De Broglie – Bohm Theorem in an underlying structure of Space and Spacetime
Salim Yasmineh ( [email protected] ) Brevalex https://orcid.org/0000-0002-9078-3306
Original Research
Keywords: simultaneity, non-locality, Hilbert space, con guration space, measurement, entanglement
Posted Date: February 9th, 2021
DOI: https://doi.org/10.21203/rs.3.rs-160859/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License 1
De Broglie – Bohm Theorem in an underlying structure of Space and Spacetime Salim YASMINEH PhD University of Paris
Email: [email protected]
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De Broglie – Bohm Theorem in an underlying structure of Space and Spacetime Abstract The concept of simultaneity is relative in special relativity whereas, it seems to have a definite meaning in quantum mechanics. On the other hand, theory and experiments in quantum mechanics have revealed the existence of non-local causal relations. We propose to use the invariant space-time interval introduced by special relativity as a benchmark for constructing a spacetime framework presenting a notion of an invariant time. We suggest that a physical system creates a non-local field within the spacetime framework and that the wavefunction of the physical system is an ontological entity that represents this non-local field. We propose to explain measurement and entanglement by using de Broglie Bohm theory in the context of the constructed spacetime framework. Key words: simultaneity, non-locality, Hilbert space, configuration space, measurement, and entanglement.
1.Introduction
A quantum system for example made up of a pair of entangled particles behaves in such a manner that the quantum state of one particle cannot be described independently of the state of the other. This quantum phenomenon was first introduced as a thought experiment in the EPR paper [1] and it was later discovered that it can be experimentally testable by using Bell’s inequality [2,3]. Standard quantum mechanics postulates that neither one of both particles has a determinate state until it is measured. As both particles are correlated, it is necessary that when the state of one particle is measured the second particle should ‘simultaneously’ acquire a determinate state. There have been numerous experiments such as Aspect's experiment [4] that proved the validity of quantum entanglement and hence of non- local causal connections. This phenomenon is discussed in detail by Jean Bricmont especially in relation to the de Broglie-Bohm theory [5,6] in his book “Making Sense of Quantum Mechanics” [7].
On the other hand, given that the laws of physics are invariant under Lorentz transformations, then there is no meaning of ‘simultaneity’ independently of any frame of reference and there should be no preferred frame of refence [8, 9]. Thus, the notions of non-locality and simultaneity are inferred by quantum mechanics while being forbidden by special relativity.
We propose a solution to this dilemma by redefining the framework of spacetime in quantum mechanics without contradicting special relativity.
In section 2, we use the formalism of Minkowski spacetime to construct an underlying framework of spacetime referred to as ‘phenomenal interface spacetime’ related to standard spacetime by Lorentz transformations and presenting a notion of an invariant time. In sections 3-5, we introduce a global space structure composed of the ‘phenomenal interface spacetime’ in addition to an underlying space which shall be referred to as ‘noumenal space’ such that a particle in noumenal space creates a non-local field whose projection onto the phenomenal interface spacetime is represented by a wavefunction. From the standpoint of the phenomenal interface spacetime, if the particle is in noumenal space, the non-local field behaves like a wave whereas, when the particle is in the phenomenal interface spacetime, the particle behaves like a corpuscle. Measurement consists in ‘bringing’ a particle from noumenal space into the phenomenal interface spacetime. In sections 6-9 we define global space in the non-relativistic limit within which we consider de Broglie-Bohm theory. In addition to Schrodinger’s equation and de Broglie Bohm’s equation, we introduce a new relation that explains measurement and entanglement.
2. Invariant spacetime structure 3
We propose to use the formalism of special relativity to construct an invariant hyperbolic spacetime structure formed out of the foliation of a family of three-dimensional hyperboloids inside a light cone with respect to which the wavefunction will be defined. 2.1 Review of Minkowski spacetime Hereafter, we shall use the formalism of Minkowski spacetime [10] as defined in a geometrical manner by Eric Gourgoulhon in his book ‘Special Relativity in General frames’ [11]. Minkowski spacetime Ԑ is defined as an affine space of dimension 4 on R endowed with a bilinear metric tensor g defined in an underlying vector space E, of signature . In the vector space E, the set C composed of the zero vector and all null vectors form a light cone C composed of two sheets C+ and C- defining the future and past light cones, respectively. (+, −, −, −)
Given the above defined spacetime Ԑ and given an arbitrary point , a family of affine subspaces is defined such that each subspace corresponds to the set of points of Ԑ that can be connected to O by a time-like vector of modulus , where : 푂 ∈ Ԑ (푆휏)휏∈푅 푆휏 푂푀⃗⃗⃗⃗⃗⃗ (2.1)휏 휏 ∈ 푅 2 We푆휏 =are {푀 henceforth, ∈ Ԑ, 푂푀⃗⃗⃗⃗⃗⃗ .interested푂푀⃗⃗⃗⃗⃗⃗ = −휏 in
and (2.9) 푢 −푢 휏푒 휏푒 The푋 = spatial√2 hyperbolic푌 = √2 coordinate is constant when which is a straight line (or a ray ) passing through the origin. Thus, all points on the −2푢 same ray share the same spatial hyperbolic coordinate . The central푢 ray (corresponding푌 = 푘푋 to= an 푒 axe푋 of symmetry with respect to the 푢 푢 slices)푅 represent the zero spatial coordinate (i.e (푢,), 휏) while those on its right푅 and left represent the 0 positive ( ) and negative푢 ( ) spatial푅 hyperbolic coordinates respectively. On the other hand, all points on the same slice share the same푢 = invariant 0 temporal coordinate . 푢 > 0 푢 < 0 Thus, the (ray푢,s 휏) and hyperbolic푆휏 slices can be used to define a hyperbolic휏 frame of reference representing the invariant spacetime structure , where a given hyperbolic point is 푢 푢 휏 휏 the intersection( 푅between) the corresponding (ray푆 ) and slice . (푂; 푅푢, 푆휏) Ԑ퐼 (푢, 휏) 푅푢 푆휏 5
On the other hand, the point can be defined with respect to the orthonormal frame of reference whose axis and axis are oriented at with respect to the axis or axis. The relations between and can(푋, thus, 푌) be expressed as follows: (푂; 푥, 푡) 푥 푡 ±45° 푋 푌 (푋, 푌) and Y(푥, 푡) (2.10) √2 √2 푋Th =erefore,2 (푡 + by 푥) combining= 2 ( relations푡 − 푥) (2.9) and (2.10), the mapping between a point in the orthonormal frame of reference and a hyperbolic point in the hyperbolic frame of reference is: (푥, 푡) (푂; 푥, 푡) (푢, 휏) 푢 휏 (푂; 푅 and, 푆 ) (2.11) 푡+푥 2 2 푢The = inverse푙푛√푡−푥 mapping휏 =of √푡(2.11)− 푥can be expressed as follows: and (2.12)
On푥 = the 휏 푠푖푛ℎ other 푢 hand, by푡 = using 휏 푐표푠ℎ expressions 푢 (2.12), the hyperbolic spatial coordinate can be expressed in function of invariant time as well as either one of the standard spatial coordinate and temporal coordinate , as follows: 푢 휏 푥 푡 where (2.13) 2 2 2 2 푢 = 푎푟푐표푠ℎ(푡⁄ 휏) = 푙푛 (푡⁄ 휏 + √푡 ⁄휏 − 1) (2.14)푡 ⁄휏 ≥ 1 2 2 푢Specifically = 푎푟푐푠푖푛ℎ, (any푥⁄ 휏given) = 푙푛 slice(푥⁄ 휏 + can√푥 be⁄ 휏parametrized+ 1) by the standard space coordinates according to expression (2.18), as follows: 푆휏 푥 (2.15)
푆As휏 =each{푢휏 slice(푥); 휏 푖푠 is 푐표푛푠푡푎푛푡, associated 푥 with ∈ 푅, a 푎푛푑corresponding 푢휏(푥) = 푎푟푐푠푖푛ℎ invariant(푥 time⁄ 휏)} , then, all points belonging to that slice are said to be simultaneous. In other words, each slice is a class of simultaneity made up of 휏 a set of points푆 that are associated to the same invariant instant 휏. 푢 푆휏 푆휏 The passage from푢휏 one slice into a subsequent slice represents휏 the ‘transition’ of the invariant- time from one invariant instant to a consequent invariant instant . The invariant time provides thus, 휏1 휏2 an invariant time ordering of푆 the set of slices. The invariant푆 time increases as the slices depart from 1 2 the origin. This invariant time ordering휏 of the set of slices implies an휏 invariant causality between points 휏 on different slices. 푆 On the other hand, the mapping between a point in the hyperbolic frame of reference and a point in another inertial frame of reference can be determined by simply using 푢 휏 the Lorentz transformations. (푢, 휏) (푂; 푅 , 푆 ) (푥′, 푡′) (푂; 푥′, 푡′) Indeed, let be another frame of reference that moves at a normalised constant velocity relative to where is a fraction of the speed of light c which is taken to be equal to 1 and hence, . The (Lorentz푂; 푥′, 푡′ )transformations are expressed as: 훽 (푂; 푥, 푡) 훽 훽 < 1 (2.16) ′ 푡 = 훾(푡 − 훽푥) (2.17) ′ 푥where= 훾(푥 − 훽푡) . Then by plugging (2.12) into the above equations, we get: 2 −1⁄ 2 = (1 − 훽 ) (2.18) 푥′ = 훾휏 (sinh 푢 − 훽 cosh 푢) 6
(2.19)
푡′In =view훾휏 (coshof the 푢above, − 훽 sinhwe postulate 푢) that the wavefunction (or state vector) is defined in a Hilbert space associated to the above constructed invariant hyperbolic spacetime structure. The arguments of the wavefunction are associated to the hyperbolic points of a corresponding slice , and are thus, invariant with respect to time . The wavefunction that we shall call ‘invariant wavefunction’ evolves 휏 according to the progression of subsequent slices. On( the푢, 휏other) hand, we will see in section푆 6 that the invariant wavefunction can be 휏approximated by the standard wavefunction with respect to standard time and space coordinates.
3. Hilbert Space In this section we propose to define the state vector of a particle and the structure of an underlying space. 3.1 Invariant state vector
Each slice can be associated to a Hilbert Space . The slice represents a position basis of the associated Hilbert Space with elements labelled by a continuous hyperbolic position variable 휏 휏 휏 normalised푆 using the Dirac -function: ℋ 푆 {|푢 ⟩} ℋ |푢휏⟩ 푢휏 (3.1) 훿 An⟨푢′휏 | invariant푢휏⟩ = 훿(푢′ unit휏 − state 푢휏) vector in the Hilbert space associated to the slice can then be expanded as an integral in function of the base elements as follows: |휓(휏)⟩ ℋ 푆휏 (3.2) |푢휏 ⟩
The|휓(휏) right-hand⟩ = ∫ 푑푢 휏side 휓(푢 of휏 the, 휏) above|푢휏⟩ equation is defined by a line integral along the piecewise smooth curve representing the slice . It describes the state vector of a physical system as a superposition of position basis elements each one of which corresponds to a definite hyperbolic position coordinate 휏 of the slice at 푆 a given invariant instant . The|휓(휏) expanding⟩ coefficients or ‘weights’ 휏 represent a complex valued|푢 ⟩ invariant wavefunction (i.e. defined with respect to an invariant instant ). 푢휏 푆휏 휏 휓(푢휏, 휏) The left-hand side belongs to Hilbert space and represents the resultant (i.e. the vector sum)휏 of the decomposed position states. It seems reasonable to suppose that there should exist an elemental resultant position |휓(휏) in ⟩ a real physical space whichℋ is associated to the resultant state vector belonging to Hilbert space . The elemental resultant position would be the resultant or at least related to the position푢̅ coordinates associated to the position basis elements . |휓(휏)⟩ ℋ 푢̅ If the resultant state vector is not푢휏 reduced to a single basis element , |then푢휏⟩ the elemental resultant position cannot belong to the slice (i.e. ) . This implies that cannot be referenced 휏 by the invariant temporal reference|휓(휏)⟩ . Indeed, as seen in section 2, a point| 푢 ⟩ if and only if 휏 휏 and푢̅ thus, if , then there푆 exists 푢̅no ∉ vector 푆 representing 푢̅such that . 휏 푀 ∈ 푆휏 Therefore, the 2elemental resultant position (if it exists and we assume it does) should belong to 2a 휏 푂푀timeless⃗⃗⃗⃗⃗⃗ . 푂푀⃗⃗⃗⃗⃗⃗ extension= −휏 of space 푢̅ such ∉ 푆 that: 푉⃗ 푢̅ 푉⃗ . 푉⃗ = −휏 푢̅ and 풩 (3.3) 푢̅The ∈ temporal 풩 reference풩 ∩ 푆휏 =of ∅the resultant state vector would then be emergent and would exist only from the standpoint of the hyperbolic slice . It will be shown in section 7 that the resultant state vector represents a non-local field generated by| 휓(휏) a corresponding⟩ particle located at an elemental 휏 resultant position . 푆 |휓(휏)⟩ 3.1 Phenomenal Spacetime푢̅ and Noumenal Space 7
In view of the above, we postulate the existence of a four-dimensional space extension whose all four dimensions are spatial. The ‘4d space extension’ and each slice can thus be part of a higher 4d structure that we shall call ‘global space’ . From the standpoint of each given slice ,풩 the global space 휏 structure may be thought of as being partitioned풩 into three space blocks푆 composed of the 3d spatial 휏 part of the given slice acting as a boundary풢 between two extradimensional 4d spaces푆 that compose the풢 timeless spatial extension . 푆휏 The pilling-up of the set of slices풩 forms an ‘interface spacetime’ which is accessible through the window of standard spacetime to which it is related via the above formalism of 휏 휏 Minkowski. However, the timeless spa{푆 ce} extension corresponds to an underlyingℐ(푢 , 휏) inaccessible space or rather not directly accessible. The standard(푥, 4d-spacetime 푡) as well as the directly related accessible interface spacetime shall be referred to as ‘phenomenal풩 standard spacetime’ and ‘phenomenal interface spacetime’ respectively, whereas the underlying non-directly accessible space extension shall be referred to as ‘noumenal space’ borrowed from푃(푥, 푡) the ‘noumenal and 휏 phenomenal worlds’ ofℐ(푢 Kant., 휏) 풩 풩(푢̅) In view of the above, the phenomenal interface spacetime (or simply ‘interface spacetime’) is defined as a set of interface spacetime points , as follows:
(푢휏,휏) (3.4)
Theℐ = {(noumenal푢휏,휏) ∈ 푆space휏 ∶ 휏 ∈ 푅, 푢휏 is∈ defined 푅, {|푢휏⟩ }as 푖푠 the푡ℎ푒 set 푏푎푠푒 of 푡ℎnoumenal푎푡 푠푝푎푛푠 points 퐻푖푙푏푒푟푡 , each푠푝푎푐푒 being ℋ} composed of an interface-like spatial coordinate and an extradimensional space coordinate , as follows: 풩(푢̅) 푢̅ 푢 휈 (3.5) The풩 = global{푢̅= space(푢, 휈) : 푢,휈 is simply ∈ 푅 ;∀ the |휓 union(휏)⟩ ∈ of ℋ, the ∃ interface푎 푐표푟푟푒푠푝표푛푑푖푛푔 spacetime 푒푙푒푚푒푛푡푎푙 and 푝표푠푖푡푖표푛 noumenal 푢̅∈ space 풩} :
(3.6)퐺 ℐ(푢휏, 휏) 풩(푢̅) To풢 = have ℐ ∪ a 풩 geometrical continuity in global space, the configuration in noumenal space should be similar to the slicing structure in interface spacetime. Thus, noumenal space can be considered as an affine space partitioned into a family of affine subspaces congruent to each interface slice . The family of affine subspaces is constituted of a family of three-dimensional풩 hyperboloids where 휈 휈∈푅 휏 each hyperboloid is made up of two sheets and (푆 associated) with the real extradimensional푆 space 휈 휈∈푅 coordinates and (푆 respectively) corresponding+ − to positive and negative invariant coordinates 휈 휈 . The piling-up of this family of three-dimensional푆 hyperboloids푆 forms an invariant hyperbolic space structure휈 ≥ of0 the 휈4d < noumenal 0 space. The two sheets and of each hyperboloid forms 휈 휈∈푅 ‘noumenal휈 slices’ where each slice is associated with an invariant+(푆 extradimensional) − space coordinate 휈 휈 . The noumenal space is a foliation of the family 푆 of three-dimensional푆 hyperboloids at all invariant space coordinates , such that, given any arbitrary interface slice , there exists a set of 휈 휈∈푅 휈 geometrically 휈 ∈ 푅 congruent noumenal slices . In noumenal(푆 ) space the extradimensional space푆 휏 coordinate is a spatial invariance휈 that plays a role similar to the temporal invariance푆 coordinate of 휈 the interface spacetime. {푆 } 휈 휏 Thus, by introducing the invariance of the extradimensional space coordinate , expression (3.5) becomes: 휈 (3.7)
In풩 view = {푢̅= of expressions(푢휈, 휈) ∈ 푆 휈(3.5),: 푢휈, 휈(3.6 ∈) 푅 and ; ∀ |(3.7),휓(휏) ⟩the∈ ℋ,global ∃ 푐표푟푟푒푠푝표푛푑푖푛푔 space can be defined푝표푠푖푡푖표푛 as (a푢 set휈, 휈 of) ∈ global 푆휈} points composed of hyperbolic spatial coordinates and invariant coordinates , as follows: 풢 푢̃ 푢훾 훾 (3.8)
풢 = {푢̃= (푢훾, 훾) ∶ 푢훾, 훾 ∈ 푅 ; 푓표푟 훾 = 휏, 푢̃ ∈ 푆휏, 푎푛푑 푓표푟 훾 = 휈, 푢̃ ∈ 푆휈} 8
4. Noumenal Space The fact that noumenal space is non-temporal does not imply that a particle in this space cannot have different positions. It entails however, that the different positions have no temporal ordering. But, as the extradimensional space coordinate is invariant for each slice, then all changes between points on different slices can be defined with respect to changes in the extradimensional coordinate . Even though, the invariant extradimensional휈 space coordinate has no specific direction, yet different positions of a particle in noumenal space can still be characterised by an undirected ordering relation휈 of the invariant coordinate . This undirected ordering can be defined휈 by a ‘betweenness relation’ as in the C-relations identified by McTaggart [12] and as explained by Baron and Miller [13]. Indeed, a ‘betweenness relation’ defines휈 an unchanging fact that a particle at an invariant coordinate in noumenal space can be considered as between its and extradimensional space coordinates without 퐵 appealing to any ordering. The relation ‘ between and ’ is an unchanging fact in noumenal 휈space 퐴 퐶 but is undirected in the sense that there is no fact of휈 matter휈 as to the direction of the sequence of positions. 휈퐵 휈퐴 휈퐶 Thus, a change in noumenal space has a sense and can conceptually be described by means of an undirected ordering relation. On the other hand, due to the lack of any temporal notion, spatially separated objects in noumenal space can be non-locally interconnected behaving as a single ‘holistic identity’. Such a holistic identity can for example be composed of a group of spatially separated particles whose features form a single state at a given invariant coordinate . In other words, all parts of a holistic identity should have the same invariant coordinate . Indeed, the invariant spatial coordinate in noumenal space plays a similar role to that of the invariant time coordinate휈 in interface spacetime such that a relation of events happening at the same invariant휈 time coordinate (i.e. simultaneity in interface휈 spacetime) corresponds to a relation of states taking place at the same invariant휏 space coordinate in noumenal space. 휏 휈
5. Relation between Noumenal Space and Phenomenal Interface Spacetime In this section we propose to treat very schematically and in a conceptual manner, the relation between noumenal space and interface spacetime and specifically, the indirect access to the former through its potential effects on the latter. 5.1 Noumenal particle and non-local noumenal field Starting from the matter of fact that theory and experiments in quantum mechanics entail non-local causal connections, we postulate the existence of a non-local noumenal field created by each physical system in noumenal space. The non-local noumenal field depends mainly on the position and movement of the system in noumenal space as well as other physical features훷 of the system. Hereafter, a system, or a particle located in noumenal space shall be called훷 ‘noumenal system’ or ‘noumenal particle’. Postulate 1: The state of a noumenal particle (i.e. a particle in noumenal space) can be indirectly specified by an invariant wavefunction representing the density distribution on the interface slice of a non-local projected noumenal field created by the noumenal particle. 휓푈̅(푢휏,휏) First, 푆we휏 postulate that a noumenal particle located휑푈̅( 푢at휏 ,휏a noumenal) position creates a non- local noumenal field at all global points . The non-local noumenal field can be expressed 푈̅ = (푈, 휈) as a function mapping the set of global points into a set of values 훷푈̅ {(푢훾, 훾)} characterising the non-local noumenal field effect at each global point: 훷푈̅ {(푢훾, 훾)} {훷푈̅(푢훾, 훾)} 9
(5.1) 훷푈̅ {The(푢훾 non-local, 훾)} → noumenal {훷푈̅(푢 field훾, 훾) } at a global point depends on the coordinates of that point as well as the noumenal position , the latter dependence being indicated by a subindex attached 푈̅ 훾 훾 to the non-local noumenal field훷 (푢. The, 훾) noumenal position(푢 is, 훾) in fact related to the set of noumenal ̅ field values by a two-way 푈relation: 훷푈̅ 푈̅ {훷푈̅(푢 훾 , 훾)(5}.2)
Any푈̅ ↔ variation {훷푈̅(푢훾 , 훾)} exacted on the noumenal field at a global point generates a variation in the noumenal position which in its turn creates a variation of the noumenal field at all other points 훿훷푈̅(푢훾, 훾) (푢훾, 훾) . In principle, any point in global space may non-locally affect any other point via the ̅ timeless noumenal space 훿푈by passing through the intermediate noumenal physical system. {훷훿푈̅(푢훾, 훾)} According to expression (3.8), a global point can be either an interface point belonging to an interface slice or a noumenal point belonging to a noumenal slice : (푢훾, 훾) (푢휏,휏) 휏 휈 휈 푆 ( 푢 ,(5 휈).3) 푆 (푢휏,휏) ∈ 푆휏 푖푛푡푒푟푓푎푐푒 푝표푖푛푡 (푢훾, 훾) = { Our main interest(푢휈, 휈) ∈ here 푆휈 is푛표푢푚푒푛푎푙 the part of 푝표푖푛푡 the noumenal field which we shall call ‘projected non-local noumenal field’ noted , that affects the phenomenal interface slice and that maps the set of interface points into a set of values characterising the projected non-local 푈̅ 휏 noumenal field effect at each휑 interface point: 푆 {(푢휏,휏) ∈ 푆휏} {휑푈̅ (푢휏,휏)} (5.4) 휑푈̅ The{(푢휏 ,휏value)} → of the non-local {휑푈̅(푢휏,휏 noumenal)} field at any interface point depends mainly on the position of the noumenal particle with respect to the interface point which can be 푈̅ 휏 휏 represented by a vector 휑 ( 푢between,휏) these two points. This(푢 vector,휏) may be ̅ 휏 defined by its푈 orientation= (푈, 휈) and its module , the latter being the( distance푢 ,휏) 휏 ̅ 푈̅ 휏 푈̅ 휏 between the interface spatial푎(푢 , 휏,coordinate 푈) ≡ 푎 (푢 and,휏) the noumenal position . The vector 푎 (푢 ,휏) can be 푈̅ 휏 푈̅ 휏 τ expressed by a complex number,휃 (푢 ,휏 as) follows: ‖푎 (푢 ,휏)‖ 푑(u − 푈̅) uτ 푈̅ 푎푈̅(푢휏,휏) (5.5) 푖θ푈̅ (푢휏,휏) 푖θ푈̅ (푢휏,휏) The푎푈̅( 푢noumenal휏,휏) = ‖푎 푈position̅(푢휏,휏) ‖푒 has no temporal= (푑(u τreference.− 푈̅))푒 Thus, we introduce the notion of a ‘projected position’ of the noumenal position on the interface slice . The projected position is ̅ referenced by the invariant푈 time associated to the interface slice where it subsists as a ‘shadow- ̅ 휏 position’ of푈(τ) the corresponding noumenal position푈 that exists in noumenal푆 space . 푈(τ) τ 푆휏 From the standpoint of any interface point 푈̅ , the noumenal position (i.e.풩 its module and orientation) can be expressed in function of the projected position and the extradimensional space 휏 coordinate . For example, the distance (푢 ,휏 )between an interface spatial푈̅ coordinate and the noumenal position can be defined in function of a first distance between푈(τ) the interface point and the τ τ projected positio휈 n and a second distance푑(u −corresponding 푈̅) to the value of the extradimensionalu space τ coordinate , as follows:푈̅ u 푈(τ) 휈 (5.6) 2 2 τ 푈̅ 휏 τ 푑Therefore,(u − 푈̅) the= ‖ vector푎 (푢 ,휏)‖ = √ (푑can(u be− expressed 푈(τ))) in+ function휈 of the interface point , the projected position , and the extradimensional space coordinate . However, for simplicity we will continue 푎푈̅(푢휏,휏) (푢휏,휏) 푈(τ) 휈 10 to use as a subindex for noting the vector by keeping in mind that it can also be expressed in function of and . 푈̅ 푎푈̅ The density푈(τ) magnitude휈 (or module) of the non-local noumenal field decreases as the distance between the interface space coordinate and the noumenal position increases. In fact, by 푈̅ 휏 supposing the noumenal field to be composed of homogeneous and휑 isotropic(푢 ,휏) field lines stemming out τ τ from푑(u −a noumenal 푈̅) particle, it is geometrically trivialu that the density of field lines푈̅ (i.e. number of field lines per unit area) decreases as the distance between the noumenal particle and a point at the interface spacetime increases. The density magnitude of the non-local projected noumenal field can thus be defined by the module of a normalised complex-valued function 푈̅ 휏 such that the module is a decreasing function with respect to the module 휑 (푢 ,휏) of 푈̅ 휏 푈̅ 휏 the complex number ‖: 푔(푎 (푢 ,휏))‖ 푔(푎 (푢 ,휏)) ‖푔(푎푈̅(푢휏,휏))‖ ‖푎(푢휏,휏)‖ 푎푈̅(푢휏,휏) (5.7) ̅ ̅ ̅ ̅ ‖푎푈(푢휏,휏)‖ < ‖푎푈(푢′휏,휏)‖ → (5.8)‖푔(푎 푈(푢휏,휏))‖ > ‖푔(푎푈(푢′휏,휏))‖ 2 The∫‖푔(푎 non푈̅-local(푢휏,휏 )projected)‖ 푑uτ = noumenal 1 field represents thus a complex-valued density distribution within the phenomenal interface slice and can be expressed in polar form, as follows: 휑푈̅(푢휏,휏) 푆휏 (5.9) 푖θ(푢휏,휏) 푖퐹푈̅ (푢휏,휏) 5.휑푈2̅ (Invariant푢휏,휏) = 푔(푎 wavefunction푈̅(푢휏,휏)) = and 푔 ((푑 projected(uτ − 푈̅) n)푒on-local) noumenal= 퐴푈̅(푢휏,휏 field)푒
As expressed in equation (3.1), each phenomenal interface slice represents a position basis of an associated Hilbert Space . It can then, be reasonably suggested that the non-local projected noumenal 휏 field within the slice is represented in the associated푆 Hilbert space by a complex-valued invariant wavefunction ℋ . The amplitude and phase of the invariant 푈̅ 휏 휏 wavefunction휑 (푢 ,휏 ) is related푆 to the amplitude and phase of the projected 푈̅ 휏 푈̅ 휏 푈̅ 휏 noumenal field 휓: (푢 ,휏) 푅 (푢 ,휏) 푆 (푢 ,휏) 휓푈̅(푢휏,휏) 퐴푈̅(푢휏,휏) 퐹푈̅(푢휏,휏) 휑푈̅(푢휏,휏) (5.10) 푖푆푈̅ (푢휏,휏) The휓푈̅( 푢invariant휏,휏) = 푅 wavefunction푈̅(푢휏,휏)푒 ↔ 휑푈̅ (or푢휏 each,휏) one of the amplitude and phase , depends on the interface point and the noumenal position (or the projected position , and 푈̅ 휏 푈̅ 휏 푈̅ 휏 the extradimensional space coordinate휓 (푢 ,휏) instead of ). Specifically, expression푅 (푢 ,휏) (5.9) indicates푆 that(푢 ,휏 the) 휏 ̅ amplitude is related(푢 to,휏 the) distance between푈 the interface point 푈(τ) and the ̅ noumenal position while the phase 휈 is related푈 to the direction of the straight line 푈̅ 휏 τ τ joining the 푅interface(푢 ,휏) point to the noumenal particle.푑(u − 푈̅) u 푈̅ 푆푈̅(푢휏,휏) 휃푈̅(푢휏,휏) Each argument ofu τ the invariant wavefunction represents a ‘shadow-position’ associated with a ‘shadow-momentum’ of the noumenal particle that has a real position in noumenal 휏 푈̅ 휏 space noting however,(푢 ,휏 that) there is no sense in associating a휓 real(푢 momentum,휏) to the noumenal particle in a timeless noumenal space. All the arguments of the invariant wavefunction are푈̅ thus, shadow- positions whose median is represented by the projected position . These shadow-positions and 푈̅ 휏 associated shadow-momenta can be considered as the positions and momenta휓 of(푢 a ,휏‘fictious) -particle’ (or ‘ghost-particle’) omnipresent all over the interface slice . Nevertheless,푈(τ) all these shadow-positions of the phenomenal interface slice are representatives of a real unique position in noumenal space. 푆휏 5.3 Implicit variables of the invariant푆휏 wavefunction 푈̅ The unique noumenal position (or equivalently, the couple ) is a hidden ‘non-local’ variable or at least a non-explicit variable of the invariant wavefunction . It is in fact implicitly defined in the expression of the wavefunction:푈̅ the interface point at(푈 which(τ), 휈) the amplitude has its 휓푈̅(푢휏,휏) 푅푈̅(푢휏,휏) 11 maximum value indicates the projected position while the dispersion of the amplitude is related to the magnitude of the interdimensional coordinate . 푈(τ) 푅푈̅(푢휏,휏) Indeed, according to equation (5.6), the distance 휈between the projected position and the noumenal position is equal to the extradimensional space coordinate and represents the minimal distance from the noumenal particle knowing that푑 (by푈(τ) definition, − 푈̅) the extradimensional space coordinate푈(τ) is equal to zero at the푈̅ interface slice . On the other hand, as the amplitude휈 is related to the distance between the interface point and the noumenal position and as this distance is 휏 푈̅ 휏 휈minimal at the projected position 푆 then, the amplitude should have푅 ( its푢 ,휏maximum) value at τ τ the projected푑(u position− 푈̅) and should decrease uas we move further away from푈̅ . For example, the 푈̅ 휏 amplitude may have the form of a푈(τ) Gaussian distribution centered푅 (푢 ,휏at )the projected position . On the other hand, if the interdimensional푈(τ) coordinate tends to zero, the noumenal position푈(τ) of the noumenal particle tends towards the latest projected position and thus, the amplitude becomes more푈(τ) and more peaked around the projected position. This suggests휈 that the interdimensional coordinate푈̅ should be implicitly related to the dispersion of the amplitude’s푈(τ) distribution. 휈 5.4 Indirect noumenal state of a noumenal휎 particle The noumenal position as well as the non-local connections are thus, inherent in the invariant wavefunction . Nevertheless, to have a complete picture of quantum phenomena especially, the non-local effects, it is푈̅ necessary to consider also the noumenal position (or equivalently, the 푈̅ 휏 projected position휓 ( 푢 ,휏) and the extradimensional space coordinate ) in the formulation of quantum rules. 푈̅ 푈(τ) 휈 Proposition 1: From the perspective of the interface spacetime, the features of a noumenal particle can be specified by an indirect noumenal state composed of the invariant wavefunction in association to the projected position on the interface slice at an invariant instant as well as the 푈̅ 푈̅ 휏 corresponding extradimensional space coordinateΩ (휏) : 휓 (푢 ,휏) 푈(τ) 푆휏 휏 (5.11) 휈 AsΩ푈̅ in(휏 )expression= (휓푈̅(푢 휏(5.2),,휏), 푈 any(휏),휈,휏 variation) in the noumenal position of a noumenal particle generates a variation in the noumenal field entailing a variation on its projected field and thus, a variation of the invariant wavefunction : 훿푈̅ 훿훷푈̅ 훿휑푈̅(푢휏,휏) 훿휓(uτ, 휏) (5.12)
훿푈The̅ ↔above 훿휑푈̅ relations(푢휏,휏) ↔ (5.12) 훿휓푈̅ define(푢휏,휏) an↔ action-reaction 훿(푅푈̅(푢휏,휏), 푆 푈̅process(푢휏,휏) )between the state of the noumenal particle and the invariant wavefunction. These relations should be formulated from the standpoint of the phenomenal interface spacetime into one or more rules. It should be noted however, that any non-local causal connection between two points in the interface slice is not conveyed within the interface slice and can only take place in noumenal space via the non-local noumenal field represented by the invariant wavefunction , by passing through the 휏 휏 corresponding푆 intermediate noumenal particle. 푆 휑푈̅ 휓(휏, 푢) 5.5 Rules related to the indirect noumenal state It is reasonable to suppose that a first rule should be a smooth evolution of the invariant wavefunction , i.e., a transformation infinitely close to identity equivalent to Schrodinger equation. This first rule would govern the evolution of the invariant wavefunction representing the non-local 푈̅ 휏 휓projected(푢 ,휏) noumenal field on an interface slice into a consequent interface wavefunction 푈̅ 휏 representing the projected noumenal field on the휓 ( subsequent푢 ,휏) interface slice 휏 according to an infinitesimal unitary transformation 푆 , as follows: 휓푈̅(푢휏,휏+훿휏) 푆휏+훿휏 풰(훿휏) 12
(5.13)
A휓푈 ̅second(푢휏,휏+훿휏 rule )could= 풰(훿휏)휓 be a relation푈̅(푢휏,휏 )between the projected position and the phase of the invariant wavefunction similar to the de Broglie Bohm law. Indeed, a variation in the direction 푈̅ 휏 of the straight line joining the interface point to the noumenal푈(τ) particle implies푆 (푢 a,휏 variation) 푈̅ 휏 in phase which휓 ( necessarily푢 ,휏) implies a variation of the first component of the noumenal 푈̅ 휏 τ position휃 (푢 ,휏 ) and consequently, a variation of theu projected position and this last variation 푈̅ 휏 can be assimilated푆 (푢 ,휏) to the momentum of a fictitious particle at the current projected푈 position : 푈̅ = (푈, 휈) 푈(τ) (5.14)푝(τ) 푈(τ) A훿푆 third(푢휏,휏 rule) ↔ should 훿푈(τ) then ↔ 푝(τ) concern the invariant interdimensional coordinate . A variation in does not necessarily imply a variation in the projected position though, it necessarily implies a variation in the distance of the noumenal particle from the interface slice which causes휈 a variation in the 휈amplitude of the invariant wavefunction : 푈(τ) 푆휏 푅푈̅(푢휏,휏) (5.15) 휓푈̅(푢휏,휏) 훿푅5.6( Measurement푢휏,휏) ↔ 훿휈 The projected position may be considered to evolve according to a sequence of projected positions matching a corresponding evolution in the first component of the noumenal position ( )푛(τ) according to a corresponding sequence of noumenal positions 푈(τ) . Any ̅ variation in noumenal position and specifically, in its first component푈 is timeless while푈 its= 푛 푛 푛 corresponding(푈, 휈) variation in the projected position takes place during a( 푈certain̅) = (time(푈) period, (휈) ) and thus, the sequence of projected positions∆푈̅ depends on the invariant∆푈 time . At any invariant instant , each projected position represents∆푈(τ) the point in the interface slice having∆휏 the (푈(τ))푛(τ) τ minimal distance from the noumenal particle which is equal to and thus, when the invariant 푛(τ) 휏 coordinateτ tends to zero, the noumenal푈 (τ) position tends to the latest projected position푆 : 휈푛 휈푛 푈̅푛 (5.16) 푈퐿(τ)
Finally,∀휀 >0,∃ when 훿 >0∶∀ becomes 휈푛, |휈 equal푛 − 0 |to< zero, 훿 → the |푈 ̅noumenal푛 − 푈퐿(τ )position| < 휀 becomes congruent to the latest projected position which in fact becomes the real interface position of the particle, as indicated in 푛 푛 푛 the following proposition.휈 (푈 , 휈 ) 푈퐿(τ) Proposition 2: Measurement is a local interaction with the projected noumenal field which in its turn acts non-locally on the corresponding noumenal particle ‘bringing’ it from noumenal space 푈̅ 휏 into the phenomenal interface slice at a definite position corresponding to the latest projected휑 (푢 ,휏 position) . Mathematically, it can be defined by an operation M acting on the invariant wavefunction 휏 and consisting in taking 푆the limit of the latter as the interdimensional coordinate tends to 퐿 zero,푈 (τ) the limit being the latest projected position : 휓푈̅(푢휏,휏) 휈 (5.17)푈 퐿(τ)
푈̅ 휏 푈̅ 휏 퐿 Measurement푀휓 (푢 ,휏) = can lim휈→0 also 휓 be(푢 defined,휏) = 푈with(τ) respect to the invariant state vector , as follows:
(5.18) |휓(휏)⟩
휏 푈̅ 휏 휏 퐿 According푀|휓(휏)⟩ = to휈→0 lim relation∫ 푑푢 (5.18),휓 (푢 ,휏 measurement) |푢 ⟩ = |푈 (휏) reduces⟩ the invariant state vector into a single observable basis element corresponding to a definite interface position and consequently, there is no more any wavefunction. |휓(휏)⟩ |푢휏⟩ = |푈퐿(휏)⟩ 푈퐿(휏) 13
Measurement is thus, a two-step process consisting of a first local act on the projected non-local noumenal field (represented by the invariant wavefunction ) and a second non-local action of the projected non-local noumenal field via noumenal space on the corresponding noumenal particle. 휓푈̅(푢휏,휏) Before measurement, all shadow-positions related to the arguments of the invariant wave function , are possible outcomes but with different probabilities depending on the distance of 휏 each shadow-position from the noumenal position (푢 of,휏 the) particle. Indeed, the 푈̅ 휏 wavefunction휓 (푢 depends,휏) on the distribution of distances of its different arguments from the noumenal 휏 ̅ 휈 particle and should thus,( 푢be,휏 related) to the probability distribution푈 = of( the푈 ,particle’s 휈) position outcome. The projected position represents the shadow position on the interface spacetime having the minimal distance from the noumenal particle and is thus associated with the highest probability amplitude which decreases as we recede푈(τ) from the current projected position . Therefore, when a measurement is to be conducted on the noumenal particle, the likelihood is maximal that it would pop out in the neighborhood of the latest projected position. 푈(τ) Moreover, once the particle is in the phenomenal interface slice and as long as it remains there, the non-local projected noumenal field and hence, its associated wavefunction exists no more, the particle 휏 behaves then as a classical corpuscular. 푆 Therefore, the particle has always a definite position which is in noumenal space before measurement and in phenomenal spacetime after measurement. The distinction between a quantum object and a classical object becomes only the location of that object being either in noumenal space or phenomenal spacetime, respectively. 5.7 Plurality of particles
More generally, a set of noumenal particles located at noumenal positions , creates a projected non-local noumenal field at each point of the 1 2 푘 interface slice . Let be a set of arguments associated to each interface푈̅ = (푈̅ ,point 푈̅ , … 푈̅ and) = 1 1 푘 푘 푈̅ 휏 corresponding((푈 , 휈 ), … , (푈 to ,the 휈 )set) of noumenal positions , then the projected휑 non-local noumenal(푢 ,휏) field 휏 휏1 휏2 휏푘 τ maps each푆 set 푢 of, 푢 arguments, … 푢 into a corresponding complex numberu 1 2 푘 characterizing the projected푈̅ , 푈̅ , … 푈̅non-local noumenal field effect at each 푈̅ 휏1 휏2 휏푘 corresponding휑 interface point, as follows: 푢 , 푢 , … 푢 휑푈̅(푢휏1, 푢휏2, … 푢휏푘, 휏) (5.19) 휑푈̅ {(where푢휏1, 푢휏2, … 푢휏푘,휏)} → is { 휑the푈̅(푢 projected휏1, 푢휏2, … non-local 푢휏푘, 휏)} noumenal field at the interface point on the interface slice with respect to each individual argument of the set of arguments . 푈̅ 휏1 휏2 휏푘 τ As in the휑 case(푢 ,of 푢 one, … particle, 푢 , 휏) the projected non-local noumenal field (generatedu ,휏) 휏 휏1 휏2 휏푘 by the plurality of 푆noumenal particles is represented by the following invariant wavefunction:푢 , 푢 , … 푢 휑푈̅(푢휏1, 푢휏2, … 푢휏푘, 휏) (5.20)
The휑푈̅( 푢invariant휏1, 푢휏2, … wave 푢휏푘,휏 function) ↔ 휓푈̅ (푢휏1, 푢휏2, … 푢휏푘,휏) is defined in a configuration space on and represents the projected noumenal field on the interface slice with respect to the different3푘 noumenal 푈̅ 휏1 휏2 휏푘 particles thus, giving an indirect휓 information(푢 , 푢 , … 푢about,휏 )their different noumenal positions. All the푅 arguments× 푅 휏 are defined at the same invariant interface instant 푆from the standpoint of the interface slice . Each set of arguments can be considered to represent a set of ‘shadow-positions’ associated 1 푘 to푢 a, …set , 푢 of ‘shadow-momenta’ corresponding to the set of noumenal휏 particles which have real noumenal 휏 1 푘 positions푆 . The푢 shadow-positions, … , 푢 and associated shadow-momenta can be considered as the positions and momenta of a set of k fictious particles omnipresent all over the interface slice . Thus, 1 2 푘 each point푈̅ of, 푈̅the, …interface 푈̅ slice can be considered to correspond to a set of k shadow positions. 푆휏 푆휏 14
Let the invariant wavefunction function be expressed in polar form in function of amplitude and phase , as follows: 휓푈̅(푢휏1, 푢휏2, … 푢휏푘,휏) 푅푈̅(푢휏1, 푢휏2, … 푢휏푘,휏) 푆푈̅(푢휏1, 푢휏2, … 푢휏푘,휏 ) (5.21) 푖푆푈̅ (푢휏1,푢휏2,…푢휏푘,휏) 휓As푈̅ (in푢 expression휏1, 푢휏2, … 푢 (5.14),휏푘,휏) = the 푅 푈̅phase(푢휏1, 푢휏2, … 푢휏푘,휏) 푒 should inherently depend on the set of projected positions related to a corresponding set of momenta . A variation in a 푈̅ 휏1 휏2 휏푘 projected position corresponds푆 to( a푢 variation, 푢 , … 푢in the,휏 )phase . On the other hand, 1 2 푘 휏1 휏2 휏푘 similarly to푈 expression, 푈 , … , 푈 (5.15), the amplitude should푝 , 푝inherently, … 푝 depend on the set 푘 푈̅ 휏1 휏2 휏푘 of extradimensional푈 space coordinates . 푆 (푢 , 푢 , … 푢 ,휏) 푅푈̅(푢휏1, 푢휏2, … 푢휏푘,휏) If the noumenal particles are correlated휈 1such, 휈2, …that , 휈 푘they are interlinked by the same non-local noumenal field , then a variation in any noumenal position generates a variation in the projected non-local noumenal field which in its turn, induces variations at all noumenal positions . 푈̅ ̅푗 Moreover,휑 in that case the invariant wavefunction 훿푈 cannot be factorized. 훿휑푈̅ 푈̅1, 푈̅2, … 푈̅푘 However, if the noumenal particles are not correlated,휓푈̅( 푢then휏1, 푢 the휏2, …projected 푢휏푘,휏) non-local noumenal field can be decomposed into a set of independent individual applications that respectively 푈̅ maps each set of arguments of each interface point into a corresponding set 휑 of 휑푈̅1, 휑푈̅2, … , 휑푈̅푘 complex numbers : 푢휏1, 푢휏2, … 푢휏푘 uτ 푈̅1 휏1 푈̅2 휏2 푈̅1 휏푘 (휑 (푢 ), 휑 (푢 ), … 휑 (푢 )) (5.22) 휑푈̅ {(In푢 that휏1, … case, 푢휏푘 ,휏a )variation } → in {any휑푈̅ (푢noumenal휏1, … 푢휏푘 position, 휏)} = { (휑푈̅1 generates(푢휏1), … 휑 a푈̅ 푘variation(푢휏푘), 휏) only } in its corresponding projected non-local noumenal field which has no effect on all other noumenal positions. Each 훿푈̅푗 individual non-local noumenal field is related to a corresponding individual invariant wavefunction 훿휑푈̅푗 and thus, the resultant invariant wavefunction can be factorized into a 휑푈̅푗 corresponding product of independent individual invariant wavefunctions: 휓푈̅푗 (푢휏푗) 휓푈̅(푢휏1, 푢휏2, … 푢휏푘,휏) (5.23)
휓The푈̅( 푢 measurement휏1, 푢휏2, … 푢휏푘 ,휏 relation) = 휓푈̅ 1 (5.17)(푢휏1, 휏)휓 can푈̅ 2 be(푢 휏2 easily, 휏) … 휓 generalized푈̅푘(푢휏푘, 휏) for numerous particles, such that a measurement conducted on one noumenal particle associated to the noumenal position is defined as follows: 푈̅푗 = (푈푗, 휈푗) (5.24)
푈̅ 휏1 휏2 휏푘 푈̅ 휏1 휏2 휏푘 If푀휓 the (noumenal푢 , 푢 , … particles 푢 ,휏) =are휈 lim 푗not→0 휓 correlated,(푢 , 푢 then, … 푢 the,휏 position) of the noumenal particle under the process of measurement will tend towards the latest projected position and the corresponding elementary wavefunction exists no more while all other elementary wavefunctions 푈퐿푗(τ) are unaffected. 휓푈̅푗 (푢휏푗, 휏) However,휓푈̅1(푢휏1,휏 If), the ⋯ , noumenal 휓푈̅푘(푢휏푘, 휏)particles are correlated, then they form a holistic identity having a single state at any given invariant space coordinate . Entangled particles are systematically kept connected to each other by the non-local noumenal field such that their invariant coordinate has constantly the same value. If the coordinate changes for one휈 noumenal particle, it changes systematically for all others. In other words, correlated noumenal particles should always have a common휈 extradimensional space coordinate . Thus, if a휈 particle is measured at one end, its coordinate becomes equal to zero and systematically via the non-local noumenal filed, the coordinate for all other entangled particles becomes also휈 equal to zero and hence, all entangled particles pop-up 휈 simultaneously on the same phenomenal interface slice . However, once the particles are localized휈 on the interface slice , there
푆휏 푆휏 15 is no more any non-local projected field or wavefunction and thus, the particles are not entangled any more though there may still exist a deterministic but local relation between them. Finally, proposition 1, also applies to a plurality of particles. Indeed, the noumenal positions of noumenal particles can be specified from the perspective of the interface spacetime, by an indirect noumenal state composed of the invariant wavefunction in association to a vector defining the projected positions of the noumenal particles on the 푈̅ 푈̅ 휏1 휏2 휏푘 interface slice Ω at(휏) an invariant instant as well as the vector 휓 of (the푢 corresponding, 푢 , … 푢 ,휏) extradimensional 1 2 푘 space coordinates푈(휏) : (푈 , 푈 , … , 푈 ,휏) 푆휏 휏 푉 휈1, 휈2, … , 휈푘 (5.25) Ω푈̅(휏) = (휓푈̅(휏), 푈(휏),V,휏) = (휓푈̅(푢휏1, … , 푢휏푘,휏), (푈1, 푈2, … , 푈푘,휏), (휈1, 휈2, … , 휈푘, 휏)) 6. Approximation of the interface spacetime and the invariant state vector The invariant state vector of equation (3.2) can also be expressed as a superposition of standard position elements labelled by the continuous position variables of the phenomenal standard spacetime . As noted|휓(휏) in section⟩ 2, there exists a bijective parametrization of the interface slice 휏 휏 , such that each interface|푥 ⟩ point defined in the interface frame푥 of reference can be mapped into푃(푥, a corresponding 푡) standard point in the orthonormal frame of reference and τ τ u τ vice-versaS . Indeed, according to ( expressionu , τ) (2.14), the points belonging to the(O; slice R ,S ) can be 휏 휏 parametrized by the standard space coordinates(푥 , 푡 as) follows: (O; x, t) 푢휏 푆휏 (6.1) 푥 Thus,푢휏(푥) the = 푎푟푐푠푖푛ℎinvariant( 푥state휏⁄휏) vector can be expressed as follows: |ψ(τ)⟩ (6.2)
|By휓(휏) differentiating⟩ = ∫ 푑푢휏 휓 푈̅(푢휏, with 휏) |푢 respect휏 ⟩ = ∫ to푑푥 휏 and푢′휏 휓as푈̅ time(푢휏(푥), is 휏) invariant|푢휏 ⟩ within each slice , we get:
푢 휏 (푥)(6.3) 푥 휏 Sτ 푑푥휏 휏 2 푢′Introducing= 휏√1+( 푥휏⁄휏 )into equation (6.2), the invariant state vector becomes:
푢′휏 (6.4) |휓(휏)⟩ 1 휏 2 2 푈̅ 휏 휏 |Let휓(휏) ⟩ = ∫ 푑푥 be √expressed휏 +푥휏 휓 as(푎푟푐푠푖푛ℎ follows:( 푥 ⁄휏), 휏) |푥 ⟩
휓푥(휏, 푥휏 ) (6.5) 퐴 푥 휏 2 2 푈̅ 휏 where휓 (휏, 푥 A ) is= a√ 휏 normalisation+푥휏 휓 (푎푟푐푠푖푛ℎ constant,(푥 ⁄ 휏 then,), 휏) the invariant state vector can be unambiguously expressed as a superposition of standard position elements , as follows: |휓(휏)⟩ (6.6) |푥휏⟩ where|휓(휏)⟩ = ∫ 푑푥휏 휓is푈 ̅a( 푥hybrid휏, 휏 ) | 푥invariant휏 ⟩ wavefunction depending on invariant time and standard spatial positions and where for simplicity, we kept the same symbol for and . 휓푈̅(푥휏,휏) 휏 However,푥 we휏 note that in the non-relativistic limit (i.e. ) each′휓푈̅ slice′ 휓푈̅ becomes(푢휏, 휏) a ‘quasi휓푈̅(푥-interface-휏,휏) line’ parallel to the x-axis. Thus, the invariant time can be approximated by the standard time . τ Indeed, expression (2.7) can be expressed as follows: /푡 ≪ 1 S 퐿푡 휏 푡 (6.7) 2 2 2 2 2 휏 = 푡 (1 − 푥 /푡 ) ≈ 푡 16
Thus, , and the phenomenal interface spacetime becomes a set of quasi-interface-lines , defined as follows: 휏 ≈ 푡 퐿푡 (6.8)
Theℐ푡 = noumenal{(푥, 푡) ∈ 퐿space푡 ∶ 푥 ∈becomes 푅, 푡 ∈ 푅,a {‘quasi|푥 ⟩} is-noumenal the base space’that spans where ℋ the} noumenal slice is approximated by a ‘quasi-noumenal-line’ , defined as follows: 푆휈 퐿휈 (6.9) The풩푡 = global{푥̅ = space(푥휈, 휈 )becomes∈ 퐿휈: 푥 휈a, ‘quasi 휈 ∈ 푅-global 푎푛푑 휈 ≠space 0} =’ {퐿 defined휈} as the union of the quasi-noumenal space and the quasi-interface spacetime . 풢푡 Therefore,풩푡 in the non-relativistic limit ℐ(i.e.푡 ), the invariant time can be replaced by the standard time in equation (6.6) and the invariant state vector becomes a standard state vector in function of standard temporal and spatial coordinates푥/푡 ≪ 1 : 휏 푡 |휓(휏)⟩ |휓(푡)⟩ (6.10) (푥, 푡 )
The|휓(푡) approximated⟩ = ∫ 푑푥 휓푈̅ (wavefunction푥, 푡 ) |푥 ⟩ represents the state of the noumenal physical system from the viewpoint of the quasi-interface-line . For simplicity, the index ‘ ’ is dropped from 푈̅ which shall be noted in the standard휓 (manner푥, 푡 ) by keeping in mind that the noumenal position is 푡 푈̅ hidden or implicitly defined in the expression퐿 of the wavefunction. 푈̅ 휓 (푥, 푡 ) 휓(푥, 푡 ) Moreover, the invariant time coordinate in expression (5.13) can be approximated by the standard time coordinate , leading to the standard unitary evolution of Schrodinger equation: 휏 푡 (6.11)
|However,휓(푡 + 훿푡) the⟩ = above 풰(훿푡 )equation|휓(푡)⟩ =should 푒푥푝(−푖퐻훿푡/ℏ) be regarded|휓(푡) as a⟩ hybrid equation of evolution where the standard time coordinate should be considered as quasi-invariant.
푡 7. De Broglie Bohm theory in the context of the quasi-global space In this section we consider the de-Broglie-Bohm theory in the formalism of the quasi-global space defined in section 6. For simplicity, we consider a two-dimensional quasi-noumenal space composed of a set of points where is replaced by and the invariant time is replaced by standard time , though considered in the non-relativistic limit to be quasi-invariant. The quasi-noumenal position of the τ noumenal particle{(푥, is 푡) }noted u , its projection푥 is noted , and휏 the non-local quasi-noumenal푡 field is noted . 푋̅ = (푋, 휈) 푋(푡) 7.1 Three rules휑푋̅ related(푥, 푡) to the indirect noumenal state As outlined in section 5 (proposition 2), from the standpoint of any quasi-interface point , the state of a noumenal particle located at a quasi-noumenal position can be specified by an indirect noumenal state . This indirect noumenal state is characterised by a wavefunction (푥, 푡) and the associated projected position of the noumenal particle on푋̅ the quasi-interface-line as well as the 푋̅ 푋̅ correspondingΩ (푡) extradimensional space coordinateΩ (푡) , at a quasi-invariant instant : 휓(푥, 푡) 푋(푡) 퐿푡 (7.1) 휈 푡 Let Ω 푋̅ (the푡) =wavefunction(휓(푥, 푡),X( t), 휈, 푡 ) be expressed in polar form in function of amplitude and phase , as follows: 휓(푥, 푡) 푅(푥, 푡) 푆(푥, 푡) (7.2) 푖푆(푥,푡) 휓(푥, 푡) = 푅(푥, 푡)푒 17
Then, the evolution of the noumenal particle obeys the following three rules: 1.The wave function is governed by Schrodinger equation:
(7.3)휓(푥, 푡) 휕휓 푖ℏ2. The휕푡 = evolution 퐻휓 of the projected position on the quasi-interface-line obeys de Broglie-Bohm law: 푋(푡) Lt (7.4) 푑 ℏ 휕푆(푥, 푡) 푑푡3. 푋(푡)Measurement = 푚 휕푥 is the limit of the wavefunction as the interdimensional coordinate tends to zero, the limit being the latest projected position : 휓(푥, 푡) 휈 (7.5) 푋퐿(t)
퐿 It푀휓 is( 푥,to 푡be) = noted lim휈→0 휓 that(푥, 푡in) =the 푋 approximation(t) of the quasi-global space, the mathematical formalism is identical to that of the de Broglie-Bohm theory except that here, is a shadow position that stands for the projected position of a noumenal particle. 푋(푡) Equation (7.4) shows that the projected position and the phase of the wave function are interrelated in an action-reaction process. To determine the velocity of the projected position , the gradient of the phase should be evaluated푋(푡) at the projectedS(x, position t) and thus, equation휓(푥, 푡) (7.4) becomes: 푋(푡) S(x, t) 푋(푡) (7.6) 푑 ℏ 휕푆(푥, 푡) 푑푡Equation푋(푡) = (7.3)푚 휕푥governs|푥=푋(푡) a deterministic evolution of the non-local noumenal field between subsequent quasi-interface-lines while considering the indirect non-local relation between the different points of each quasi-interface-line . Each point (i.e. shadow position) of the quasi-interface-line which corresponds to an argument of the wavefunction , relates directly to the noumenal particle and 푡 푡 only indirectly via the noumenal퐿 particle(푥, to its 푡) neighbouring points (i.e. neighbouring shadow positions).퐿 The indirect relations between the arguments of the휓(푥, wavefunction 푡) take place via non-local field lines across the noumenal space and does not depend on the separation between the corresponding points on the quasi-interface-line .
7.2 Amplitude of the wavefunction퐿푡 As seen in section 5, the density of the non-local quasi-noumenal field decreases as the distance between the interface point and the noumenal position increases. The amplitude- 푋̅ square can thus be represented by a uniform distribution of휑 a( 푥, real-valued 푡) random variable ̅ ̅ characterized푑(푥(푡) −2 푋) by the position on the 푥(푡)quasi-interface-line and the projected푋 position as the median,푅 knowing(푥) that is the nearest to the noumenal position . 푥(푡) 퐿푡 푋(푡) Moreover, by supposing푋(푡) the non-local noumenal field to be composed푋̅ of homogeneous and isotropic field lines stemming out from a noumenal particle, it is geometrically trivial that when the interdimensional coordinate decreases, the density of noumenal field lines increases at the neighbourhood of the projected position . In other words, when decreases, becomes more peaked around . Conversely휈 when increases the distribution of becomes2 uniform or flattened. This entails that the dispersion 푋(푡) of the noumenal distribution휈 should2 depend푅 ( 푥on) the coordinate while preserving푋(푡) its order (i.e. should휈 be a monotonically increasing푅 function).(푥) 휎 The휈 amplitude can therefore휎(휈) be described by a normal distribution related to the position on the quasi-interface-line , as follows: 푅(푥) 푥 퐿푡 18
1 푥−푋(푡) 2 (7.7) 1 −4( 휎(흂) ) 2 1⁄ 4 The푅(푥 )median= (2휋휎 is(흂) represented) 푒 by the projected position of the noumenal particle from the perspective of the quasi-interface-line at a given quasi-invariant instant . The difference indicates the distance of each point of the quasi-interface-line푋(푡) at the instant , from the projected position 푡 . The density distribution퐿 related to the wavefunction푡 gives a probabilistic푥 − 푋(푡) description of the 푡 location where the noumenal(푥, 푡) particle2 would most likely퐿 settle down on the푡 quasi-interface-line , the 푋(푡)probability being larger at the 푅interface(푥) points which are nearest to the noumenal particle and thus, to 푡 the projected position . 퐿
The field lines are denser푋(푡) (i.e. have greater number of lines) between the noumenal particle and the nearest points on the interface. However, the intensity of each link is unaffected by the distance. Indeed, in the timeless noumenal space a non-local link between two points does not depend on their separation. It is explained by Tim Maudlin in his book ‘Quantum non-locality and relativity’ [14] that a quantum connection between entangled particles should be unattenuated, discriminating and instantaneous. 7.3 Rough estimate of the dispersion with respect to the interdimensional coordinate
It is known [15] that a free particle, whose initial probability density is represented by흂 a Gaussian distribution of dispersion , evolves according to Schrodinger’s equation such that at a later time t, the square of its Gaussian dispersion evolves as follows: 휎 (7.8) 휎(푡) 2 2 2 ℏ푡 휎This(푡 )seems= 휎 to+ suggest(2푚휎) that when a particle is emitted from a source, it fades away into noumenal space becoming a noumenal particle and the interdimensional coordinate keeps increasing until measurement is conducted that makes the opposite process bringing back the particle into the phenomenal interface spacetime. 휈
As the square of the dispersion increases continuously with time, the initial dispersion-square can be neglected from the above equation2 and thus, an estimate of the dispersion at an instant t far2 from the initial instant can be expressed휎 (푡) as follows: 휎 휎(푡) (7.9) ℏ푡 As휎(푡) suggested ≈ 2푚휎 in section 3, the extradimensional spatial coordinate is an invariant spatial coordinate equivalent to the invariant time coordinate . To have a rough evaluation of the interdimensional coordinate , the variable in the above expression can reasonably휈 be replaced by . This implies that the dispersion is approximately proportional휏 to the interdimensional coordinate , as follows: 휈 푡 휈⁄ 푐 휎(휈) (7.10) 휈 ℏ휈 To휎(휈 have) ≈ 2푚푐휎 a very rough estimate of the interdimensional coordinate , we ascribed Planck’s length to the initial minimal dispersion, then expression (7.10) becomes: 휈 푙푝 (7.11) ℏ휈 휎For(휈 )example,≈ 2푚푐푙푝 a very rough estimate of the dispersion of an electron in function of the interdimensional coordinate , would be of the order:
휈 (7.12) 23 휎(휈) ≈ 10 휈 19
The above expression (7.12) entails that the slightest change in greatly amplifies the dispersion suggesting that the noumenal particle keeps ‘hovering’ faintly off the phenomenal interface spacetime. 휈 휎 7.4 Measurement in relation to a standard state vector Equation (7.5) illustrates that measurement consists in bringing a noumenal particle from noumenal space into the quasi-interface-line and once the particle is on that line, it behaves as a point particle governed by classical laws and there in no more any wavefunction. 퐿푡 When the interdimensional coordinate tends to zero, the dispersion tends to zero and the normal distribution becomes a Dirac delta function: 휈 휎(휈)
1 푥−푋(푡) 2 (7.13) 1 −4( 휎(흂) ) 2 1⁄ 4 휈→0lim 푅(푥, 푡) =휎( lim흂)→0 푅(푥, 푡) =휎( lim흂)→0 ((2휋휎 (흂)) 푒 )~훿(푥 − 푋(푡)) The operation of measurement on the state vector can be described, as follows:
|휓(푡)⟩ (7.14) 푖푆(푥,푡) However,푀|휓(푡)⟩ =as 휈→0 limindicated∫ 푑푥 휓(푡, by expression 푥 ) |푥 ⟩ = 휎(7.13)( lim흂)→0,∫ 푑푥 푅( 푥,converges 푡)푒 to|푥 ⟩ when (hence, ) tends to zero. The noumenal position tends to where is equal to the latest projected position . Moreover, is 푅a (continuous푥, 푡) bounded훿(푥 function, − 푋(푡)) thus: 휈 휎 ̅ 푋 =푖푆( (푋,푥,푡) 휈) (푋, 0) 푋 푋(푡) ≡ 푋퐿(푡) 푒 (7.15) 푖푆(푥,푡) 푖푆(푥,푡) 푖푆(푋,푡) 퐿 The휈→0lim ∫above푑푥 푅 (equation푥, 푡)푒 shows|푥 ⟩that= ∫ the푑푥 particle 훿(푥 − 푋(푡))푒pops out at a| definite푥 ⟩ = 푒 position|푋⟩ ≡|푥 ⟩ corresponding to the latest projected position. Indeed, the projected position may vary during the convergence of 퐿 퐿 towards zero and hence, the position of the particle in the interface spacetime푥 ≡ corresponds 푋 (푡) to the latest projected position. The measured position is represented by a unique base vector and is thus휈 observable. |푥퐿⟩ It is to be noted that the going back and forth of the particle between noumenal space and phenomenal spacetime should not be restricted to measurement and should be considered also in many other quantum phenomena such as quantum tunnelling. In his book ‘The Order of Time’, Carlo Rovelli [16] considers that spacetime has a granular nature and a particle exists in a cloud of probabilities and jumps from one point to another. Here noumenal space can be considered to correspond to the cloud of probability of Rovelli and the jumps take place between noumenal space and phenomenal spacetime. To sum up, the wave function is an ontological entity representing the projection of a non-local noumenal field on the quasi-interface-line created by a noumenal particle and is thus, an indirect representation of the noumenal휓 particle.(푥, 푡) 퐿푡 According to this model, when a particle is in noumenal space, it is not observable but, its projected non-local noumenal field is detectable and behaves in a wave-like manner. On the other hand, when it is in the interface spacetime, it becomes observable and behaves as a point particle. For example, in the double-slit experiment, interference is the result of the wave (i.e. the projected non- local noumenal field) going through both slits though the particle being in noumenal space, goes through neither. However, the movement of the noumenal particle is non-locally affected by the interference such that when the interfering wave finally interacts with the detecting screen, the noumenal particle is brought into the interface spacetime as an impact on the screen whose position has been already configured by the interfering wave before the impact. When detectors are placed at the slits and the particle is detected (i.e. measured) at one of the slits means that the particle has popped out into the interface spacetime at the detected position and the associated wave does not exist any longer. The particle produces then an impact on the screen according to a classical trajectory. 20
8. Plurality of particles in the quasi-global space
Let a set of two noumenal particles be at a definite position vector :
(8.1) 푋̅
푋The̅ = wavefunction(푋̅1, 푋̅2) = ((푋 representing1, 휈1), (푋2, 휈the2)) projected non-local noumenal field on the quasi-interface-line is defined in a configuration space , as follows: 푡 2 퐿 푅 × 푅 (8.2) 푖푆(푥1,푥2,푡) From휓(푥1 ,the 푥2 ,perspective 푡) = 푅(푥1, 푥of2, the 푡)푒 quasi-interface-line , the dynamics of the noumenal particles of masses and at the projected positions and are related to the phase of 푡 the wave function , as follows: 퐿 푚1 푚2 푋1 ≡ 푋1(푡) 푋2 ≡ 푋2(푡) 푆(푥1, 푥2, 푡) 휓(푥1, 푥 2 , 푡 ) (8.3) 푑푋푘 ℏ 휕푆(푥1,푥2,푡) 푑푡 푚푘 휕푥푘 where= the phase |푥푘=푋푘 is evaluated at the corresponding projected position. If the noumenal particles are correlated, then any change in the projected position of the first noumenal particle 1 2 changes the value 푆of(푥 the, 푥 phase, 푡) which automatically changes the projected position 1 1 of the second particle according to equation훿푋 (1b) as explained by Bricmont푋 [7]. 푆(푋1 + 훿푋1, 푋2, 푡) According to this model, the entanglement between both particles is achieved by means of the non-local noumenal field. Similarly, any action on the projected non-local noumenal field within the interface spacetime has a non-local action on both particles. 8.1 Bivariant normal distribution
The amplitude can be described by a bivariant normal distribution of the positions at a given instant , on the interface spacetime, as follows: 푅(푥1, 푥2, 푡) 푥1, 푥2 푡 (8.4) 2 2 1 1 (푥1−푋1) (푥2−푋2) 2휌(푥1−푋1)(푥2−푋2) 1 2 2 2 2 2 where푅(푥 , 푥 )is= a 2constant휋휎(흂1)휎(흂 representing2)√1−휌 푒푥푝 a( −correlation2(1−휌 ) [ coefficient휎 (흂1) + .휎 (흂2) − 휎(흂1)휎(흂2) ])
The medians휌 are represented by the projected positions and of the noumenal particles from the perspective of the quasi-interface-line at a given quasi-invariant instant . 푋1 ≡ 푋1(푡) 푋2 ≡ 푋2(푡) 8.2 Entangled particles 퐿푡 푡 If the particles are entangled, the wave function does not factorise into a product of independent wavefunctions. In particular, the correlation coefficient in the expression (8.4) of the 1 2 bivariant normal distribution is not equal to zero ( 휓(푥 ,), 푥 and, 푡) systematically as indicated in section 5.7, , hence, 휌 휌 ≠ 0 The 휈above1 = 휈 2expression= 휈 (8.4)휎( 휈becomes:1) = 휎(휈 2) = 휎(휈)
(8.5) 2 2 1 (푥1−푋1) +(푥2−푋2) −2휌(푥1−푋1)(푥2−푋2) 1 2 2 2 2 2 Let푅(푥 , 푥 ) = 2휋휎 (흂) and√1−휌 푒푥푝 (− , then 2휎 (흂)(1−휌 transforms) into) which we shall simply note , expressed as follows: 푥′1 = 푥1 − 푋1 푥′2 = 푥2 − 푋2 푅(푥1, 푥2) 푅(푥′1 + 푋1 , 푥′2 + 푋2) 푅(푥′1 , 푥′2) (8.6) 2 2 1 (푥′1) +(푥′2) −2휌(푥′1)(푥′2) 2 2 2 2 푅(푥′1 , 푥′2) = 2휋휎 (흂)√1−휌 푒푥푝 (− 2휎 (흂)(1−휌 ) ) 21
To consider the amplitude from the perspective of the first particle, we apply to the above equation (8.6), the following transformation: (8.7) where푥′2 = 푥′3 +is 휌푥′orthogonal1 to , then, 푥′3 transforms푥′1 into which we shall simply note , expressed as follows: 푅(푥′1 + 푋1 , 푥′2 + 푋2) 푅(푥′1 + 푋1 , 푥′3 + 휌푥′1 + 푋2) R(푥′1 , 푥′3) (8.8) 2 2 1 −(푥′1) 1 −(푥′3) 2 2 2 2 2 2 R(푥′1 , 푥′3) = (√2휋휎 (흂) 푒푥푝 ( 2휎 (흂) ))(√2휋휎 (흂)(1−휌 ) 푒푥푝 (2휎 (흂)(1−휌 ))) The above equation can be expressed as a product of two normal distributions and .
(8.9) 푅1(푥′1) 푅2(푥′3) ByR(푥′ reintroducing1 , 푥′3) = 푅1 (푥′1)푅2(푥′3) = and 푅1 (푥′1)푅2(푥′2 − 휌푥′, into1) (8.9), it becomes: 푥′1 = 푥1 − 푋1 푥′2 = 푥2 − 푋2 (8.10)
TheR(푥′ above1 , 푥′3) expression= 푅1(푥1 − clearly 푋1)푅2 shows(푥2 − 푋that2 − when 휌(푥1 the− 푋particles1) are correlated (i.e. ), the amplitude of the second particle depends on the coordinates of the first particle. 휌 ≠ 0 On the other hand, by applying the above transformations to the phase , it can be expressed as follows: 푆(푥1, 푥2, 푡) (8.11)
Thus,푆(푥1, 푥when2, 푡) =a measurement 푆(푥′1 + 푋1 , 푥′ is2 conducted+ 푋2, 푡) = on 푆( the푥′1 first+ 푋1 particle, , 푥′3 + 휌푥′we1 get:+ 푋 2) (8.12) 푖푆(푥1,푥2,푡) 1 2 1 2 1 2 휎(흂)→0 ( ) | ⟩ 푀 = lim ∬ 푑푥′ 푑푥 ′ 푅 푥 , 푥 푒 푥 , 푥 (8.13) 푖푆(푥′1+푋1 ,푥′3+휌푥′1+푋2,푡) 1 3 1 1 2 3 1 1 3 1 2 휎(흂)→0 | ⟩ 푀 = lim ∬ 푑푥 푑푥 ′ 푅 (푥′ )푅 (푥′ )푒 ′ 푥′ + 푋 , 푥′ + 휌푥′ + 푋 (8.14) 푖푆(푥′1+푋1 ,푥′3+휌푥′1+푋2,푡) 2 3 3 1 1 1 1 1 3 1 2 However,푀 =휎(흂)→0 lim for∫ 푅 (푥′ )푑푥; the∫ first푅 (푥′ and)푒 second normal distribution푑푥 s |become푥′ + 푋 Dirac’s , 푥′ + delta 휌푥′ +functions: 푋 ⟩
휎(흂)′ → 0 (8.15)
푅1(푥′1) → 훿(푥′1 − 0) (8.16) By푅2(푥′ introducing3) → 훿(푥 relations3 − 0) (8.15) and (8.16) into (8.14), it becomes: M= ′ ′ ′ ′ (8.17) 푖푆(푥′1+푋1 ,푥′3+휌푥′1+푋2,푡) Finally,∬ 푑푥 relation1 푑푥 3 (8.17)훿(푥 1 −reduces 0)훿(푥 into:3 − 0)푒 |푥′1 + 푋1 , 푥′3 + 휌푥′1 + 푋2 ⟩ M= (8.18) 푖푆(푋1 ,푋2,푡) Thus,푒 both particles|푋1 , 푋 settle2 ⟩ at the corresponding projected positions and . 8.3 Non-entangled particles 푋1(t) 푋2(푡) When the particles are not correlated, the wavefunction factorises into a product of independent wavefunctions: 휓(푥1, 푥2, 푡) (8.19) 푖푆(푥1,푥2,푡) 휓(푥1, 푥2, 푡) = 푅(푥1, 푥2, 푡)푒 = 휓1(푥1, 푡)휓2(푥2, 푡) 22
The above relation (8.19) can also be expressed in polar form:
(8.20) 푖푆(푥1,푥2,푡) 푖푆1(푥1,푡) 푖푆2(푥2,푡) such휓(푥1 that;, 푥2, 푡) = 푅(푥1, 푥2, 푡)푒 = 푅1(푥1, 푡)푒 푅2(푥2, 푡)푒 (8.21)
푆(푥1, 푥2, 푡) = 푆1(푥1, 푡) + 푆2(푥2, 푡 ) (8.22) Applying푅(푥1, 푥2, 푡de) = Broglie 푅1(푥1 ,Bohm 푡)푅2( 푥law2, 푡 )(8.3) to the phase as expressed in relation (8.21), gives two independent equations: 푆(푥1, 푥2, 푡) (8.23) 푑푋1 ℏ 휕푆1(푥1,푡) 1 1 푑푡 = 푚 휕푥 |푥1=푋1 (8.24) 푑푋2 ℏ 휕푆2(푥2,푡) 2 2 The푑푡 =projected푚 휕푥 position|푥2=푋 2of each particle follows its own law independently from the other particle. On the other hand, the amplitude is a product of two independent amplitudes such that when the dispersion of the first particle tends to zero, the other particle is not affected.
Indeed, when휎(휈 1 the) particles are not entangled, then the correlation coefficient , and and (hence, and ) are not related. As , the amplitude of the bivariant normal 1 2 distribution at a given instant , factorises into a product of independent amplitudes,휌 = as 0 follows:휈 휈 휎(휈1) 휎(휈2) 휌 = 0 푅(푥1, 푥2, 푡) 푡 (8.25) 2 2 1 (푥1−푋1) 1 (푥2−푋2) 2 2 ( 1 2) 2휋휎(흂1) ( 2휎 (흂1) ))(휎(흂2) ( 2휎 (흂2) )) = 푅1(푥1, 푡)푅2(푥2, 푡) 푅When푥 , 푥a measurement= ( 푒푥푝is conducted− on the first particle,푒푥푝 − we get:
(8.26) 푖푆(푥1,푥2,푡) 1 2 1 2 1 2 푀 =휎(흂 lim1)→0 ∬ 푑푥 푑푥 푅(푥 , 푥 )푒 |푥 푥 ⟩ (8.27) 푖푆1(푥1,푡) 푖푆2(푥2,푡) 1 1 1 2 2 2 1 2 When푀 =휎(흂 lim1)→0 ∫ 푅 ;( the푥 , 푡first)푒 normal푑푥 distribution∫ 푅 (푥 , 푡)푒 푑푥 |푥 푥 ⟩, thus:
휎(휈1) → 0 푅1(푥1) → 훿(푥 1 −(8.28) 푋1) 푖푆1(푥1,푡) 푖푆2(푥2,푡) 푀 = ∫ 훿(푥1 − 푋1)푒 ∫ 푅2(푥2, 푡)푒 (8.29) 푑푥2 |푥1푥2 ⟩ 푖푆1(푋1,푡) Th푀e = first 푒 particle|푋 1is⟩ ∫thus휓2 (observed푥2, 푡)푑푥 2at|푥 a2 position ⟩ corresponding to while the second particle continues to evolve independently. |푋1⟩
9. Uncertainty Principle Here, we should consider the location of the particle whether it is in noumenal space or in phenomenal spacetime:
1.When the particle is in noumenal space, it has in principle, a definite noumenal position and thus, has no uncertainty in position. However, the momentum of the particle has no sense in a timeless noumenal space. 푈̅ = (푈, 휈) 2.When the particle is in the phenomenal interface spacetime, it behaves as a classical point particle and has well determined position and momentum. 23
3.When the particle is in noumenal space, it creates a non-local noumenal field whose projection on the interface spacetime is represented by a corresponding wavefunction. As seen in previous sections, the arguments of the wavefunction can be considered to represent shadow positions on the interface slice of a ubiquitous fictious particle presenting also shadow momenta. Spatial dispersion depends on the interdimensional coordinate and thus, the uncertainty in position depends on such that when the value of increses, the uncertainty in position becomes larger. The relation between the dispersion in shadow positions and the dispersion휈 in shadow momenta of the fictitious particle is휈 simply derived from Fourier transformations휈 between position and momentum representations of the wavefunction. For a wavefunction defined in a standard one-dimensional spacetime, Fourier analysis shows that the dispersion for the distribution of position and the dispersion for the distribution of momenta satisfy the following well-known classical mathematical relation [15, 17]: 휎푥 푥 휎푝 푝
휎When푥휎푝 ≥the ℏ interdimensional⁄ 2 coordinate tends to zero or even becomes equal to zero, the uncertainty in position diminishes and even ceases to exist as the particle becomes localized in the phenomenal spacetime (above case 2). 휈 In view of the above, it is clear that the uncertainty principle does not concern the physical particle itself whether it is located in noumenal space or in phenomenal spacetime. Nevertheless, it concerns the fictitious particle as represented by the interface wavefunction created by the noumenal particle. In other words, it concerns the density distribution of a physically real non-local projected 푈̅ 휏 noumenal field within the phenomenal interface slice휓 ( 푢 emanating,휏) from the noumenal particle. 휑푈̅(푢휏,휏) 푆휏
10.Conclusion The invariant space-time interval of special relativity is considered as an invariant time in a phenomenal interface spacetime accounting for the simultaneity in the arguments of the wavefunction. By extrapolation, a timeless noumenal space is constructed presenting a notion of an invariant space accounting for non-locality between the arguments of the wavefunction. Non-locality and simultaneity go in pair, there is indeed, no sense of non-local events between two points without the events being simultaneous. The wavefunction is an ontological object that represents a non-local projected noumenal field within the phenomenal interface spacetime created by a particle located in the timeless noumenal space. The non-local connection between the different arguments of the wavefunction takes place within the timeless noumenal space through the intermediary of the particle located in the noumenal space and not within the phenomenal interface spacetime. Measurement is the passage of a particle from the noumenal space into the phenomenal spacetime. The particle has always a definite position either in noumenal space or in the phenomenal spacetime. From the standpoint of the phenomenal spacetime, if the particle is in noumenal space, the non-local field behaves like a wave whereas, if it is in the phenomenal spacetime, the particle behaves like a classical point particle since there is no more any non-local field. There are no different ontologies between quantum systems and classical systems. Instead, the demarcation between the quantum realm and the classical realm is based on the location of the physical system and not on its nature. From the perspective of phenomenal spacetime, any physical system located in noumenal space can only be described via its wavefunction according to quantum rules but when the same physical system is in the phenomenal spacetime it can then be described by the dynamical rules of classical mechanics. 24
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