physical Space"--ATPS). The universe is an a-temporal phenomenon where ATPS and matter are in a permanent dynamic equilibrium. There was no beginning of the universe and there will be no end.

4. The fundamental interactions and physical fields represent special states of ATPS and the different ambient situations existing in ATPS in presence of certain material particles. All interactions produce and determine modifications in the properties of ATPS. In particular, gravity has the effect to produce modifications in the geometrical properties (i.e., in the curvature) of ATPS. The other three interactions (electromagnetic, weak and strong) determine modifications in the vibrations of QS. They can change the frequencies of QS from the "basic frequency" (characterizing empty ATPS) to other appropriate frequencies less than the basic one (which can be the frequencies characteristic of material particles and/or of electromagnetic waves), or they can change the frequencies characteristic of some material particles into the frequencies characteristic of other material particles (and/or of electromagnetic waves). Each interaction is characterized by its own strength parameter, which indicates the intensity of the modifications induced in the region of ATPS under study, and its own particular range, i.e., the range in which the modifications of ATPS determined by the interaction happen. The density of ATPS transmits gravitational force; material objects move in the direction in which density of ATPS is increasing. The other forces are each mediated by the exchange of a particular boson: The photon for the electromagnetic interaction, the intermediate bosons [W.sup.[+ or -]] and [Z.sup.0] for the weak interaction, and the gluons for the strong interaction.

5. The description of a force is not altered by any modification of the length scales of rulers and of temporal scales of clocks utilized as measurement instruments (gauge invariance principle). Each interaction satisfies its peculiar gauge symmetry. (For example, in the case of the electromagnetic interaction, the quantum-mechanical description of experiments on charged particles is invariant under local phase transformations on the particle wavefunction, if one introduces a long-range field coupled to the charge-the electromagnetic field-and one makes simultaneously a suitable local gauge transformation on the electromagnetic potential. In the case of the strong interaction, the gauge symmetry is the isospin symmetry--the strong interactions are invariant under rotations in the isospin space). There are solid experimental results (for example, the fact that the Active Galactic Nucleus of the Milky Way Galaxy is "eating" the galaxy of Sagittarius, and at same time, continuously emits fresh gas) and theoretical results (a-temporal gravitation theory and loop quantum gravity), which support the ideas contained in some of these postulates. (6-8)

In order to illustrate the new interpretation of subatomic particles inside ATPS, consider an electron being in a stationary state of hydrogen atom. According to this model, the electron in a stationary state of hydrogen atom, i.e., the ambient situation existing in ATPS represented by the coulombian field created by a proton, can be seen as the result of the interaction of entropic energy, given by one of the eigenvalues of the energetic spectrum

[E.sub.n] = - 1/2 [m.sub.e] [e.sup.4] / [<begin strikethrough>h<end strikethrough>.sup.2][n.sup.2]

(where is [m.sub.e] the mass of electron and n is an integer positive number), with one quantum of space. This quantum of space becomes seat of this entropic energy as a consequence of the vibration at an appropriate frequency [v.sub.n], which is obtained by the relation:

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

and thus is given by

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

The particular value of the frequency of vibration determined by the coulombian potential in a quantum of space depends on the position of this quantum of space. One can say in fact that the coulombian potential created by a proton changes the frequency of a quantum of space being at a distance

[r.sub.n] = [n.sup.2][<begin strikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton, from the value given by the "basic frequency" to the value

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

(and therefore determines the appearance in it of the entropic energy).

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

For example, in the case in which the quantum of space is at a distance

[r.sub.1] = [<begin strikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton the coulombian potential produces the change of the frequency of this quantum of space from the value given by the "basic frequency" to the value

[v.sub.1] = - 2[pi] [sup.2][m.sub.e.][e.sup.4] / [h.sup.3]

(and therefore determines the appearance in this quantum of space of the entropic energy

[E.sub.1] = - [m.sub.e][e.sup.4] / 2[<begin srikethrough>h<end strikethrough>.sup.2]

equal to the first eigenvalue of the energetic spectrum of an hydrogen atom). A quantum of space at a distance

[r.sub.1] = [<begin srikethrough>h<end strikethrough>.sup.2] / [m.sub.e][e.sup.2]

from the proton, under the action of the coulombian potential,

V(r) = -[e.sup.2] / r

vibrates at the frequency

[v.sub.1] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3]

and becomes, therefore, seat of a quantity of entropic energy equal to the first eigenvalue of the energetic spectrum of an hydrogen atom, i.e., gives origin to an electron being in the first stationary state of the hydrogen atom.

Therefore, in this model, the electron being in a stationary state of the hydrogen atom derives from a quantum of space vibrating at appropriate frequencies less than the basic one. It's the vibration of a quantum of space at an appropriate frequency that causes the change of energy of this quantum of space from the "non-entropy state" to the "entropy state" and, therefore, creates the appearance of an electron in a stationary state of the hydrogen atom. Therefore, the following reading of the mathematical formalism, which concerns the electron of an hydrogen atom turns out to be lawful: In a given region of ATPS, the ambient situation represented by the coulombian field created by a proton determines a modification in the properties of that region--more precisely, it produces the change of the frequency of a quantum of space surrounding the proton from the value given by the "basic frequency" to one of the values

[v.sub.n] = - 2[pi] [sup.2][m.sub.e.][e.sup.4] / [h.sup.3][n.sup.2]

This quantum of space, vibrating at one of these frequencies [v.sub.n] becomes seat of a discrete quantity of entropic energy given by

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

and this means that it has given origin to an electron being in a stationary state of the hydrogen atom.

On the grounds of this interpretation of the electron being in a stationary state of the hydrogen atom, it also derives the following important consequence: One can say that it's the special "state" of ATPS (in this case represented by the coulombian potential) to "create" matter (in this case the electron being in a stationary state of hydrogen atom). In other words, one can also say that the presence, in a given point of ATPS, of a mass and a charge equal to the mass and the charge of electron is an effect of the vibration of a quantum of space at a frequency given by one of the values

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2],

caused by the ambient situation existing in the region of ATPS in examination.

Also, the quantum wave associated with the electron being in a stationary state of the hydrogen atom can be considered an effect of the vibration at the frequencies

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2];

in fact, since the appearance in a quantum of space of an entropic energy given by

[E.sub.n] = [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

is determined by its vibration at the frequency

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

and since an electron having energy equal to one of these values

[E.sub.n] [hv.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.2][n.sup.2]

is described by an eigenfunction of the hydrogen atom, i.e.,

[[psi].sub.nlm] (r, v, [phi]) [<begin strikethrough>m<end strikethrough>.sub.e] [R.sub.nl](r)[Y.sup.m.sub.l](v, [phi]),

one can say that the quantum wave-function of the electron being in a stationary state of the hydrogen atom can itself be seen as a consequence of the vibration at the frequency

[v.sub.n] = - 2[pi] [sup.2][m.sub.e][e.sup.4] / [h.sup.3][n.sup.2]

This particular example of the electron in a hydrogen atom shows the ordinary stationary states of energy predicted by quantum mechanics (for the particles devoid of internal structure) can be seen as the effect of the vibration of a quantum at appropriate frequencies characteristic of that particle; the fact that the frequency is quantized then implies the energy (in the entropy state) that a quantum of space acquires (as a consequence of that vibration) is quantized, and to each of these values there will correspond a different quantum wavefunction.

Therefore, on the grounds of this model, one can say that it's just the vibrations of QS at the frequencies characteristic of subatomic particles, which give place to the quantum waves associated to the material particles. The wave behavior of the subatomic particles arises from the vibrations of QS constituting them. In short, one can say that all derives from the vibrations at appropriate frequencies (characteristic of material particles). The vibrations of QS at appropriate frequencies create the appearance of material particles (in the sense that, because of them, these QS become seat of a discrete quantity of energy in the "entropy state") and, at the same time, create the wave behavior, the quantum waves associated to such particles. These quantum waves can be interpreted both in standard sense (as mathematical tools to compute certain probabilities) and in more "realistic" senses. We emphasize, however, that we prefer to interpret them in a realistic sense. We suggest, in particular, that these quantum waves can be interpreted like in Bohm's pilot wave theory.

Now, let's consider and compare between them the wave equation

[psi] = - [delta]Q[[psi], t] / [delta][psi]

of Bohm's quantum field theory concerning a massless quantum field and the classical massive Klein-Gordon equation [psi] = -[m.sup.2][psi]. In this regard, it has been shown that for particular solutions of the Klein-Gordon equation, the quantum potential associated with a one-quantum state acts so that the mass-less quantum field behaves as if it were a classical field with mass. As a consequence of this fact, in Bohm's quantum field theory, there is the possibility that the attribute of mass can be seen as an effect of quantum potential, i.e., that quantum potential can be viewed as the origin of the mass of a particle. (2)

If we take into account that in Bohm's quantum field theory, there is a possible link between the mass of a particle and the quantum potential, and that (in the model suggested here) the appearance of a mass means the appearance of an entropic energy, it seems lawful to think that the quantum waves actually guide the corresponding particles, through the action of quantum potential, during their motion, in the different points of ATPS. Taking into account the possible link between the mass of a particle and the quantum potential, on the grounds of the model proposed here, it appears lawful to suggest the idea that the role of quantum potential is just to transfer a discrete quantity of entropic energy from one quantum to another of ATPS (thus making a particle appear in the QS, which compose its trajectory). One can say that the quantum waves associated with material particles guide the corresponding particles in the regions where the wavefunction is more intense, through the action of quantum potential, which is responsible for the entropic energy shifting between the different QS composing the trajectory described by the particle in ATPS.

Quantum Potential as "Special State of A-temporal Physical Space in Presence of Microscopic Processes"

Studying the motion of a subatomic particle on the basis of the laws of Bohm's version of quantum mechanics and of our a-temporal model in quantized ATPS, we have the following results: On one side, in Bohm's version of quantum mechanics, the movement of the particle is determined by the sum of a classic potential and a quantum potential and, on the other side, in quantized ATPS, it's tied to the interaction of a discrete quantity of entropic energy with the various QS composing the trajectory described by the particle itself (interaction produced by the vibration of these QS at certain appropriate frequencies). In the model here, the vibration at appropriate frequencies of a quantum of space determines the interaction of energy in the "entropy-state" with this quantum of space and the appearance in it of a subatomic particle (or part of it); this vibration also produces a quantum wave that guides the particle during its motion, making it appear in the different points of ATPS. Therefore, the vibration of certain QS at appropriate frequencies always causes the motion of a subatomic particle.

The evolution of the wavefunction of a physical system happens in space-time only in the mathematical models of reality. According to this model, the wavefunction [psi] describing the state of a given physical system doesn't vary in time, but varies in a four-dimensional ATPS and the stream of changes that it has in space is itself time. This means that the coordinate t, which appears in Schrodinger's equation

i<begin strikethrough>h<end strikethrough> [partial derivative][psi] / [partial derivative]t = H[psi]

(H being the Hamiltonian of the system), doesn't represent time but rather the stream of changes that the physical system has in ATPS and, thus,

[partial derivative][psi] / [partial derivative]t

is the partial derivative of the wavefunction with respect to the stream of changes of the system in ATPS.

Also, the law of motion of the particle in Bohm's version of quantum mechanics

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

can receive an analogous interpretation; here, t doesn't represent a "real" physical time, but rather the stream of changes of the particle in examination in ATPS. The acceleration

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

isn't the variation of the speed in time, but the stream of changes in which the speed is subjected in ATPS. Then as written above, the law of motion says that the total force (classic + quantum) acting on a physical system is tied to the stream of changes of the system's speed in ATPS (and thus, if the particle in examination is still, there isn't stream of changes in space and, therefore, no force acts on the particle).

Now, we can understand well the connection between the a-temporal interpretation of the law of motion in Bohm's pilot-wave theory and the interpretation of a subatomic particle in quantized ATPS. As we have said in the previous chapter, it's the vibration of the QS at appropriate frequencies that determines the appearance of a particle in ATPS and creates the wave that guides the particle during its motion. It then derives that the motion, the stream of changes to which a particle is subjected in ATPS, is itself tied to the vibration at appropriate frequencies of the QS composing its trajectory. Therefore, quantum potential itself is determined, in short, by the vibration at appropriate frequencies of some QS of ATPS.

The following idea is therefore proposed: It's completely lawful to think that there may be in some way a connection, a correspondence, between the entropic energy movement among the various QS, these elementary grains of ATPS (movement that, as said before, materializes a particle, making it appear in the different points of space), and the action of the total force, which produces the motion of that particle, as we know it from Bohm's pilot wave theory. The movement of a particle under the action of the total force (classic + quantum) can be seen as the effect of the energy transfer among the different QS in which the trajectory described by the particle can be decomposed. In other words, one can think that the total force (classic + quantum), which acts on a particle represents the immediate effect, i.e., the immediate manifestation of the entropic energy shifting between the various QS composing the trajectory described by this particle.

In this way, a new interpretation of quantum potential proposes itself: The idea that quantum potential has origin from this deeper level of reality, in which one assumes that physical space is a-temporal, composed by a net of QS having the size of Planck's length (which represent its elementary grains) and in which one assumes that subatomic particles appear in virtue of the interaction of a discrete quantity of entropic energy with one or more QS. In fact, this new interpretation implies a correspondence between Bohmian quantum potential and the energy shifting among the QS (responsible for the materialization of an elementary particle in different points of space). In regards to microscopic processes, ATPS assumes the special "state" represented by quantum potential in consequence of the energy shifting between certain QS, and, therefore, in consequence of the vibration of such QS at appropriate frequencies. In short, the movement of atomic and subatomic particles is determined by the energy shifting among the various QS (caused by the vibration at certain frequencies) and its in consequence of this energetic shifting that ATPS assumes the special "state" represented by quantum potential.

The interpretation of quantum potential as the "state" of ATPS in presence of microscopic processes can also be seen as a natural consequence that derives from quantum nonlocality. Nonlocality provides us with another element for which this new interpretation of quantum potential appears lawful. In Bohm's version of quantum mechanics, the instantaneous connection between two subatomic particles separated by big distances is explained by quantum potential. But at the same time, the nonlocality of quantum phenomena can also be seen as an effect of ATPS. One can think that it's ATPS transmitting the information between two particles--before being joined and then removed and carried at big distances one from the other--to let these two particles communicate instantaneously. Just as gravitational interaction is expected in a-temporal gravitation theory, the information between two quantum particles also is a-temporal (and has no speed). Hence, it immediately derives the possibility that there may be a sort of correspondence between quantum potential and ATPS, in particular that quantum potential can be interpreted as the special "state" of ATPS in presence of microscopic processes, and thus, of quantum particles. When one takes into consideration an atomic or subatomic process (such as, for example, the case of an EPR-type experiment, of two subatomic particles, before being joined and then separated and carried away at big distances one from the other), ATPS assumes the special "state" represented by quantum potential, and this allows an instantaneous communication between the two particles under study.

It's important to point out, finally, that this new interpretation of quantum potential as "special state of ATPS in presence of microscopic processes" is substantially compatible with the idea, originally proposed by Bohm and Hiley, to consider quantum potential as "information potential" and sheds new light on it. In fact, saying that quantum potential contains active information about the region of the experiment (and that this information manifests itself in the movement of the particles) is the equivalent of saying that it includes the whole of features that the region of experiment presents, and that such features determine the movement of the particles. Quantum potential contains active information just because it represents, at the deepest level, the state of ATPS in presence of microscopic processes; its way of acting derives from what happens at the deepest level of reality and it's from here that it arises its feature to contain an active information, in regards to the region of experiment.

After having illustrated and provided some important indications toward this new interpretation of quantum potential, in order to understand how it allows to throw new light on the behaviour of subatomic particles, let's see now in detail some fundamental experiments of atomic physics.

The Double-Slit Interference

The first important experiment that we confront is double-slit interference. It's the classic experiment in which a beam of electrons are sent, one at a time, to a double-slit, and then are revealed on a screen, where an interference figure is observed. As underlined by Feynman, the understanding of what happens in the double-slit experiment takes a part of the heart of the problems of quantum mechanics. In fact, standard quantum mechanics (Copenhagen interpretation) hardly enables us to understand what happens when electrons are sent toward a double-slit apparatus, or to understand why these electrons reproduce the interference figure. In the standard version of quantum mechanics, we can only say what the probability is--that the electron finishes in a certain point of the screen, but we cannot follow its motion, and we cannot speak of the electron's trajectory after it has passed through the slits. In short, we cannot provide a lecture in visualizable terms on this process.

Instead, Bohm's theory (on the grounds of calculations made by Philippidis, Dewdney and Hiley) explains what happens in the double-slit experiment in agreement with a principle of causality. (10) It recovers some form of visualizability; one can keep the concept of well-defined trajectories of the particles and the final position of the particle on the screen. This allows us to deduce through which slit it has passed. Bohm's theory shows how an accurate calculation of quantum potential for the double-slit experiment can generate the interference figure without leaving the concept of well-defined trajectories of the particles. By analyzing the double-slit experiment on the grounds of this theory, namely, one obtains the following results: Each single particle follows a precise and calculable trajectory, which comes from one slit or the other and this aggregate of trajectories produces the requested interference figure and, at the same time, shows that the final position of the particle on the screen permits us to deduce through which slit it has passed. Besides, if the distribution of the arrivals on the two slits is uniform, the action of quantum potential assures that the presence probability density of the particles remain equal to the modulus square of the wavefunction in all the following instances.

Now, utilizing the results displayed in the previous chapters about quantized ATPS, one can shed new light on double-slit experiment and provide a significant justification--at least in an interpretative way--of how the particles emitted by the source move in the region between the slits and the screen, and of their trajectories (which are considered "strange" by some physicists of Copenhagen interpretation).

The trajectory described by each subatomic particle can be interpreted as the result of the energy transfer between some QS; the QS occupied by a particle during its motion vibrate at appropriate frequencies and therefore assume a discrete quantity of entropic energy, perceivable to our senses. In consequence of the energy shifting between the different QS composing the trajectory described by each particle coming from the beam, ATPS of the double-slit experiment assumes the special state represented by quantum potential. If in Bohm's original theory, the trajectory described by each particle in the double-slit experiment is tied to quantum potential, we can now introduce our interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes." In fact, quantum potential, in our interpretation, draws its origin at the deepest level of reality in which physical space is a-temporal and endowed with a granular structure. The materialization of each subatomic particle in the different points is tied to an entropic energy transfer among the QS composing its trajectory. The region of double-slit assumes the special "state" represented by quantum potential--whose actions establish, according to Bohm's theory, one determined trajectory for each particle of the beam--by virtue of what happens at the deepest level of reality (i.e., by virtue of the interaction of entropic energy with certain QS, which are caused by the vibration at certain frequencies). It's the energy shifting between certain QS that produces the movement of each particle of the beam that is responsible for its following appearance in different points and thus, explains the particular way in which quantum potential acts. It's just in this deeper level of reality that quantum potential arises and emerges.

Therefore, if one utilizes this interpretation of quantum potential as "special state of a-temporal physical space in presence of microscopic processes," the aggregate of trajectories described by the electrons in double-slit experiment must not be considered devoid of sense, because they are produced by the events concerning the most fundamental level of reality. The following appearance of a particle in different points of ATPS is tied to the interaction of entropic energy with determined QS of the region of ATPS in exam (and which is determined by the vibration of these QS at appropriate frequencies). On the basis of this new interpretation of quantum potential, the following perspectives opens: When a single electron arrives on the slits, it's the particular "configuration" of QS composing that region of ATPS that determines what will happen to such electrons successively (i.e. after the passing through the slits). When the first electron is sent to the slits, the configuration of quantized ATPS is characterized by a net of QS with certain quantum numbers, and this net--which has quantum potential--acts in a certain way on that electron; it makes energy appear in certain points instead of others, which produces a determined trajectory for such electron. Thus, when the second electron arrives on the double-slit, we may conclude that the "configuration" of QS (composing that region of ATPS) is changed regarding the beginning; in the sense now that it will be characterized by a net of QS and characterized by different quantum numbers. This different configuration of ATPS determines a "different" action of quantum potential by allowing the electron to follow another trajectory (different from that of the first electron). This time, there is the vibration, and thus the creation of entropic energy, in different QS (in regards to those characterizing the state of space at the beginning of the experiment). These considerations can then be extended to all the particles, which are sent, one at a time, toward the double-slit. On the grounds of the interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes," it's possible to provide a significant justification--at least on interpretative levels--of the trajectory described by each particle of the beam.

Tunnelling

The second experiment we consider is the overcoming of a potential barrier by tunnelling. (11) If a beam of particles is sent, one at a time, toward a potential barrier, one observes experimentally that a given particle has a finite probability to overcome the potential barrier when its initial energy is less than the barrier. In the standard version of quantum mechanics, it's not possible to provide a reading in visualizable terms for this typical quantum phenomenon. Instead, according to Dewdney's results, Bohm's theory can provide a clear explanation, in terms of motions of single particles, of what happens when a single particle arrives on the potential barrier. This theory shows that some particles are reflected and some manage to overcome the barrier, according to their initial position. In particular, by applying Bohm's theory to this problem, one finds that in the barrier region, quantum potential is negative; this determines an effective lowering of the barrier, which, therefore, can be overcome by a particle (if it possesses sufficient energy) without violation of energy conservation law.

The interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes" also brings new light to this experiment about tunnelling. In virtue of the link between quantum potential and the energy shifting between the QS composing a region of ATPS, it's legitimate to think that the negative value of quantum potential in the barrier region is tied to the particular "configuration" of QS composing such region. Particles having bigger initial energy (and therefore speed) possess bigger probability to overcome the barrier. We can explain this by taking into account these particles pass more rapidly from one point to another of their trajectory. This means that for these particles endowed with bigger speed, the energy transfer between the different QS composing their trajectory happens more rapidly than for those particles having smaller initial speed; determining for particles endowed with bigger initial speed, a bigger probability to overcome the barrier.

In conclusion, according to the idea that at the most profound and fundamental level of reality, space is a-temporal and quantized, and that the motion of each subatomic particle in this space is tied to the entropic energy shifting between the various QS composing its trajectory, it's possible to provide a clear reading, a convincing explanation (at least at an interpretative level) of what happens in tunnelling experiment. It's possible to clearly understand why a particle endowed with sufficient energy can overcome the potential barrier.

Conclusions

The considerations addressed in this article about quantum potential introduce perspectives regarding the description of atomic processes. With the new interpretation of quantum potential as a "special state of a-temporal physical space in presence of microscopic processes," it's possible here to obtain a more profound view, at a more fundamental and "microscopic" level, of atomic and subatomic processes. The trajectory described by a subatomic particle in a given quantum experiment can be seen as the consequence of the state of ATPS when that experiment is performed. ATPS assumes that state, which is represented by quantum potential, in virtue of the configuration of QS composing the region of ATPS in examination. It's the energy shifting between these particular QS composing a given region of ATPS--determined in turn by the vibration at appropriate frequencies--that materializes a subatomic particle in the different points of its trajectory. Therefore, we suggest that it is lawful to think that the way of acting of quantum potential emerge from here.

References

(1.) Fiscaletti, D. (2003). I fondamenti nella meccanica quantistica. Un'analisi critica dell'interpretazione ortodossa, della teoria di Bohm e della teoria GRW. Padova: CLEUP.

(2.) Holland, P. R. (1993). The quantum theory of motion. Cambridge, Massachusetts: Cambridge University Press.

(3.) Fiscaletti, D. (2004). "L'evoluzione dei concetti di onda pilota e potenziale quantistico dalle idee originali di de Broglie fino agli anni Ottanta." Quaderni di storia della Fisica, 12, 35-44.

(4.) Hiley, B. J. (2002). "From the Heisenberg Picture to Bohm: a New Perspective on Active Information and its relation to Shannon Information." In: Proc. Conf. Theory: reconsiderations of foundations. Vaxjo, Sweden: Vaxjo University Press, 141-162.

(5.) Sorli, A. and Sorli, I. (2004). "A-Temporal Gravitation." Electronic Journal of Theoretical Physics, 1 (2), 1-3. www.ejtp.com

(6.) Sorli, A. and Fiscaletti, D. (2005). "Active Galactic Nucleus as A Renewing System of the Universe." Electronic Journal of Theoretical Physics, 2 (6), 7-13. www.ejtp.com

(7.) Rovelli, C. (2003). "Loop quantum gravity." Physics World, 7 (11), 1-5.

(8.) Sorli, A. and Sorli, I. K. (2005). "From Space-Time to A-Temporal Physical Space." Frontier Perspectives, 14 (1), 38-40.

(9.) Fiscaletti, D. (2005). "A-Temporal Physical Space and Quantum Nonlocality." Electronic Journal of Theoretical Physics, 2 (6), 15-20. www.ejtp.com

(10.) Philippidis, C., Dewdney, C. and Hiley, B. (1979). Il Nuovo Cimento, 52B, 15.

(11.) Bergia, S. (2004). "La versione di Bohm della meccanica quantistica: variazioni sul tema." In: Quanti Copenaghen? Bohr, Heisenberg e le interpretazioni della meccanica quantistica, a cura di I. Tassani. Cesena: Il Ponte Vecchio, 179-199.

(12.) Dewdney, C. (1987). "Calculations in the causal interpretation of quantum mechanics." In: Quantum Uncertainties--Recent and Future Experiments and Interpretations. New York: Plenum Press, 19-40.

Davide Fiscaletti and Amrit Sorli SpaceLife Institute, Via Roncaglia, 35 61047 San Lorenzo in Campo (PU), Italy

FiscalettiDavide@libero.it; spacelife@libero.it

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Publication: | Frontier Perspectives |

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Date: | Sep 22, 2005 |

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