Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Three-dimensional space as a medium of quantum entanglement

Three-dimensional space as a medium of quantum entanglement Most physicists today still conceptualize time as a part of the physical space in which material objects move, although time has never been observed and measured as a part of the space. The concept of time here presented is that time measured with clocks is merely the numerical order of material change, i.e. motion in a three-dimensional space. In special relativity the Minkowskian four-dimensional space-time can be replaced with a three-dimensional space where time does not represent a fourth coordinate of space but must be considered merely as a mathematical quantity measuring the numerical order of material changes. By quantum entanglement the three-dimensional space is a medium of a direct information transfer between quantum particles. Numerical order of non-local correlations between subatomic particles in EPR-type experiments and other immediate quantum processes is zero in the sense that the three-dimensional space acts as an immediate information medium between them. Key words: space, time, numerical order of material change, run of clocks, photon motion, quantum entanglement, quantum entropy, symmetrized quantum potential 1. INTRODUCTION In Newtonian physics, as well as in standard quantum mechanics, time is postulated as a special physical quantity and plays the role of the independent variable of physical evolution. Newton or Hamilton equations, as well as the Schrödinger equation, are introduced on the basis of the underlying assumption * corresponding author DAVIDE FISCALETTI, AMRIT S. SORLI that an idealized, absolute time t in which the dynamics is de¿ned, exists. However, it is an elementary observation that we never really measure this idealized time t, that this idealized, absolute time does not ever appear in laboratory measurements: we rather measure the frequency, speed and numerical order of material changes. What experimentally exists is only the motion of a system and the tick of a clock. What we realize in every experiment is comparing the motion of the physical system under consideration with the motion of a peculiar clock described by a peculiar tick T. This means that the duration of material motions has not a primary physical existence, that time as humans perceive it does not exist as an absolute quantity, that time does not Àow on its own as an independent variable and thus does not exist as a primary physical reality. Changes of the state of the universe and, at the same time, changes of the state of any physical system can be considered the primary phenomena which generate evolution of the universe. This evolution can be described by introducing a mathematical parameter, which provides only the ordering of events. In the article Projection evolution and delayed choice experiment, A. Gó d and K. Stefa ska have shown that an evolution parameter, "numerical order", which provides only the order of events, can be easily introduced [1]. In the reference [2], the authors of this article have gone beyond by suggesting the following concept of time: according to this view, the symbol of time t in all mathematical formalisms of physics is a number which represents the numerical order of material change, i.e. motion and therefore, only the numerical order of the motion of the system under consideration, which is obtained by the clock under consideration, exists. In the Minkowskian arena of the Special Theory of Relativity the fourth coordinate X 4 of space is spatial, too. X 4 is a product of imaginary number i , light speed c , and the numerical order t of a physical event: X4 = i · c · t. On the basis of the mathematical expression of the fourth coordinate, the Minkowski arena is a four-dimensional (4D) space [3]. In the recent article Special theory of relativity in a three-dimensional Euclidean space [4] the authors have shown that Minkowski 4D space can be replaced with a three-dimensional Euclidean space with Galilean transformations X ' X v t Y' Y Z' Z (1) for the three spatial dimensions and Selleri's transformation t' v2 1 2 t c (2) for the rate of clocks. The Galilean transformations are valid for both the observers O and O' in inertial systems o and o'. The transformation of the speed of clocks given by Selleri's formalism [5, 6, 7] shows clearly that the speed of the moving clock does not depend on the spatial coordinates but is linked only with the speed v of the inertial system o'. In the formalisms (1) and (2), time and space are two separated entities. Equations (1) and (2) determine an arena of Special Relativity in which the temporal coordinate must be clearly considered as a different entity with respect to the spatial coordinates just because the transformation of the speed of clocks between the two inertial systems does not depend on the spatial coordinates. Selleri's results seem thus to suggest that the three spatial coordinates of the two inertial systems turn out to have a primary ontological status, de¿ne an arena that must be considered more fundamental than the standard space-time coordinates interpreted in the sense of Einstein. On the basis of equations (1) and (2) one can assume that the real arena of Special Relativity is not a mixed 3D+T space-time but rather a three-dimensional (3D) space and that time does not represent a fourth coordinate of space but exists merely as a mathematical quantity measuring the numerical order of material changes. The main idea which is at the basis of this article is that evolution in the universe occurs in a 3D space. The article is structured in the following manner. In chapter 2 we will illustrate in what sense a timeless 3D space (where time exists only as a numerical order of events) is the fundamental arena of physical processes (and we will indicate some current research which point in this direction). In chapter 3 we will mention some predictions of our model in the relativistic domain and propose a way in which our theory of space and time could be falsi¿ed. Finally, in chapter 4 we will show that, as regards non-local correlations between subatomic particles, the 3D space acts as a direct medium of quantum information transfer (in the picture of Bohm's quantum potential and then of a symmetrized quantum potential). 2. THREE-DIMENSIONAL SPACE AS A FUNDAMENTAL ARENA OF PHYSICS In his paper Time and Classical and Quantum Mechanics: Indeterminacy versus Discontinuity Lynds argues that between time and space there is always a difference: "The fact that imaginary numbers when computing space-time intervals and path integrals do not facilitate that when multiplied by i , that time intervals become basically identical to dimensions of space. Imaginary numbers show up in space-time intervals when space and time separations are combined at near the speed of light, and spatial separations are small, comparing to time intervals. What this illustrates is that although space and time are interwoven in Minkowski space-time, and time is the fourth dimension, time is not spatial dimension: time is always time, and space is always space, as those i ' s keep showing us. There is DAVIDE FISCALETTI, AMRIT S. SORLI always a difference. If there is any degree of space, regardless of how microscopic, there would appear to be inherent continuity i.e. interval in time" [8]. Although Lynds' conclusion that time is time and space is always space may appear a little questionable (the imaginary space-time interval, in fact, means only an impossibility to connect the points under consideration by a signal equal or slower than the speed of light), according to the authors there is nothing wrong in assuming that time and space are different in their nature, that time is a different entity from space, that time is not a spatial dimension. The crucial starting hypothesis of the view suggested in this paper is that time and space are different in their nature and that the difference between space and time is the following: the fundamental arena of the universe is a 3D space and time is a numerical order of material changes that take place in space. On the other hand, many researchers are challenged with the view that time is not a fundamental arena of the universe. For example, in their paper The Mathematical Role of Time and Space-Time in Classical Physics, Newton C. A. da Costa and Adonai S. Sant'Anna show that time as a fundamental physical arena in which material changes take place can be eliminated: "We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge ¿elds, and general relativity" [9]. According to several current studies, the mathematical model of space-time does not correspond indeed to a physical reality and a "state space" or a "timeless space" can be proposed as the fundamental arena. In particular, Girelli, Liberati and Sindoni have developed a toy model in which they have shown how the Lorentzian signature and a dynamical space-time can emerge from a non-dynamical Euclidean space, with no diffeomorphisms invariance built in. In this sense this toy-model provides an example where time is not fundamental, but simply an emerging feature [10]. In more detail, this model suggests that at the basis of the arena of the universe there is some type of "condensation", so that the condensate v Q is described by a manifold R 4 equipped with the Euclidean metric G P . Both the condensate and the fundamental theory are timeless. The condensate is characterized by a set of scalar ¿elds <i x P , i=1, 2, 3. Their emerging Lagrangian L is invariant under the Euclidean Poincarè group ISO(4) and has thus the general shape F X1, X 2 , X 3 f X1 f X 2 f X 3 ; X i P Q G v w P <i wQ <i . (3) The equations of motion for the ¿elds <i x P are given by § wF P · wP ¨ ¸ ¨ wX w <i ¸ ¹ © i wF w F ¦ ¨ wX wX w P X w P < wX ¨ 2 j § © The ¿elds <i x P can be expressed as <i \ i M i where M i are perturbations which encode both the gravitational and matter degrees of freedom and the functions \ i are classical solutions (of the above Eq. (4)). The Lagrangian for the perturbations M i is given by F X 1 , X 2 , X 3 ¦ j 2 3 wF X GX j 1 ¦ w F X GX jGX k 1 ¦ w F X GX jGX kGX l wX j 2 jk wX j wX k 6 jkl wX j wX k wX l j k · w P w P i Q x1 , x2 ,..., xn iVi xi @ (24) which is a quantum Newton law for a many-body system. Equation (24) shows that the contribution to the total force acting on the i-th particle coming from the quantum potential, i.e. i Q , is a function of the positions of all the other particles and thus in general does not decrease with distance. The quantum potential is the crucial entity which allows us to understand the features of the quantum world determined by Bohm's version of quantum mechanics. The mathematical expression of quantum potential shows that this entity does not have the usual properties expected from a classic potential. Relations (21) and (23) tell us clearly that the quantum potential depends on how the amplitude of the wave function varies in the 3D space. The presence of Laplace operator indicates that the action of this potential is space-like, namely creates onto the particles a non-local, instantaneous action. In relations (21) and (23) the appearance of the absolute value of the wave function in the denominator also explains why the quantum potential can produce strong long-range effects that do not necessarily fall off with distance and so the typical properties of entangled wave functions. Thus even though the wave function spreads out, the effects of the quantum potential need not necessarily decrease (as the equation of motion (24) of the many-body systems shows clearly, the total force acting on the i-th particle coming from the quantum potential, i.e. 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 \ may decrease in this limit). This is just the type of behaviour required to explain EPR-type correlations. If we examine the expression of the quantum potential in the two-slit experiment, we may ¿nd that it depends on the width of the slits, their distance apart and the momentum of the particle. In other words, the quantum potential has a contextual nature, namely brings global information on the process and its environment. It contains instantaneous information about the overall experimental arrangement. Moreover, this information can be regarded as being active in the sense that it modi¿es the behaviour of the particle. In a double-slit experiment, for example, if one of the two slits is closed the quantum potential changes, and this information arrives instantaneously to the particle, which behaves as a consequence. Now, the fact that the quantum potential produces a space-like and active information means that it cannot be seen as an external entity in space but as an entity which contains spatial information, as an entity which represents space. On the basis of the fact that the quantum potential has an instantaneous action and contains active information about the environment, one can say that it is space which is the medium responsible for the behaviour of quantum particles. Considering the double-slit experiment, the information that quantum potential transmits to the particle is instantaneous just because it is spatial information, is linked to the 3D space. In virtue of its features, the quantum potential can be considered a geometric entity, the information determined by the quantum potential is a type of geometric information "woven" into space. Quantum potential has a geometric nature just because it has a contextual nature, contains global information on the environment in which the experiment is performed and at the same time it is a dynamical entity just because its information about the process and the environment is active, determines the behaviour of the particles. In this geometric picture one can say that the quantum potential indicates, contains the geometric properties of space from which the quantum force, and thus the behaviour of quantum particles, derive. Considering the double-slit experiment, the fact that the quantum potential is linked with the width of the slits, their distance apart and the momentum of the particle, namely that brings global information on the environment means that it describes the geometric properties of the experimental arrangement (and therefore of space) which determine the quantum force and the behaviour of the particle. And the presence of Laplace operator (and of the absolute value of the wave function in the denominator) DAVIDE FISCALETTI, AMRIT S. SORLI indicates that the geometric properties contained in the quantum potential determine a non-local, instantaneous action. We can say therefore that Bohm's theory manages to make manifest this essential feature of quantum mechanics, just by means of the geometric properties of space described and expressed by the quantum potential. As regards the geometric nature of the quantum potential and the non-local nature of the interactions in physical space, one can also say, by paraphrasing J. A. Wheeler's famous saying about general relativity, that the evolution of the state of a quantum system changes active global information, and this in turn inÀuences the state of the quantum system, redesigning the non-local geometry of the universe. In synthesis, according to the authors, in virtue of the space-like action of the quantum potential, the medium of the 3D space has a crucial role in determining the motion and the behaviour of subatomic particles. On the basis of the equations (21) and (23), one can say that it is space which is the medium responsible for the behaviour of quantum particles. One can say that equations (21) and (23) of the quantum potential contain the idea of space as an immediate information medium in an implicit way. In particular, if we consider a many-body quantum process (such as for example the case of an EPR-type experiment, of two subatomic particles, ¿rst joined and then separated and carried away at big distances one from the other), we can say that the 3D physical space assumes the special "state" represented by the quantum potential (23), and this allows an instantaneous communication between the particles under consideration [29]. If we examine the situation considered by Bohm in 1951 (illustrated before) we can say that it is the state of space in the form of the quantum potential (23) which produces an instantaneous connection between the two particles as regards the spin measurements: by disturbing system 1, system 2 may indeed be instantaneously inÀuenced despite a big distance between the two systems thanks to the features of space which put them in an immediate communication. In synthesis, one can say that in EPR-type experiments the quantum potential (23) makes the 3D physical space an "immediate information medium" between elementary particles. In EPR-type experiments the behaviour of a subatomic particle is inÀuenced instantaneously by the other particle thanks to the 3D space which functions as an immediate information medium in virtue of the geometric properties represented by the quantum potential (23). However, what makes indeed the 3D space an immediate information medium in EPR-type correlations? If the space that we perceive seems to be characterized by local features, from which fundamental entity or structure the property of the quantum potential to determine the action of the 3D space as an immediate information medium derives? In this regard, according to the authors, it is important to mention that in the recent article Bohmian split of the Schrödinger equation onto two equations describing evolution of real functions, Sbitnev [30] has shown that the quantum potential can be determined as an information channel into the movement of the particles as a consequence of the fact that it determines two quantum correctors into the energy of the particle depending on a more fundamental physical quantity that can be appropriately called "quantum entropy". This new way of reading Bohmian mechanics can be called as the "entropic version" of Bohmian mechanics or, more brieÀy, "entropic Bohmian mechanics". In the case of a one-body system, the quantum entropy is de¿ned by the logarithmic function SQ & 1 ln U l n 2 (25a) where U \ x ,t is the probability density (describing the space-temporal distribution of the ensemble of particles, namely the density of particles in the & element of volume d 3 x around a point x at time t) associated with the wave & function \ x, t of an individual physical system. In the case of a many-body system, the quantum entropy is always de¿ned by the logarithmic function where here U space-temporal distribution of the ensemble of particles, namely the density of & particles in the element of volume d 3 x around a point x at time t) associated & & & with the wave function \ x1 , x2 ,..., x N , t of the many-body system under consideration. In the entropic version of Bohmian mechanics, one can assume that the 3D space distribution of the ensemble of particles describing the physical system under consideration generates a modi¿cation of the background space characterized by the quantity given by equation (25a) (or (25b)). The quantum entropy ((25a) and (25b)) can be interpreted as the physical entity that, in the quantum domain, characterizes the degree of order and chaos of the vacuum ­ a storage of virtual trajectories supplying optimal ones for particle movement ­ which supports the density U describing the space-temporal distribution of the ensemble of particles associated with the wave function under consideration. By introducing the quantum entropy, for one-body systems, the quantum potential can be expressed in the following convenient way 1 l U (25b) n ln 2 & & & 2 \ x1 , x2 ,..., x N , t is the probability density (describing the SQ !2 !2 2 2 SQ SQ 2m 2m (26). and we obtain the following equation of motion for the corpuscle associated with & the wave function \ x, t : DAVIDE FISCALETTI, AMRIT S. SORLI 2m 2 !2 SQ 2 V ! 2 SQ 2m 2m wS wt (27) !2 2 which provides an energy conservation law where the term 2m S Q can be 2 interpreted as the quantum corrector of the kinetic energy S of the particle, 2 2m while the term ! 2 S Q can be interpreted as the quantum corrector of the 2m potential energy V. In the case of many-body systems, the quantum potential is given by the following expression 2 !2 i SQ 2 ! i 2 SQ 2mi 2mi (28) and the equation of motion is i 1 i S 2mi i 1 2 N !2 i SQ 2 V ¦ ! i 2 SQ 2mi i 1 2mi wS wt (29) which provides an energy conservation law where the term body system, while the term ¦ !2 i SQ 2 2mi can be interpreted as the quantum corrector of the kinetic energy of the many!2 2 i SQ 2mi can be interpreted as the quantum corrector of the potential energy. On the ground of Sbitnev's results, it becomes thus permissible the following reading of the quantum potential and of the energy conservation law in quantum mechanics. The quantum potential derives from the quantum entropy describing the degree of order and chaos of the background space (namely the modi¿cation in the background space) produced by the density of the ensemble of particles associated with the wave function under consideration. And, on the basis of equations (27) and (29), we can say that the quantum entropy determines two quantum correctors in the energy of the physical system under consideration (of the kinetic energy and of the potential energy respectively) and without these two quantum correctors (linked just with the quantum entropy) the total energy of the system would not be conserved. Moreover, in this entropic approach to Bohmian mechanics, the classical limit can be expressed by the conditions o 2 SQ (30) for one-body systems and i Q o i SQ (31) for many-body systems. The quantum dynamics will approach the classical dynamics when the quantum entropy satis¿es conditions (30) (for one-body systems) or (31) (for many-body systems) which can be considered as the expression of the correspondence principle in quantum mechanics. With the introduction of the quantum entropy ((25a) for one-body systems and (25b) for many body systems) which leads to the energy conservation law (equation (27) for one-body system and equation (29) for many-body system), now new light can be shed on the interpretation of the action of the 3D space as an immediate information medium in EPR-type correlations. In fact, on the basis of equation (29), one can say that the action of the 3D space as an immediate information medium derives just from the two quantum correctors to the energy of the system under consideration, namely from the quantum corrector to the potential energy ¦ N ! i SQ 2 2mi i 1 !2 2 i SQ 2mi and the quantum corrector to the kinetic energy (while the other two terms ¦ i 1 i S 2mi and V on the right-hand of equation (29) determine a local feature of space). The feature of the quantum potential to make the 3D space an immediate information channel into the behaviour of quantum particles derives just from the quantum entropy. In other words, one can see that by introducing the quantum entropy given by equation (25b), it is just the two quantum correctors to the energy of the system under consideration, depending on the quantity describing the degree of order and chaos of the vacuum supporting the density U (of the particles associated with the wave function under consideration) the fundamental element, which at a fundamental level produces an immediate information medium in the behaviour of the particles in EPR-type experiments. The space we perceive seems to be characterized by local features because in our macroscopic domain the quantum entropy satis¿es conditions (30) or (31). In synthesis, in the entropic version of Bohmian mechanics, one can say that the quantum entropy, by producing two quantum corrector terms in the energy, can be indeed interpreted as a sort of intermediary entity between space and the behaviour of quantum particles, and thus between the non-local action of the quantum potential and the behaviour of quantum particles. The introduction of the quantum entropy (given by equation (25a) or (25b)) as the fundamental entity that determines the behaviour of quantum particles leads to an energy conservation law in quantum mechanics (expressed by equations (27) and (29)) which lets us realize what makes indeed the 3D space an immediate information medium in EPR-type DAVIDE FISCALETTI, AMRIT S. SORLI correlations, which in turn lets us realize from which property of the quantum potential one can derive the action of the 3D space as an immediate information medium. The ultimate source, the ultimate visiting card which determines the action of the 3D space as an immediate information medium between quantum particles is a fundamental vacuum de¿ned by the quantum entropy ((25a) or (25b)). The quantum entropy, by producing two quantum corrector terms in the energy, is the fundamental element which gives origin to the non-local action of the quantum potential. Now, as regards the instantaneous communication between quantum particles in EPR-type experiments and the role of the 3D space as a direct information medium between them, if one imagines to exchange, to invert the roles of the two particles what happens is always the same type of process, namely an instantaneous communication between the two particles. In other words, the instantaneous communication between two particles in EPR experiment is characterized by a sort of symmetry: it occurs both if one intervenes on one and if one intervenes on the other. In both cases the same type of process happens and ­ we can say ­ always owing to space which functions as an immediate information medium. Moreover, if we imagine to ¿lm the process of an instantaneous communication between two subatomic particles in EPR-type experiments backwards, namely inverting the sign of time, we should expect to see what really happened. Inverting the sign of time, we have however no guarantee that we obtain something that corresponds to what physically happens. Although the quantum potential ((21) for one-body systems and (23) for many-body systems) has a space-like, an instantaneous action, however it comes from Schrödinger equation which is not time-symmetric and therefore its expression cannot be considered completely satisfactory just because it can meet problems inverting the sign of time. On the basis of these considerations, in order to interpret in the correct way, also in symmetric terms in exchange of t for ­t, the instantaneous communication between subatomic particles and thus the interpretation of 3D space as an immediate information medium, in quantum theory in line of principle a symmetry in time is required. For this reason the authors of this article have recently introduced a research line based on a symmetrized version of the quantum potential. The symmetrized quantum potential can explain a symmetric and instantaneous communication between subatomic particles and thus can be considered as a better candidate for the state of the 3D space as an immediate information medium in EPR-type experiments (or, more generally, in each immediate physical phenomenon). In the case of a system of N particles the symmetrized quantum potential assumes the form § i 2 R1 · ¨ ¸ N ! 2 ¨ R1 ¸ (32) ¦ 2m ¨ 2 R ¸ i 1 i ¨ i 2 ¸ ¨ R2 ¸ © ¹ where R1 is the absolute value of the wave-function = R1eiS1 / describing the forward-time process (solution of the standard Schrödinger equation) and R2 is the absolute value of the wave-function I R2 e iS 2 / ! describing the time-reverse process (solution of the time-Schrödinger equation). On the basis of equation (32), we can explain non-local correlations in many-body systems ­ and thus EPR experiments ­ in the correct way (that is, also if one would imagine to ¿lm back the process of these correlations). The symmetrized quantum potential (32) can be considered the most appropriate candidate to provide a mathematical reality to a 3D space intended as a direct information medium [31]. In fact, the symmetrized quantum potential is characterized by two components, the one regarding the forward-time process, the other regarding the time-reverse process. The ¿rst component of the symmetrized quantum potential, Q1 ! 2 i R1 ¦ 2m R i 1 i 1 N 2 (33), which is related to the forward-time process and coincides with the original Bohm's quantum potential, is the real physical component which produces observable effects in the quantum world. As regards the observable effects of Bohm's quantum potential, the reader can ¿nd details in the results obtained, for example, by Philippidis, Dewdney, Hiley and Vigier about the classic double-slit experiment, tunnelling, trajectories of two particles in a potential of harmonic oscillator, EPR-type experiments, experiments of neutron-interferometry [32, 33]). The ¿rst component (33) expresses the instantaneous action on quantum particles and thus the immediate action of space on them. The second component, Q2 ! 2 i R2 ¦ 2m R i 1 i 2 N 2 (34), is introduced to reproduce in the correct way the time-reverse process of the instantaneous action and thus it guarantees that the quantum world can be interpreted correctly with the idea of space as an immediate information medium if one would imagine to ¿lm the process backwards: it must be introduced in order DAVIDE FISCALETTI, AMRIT S. SORLI to recover a symmetry in time in quantum processes, to interpret in the correct way quantum processes if one would imagine to ¿lm that process backwards. The opposed sign of the second component with respect to the physical ¿rst component (that is, with respect to the original Bohm's quantum potential) can be interpreted as a consequence of the idea of the measurable time as a measuring system of the numerical order of material change: the mathematical features of the second component of the symmetrized quantum potential imply that it is not possible to go backwards in the physical time intended as the numerical order of physical events. Both the components (33) and (34) of the symmetrized quantum potential can be considered as physical quantities deriving from the quantum entropy (25b). The ¿rst component can be expressed as i 1 2 !2 i SQ1 2 ! i 2 SQ1 2mi 2mi (35), while the second component can be expressed as i 1 2 !2 i SQ 2 2 ! i 2 SQ 2 2mi 2mi (36), l n where S Q1 2 ln U1 is the quantum entropy de¿ning the degree of order and & & & 2 chaos of the vacuum for the forward-time processes (where U1 \ x1 , x2 ,..., x N , t , & & & i S \ x1 , x2 ,..., x N , t R1e iS 1 / ! being the forward-time may-body wave function, so1 l n lution of the standard Schrödinger equation) and S Q 2 ln U 2 is the quantum 2 entropy de¿ning the degree of order and chaos of the vacuum for the time-reverse & & & & & & processes (where U 2 I x1 , x2 ,..., x N , t 2 , I x1 , x2 ,..., x N , t R2 e iS 2 / ! being the time-reverse many-body wave function, solution of the time-reversed Schrödinger equation). The energy conservation law for the forward-time process is i 1 i S1 2mi i 1 2 N !2 i SQ1 2 V ¦ ! i 2 SQ1 2mi 2mi i 1 wS1 , wt (37) while the energy conservation law for the reversed-time process is i 1 i S2 2 mi i 1 2 N !2 i S Q 2 2 V ¦ ! i 2 S Q 2 2 mi 2 mi i 1 wS 2 wt (38). On the basis of its mathematical features, the symmetrized quantum potential implies that in the quantum domain a timeless 3D space has a crucial role in determining the motion of a subatomic particle because the symmetrized quantum potential produces a like-space and instantaneous action on the particles under consideration and contains active information about the environment and, on the other hand, implies the concept of time as a numerical order of material change. In EPR-type experiments (and, more generally, in all immediate physical phenomena regarding the quantum domain) the 3D timeless space acts as an immediate information medium in the sense that the ¿rst component of the symmetrized quantum potential makes physical space an "immediate information medium" which keeps two elementary particles in an immediate contact (while the second component of the symmetrized quantum potential reproduces, from the mathematical point of view, the symmetry in time of this communication and the fact that time exists only as a numerical order of material change). We can call this peculiar interpretation of quantum non-locality as the "immediate symmetric interpretation" of quantum non-locality. 5. CONCLUSIONS This article shows that a 3D space where time t is exclusively a numerical order of material changes can be considered a fundamental arena of physical processes. At a fundamental level, we live in a universe where time measured by clocks exists exclusively as a numerical order of material changes. Nonlocal correlations in EPR-type experiments are carried directly by the 3D space, the numerical order t of quantum entanglement is zero in the sense that the 3D space functions as an immediate information medium. The action of the 3D space as an immediate information medium derives from the quantum entropy describing the degree of order and chaos of the vacuum supporting the density of the particles associated with the wave function under consideration. The symmetrized quantum potential characterized by the two components (where the ¿rst component coincides with the original Bohm's quantum potential and the second component is endowed with an opposed sign with respect to it) seems to be the most appropriate candidate to represent the mathematical state of the 3D space as an immediate information medium between subatomic particles that accounts for entanglement and non-locality (and more generally, for all immediate physical phenomena in the quantum domain). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Physica de Gruyter

Three-dimensional space as a medium of quantum entanglement

Fiscaletti, Davide; Sorli, Amrit S.
Annales UMCS, Physica , Volume 67 (1) – Jan 1, 2012
Download PDF
26 pages

Loading next page...
 
/lp/de-gruyter/three-dimensional-space-as-a-medium-of-quantum-entanglement-SrhXVQfmA0
Publisher
de Gruyter
Copyright
Copyright © 2012 by the
ISSN
0137-6861
DOI
10.2478/v10246-012-0014-5
Publisher site
See Article on Publisher Site

Abstract

Most physicists today still conceptualize time as a part of the physical space in which material objects move, although time has never been observed and measured as a part of the space. The concept of time here presented is that time measured with clocks is merely the numerical order of material change, i.e. motion in a three-dimensional space. In special relativity the Minkowskian four-dimensional space-time can be replaced with a three-dimensional space where time does not represent a fourth coordinate of space but must be considered merely as a mathematical quantity measuring the numerical order of material changes. By quantum entanglement the three-dimensional space is a medium of a direct information transfer between quantum particles. Numerical order of non-local correlations between subatomic particles in EPR-type experiments and other immediate quantum processes is zero in the sense that the three-dimensional space acts as an immediate information medium between them. Key words: space, time, numerical order of material change, run of clocks, photon motion, quantum entanglement, quantum entropy, symmetrized quantum potential 1. INTRODUCTION In Newtonian physics, as well as in standard quantum mechanics, time is postulated as a special physical quantity and plays the role of the independent variable of physical evolution. Newton or Hamilton equations, as well as the Schrödinger equation, are introduced on the basis of the underlying assumption * corresponding author DAVIDE FISCALETTI, AMRIT S. SORLI that an idealized, absolute time t in which the dynamics is de¿ned, exists. However, it is an elementary observation that we never really measure this idealized time t, that this idealized, absolute time does not ever appear in laboratory measurements: we rather measure the frequency, speed and numerical order of material changes. What experimentally exists is only the motion of a system and the tick of a clock. What we realize in every experiment is comparing the motion of the physical system under consideration with the motion of a peculiar clock described by a peculiar tick T. This means that the duration of material motions has not a primary physical existence, that time as humans perceive it does not exist as an absolute quantity, that time does not Àow on its own as an independent variable and thus does not exist as a primary physical reality. Changes of the state of the universe and, at the same time, changes of the state of any physical system can be considered the primary phenomena which generate evolution of the universe. This evolution can be described by introducing a mathematical parameter, which provides only the ordering of events. In the article Projection evolution and delayed choice experiment, A. Gó d and K. Stefa ska have shown that an evolution parameter, "numerical order", which provides only the order of events, can be easily introduced [1]. In the reference [2], the authors of this article have gone beyond by suggesting the following concept of time: according to this view, the symbol of time t in all mathematical formalisms of physics is a number which represents the numerical order of material change, i.e. motion and therefore, only the numerical order of the motion of the system under consideration, which is obtained by the clock under consideration, exists. In the Minkowskian arena of the Special Theory of Relativity the fourth coordinate X 4 of space is spatial, too. X 4 is a product of imaginary number i , light speed c , and the numerical order t of a physical event: X4 = i · c · t. On the basis of the mathematical expression of the fourth coordinate, the Minkowski arena is a four-dimensional (4D) space [3]. In the recent article Special theory of relativity in a three-dimensional Euclidean space [4] the authors have shown that Minkowski 4D space can be replaced with a three-dimensional Euclidean space with Galilean transformations X ' X v t Y' Y Z' Z (1) for the three spatial dimensions and Selleri's transformation t' v2 1 2 t c (2) for the rate of clocks. The Galilean transformations are valid for both the observers O and O' in inertial systems o and o'. The transformation of the speed of clocks given by Selleri's formalism [5, 6, 7] shows clearly that the speed of the moving clock does not depend on the spatial coordinates but is linked only with the speed v of the inertial system o'. In the formalisms (1) and (2), time and space are two separated entities. Equations (1) and (2) determine an arena of Special Relativity in which the temporal coordinate must be clearly considered as a different entity with respect to the spatial coordinates just because the transformation of the speed of clocks between the two inertial systems does not depend on the spatial coordinates. Selleri's results seem thus to suggest that the three spatial coordinates of the two inertial systems turn out to have a primary ontological status, de¿ne an arena that must be considered more fundamental than the standard space-time coordinates interpreted in the sense of Einstein. On the basis of equations (1) and (2) one can assume that the real arena of Special Relativity is not a mixed 3D+T space-time but rather a three-dimensional (3D) space and that time does not represent a fourth coordinate of space but exists merely as a mathematical quantity measuring the numerical order of material changes. The main idea which is at the basis of this article is that evolution in the universe occurs in a 3D space. The article is structured in the following manner. In chapter 2 we will illustrate in what sense a timeless 3D space (where time exists only as a numerical order of events) is the fundamental arena of physical processes (and we will indicate some current research which point in this direction). In chapter 3 we will mention some predictions of our model in the relativistic domain and propose a way in which our theory of space and time could be falsi¿ed. Finally, in chapter 4 we will show that, as regards non-local correlations between subatomic particles, the 3D space acts as a direct medium of quantum information transfer (in the picture of Bohm's quantum potential and then of a symmetrized quantum potential). 2. THREE-DIMENSIONAL SPACE AS A FUNDAMENTAL ARENA OF PHYSICS In his paper Time and Classical and Quantum Mechanics: Indeterminacy versus Discontinuity Lynds argues that between time and space there is always a difference: "The fact that imaginary numbers when computing space-time intervals and path integrals do not facilitate that when multiplied by i , that time intervals become basically identical to dimensions of space. Imaginary numbers show up in space-time intervals when space and time separations are combined at near the speed of light, and spatial separations are small, comparing to time intervals. What this illustrates is that although space and time are interwoven in Minkowski space-time, and time is the fourth dimension, time is not spatial dimension: time is always time, and space is always space, as those i ' s keep showing us. There is DAVIDE FISCALETTI, AMRIT S. SORLI always a difference. If there is any degree of space, regardless of how microscopic, there would appear to be inherent continuity i.e. interval in time" [8]. Although Lynds' conclusion that time is time and space is always space may appear a little questionable (the imaginary space-time interval, in fact, means only an impossibility to connect the points under consideration by a signal equal or slower than the speed of light), according to the authors there is nothing wrong in assuming that time and space are different in their nature, that time is a different entity from space, that time is not a spatial dimension. The crucial starting hypothesis of the view suggested in this paper is that time and space are different in their nature and that the difference between space and time is the following: the fundamental arena of the universe is a 3D space and time is a numerical order of material changes that take place in space. On the other hand, many researchers are challenged with the view that time is not a fundamental arena of the universe. For example, in their paper The Mathematical Role of Time and Space-Time in Classical Physics, Newton C. A. da Costa and Adonai S. Sant'Anna show that time as a fundamental physical arena in which material changes take place can be eliminated: "We use Padoa's principle of independence of primitive symbols in axiomatic systems in order to discuss the mathematical role of time and space-time in some classical physical theories. We show that time is eliminable in Newtonian mechanics and that space-time is also dispensable in Hamiltonian mechanics, Maxwell's electromagnetic theory, the Dirac electron, classical gauge ¿elds, and general relativity" [9]. According to several current studies, the mathematical model of space-time does not correspond indeed to a physical reality and a "state space" or a "timeless space" can be proposed as the fundamental arena. In particular, Girelli, Liberati and Sindoni have developed a toy model in which they have shown how the Lorentzian signature and a dynamical space-time can emerge from a non-dynamical Euclidean space, with no diffeomorphisms invariance built in. In this sense this toy-model provides an example where time is not fundamental, but simply an emerging feature [10]. In more detail, this model suggests that at the basis of the arena of the universe there is some type of "condensation", so that the condensate v Q is described by a manifold R 4 equipped with the Euclidean metric G P . Both the condensate and the fundamental theory are timeless. The condensate is characterized by a set of scalar ¿elds <i x P , i=1, 2, 3. Their emerging Lagrangian L is invariant under the Euclidean Poincarè group ISO(4) and has thus the general shape F X1, X 2 , X 3 f X1 f X 2 f X 3 ; X i P Q G v w P <i wQ <i . (3) The equations of motion for the ¿elds <i x P are given by § wF P · wP ¨ ¸ ¨ wX w <i ¸ ¹ © i wF w F ¦ ¨ wX wX w P X w P < wX ¨ 2 j § © The ¿elds <i x P can be expressed as <i \ i M i where M i are perturbations which encode both the gravitational and matter degrees of freedom and the functions \ i are classical solutions (of the above Eq. (4)). The Lagrangian for the perturbations M i is given by F X 1 , X 2 , X 3 ¦ j 2 3 wF X GX j 1 ¦ w F X GX jGX k 1 ¦ w F X GX jGX kGX l wX j 2 jk wX j wX k 6 jkl wX j wX k wX l j k · w P w P i Q x1 , x2 ,..., xn iVi xi @ (24) which is a quantum Newton law for a many-body system. Equation (24) shows that the contribution to the total force acting on the i-th particle coming from the quantum potential, i.e. i Q , is a function of the positions of all the other particles and thus in general does not decrease with distance. The quantum potential is the crucial entity which allows us to understand the features of the quantum world determined by Bohm's version of quantum mechanics. The mathematical expression of quantum potential shows that this entity does not have the usual properties expected from a classic potential. Relations (21) and (23) tell us clearly that the quantum potential depends on how the amplitude of the wave function varies in the 3D space. The presence of Laplace operator indicates that the action of this potential is space-like, namely creates onto the particles a non-local, instantaneous action. In relations (21) and (23) the appearance of the absolute value of the wave function in the denominator also explains why the quantum potential can produce strong long-range effects that do not necessarily fall off with distance and so the typical properties of entangled wave functions. Thus even though the wave function spreads out, the effects of the quantum potential need not necessarily decrease (as the equation of motion (24) of the many-body systems shows clearly, the total force acting on the i-th particle coming from the quantum potential, i.e. 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 \ may decrease in this limit). This is just the type of behaviour required to explain EPR-type correlations. If we examine the expression of the quantum potential in the two-slit experiment, we may ¿nd that it depends on the width of the slits, their distance apart and the momentum of the particle. In other words, the quantum potential has a contextual nature, namely brings global information on the process and its environment. It contains instantaneous information about the overall experimental arrangement. Moreover, this information can be regarded as being active in the sense that it modi¿es the behaviour of the particle. In a double-slit experiment, for example, if one of the two slits is closed the quantum potential changes, and this information arrives instantaneously to the particle, which behaves as a consequence. Now, the fact that the quantum potential produces a space-like and active information means that it cannot be seen as an external entity in space but as an entity which contains spatial information, as an entity which represents space. On the basis of the fact that the quantum potential has an instantaneous action and contains active information about the environment, one can say that it is space which is the medium responsible for the behaviour of quantum particles. Considering the double-slit experiment, the information that quantum potential transmits to the particle is instantaneous just because it is spatial information, is linked to the 3D space. In virtue of its features, the quantum potential can be considered a geometric entity, the information determined by the quantum potential is a type of geometric information "woven" into space. Quantum potential has a geometric nature just because it has a contextual nature, contains global information on the environment in which the experiment is performed and at the same time it is a dynamical entity just because its information about the process and the environment is active, determines the behaviour of the particles. In this geometric picture one can say that the quantum potential indicates, contains the geometric properties of space from which the quantum force, and thus the behaviour of quantum particles, derive. Considering the double-slit experiment, the fact that the quantum potential is linked with the width of the slits, their distance apart and the momentum of the particle, namely that brings global information on the environment means that it describes the geometric properties of the experimental arrangement (and therefore of space) which determine the quantum force and the behaviour of the particle. And the presence of Laplace operator (and of the absolute value of the wave function in the denominator) DAVIDE FISCALETTI, AMRIT S. SORLI indicates that the geometric properties contained in the quantum potential determine a non-local, instantaneous action. We can say therefore that Bohm's theory manages to make manifest this essential feature of quantum mechanics, just by means of the geometric properties of space described and expressed by the quantum potential. As regards the geometric nature of the quantum potential and the non-local nature of the interactions in physical space, one can also say, by paraphrasing J. A. Wheeler's famous saying about general relativity, that the evolution of the state of a quantum system changes active global information, and this in turn inÀuences the state of the quantum system, redesigning the non-local geometry of the universe. In synthesis, according to the authors, in virtue of the space-like action of the quantum potential, the medium of the 3D space has a crucial role in determining the motion and the behaviour of subatomic particles. On the basis of the equations (21) and (23), one can say that it is space which is the medium responsible for the behaviour of quantum particles. One can say that equations (21) and (23) of the quantum potential contain the idea of space as an immediate information medium in an implicit way. In particular, if we consider a many-body quantum process (such as for example the case of an EPR-type experiment, of two subatomic particles, ¿rst joined and then separated and carried away at big distances one from the other), we can say that the 3D physical space assumes the special "state" represented by the quantum potential (23), and this allows an instantaneous communication between the particles under consideration [29]. If we examine the situation considered by Bohm in 1951 (illustrated before) we can say that it is the state of space in the form of the quantum potential (23) which produces an instantaneous connection between the two particles as regards the spin measurements: by disturbing system 1, system 2 may indeed be instantaneously inÀuenced despite a big distance between the two systems thanks to the features of space which put them in an immediate communication. In synthesis, one can say that in EPR-type experiments the quantum potential (23) makes the 3D physical space an "immediate information medium" between elementary particles. In EPR-type experiments the behaviour of a subatomic particle is inÀuenced instantaneously by the other particle thanks to the 3D space which functions as an immediate information medium in virtue of the geometric properties represented by the quantum potential (23). However, what makes indeed the 3D space an immediate information medium in EPR-type correlations? If the space that we perceive seems to be characterized by local features, from which fundamental entity or structure the property of the quantum potential to determine the action of the 3D space as an immediate information medium derives? In this regard, according to the authors, it is important to mention that in the recent article Bohmian split of the Schrödinger equation onto two equations describing evolution of real functions, Sbitnev [30] has shown that the quantum potential can be determined as an information channel into the movement of the particles as a consequence of the fact that it determines two quantum correctors into the energy of the particle depending on a more fundamental physical quantity that can be appropriately called "quantum entropy". This new way of reading Bohmian mechanics can be called as the "entropic version" of Bohmian mechanics or, more brieÀy, "entropic Bohmian mechanics". In the case of a one-body system, the quantum entropy is de¿ned by the logarithmic function SQ & 1 ln U l n 2 (25a) where U \ x ,t is the probability density (describing the space-temporal distribution of the ensemble of particles, namely the density of particles in the & element of volume d 3 x around a point x at time t) associated with the wave & function \ x, t of an individual physical system. In the case of a many-body system, the quantum entropy is always de¿ned by the logarithmic function where here U space-temporal distribution of the ensemble of particles, namely the density of & particles in the element of volume d 3 x around a point x at time t) associated & & & with the wave function \ x1 , x2 ,..., x N , t of the many-body system under consideration. In the entropic version of Bohmian mechanics, one can assume that the 3D space distribution of the ensemble of particles describing the physical system under consideration generates a modi¿cation of the background space characterized by the quantity given by equation (25a) (or (25b)). The quantum entropy ((25a) and (25b)) can be interpreted as the physical entity that, in the quantum domain, characterizes the degree of order and chaos of the vacuum ­ a storage of virtual trajectories supplying optimal ones for particle movement ­ which supports the density U describing the space-temporal distribution of the ensemble of particles associated with the wave function under consideration. By introducing the quantum entropy, for one-body systems, the quantum potential can be expressed in the following convenient way 1 l U (25b) n ln 2 & & & 2 \ x1 , x2 ,..., x N , t is the probability density (describing the SQ !2 !2 2 2 SQ SQ 2m 2m (26). and we obtain the following equation of motion for the corpuscle associated with & the wave function \ x, t : DAVIDE FISCALETTI, AMRIT S. SORLI 2m 2 !2 SQ 2 V ! 2 SQ 2m 2m wS wt (27) !2 2 which provides an energy conservation law where the term 2m S Q can be 2 interpreted as the quantum corrector of the kinetic energy S of the particle, 2 2m while the term ! 2 S Q can be interpreted as the quantum corrector of the 2m potential energy V. In the case of many-body systems, the quantum potential is given by the following expression 2 !2 i SQ 2 ! i 2 SQ 2mi 2mi (28) and the equation of motion is i 1 i S 2mi i 1 2 N !2 i SQ 2 V ¦ ! i 2 SQ 2mi i 1 2mi wS wt (29) which provides an energy conservation law where the term body system, while the term ¦ !2 i SQ 2 2mi can be interpreted as the quantum corrector of the kinetic energy of the many!2 2 i SQ 2mi can be interpreted as the quantum corrector of the potential energy. On the ground of Sbitnev's results, it becomes thus permissible the following reading of the quantum potential and of the energy conservation law in quantum mechanics. The quantum potential derives from the quantum entropy describing the degree of order and chaos of the background space (namely the modi¿cation in the background space) produced by the density of the ensemble of particles associated with the wave function under consideration. And, on the basis of equations (27) and (29), we can say that the quantum entropy determines two quantum correctors in the energy of the physical system under consideration (of the kinetic energy and of the potential energy respectively) and without these two quantum correctors (linked just with the quantum entropy) the total energy of the system would not be conserved. Moreover, in this entropic approach to Bohmian mechanics, the classical limit can be expressed by the conditions o 2 SQ (30) for one-body systems and i Q o i SQ (31) for many-body systems. The quantum dynamics will approach the classical dynamics when the quantum entropy satis¿es conditions (30) (for one-body systems) or (31) (for many-body systems) which can be considered as the expression of the correspondence principle in quantum mechanics. With the introduction of the quantum entropy ((25a) for one-body systems and (25b) for many body systems) which leads to the energy conservation law (equation (27) for one-body system and equation (29) for many-body system), now new light can be shed on the interpretation of the action of the 3D space as an immediate information medium in EPR-type correlations. In fact, on the basis of equation (29), one can say that the action of the 3D space as an immediate information medium derives just from the two quantum correctors to the energy of the system under consideration, namely from the quantum corrector to the potential energy ¦ N ! i SQ 2 2mi i 1 !2 2 i SQ 2mi and the quantum corrector to the kinetic energy (while the other two terms ¦ i 1 i S 2mi and V on the right-hand of equation (29) determine a local feature of space). The feature of the quantum potential to make the 3D space an immediate information channel into the behaviour of quantum particles derives just from the quantum entropy. In other words, one can see that by introducing the quantum entropy given by equation (25b), it is just the two quantum correctors to the energy of the system under consideration, depending on the quantity describing the degree of order and chaos of the vacuum supporting the density U (of the particles associated with the wave function under consideration) the fundamental element, which at a fundamental level produces an immediate information medium in the behaviour of the particles in EPR-type experiments. The space we perceive seems to be characterized by local features because in our macroscopic domain the quantum entropy satis¿es conditions (30) or (31). In synthesis, in the entropic version of Bohmian mechanics, one can say that the quantum entropy, by producing two quantum corrector terms in the energy, can be indeed interpreted as a sort of intermediary entity between space and the behaviour of quantum particles, and thus between the non-local action of the quantum potential and the behaviour of quantum particles. The introduction of the quantum entropy (given by equation (25a) or (25b)) as the fundamental entity that determines the behaviour of quantum particles leads to an energy conservation law in quantum mechanics (expressed by equations (27) and (29)) which lets us realize what makes indeed the 3D space an immediate information medium in EPR-type DAVIDE FISCALETTI, AMRIT S. SORLI correlations, which in turn lets us realize from which property of the quantum potential one can derive the action of the 3D space as an immediate information medium. The ultimate source, the ultimate visiting card which determines the action of the 3D space as an immediate information medium between quantum particles is a fundamental vacuum de¿ned by the quantum entropy ((25a) or (25b)). The quantum entropy, by producing two quantum corrector terms in the energy, is the fundamental element which gives origin to the non-local action of the quantum potential. Now, as regards the instantaneous communication between quantum particles in EPR-type experiments and the role of the 3D space as a direct information medium between them, if one imagines to exchange, to invert the roles of the two particles what happens is always the same type of process, namely an instantaneous communication between the two particles. In other words, the instantaneous communication between two particles in EPR experiment is characterized by a sort of symmetry: it occurs both if one intervenes on one and if one intervenes on the other. In both cases the same type of process happens and ­ we can say ­ always owing to space which functions as an immediate information medium. Moreover, if we imagine to ¿lm the process of an instantaneous communication between two subatomic particles in EPR-type experiments backwards, namely inverting the sign of time, we should expect to see what really happened. Inverting the sign of time, we have however no guarantee that we obtain something that corresponds to what physically happens. Although the quantum potential ((21) for one-body systems and (23) for many-body systems) has a space-like, an instantaneous action, however it comes from Schrödinger equation which is not time-symmetric and therefore its expression cannot be considered completely satisfactory just because it can meet problems inverting the sign of time. On the basis of these considerations, in order to interpret in the correct way, also in symmetric terms in exchange of t for ­t, the instantaneous communication between subatomic particles and thus the interpretation of 3D space as an immediate information medium, in quantum theory in line of principle a symmetry in time is required. For this reason the authors of this article have recently introduced a research line based on a symmetrized version of the quantum potential. The symmetrized quantum potential can explain a symmetric and instantaneous communication between subatomic particles and thus can be considered as a better candidate for the state of the 3D space as an immediate information medium in EPR-type experiments (or, more generally, in each immediate physical phenomenon). In the case of a system of N particles the symmetrized quantum potential assumes the form § i 2 R1 · ¨ ¸ N ! 2 ¨ R1 ¸ (32) ¦ 2m ¨ 2 R ¸ i 1 i ¨ i 2 ¸ ¨ R2 ¸ © ¹ where R1 is the absolute value of the wave-function = R1eiS1 / describing the forward-time process (solution of the standard Schrödinger equation) and R2 is the absolute value of the wave-function I R2 e iS 2 / ! describing the time-reverse process (solution of the time-Schrödinger equation). On the basis of equation (32), we can explain non-local correlations in many-body systems ­ and thus EPR experiments ­ in the correct way (that is, also if one would imagine to ¿lm back the process of these correlations). The symmetrized quantum potential (32) can be considered the most appropriate candidate to provide a mathematical reality to a 3D space intended as a direct information medium [31]. In fact, the symmetrized quantum potential is characterized by two components, the one regarding the forward-time process, the other regarding the time-reverse process. The ¿rst component of the symmetrized quantum potential, Q1 ! 2 i R1 ¦ 2m R i 1 i 1 N 2 (33), which is related to the forward-time process and coincides with the original Bohm's quantum potential, is the real physical component which produces observable effects in the quantum world. As regards the observable effects of Bohm's quantum potential, the reader can ¿nd details in the results obtained, for example, by Philippidis, Dewdney, Hiley and Vigier about the classic double-slit experiment, tunnelling, trajectories of two particles in a potential of harmonic oscillator, EPR-type experiments, experiments of neutron-interferometry [32, 33]). The ¿rst component (33) expresses the instantaneous action on quantum particles and thus the immediate action of space on them. The second component, Q2 ! 2 i R2 ¦ 2m R i 1 i 2 N 2 (34), is introduced to reproduce in the correct way the time-reverse process of the instantaneous action and thus it guarantees that the quantum world can be interpreted correctly with the idea of space as an immediate information medium if one would imagine to ¿lm the process backwards: it must be introduced in order DAVIDE FISCALETTI, AMRIT S. SORLI to recover a symmetry in time in quantum processes, to interpret in the correct way quantum processes if one would imagine to ¿lm that process backwards. The opposed sign of the second component with respect to the physical ¿rst component (that is, with respect to the original Bohm's quantum potential) can be interpreted as a consequence of the idea of the measurable time as a measuring system of the numerical order of material change: the mathematical features of the second component of the symmetrized quantum potential imply that it is not possible to go backwards in the physical time intended as the numerical order of physical events. Both the components (33) and (34) of the symmetrized quantum potential can be considered as physical quantities deriving from the quantum entropy (25b). The ¿rst component can be expressed as i 1 2 !2 i SQ1 2 ! i 2 SQ1 2mi 2mi (35), while the second component can be expressed as i 1 2 !2 i SQ 2 2 ! i 2 SQ 2 2mi 2mi (36), l n where S Q1 2 ln U1 is the quantum entropy de¿ning the degree of order and & & & 2 chaos of the vacuum for the forward-time processes (where U1 \ x1 , x2 ,..., x N , t , & & & i S \ x1 , x2 ,..., x N , t R1e iS 1 / ! being the forward-time may-body wave function, so1 l n lution of the standard Schrödinger equation) and S Q 2 ln U 2 is the quantum 2 entropy de¿ning the degree of order and chaos of the vacuum for the time-reverse & & & & & & processes (where U 2 I x1 , x2 ,..., x N , t 2 , I x1 , x2 ,..., x N , t R2 e iS 2 / ! being the time-reverse many-body wave function, solution of the time-reversed Schrödinger equation). The energy conservation law for the forward-time process is i 1 i S1 2mi i 1 2 N !2 i SQ1 2 V ¦ ! i 2 SQ1 2mi 2mi i 1 wS1 , wt (37) while the energy conservation law for the reversed-time process is i 1 i S2 2 mi i 1 2 N !2 i S Q 2 2 V ¦ ! i 2 S Q 2 2 mi 2 mi i 1 wS 2 wt (38). On the basis of its mathematical features, the symmetrized quantum potential implies that in the quantum domain a timeless 3D space has a crucial role in determining the motion of a subatomic particle because the symmetrized quantum potential produces a like-space and instantaneous action on the particles under consideration and contains active information about the environment and, on the other hand, implies the concept of time as a numerical order of material change. In EPR-type experiments (and, more generally, in all immediate physical phenomena regarding the quantum domain) the 3D timeless space acts as an immediate information medium in the sense that the ¿rst component of the symmetrized quantum potential makes physical space an "immediate information medium" which keeps two elementary particles in an immediate contact (while the second component of the symmetrized quantum potential reproduces, from the mathematical point of view, the symmetry in time of this communication and the fact that time exists only as a numerical order of material change). We can call this peculiar interpretation of quantum non-locality as the "immediate symmetric interpretation" of quantum non-locality. 5. CONCLUSIONS This article shows that a 3D space where time t is exclusively a numerical order of material changes can be considered a fundamental arena of physical processes. At a fundamental level, we live in a universe where time measured by clocks exists exclusively as a numerical order of material changes. Nonlocal correlations in EPR-type experiments are carried directly by the 3D space, the numerical order t of quantum entanglement is zero in the sense that the 3D space functions as an immediate information medium. The action of the 3D space as an immediate information medium derives from the quantum entropy describing the degree of order and chaos of the vacuum supporting the density of the particles associated with the wave function under consideration. The symmetrized quantum potential characterized by the two components (where the ¿rst component coincides with the original Bohm's quantum potential and the second component is endowed with an opposed sign with respect to it) seems to be the most appropriate candidate to represent the mathematical state of the 3D space as an immediate information medium between subatomic particles that accounts for entanglement and non-locality (and more generally, for all immediate physical phenomena in the quantum domain).

Journal

Annales UMCS, Physica – de Gruyter

Published: Jan 1, 2012

There are no references for this article.