A Neuronal Interpretation of Quantum Mechanics Part I: Some Basic Features
Abstract
Part one of the QM interpretation based on a brain-like universe-generating being.
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1 A Neuronal Interpretation of Quantum Mechanics Part I: Some Basic Features Manuel Blanco ABELLO https://whizkod.com Email: [email protected] Abstract—All interpretations of Quantum Mechanics (QM)1 still lack consensus on their sufficiency, are ill-defined, produce2 paradoxes, and/or yield scientific problems. An interpretation3 of some mechanics of the non-relativistic Schr¨ odinger equation-4 abiding particles devoid of the drawbacks is presented in this5 paper, which is the first in a series. It considers every physical6 particle as information emerging from the metaphysical neuro-7 morphic entity referred to as 7.1M.The latter is formulated based8 on the human neuronal dynamics and the Artificial Intelligence9 agent referred to as 3.1M. Its well-defined ontological features do10 not yield scientific problems (e.g., measurement problem) and/or11 paradoxes (e.g., Wigner’s friend). Due to scope limitation, some12 of the features are applied in this paper only to the following13 quantum mechanical-related topics: (a) particle ontology and14 tunnelling; (b) nature of spacetime, and time flow direction in15 relation to computation;(c) the interpretation of the wave func-16 tion and its collapse; (d) the interpretation of the non-relativistic17 Schr¨ odinger equation, the Heisenberg uncertainty principle, and18 the atom-bound electron dynamics;(e) the Statistics-rooted justi-19 fication on the use of Hilbert space in defining some 7.1Mfeatures;20 (f) the 7.1Mexplanation of some paradoxical Quantum Mechani-21 cal concepts, e.g., the wave-particle duality, the Schr¨ odinger’s cat,22 and the which-way information;(g) and the 7.1Mexplanation of23 the electron double slit interference experiment.24 I. Introduction25 In 2015, Weinberg stated that, “My own conclusion is that26 today there is no interpretation of quantum mechanics that27 does not have serious flaws.” p102 [1]. Despite more than a28 century of investigations on Quantum Mechanics (QM), all29 of its interpretations still lack consensus on their sufficiency,30 are ill-defined, produce paradoxes, and/or yield scientific prob-31 lems. A few of them are explored in the following: (a) The32 Copenhagen interpretation implies that any quantum system33 being measured is independent of the device that measures34 it, despite the device being a quantum system as well [2],35 [3]. (b) The Everett interpretation implies that every quantum36 event generates new physically real universes [2]. In one of the37 multiverse for example, the author is an Australian labourer;38 in another, his identical version is simultaneously an American39 scientist; et cetera. The various physically real universes can40 be viewed as unnecessary additional complexities, and hence41 defy the Occam’s razor principle [4]. (c) The Pilot wave42 theory is deterministic, and hence does not bode well with43 the inherent randomness in quantum mechanics. [5]. (d) The44 Schr¨ odinger equation transformed to become time-symmetric45 yields retrocausality[6], and hence, also the grandfather’s46 paradox. (e) The quantum mechanical interpretations that give47 special and essential role of human observer of any quantum48 phenomena yield the Wigner’s friend paradox. Their examples 49 are the Heisenberg interpretation [7], Quantum information 50 theories [8], and Relational quantum theories [9], to name a 51 few. (f) The essential observer role is absent from the Ghirardi52 Rimini-Weber theory which, however, is incompatible with the 53 relativistic principles [10]. Also observer independent is the 54 indivisible stochastic processes interpretation [11], [12] which, 55 among its several issues, does not facilitate the definition of 56 statistical and measurable quantities of particle trajectories. 57 (g) The interpretation of Penrose [13] on wave function 58 collapse based on the nature of the QM and General Theory 59 of Relativity (GTR) conflict is ill-defined [14]. 60 The scientific problems and/or the paradoxes consequent to 61 the interpretations of QM in the literature are absent from 62 the well-defined ontological interpretation established from 63 the metaphysical neuromorphic entity referred to as 7.1M, as 64 justified beginning with this paper — the first in a series. 65 7.1Mis formulated based on human neuronal dynamics and 66 the Artificial Intelligence (AI) agent referred to as 3.1M[15]. 67 Its features are used for the ontological interpretation of 68 all particles in the physical universe as the information it 69 generates. The universe is defined with a mathematical entity 70 analogous to the universal wave function postulated in the Von 71 Neumann—Wigner interpretation [16]. In the universe, the 72 information/particles is either consciously or subconsciously 73 perceived by 7.1M, where the consciously perceived constitute 74 a single reality, and the subconsciously perceived are in the 75 state analogous to that of particles in the transitional reality 76 proposed by Heisenberg [7]. The conscious perception is 77 related to the proposal of Chalmers that consciousness is a 78 fundamental particle attribute [17]. 79 The 7.1Mfeature crucially applied to interpret the quantum 80 mechanics is its ability to imagine, an ability also of the 81 human mind. The 7.1Mfeatures account the interactions of 82 particles comprising any given quantum system with those of 83 any device that measures this system; and give no essential role 84 of any observer of any quantum phenomenon. Despite being 85 observer-independent, they do not defy the no-go theorem of 86 [18] considering that 7.1Mgenerates a non-local universe, as 87 will be justified in the series successors of this paper. 88 In the 7.1Minterpretation, neuronal and physical dynamics 89 are related, an approach that is also applied in the following: 90 (A) A neuronal dynamics is modelled by an expression remi91 niscent of the Schr¨ odinger equation in [19], and some neuronal 92 dynamics are investigated using the Feynman diagrams [20], 93 [21]. (B) There are even quantum systems formulated using 94
2 fields of neurons [22], [23], [24]. However, this formulation95 was not applied to resolve the scientific problems and/or96 paradoxes brought about by the interpretations of QM in the97 literature. (C) Reality is defined in the Cognitive-Theoretic98 Model of the Universe (CTMU) as having two aspects: one is99 the observable universe described as a language (analogous to100 the information/particle in 7.1M), and the other is the cognitive101 process that interprets the universe/language [25]. CTMU102 is incorporated in the Quantum Metamechanics (QMM) to103 interpret QM [26]. However, QMM gives special and essential104 role to the observer of the universe, a role that leads to the105 Wigner’s friend paradox which is absent in 7.1M.106 Although 7.1Mis immaterial (as implied above), but the107 metaphysical approach on QM interpretation is justified as108 necessary in [27]. It is interesting to note that Anaxagoras109 (born 500 BC) theorized that the metaphysical Nous (mind)110 interacts with matter to form the universe [28]. Thus, Nous111 and 7.1Mshare the attribute of being theorized as a cause of112 universe formation.113 The 7.1Mattributes, their background knowledge, and their114 applications to the interpretation of some quantum mechanics115 are discussed in the following: Sections II and III explore,116 respectively, the set of neurons referred to as the neuronal117 group, and the AI agent 3.1M.Some neuronal group and 3.1M 118 characteristics are the foundational concepts of 7Mwhich is the119 parent (or base class in Computer Science lingua) of 7.1Mand120 investigated in Section IV. Some 7.1Mattributes presented in121 Section V are on the following: (i) the ontological definition of122 particles, particle tunnelling, the nature of spacetime, and the123 time flow direction; (ii) the interpretation of the wave function124 and its collapse, the non-relativistic Schr¨ odinger equation, the125 Heisenberg uncertainty principle, and the atom-bound electron126 dynamics;(iii) the Statistics-rooted justification on the use of127 Hilbert space in defining some 7.1Mfeatures; (iv) and the128 7.1Mexplanation of some paradoxical Quantum Mechanical129 concepts, e.g., the wave-particle duality, the Schr¨ odinger’s130 cat, and the which-way information. The 7.1Mfeatures are131 also used in Section VI to interpret the electron double-132 slit interference experiment. Lastly, the publications on 7.1M 133 succeeding this paper are described in Section VII.134 II. Neuronal Characteristics135 The fundamental functional unit of the human brain is the136 neuron, illustrated in Figure 1(a) and schematically represented137 in Figure 1(b). It has short branches called dendrites and one138 long branch, relative to the dendrites, called the axon which139 is connected to the dendrites of the other neurons, such as140 illustrated in Figure 1(c). Let it be denoted as neuron nand141 indicated as a circle in Figure 1(b). Its features foundational142 to 7Mare the following: (a) It fires electrical impulses which143 travel across its axon, and received by dendrites of other144 neurons. (b) The electrical impulses its dendrites receive from145 the axons of other neurons m=1,2,3,are weighted by the146 connection weight wm,nbetween its dendrites and the axons,147 as indicated in Figure 1(b). (c) The weighing outcomes are148 summed and then added to its bias parameter bn.The addition149 result is feed to its activation function (e.g., sigmoid or unit150 step function) thereby outputting its activation potential (AP). 151 [19], [29]. (d) It fires an electrical impulse once its AP is 152 at or above a certain threshold THR [30]. Even if it is not 153 firing, the AP is stochastic. The variation of its AP due to 154 its dendrite-connected neurons is defined as its encounter 155 with them. And, it does not need to fire in order to be 156 referred to as encountering with them. (e) Some neuroscientists 157 theorized that the firing of neurons triggers the mind to be 158 conscious of the mental information associated with these 159 neurons. However, not all neuroscientists are in consensus on 160 this theory [31]. 161 Axon Dendrites Dendrites Axon w 1,n (c) (a) (b) w 2,n w 3,nb n n 0 n n 2 n 3 m x 1 x 2 (d) v 1,0 w 0,0 v 1,3 w 3,2 n 1 x 0 Fig. 1. Neuron and neural network diagrams for sub-figures (a) and (c), respectively: through vecteezy, retrieved from https://www.vecteezy.com/ vector-art/431354-stem-cell-diagram-on-white-background: (a) A representation of the physical neuron. (b) A schematic presentation of a neuron where the connection weight from neuron mto n,mis denoted as wm,n. (c) A sample neuronal network representation. (d) A schematic presentation of a sample neural network GR with feedforward architecture. The connection weights from neuron mto neuron nk,k=0,1,2,3, is v1,k; and from neuron nkto neuron xl,l=0,1,2 is wk,l. Now, any set of connected neurons (such as illustrated in 162 Figure 1(c)) that encodes a mental concept is referred to 163 as the neuronal group [32]. Its features foundational to 7M164 are the following: (A) Its intrinsic parameters (e.g., axon 165 electrical capacitance, firing threshold) are all real-valued. 166 (B) The firing dynamics of its member neurons is characterized 167 by the neuronal encountering, architecture, and connection 168 weights [33], [34], [35]. (C) Some of its member neurons, 169 but not all, may belong to other neuronal groups [32], and 170 these members may be temporary and can be replaced by 171 non-member neurons. It can even split into multiple neuronal 172 groups, and may emerge from, or transform to, a set of 173 non-information-encoding neuronal set as supported by neural 174 plasticity [36]. 175 A. Concept-particle correspondence 176 Suppose the schematic diagram in Figure 1(d) is that of 177 a neural network GR where: (a) The neuron moutputs the 178 Euclidean time t √−1 with tas the physical time. (b) The 179 neuron layer referred to as the input layer is comprised of 180 the solitary neuron mwhose output is weighted (multiplied) 181 by the deterministic v1,k,k=0,1,2,3. The weighting results 182
3 are then fed respectively to the neurons nk, which have183 deterministic bias bk, have rectified linear unit function as184 activation function, and comprised the neuron layer referred185 to as the hidden layer.(c) The output of every neuron nkis186 weighted by the stochastic wk,l,l=0,1,2,that have joint187 probability JP, and then fed to the neurons xl. The latter188 have deterministic bias cl, have linear function as activation189 function, and constitute the neuron layer referred to as the190 output layer.(d) Its output, labelled as xl, are the neural191 network GR outputs and, in preparation for the succeeding192 sections, are defined as the space x(t)≡hx1(t),x2(t)iand time193 t=x0attributes of a particle PN.This type of neural network194 whereby information is successively feed from one layer to195 the next but not to the previous layers is referred to as a196 feedforward architecture, e.g., considered in [22].197 Hashimoto, et al., [23] investigated a neural network similar198 to that in Figure 1(d), but with neurons n0and x0, and the199 connections to and from them, removed; and the hidden layer200 has NN +1 neurons, ∃NN ∈Z+, prior to the removal,201 i.e., the rightmost neuron in this layer is the neuron nNN.202 In their investigation, the transition amplitude ⟨x(tini)||xtfin⟩203 (a correlation) has the state vectors |x(tini)⟩and |xtfin⟩as204 the initial and final states of particle PN at times tini and205 tfin, respectively. They show that from the correlation, the206 joint probability density function (PDF) JP is found using the207 particle PN Lagrangian expressed in terms of the stochastic208 weights wk,l,k=1,...,NN,l=1,2,of the neural network GR.209 Using the joint probability JP, they also showed the particle210 PN path described by x(t) as a random walk.211 Now, if the neural network GR — defined in terms of212 the activation function, weights, and biases — is indeed213 a neuronal group then there is a correspondence between214 the neuronal group-encoded concept and particle PN whose215 dynamics is defined by the Lagrangian expressible by the same216 terms. The concept-particle correspondence is foundational217 to 7M.As concepts are information, it is analogous to the218 information-particle correspondence in [37].219 III. The 3.1MAI agent220 The presented neuronal group and neuron features were221 used to develop the AI agent referred to as 3.1Mwhich is foun-222 dational to 7M, is equipped with sensors of its environment,223 and has the following attributes: (a) It contains and processes224 information, analogous to the neuronal group-encoded mental225 concepts. (b) Considering the stochastic AP expressed in Item226 II.c, some attributes endowed to the information are defined as227 being stochastic. (c) Advocated by the concept of conscious-228 ness cited in Item II.e, one of the stochastic attributes is the229 consciousness level, defined as the measure of how conscious230 3.1Mis of the information. (d) Related to the threshold cited231 in Item II.d, the consciousness level of any information lower232 than, or otherwise, a given consciousness threshold is defined233 as indicating the information as being subconsciously, or234 consciously perceived, by 3.1M, respectively [15].235 The information processing components of 3.1Magent are236 as follows: (A) Transformer creates information by changing237 and/or combining some of the last 3.1Moutput information238 (defined in Item III.E below) and the current 3.1Msensor 239 output information. The transformation ability is established 240 from the creativity of humans and other AI agents, e.g., those 241 in [38], [39], [40]. After the creation, Transformer then reduces 242 every most numerous set (could be a singleton) of similar 243 information — from the union of the information it created 244 and those it admitted as input — into a single information 245 with consciousness level being the consciousness level sum of 246 all the set elements, i.e., analogous to the summation made 247 to obtain AP mentioned in Item II.b. Its output is the set of 248 all of the reduction outcomes. (B) Contrector yields attribute249 endowed information being the responses to the Transformer 250 actions on information, except the set reduction and the 251 attribute endowment actions. Its inclusion as 3.1Mcomponent 252 is motivated by the issuance of sensations (e.g., sense of 253 accomplishment) triggered by human mental processes. (C) In254 teractor produces attribute endowed-information as the product 255 of the interactions between the Transformer input information, 256 and the union of the Contrector and Transformer output 257 information. (D) Selector segregates the Interactor product 258 information according to their consciousness level using a 259 certain threshold, i.e., analogous to the neuronal firing based 260 on a threshold mentioned in Item II.d. (E) The 3.1Moutput 261 information is all the segregated Interactor output information 262 with consciousness level not lesser than the threshold [15]. 263 IV. The 7Mentity 264 The explored 3.1Mand neuronal group characteristics are 265 used in the generic formulation of the metaphysical neuromor266 phic entity referred to as 7Mwhich is a base class in Computer 267 Science lingua. Thus: (a) 7Mis defined as constituted of the 268 neuronal group-like entities, called 7Mneuronal groups, which 269 in turn are constituted of the neuron-like entities, called 7M270 neurons. (b) Inside 7M, every 7Mneuronal group is connected 271 to all other 7Mneuronal groups. (c) The 7Mneuronal groups 272 are defined as capable of outputting their respective encoded 273 information which — motivated by the concept-particle 274 correspondence explored in Section II-A — are defined as 275 the particles that constitute the universe (defined as physical 276 from here onwards except when indicated otherwise) which is 277 therefore a product of the neuromorphic 7M.(d) Conforming 278 with Item II.C, any 7Mneuronal group may have temporary 279 7Mneurons, and/or belonging to other 7Mneuronal groups, 280 i.e., the neurons to particle correspondence may not be one281 to-one. 282 The features of particles in the universe are defined in the 283 following: (e) Analogous to the concept presented in Item III.c, 284 any particle has an attribute referred to as the consciousness 285 level which is the measure of how conscious 7Mof it. In rhyme 286 with Item III.d, it is defined as subconsciously or consciously 287 perceived by 7Mif its consciousness level is lesser than, or 288 otherwise, to a given consciousness threshold, respectively. 289 (f) The particles/information are cyclically processed by 7M290 where the cycle index τ∈Z+ 0is referred to as the Universal 291 Rhythm (UR). The latter is not necessarily the physical time, 292 and could be related to something else, e.g., the entropy of 293 the universe, or the order of spin network state evolution as 294 define in Quantum Loop Gravity (QLG) [41]. 295
4 At a given UR τ > 0, all particles in the universe are pro-296 cessed analogous to the 3.1Minformation processing through297 the following generically defined 7Mcomponents: (A) As a298 counterpart to the 3.1MTransformer described in Item III.A,299 the 7MImaginator creates myriad non-physical versions of300 its input universe. Then, it evolves sequentially over non-301 temporal parameter each of the versions, whereby each of302 the evolved non-physical universe is constituted by particles303 defined as subconsciously perceived by 7M.It may or may304 not make particles to interact in the evolution series, such305 that, it incorporates an analogue of the Item III.C-defined 3.1M 306 Interactor process. Further, it reacts to its last actions on, and307 to the nature of, the particles in the evolutionary process,308 i.e., it includes the analogue of the Item III.B-cited 3.1M 309 Contrector process. (B) Inspired by the 3.1MSelector cited310 in Item III.D, the 7MSelector randomly chooses, according to311 some parameters, one of the 7MImaginator evolution series.312 Extracted from the chosen series is a universe, defined as313 containing reality at UR τ+1, that is then fed to the 7M 314 Imaginator at the next 7Mdynamical cycle, i.e., at UR τ+1.315 At UR τ=0, there is no 7MSelector output, and the 7M 316 Imaginator input is only a given universe labelled as the317 primordial universe represented as an ellipse labelled PUV318 in Figure 2 which illustrates the 7Mprocess block diagram.319 Imaginator Selector universe PUV evolutionary sequences Fig. 2. Block diagram of 7Mcomponents: 7MImaginator creates myriad nonphysical versions of its input universe. It then evolves sequentially over nontemporal parameter each of the versions. One of the evolutionary sequences is chosen by 7MSelector from where a new universe is extracted containing the new solitary reality. The new universe is then feedback to 7MImaginator in the next cycle. At UR τ=0, there is no 7MSelector output, and the 7M Imaginator input is only a given universe PUV. V. The 7.1MImplementation320 As aforementioned, the base class 7Mis generically defined.321 It can be made specific in several manners; The specification322 presented in this paper yields the entity referred to as 7.1M 323 which is a descendant/derived class (in Computer Science324 lingua) of 7M. Being a descendant: (1) It is defined as325 metaphysical, neuromorphic, and constituted of neuron-like326 entities forming neuronal group-like entities — the coun-327 terparts to similar terms on 7M.(2) Being metaphysical, its328 constituents are not situated in spacetime. (3) It inherits, but329 does not make specific, the 7Mattributes described in Items330 IV.a to f. However, in inheriting, the term “7M” in the items is331 replaced by “7.1M”. ■From here onward, the terms “neuronal332 group”, “neuron” are assumed to be of 7.1M, and the terms333 “consciously” and “subconsciously perceived” as by 7.1M,334 except when indicated otherwise. Further, the filled square335 symbol “■” indicates the termination or suspension of the 7.1M 336 feature enumeration whose item label is a parenthesis-enclosed337 positive integer.338 A. Preliminary definitions and approaches 339 Before exploring the rest of 7.1Mfeatures, consider the fol340 lowing preliminary definitions: (a) An ensemble of a quantity 341 is defined as the set of all possible values of this quantity at 342 a given UR. An ensemble average of a discrete (continuous) 343 quantity is defined as the weighted sum (integral) of all the 344 ensemble elements of the quantity, at a given UR, whose 345 respective occurrence probability being the weight. From here 346 onward, any stochastic attribute term not preceded by the term 347 “average” refers to a particular element in the attribute en348 semble, e.g., the term “momentum” not preceded by the term 349 “average” refers to an element of the momentum ensemble. 350 Further, the mass of any particle is assumed relative to itself 351 (i.e., a rest mass), except when stated otherwise. (b) The inter352 action of particles (or the self-interaction of a single particle) 353 is defined as the process in which at least one of the particles 354 changes its mass, ensemble average kinetic or potential energy, 355 or is lost, created, or absorbed to, or emitted by at least one of 356 the other particles. For example, the absorption of an electron 357 by an atom, and the consequent emission by the atom of a 358 different electron are two interactions. 359 (c) Afree particle is a non-interacting particle. However, 360 the particles investigated in this paper are assumed to be in 361 an Earth-bound gedankin experiment and hence may inter362 act (slight change in ensemble average kinetic energy) due 363 to Earth’s gravity for example. (d) Thus, any particle that 364 becomes non-interacting when gravitational, dark matter, and 365 dark energy acting on it are removed is referred to as an 366 approximated-free particle.(e) Any particle, free or otherwise, 367 can be designated as a reference and, as such, is referred to 368 as the reference particle (RP), and assumed to be at the origin 369 (i.e., at vector 0) of the coordinate system attached to it. For 370 example, a particle at the retina of the physicist conducting an 371 experiment can be a RP. 372 After enumerating the preliminary definitions, consider the 373 following preliminary approaches: (i) When each of the human 374 neurons is modeled using a non-linear stochastic differential 375 equation, many human brain activities cannot analytically 376 be proven to emerge from the neurons due to the extreme 377 challenge of analytically solving the myriad simultaneous 378 equations. This challenge is evident in neuronal dynamics 379 literature where most authors, instead of analytically solv380 ing the equations, resort to apply approximations, such as, 381 the mean field theory applied in [19], [29], [42]. Thus, 382 the foregoing enumerated 7.1Mattributes are only defined, 383 but neurologically-accordant, and not analytically proven to 384 emerge from the 7.1Mneuronal groups. (ii) Due to the scope 385 limitation of this paper, some dynamical properties of fields 386 and particles (e.g., energy conservation) considered in QM, 387 STR and GTR are simply adopted, i.e., not interpreted nor 388 justified using the 7.1Mfeatures. 389 B. The particle model 390 Let us now resume exploring the 7.1Mattributes. In accord 391 with Item IV.c, let the neuronal groups GR and GN be the 392 generators of particles PR and PN, respectively, where PN is 393 defined as having spacetime coordinate xR,N≡tR,N,xR,N394
5 relative to the RP PR.(4) There are two placeholder particles395 PN+and PN−(or PR+and PR−) both generated by the neuronal396 group GN (or GR). At every given UR τ, only one of the397 placeholder particles is randomly chosen to be substituted by398 particle PN (or PR). ■Except when stated otherwise, any399 particle PN attribute applies to any massive particle; and the400 attribute is relative to particle PR assumed as the RP.401 Now, consider the neuronal group GR schematically illus-402 trated in Figure 3 as the thickest rectangle labelled as GR403 and encloses polygons and arrows. Without loss of generality,404 the figure is illustrate to be in a two-dimensional universe405 with orthogonal bases vectors ˆ xand ˆ y, and origin at the RP406 PR.(5) The neuronal group GR has input, hidden, and output407 layers represented in the figure as circle IR, rectangle HR, and408 three circles n+R,n−Rand OR, respectively. Each of the circles409 n+Rand n−Rrepresents a neuronal set and referred to as the410 spacetime-outputting neuronal set. In congruence with Item411 V-B.4, one of them is randomly chosen at every UR τto412 output the spacetime coordinate xR,N(τ)≡tR,N(τ),xR,N(τ)of413 particle PN relative to the RP PR.If the spacetime-outputting414 neuronal set n±Ris chosen at UR τit (and not n∓R) outputs415 the location xR,N(τ) of particle PN (i.e., of the placeholder416 PN±) relative to particle PR.■Any sentence with the “±”417 and/or “∓” symbols is true when all the upper signs are418 the only ones considered and also true when all the lower419 signs are the only ones considered. Further, the superscript420 or subscript “+” (or “-”) in the attribute nomenclature (e.g.,421 n+R) of a given neuronal group GX,X∈{R,N},component422 indicates the component being located to the right (or left) of423 the GX-generated particle PX travel direction — The neuronal424 group component location is referred to as its handedness. The425 spacetime-outputting neuronal sets n±Rand n±Nwith the same426 handedness form a pair denoted as [n±R,n±N].427 As a computational tool, a spacetime-outputting neuronal428 set ns is devised to have a configuration of connections to the429 hidden layer HRidentical to those of the spacetime-outputting430 neuronal set n±R. It then be defined that: (6) The joint PDF431 of the stochastic connection weights between the spacetime-432 outputting neuronal set ns and the hidden layer HRare derived433 from the Lagrangian LNof the particle PN relative to PR,434 analogous to the derivation explained in Section II-A. Then, it435 is endowed to the stochastic connection weights between the436 spacetime-outputting neuronal set n±Rand the hidden layer437 HR.(7) The Lagrangian LNis created from UR τand the438 information cN,R(e.g., the particle PN mass), and provided439 by the input layer IR.The latter receives the information cN,R 440 created by its encounter with the GN output layer neuronal441 set ON. The IR(or IN) received information cN,R(or cR,N) is442 conveyed by the axonomorphic connection between the two443 encountering neuronal sets IRand ON(or INand OR), and444 referred to as the Inter-Neuronal Group Information (INGI).445 ■INGI is analogous to the AP defined in Item II.c, and to the446 mutual information between the two products of the Hilbert447 space decomposition described in [43]. (8) Similar to 7.1M,448 the INGI axonomorphic conveyor is defined as metaphysical.449 (9) Every neuronal group is defined as connected to the rest450 of the neuronal groups in 7.1Mthrough the axonomorphic451 conveyors. ■452 GR GN x xR,N ^ y ^ vR,N x+ R,N xR,N IR n-R n+N n+R n-N IN c R,N OR ONHN c N,R τ τ LN LR Fig. 3. Schematic diagram of the neuronal groups GN and GR represented as the thickest bounded rectangles and respectively generate the particles PN and PR relative to which PN has spacetime coordinate xR,Nand velocity vR,N in a sample two-dimensional universe with orthonormal basis ˆ xand ˆ y. The neuronal group GR has a hidden layer represented as a rectangle labelled HR and neuronal sets indicated as circles, such as in its output layer sets labelled n+Rand n−Rthat provide the spacetime coordinate xR,Nof the particle PN relative to PR. The information cN,Ris created by the encounter (defined in Item II.d) of IRand the GN output layer neuronal set ON, and conveyed by the axonomorphic connection between these two neuronal sets. 1) Multi-neuronal group system: In case of multi-neuronal 453 group (multi-particle) system: (10) There is one GR input 454 layer neuronal set of type IRfor every other neuronal group 455 GX,X,R,in the system. Further, there is one GR output 456 layer spacetime-outputting neuronal set of type n±Rfor every 457 placeholder particle PX±generated by the neuronal group GX.458 All the IR-type neuronal set outputs are lumped into a single 459 multi-particle Lagrangian LMfed to the hidden layer HRand 460 used to find the joint PDF of the stochastic connection weights 461 between the ns-type spacetime-outputting neuronal sets and the 462 hidden layer HR.As in Item V-B.6, the PDF is endowed to the 463 stochastic connection weights between the n±R-type spacetime464 outputting neuronal sets and the hidden layer HR.■465 The axonomorphic link-mediated encounter between the 466 neuronal sets IRand ONsignifies the encounter between their 467 affiliated neuronal groups GR and GN.(11) Some neuronal 468 groups GX,X,R,N,can also encounter with neuronal 469 group GN through the axonomorphic connections (which exist 470 according to Item V-B.9) between them, thereby the neuronal 471 group GR — using the GN-outputted INGI cN,R— is defined 472 to form Lagrangian LMinfluenceable by them as well. ■473 This multi-neuronal group encounter is analogous to a spring 474 SN (corresponds to GN) in a mattress that interacts with its 475 adjacent springs SX,X,N(corresponds to GX). Thereby, the 476 dynamics, and hence the Lagrangian LM, of spring SN can be 477 influenced also by springs SX, i.e., the Lagrangian LMhas the 478
6 parameters of SX not only of SN.479 (12) In formulating the Lagrangian LM, the force exerted480 to the RP particle PR must be cancelled. ■Consequently, if481 particle PR is accelerated by a net force FR while particle482 PN is not experiencing a net force then PN is not considered483 (due to the force FR cancellation) in the formulation as the484 one on ensemble average accelerating relative to the RP PR.485 Thus, the charged particle PN should not be considered in486 the formulation as radiating electromagnetic waves (unjustified487 according to the 7.1Mprinciples in accord with Item V-A.ii)488 even if it is accelerating relative to particle PR.Further, with489 respect to a person standing still at the spinning Earth’s north490 pole, the zero net force-exerted particle 13 billion light-years491 from Earth should be considered being not revolving around492 the Earth. The force exerted to any RP is assumed being493 cancelled from here onwards.494 C. Spacetime illusion495 Let us now explore the interpretation of spacetime using the496 7.1Mattributes. To begin, imagine the spacetime-outputting497 neuronal sets n−Rand n+Rin Figure 3 as the ears of a blind498 man listening to the stereophonic headphones sounding their499 received electromagnetic information from a mobile phone500 static relative to him. Thereby, he senses a sounding object OB501 — unknown to him as an illusion only — ostensibly located502 at orelative to him, i.e., the location is also an illusion. Further,503 suppose that the change in the electromagnetic information504 over time changes the headphone sound thereby the blind505 man senses the illusive object OB as if running with constant506 “velocity” v.507 Now, the arrows representing velocity vR,Nand location xR,N 508 in Figure 3 are for illustration purposes only, i.e., their ends do509 not indicate actual location and velocity; there is no connection510 between the spacetime-outputting neuronal sets n±Rand n±N;511 and particle PN is non-physical but the information generated512 by the neuronal group GN (according to Item IV.c). Further,513 the neuronal group GR outputted location xR,Nis derived from514 (based on Items V-B.5 and 6) the non-physical INGI cN,R 515 (as implied in Item V-B.8) that, in turn, produced by the516 encounters between the metaphysical neuronal groups GR and517 GN, which are therefore not spacetime situated (in accord with518 Item V.2). Thus, the location xR,Nis analogous to the illusive519 location osensed by the blind man; INGI cN,Ris analogous to520 the electromagnetic information received by the headphones;521 and the information/particle PN is analogous to the running522 illusive object OB.Therefore, the “velocity” and “location”523 indicated in Figure 3 are illusions only, thus the quotation.524 To generalize: (13) The “location” of any particle in 7.1M 525 interpretation is an illusion and hence, there is no physical526 space in the universe. ■This interpretation will be used in the527 series successors of this paper to justify non-locality in the528 universe.529 By the definition in Item V-A.e, any particle in the universe530 can be designated as the RP in the illusive “space”. Thus:531 (14) There is no absolute “frame of reference” in the universe,532 i.e., akin to that in STR. ■533 1) Illusive time: Considering that the time t± R,Nat place534 holder particle PN±relative to particle PR±is also the output 535 of the spacetime-outputting neuronal set n±R, it is also an 536 illusion based on the above reasoning scheme proving space as 537 illusion. Thus: (15) “Spacetime” (i.e., and “vacuum”) in the 538 universe is illusion. ■For convenience, the quotation on any 539 “spatiotemporal” term is removed from here onward but this 540 term still expresses itself being illusive, except when preceded 541 by the term “physical” which in this case the spatiotemporal 542 term is physical. 543 The physical time being illusion is supported by Julian 544 Barbour who considers its human conception as arising from 545 a set of information that conveys dynamics [44], e.g., a set 546 of snapshots of a flying leaf blown by the wind relative to its 547 background. It is viewed as emergent, and may be extractable, 548 from the spin network state evolution defined in QLG. How549 ever, the definite method for its extraction is not yet made and 550 not even fully understood [45]. This methodological challenge 551 is referred to as the “problem of time”. The physical spacetime 552 is viewed by Leibniz as an abstract structure of relationships 553 among physical objects [46]. Also, physical space is supported 554 as emergent from particle entanglement in [43], [47], [48]. On 555 the contrary, Lindner justified the relevance of the realness of 556 physical space [49]. 557 Time ris defined in Section V-B as relative to a given RP, 558 such that, it is referred to as the relative time (RT). (16) It is 559 also defined as, 560 r≡FT (τ, U)(V.1) where FT is a function which can be non-linear, and Uis the 561 set of neuronal groups that produce particles constituting the 562 universe at UR τ.For this first paper in the series that deals 563 with non-relativistic particles only, FT has a form such that 564 RT is identical at all RPs, and proportional to UR, i.e. 565 r=δτ ∈R≥0(V.2) where the assumed minuscule δ∈R>0is referred to as the 566 Chronos Proportionality Constant (CPC), and could be equal 567 to the planck time. ■Eq. V.2 implies that RT r∈R>0is 568 discrete. Based on this, UR τis also referred to as the relative 569 time index (RTI). Consistent with Item V.a, the coordinate 570 (x,r)of the particle at location x(i.e., a space ensemble 571 element) and at RT rrelative to a given RP is referred to as the 572 space ensemble element and relative time (SEERT) coordinate. 573 However, for convenience, the term “coordinate” is assumed 574 from here onward to be “SEERT coordinate”, except when 575 stated otherwise. 576 D. Stochastic nature 577 Let us now consider particle identity. (17) Every parti578 cle must have a set of non-stochastic identifying attributes 579 (e.g., electronic mass and charge) unique among all types of 580 particles. ■These attributes are referred to as the essential 581 attributes. 582 Now, as mentioned in Item II.d, the AP of any human 583 neuron is stochastic. Then, any human neuronal group (com584 prised of human neurons) is expected to have stochastic at585 tributes. Considering the correspondence between the concept 586
7 encoded in, and the particle generated by, a neuronal group587 (explored in Section II-A), one can expect any particle to588 have stochastic attributes. Thus: (18) All the multi-valued589 non-essential attributes (e.g., energy, momentum, etc.) of any590 non-interacting particle are derived from the stochastic features591 (e.g., weights) of the neuronal group generating it, and defined592 as stochastic, except its time. However, by the definitions in593 Items V-A.b and c and Item V-D.17, the mass and ensemble594 average energy of any type of any non-interacting particle are595 constant. (The particle mass is relative to itself as mentioned596 in Item V-A.a) Further: (19) Given any Classical Physics non-597 identifying particle attribute, there exist a stochastic particle598 attribute (i.e., non-essential) whose ensemble average is the599 given. ■For example, the ensemble average of a particle600 stochastic kinetic energy is its Classical Physics kinetic energy.601 Due to the randomness of some neuronal group attributes,602 the concept (or particle) it bears can also vary randomly.603 Thus: (20) For some particles, their identity (e.g., being a604 neutrino at a certain flavour [50]) may randomly change even605 when not externally triggered. And based on Item II.C, a non-606 information-encoding neuronal set may randomly become a607 neuronal group (i.e., encodes an information) at a certain UR608 τ.Further, any neuronal group may randomly split into two609 neuronal groups (i.e., each encoding information), without610 any cause at UR υ, that then may randomly turn to a non-611 information-encoding neuronal set at a later UR ϕ > υ.612 ■If the two splitting products generate a particle and an613 antiparticle pair, these particles therefore may randomly appear614 without any cause at υand then may annihilate at ϕ, i.e., an615 event analogous to that in vacuum fluctuation.616 1) Stochastic movements: To provide a concrete illustra-617 tion on the stochastic free particle PN (illusive) movements,618 consider again Figure 3 where suppose: (a) The spacetime-619 outputting layer neuronal sets n+Rand n−Rare replaced with620 one neuronal set nRconstituted of neurons xl,l=0,1,2,that621 outputs xl R,N.(b) The hidden layer HRis constituted of neurons622 nk,k=0,1,2,...,NN,∃NN ∈Z+.(c) All connections from623 neurons nk,k,0,to neuron x0, and from neuron n0to624 neurons xl,l=1,2, are absent. (d) The input layer neuronal625 set IRis replaced with a single neuron mbut its supposed626 output Lagrangian LNis still used to determine the joint627 PDF (as explained in Section II-A) of the present stochastic628 connection weights wk,l,k=1,2,...,NN,l=1,2,between629 the output layer neurons and the hidden layer neurons. (e) The630 latter no longer receive UR τbut instead the Euclidean UR631 τ√−1 from the input layer neuron m.(f) The hidden layer632 neuron n0outputs time t, along the connection with weight633 w0,0=1, and received by neuron x0which outputs time634 t≡x0 R,Nin response. (g) The output layer neurons x1and635 x2output the particle PN two-dimensional spatial components636 x1 R,Nand x2 R,N, respectively; thereby the overall output of the637 neuronal group GR is the particle PN spacetime coordinate638 xR,N=hx0 R,N,x1 R,N,x2 R,Ni.(h) All undefined weight connections639 and bias parameters are deterministic. (i) Lastly, the neuronal640 set ORis removed resulting to the neuronal group GR almost641 identical (except for the absent connections mentioned in Item642 V-D1.c) to the neural network illustrated in Figure 1(d).643 Given the enumerated suppositions, it is straightforward to 644 show that by using the approach of Hashimoto, et al., [23] 645 explained in Section II-A for a free massive particle: (i) One 646 can obtain the particle PN mass-parametrized joint PDF of the 647 stochastic connection weights wk,l,k=1,2, . . . NN,l=1,2, 648 between the neurons nkand xlof the neuronal group GR.649 (ii) Using the PDF, one can obtain the time tat, and the 650 random location xR,N(t)≡hx1 R,N(t),x2 R,N(t)iof, particle PN 651 relative to particle PR.(iii) If the particle PN mass is given 652 certain concrete value in the PDF expression, and the initial 653 spacetime coordinate is set at (0,0), one can obtain two 654 samples of the random process xR,N(t)as the two-dimensional 655 arrow sequences illustrated in Figure 4 (akin to random walks 656 explored in [23]) with different arrow types under a coordinate 657 system with orthonormal basis vectors ˆ x1and ˆ x2.658 x1 ^ x2 ^ Fig. 4. Two arrow type-differentiated free particle PN movements relative to particle PR in a two-dimensional universe with orthonormal basis vectors ˆ x1 and ˆ x2. In each arrow sequence illustrated in Figure 4: (A) The 659 beginning and tip of each arrow indicate, respectively, the 660 current x(a) and next x(b) locations of particle PN relative to 661 particle PR. Without loss of generality, let b>a,a=δa′,b=662 δb′,a′,b′∈Z+(δis affiliated with Eq. V.2); (B) And, particle 663 PN does not exist from location x(a) to x(b), exclusive, over 664 the period ato b, exclusive. Thus, it successively pops at, but 665 does not travel in between, the locations. Its non-existence 666 between the locations may allow it to not interact with an 667 object (a set of neuronal group-generated particles) between 668 its successive location occurrences, i.e., a tunnelling effect. 669 Considering its random movement, why does the particle PN,670 say charged, not interacting? 671 2) Conscious perception: To answer the last question, con672 sider that any human neuron is described in Item II.d to fire 673 an electrical impulse when its AP is at or above the thresh674 old THR, and is defined as encountering with its dendrite675 connected neurons at its AP variation. Thus, by analogy, the 676 neuronal group GN is defined as encountering (or interacting) 677 with the neuronal group GR when its INGI cN,Rbelongs (or 678 does not belong) to a subset EN of the INGI space IS . The 679 neuronal group interaction is also defined as the interaction of 680 the particles they generated. 681 Now, as theorized by several neuroscientists, the electrical 682 impulse firing of human neurons (corresponds to the neuronal 683 group-generated particle interaction) makes their encoded 684
8 information consciously perceived by the human mind as685 mentioned in Item II.e. It is then proposed that: (21) Any686 particle is consciously (or subconsciously)perceived by 7.1M 687 iffit (and hence its neuronal group generator) is interacting688 or is created (or is not interacting). (22) Any free particle689 is defined as subconsciously perceived. ■Thus, despite its690 random motion, the charged particle PN is not interacting as691 it is subconsciously perceived. Notice that the definition in692 Item V-D2.22 is meant to justify that in Item V-A.c. Thus, the693 former supersedes the latter.694 3) Redefining non-essentials: A stochastic free particle695 particle movement may form a discontinuous path (e.g., such696 as those illustrated in Figure 4) which, in Classical Physics,697 entails infinite kinetic energy. According to Item V-C.15, the698 particle location is just an illusion (perceived by the blind699 man). By this illusion, it (or its mass) is not in reality jumping700 (or carried along) from one location to another. Thus, it is701 inappropriate to use any of its two successive illusive locations702 and the time to transfer between these locations to obtain its703 speed and then its kinetic energy using its mass. But rather,704 its energy should be defined as its ability to do work in705 reality, i.e., not in illusion. As posited in Item V-J.48 below,706 reality is constituted of interacting particles. Thus: (23) All707 non-essential attributes of any particle are defined based only708 on their role in its interactions. ■Consequently, a particle709 attribute at a spacetime coordinate is assumed from here710 onwards to mean the particle has this attribute if it interacts711 at this coordinate.712 E. Quaternionic wavefunction713 By the stochastic nature of the non-essential attributes of714 any particle (according to Item V-D.18), it is appropriate to715 investigate their PDF. To begin, Item V-B.4, cited that only716 one of the locations x+ R,Nand x− R,Nindicated in Figure 3 of the717 placeholder particles PN+and PN−, respectively, is randomly718 chosen as the particle PN location at every given UR τ.For719 notational convenience, let the subscript R,Nbe dropped, i.e..,720 xR,N→x.Thus, using Statistics and Eq. V.2, the probability721 for the particle PN to be at coordinate (x+,t)or (x−,t)is722 PX(x,t)=P+x+,t+P−x−,t≡Ψ(x,t)Ψ∗(x,t)≥0 (V.3) where P±(x±,t)is the probability of particle PN to be at the723 coordinate (x±,t);724 Ψ(x,t)=pP+(x+,t)+˜ ipP−(x−,t)(V.4) ˜ i=0+˜ i+0˜ j+0˜ k∈Qis a quaternion and not an imaginary value,725 although behaves as such [51]; the symbol (∗) is the quaternion726 conjugation operation; and Qis the space of quaternions with727 zero ˜ jand ˜ kcomponents.728 Using Eq. V.4, let us provide an interpretation of the wave-729 function in QM. To begin, Items V-A.b and V-D2.22 imply730 that a free particle PN moves with time-invariant ensemble731 average speed |¯ v|, also referred to as the group speed, where732 ¯ vis its ensemble average velocity. In its movement, let it733 start at spacetime coordinate (x0,t0), runs at ensemble average734 velocity ¯ vover the period t−δ, and then after a CPC δ(defined735 in Item V-C1.16) randomly displaced by dat RT t0+tto,736 x=d+¯ v(t−δ)+x0≈d+¯ vt+x0.(V.5) Being a free particle, it is defined to have a greater probability 737 of being displaced along, or closer to, the direction ˆ vof ¯ vthan 738 otherwise. Thus, its probability to be displace by dis defined 739 to have density,740 PD(t)(d)=PD (WN ˆ v·d,t)(V.6) where PD is a function that globally maximizes when ˆ v·ˆ d=1; 741 ˆ dis the unit direction of d;WN =2π λis referred to as the wave 742 number; and λis an INGI cN,Rcomponent, and referred to as 743 the wavelength. Applying the technique of variable change, the 744 probability of PN to exist at the spacetime coordinate (x,t)has 745 PDF, 746 PX(x,t)=PD (WN ˆ v·(x−¯ vt−x0),t).(V.7) Let WN ˆ v≡k,ω≡k·¯ v, and k·x0≡c0to yield, 747 PX(x,t)=PD (k·x−ωt−c0,t),(V.8) i.e., a PDF with running envelope analogous to the dispersive 748 Gaussian wave packet defined in equation 17 in page 60 749 of [52]. Although any massive free particle has dispersive 750 QM wavefunction, however for convenience, its quaternionic 751 wavefunction is approximated as a travelling non-dispersive 752 Gaussian wave packet, from here onwards. Further, the con753 stant c0in Eq. V.8 is dropped thereby, 754 PX(x,t)=PD (S,t) where 755 S=k·x−ωt.(V.9) From the definition of PX(x,t):(24) The free particle PN has 756 probability PX(x,t)to exist at the spacetime coordinate (x,t)757 relative to particle PR.■This is one of the major differences 758 between 7.1Mand Copenhagen interpretation of QM. 759 Now, knowing that the PDF PX(x,t)term in Eq. V.8 is a 760 running function, let Eq. V.4 be express as 761 Ψ(x,t)=√P+(x+,t)+˜ i√P−(x−,t) =hcos (S)+˜ isin (S)i√PX(x,t) =exp ˜ iS√PX(x,t) ≡Ψ(+)(x,t)+˜ iΨ(−)(x,t). (V.10) Thus, Ψ(±)(x,t)∈R, and Ψ(x,t) is quaternionic and not 762 a complex value, in accord with Item II.A. Similar to the 763 wavefunction concept in QM, Ψ(x,t) is a wavelike travelling 764 function as evident in Eq. V.10, a component of the PDF 765 expressed in Eq. V.3, and has a quaternionic term which 766 behaves as, but not, a complex value. Thereby, it is referred 767 to as the quaternionic wavefunction, where in this case is for 768 the free particle PN relative to particle PR. The quaternionic 769 wavefunctions of equi-essence particles, and having similar 770 ensemble average group speed, wavelength, and frequency are 771 referred to as of similar type.772 1) Relationship with the neuronal group the components: 773 Based on Eq. V.10, the quaternionic wavefunction component 774 Ψ(+)(x,t)(Ψ(−)(x,t)) corresponds to the placeholder particle 775 PN+(PN−) spacetime coordinate (x+,t)((x−,t)) outputted by 776 the spacetime-outputting neuronal set n+R(n−R) (as cited in 777 Item V-B.5) which is indicated in Figure 3 as being at the right 778 (left) side of the particle PR travel direction. Thus, Ψ(+)(x,t)779
9 (Ψ(−)(x,t)) is referred to as the right (left)component of780 Ψ(x,t).781 Next, although there is no connection between every782 spacetime-outputting neuronal set pair [n±R,n±N](defined in783 Section V-B), but for discussion convenience: (25) Let there784 be an imaginary connection, referred to as the pair imaginary785 axon (PIA), from the spacetime-outputting neuronal set n±R 786 to n±Nof the pair [n±R,n±N]which has non-commutative787 elements according to the connection direction. PIA is defined788 as a metaphorical lossless elastic entity which reacts to the en-789 counters (not necessarily firing based on Item V-A.b) between790 the neuronal groups GR and GN leading to its eternal vibration791 (in accord with the Eq. V.10 oscillatory terms) with waveform792 being the quaternionic wavefunction component Ψ(±)(x,t) 793 corresponding to the particle placeholder PN±relative to PR.794 ■As discussed in Item V-B.11, the encounters between the795 neuronal groups GN and GX,X,N,R,yield the Lagrangian796 LMused to determine the connection weights between the797 spacetime-outputting neuronal set n±Rand the hidden layer HR 798 of the neuronal group GR that then influences the n±Routputted799 spacetime coordinates from which the component Ψ(±)(x,t) 800 can be obtained. Thus: (26) The encounters are influential801 to the quaternionic wavefunction (on PIAs between the pairs802 [n+R,n+N]and [n−R,n−N]) of particle PN relative to PR in the803 multi-particle system. ■804 The particle PR is generated by the neuronal group GR805 whose components are described in Item V-B.5. Not explored806 in the latter, however, are the specific internal details of, and807 the difference between, the spacetime-outputting neuronal sets808 n+Rand n−R(except their similar connection weights to the809 hidden layer HR as cited in Item V-B.6), and the specific810 details of the INGI cN,R. The details are such that the resulting811 stochastic outputs (xR,N(τ)≡tR,N(τ),xR,N(τ)as mentioned in812 Item V-B.5) of the spacetime-outputting neuronal sets have813 PDF PX(x,t)that of the free particle PN and relative to814 PR,where PX(x,t)has component quaternionic wavefunction815 Ψ(x,t)(according to Eq. V.3) which has different components816 Ψ(+)(x,t)and Ψ(−)(x,t)(that correspond to the spacetime-817 outputting neuronal sets n+Rand n−R). Thus, the details need818 not be specified as only their resulting quaternionic wavefunc-819 tion is investigated in the following.820 2) Collapse: Using the definition of the PDF PX(x,t) 821 affiliated to Eq. V.3: (27) The quaternionic wavefunction822 is interpreted as the existence probability component of its823 associated particle/s. ■As the PDF PX(x,t), and hence its824 quaternionic wavefunction component, are non-physical, the825 interpretation is contrary to the Everett interpretation of the826 QM wavefunction being physical [2].827 Considering quaternionic wavefunction being derived from828 the existence probability of its associated particle: The banish-829 ment of any particle (e.g., the electron in the electron–positron830 annihilation) implies that its probability of existence becomes831 zero, and hence, its quaternionic wavefunction as well. This is832 the interpretation of the wave function collapse in QM for ban-833 ished particles. Further: (28) The quaternionic wavefunction834 of any massive particle, still existing despite being absorbed by835 another particle, does not collapse but transformed according836 to QM. ■For example, the quaternionic wavefunction of a837 previously free electron absorbed by an atom is transformed 838 (e.g., to the quaternionic wavefunction analogous to QM 839 eigenfunction) according to the state of the bound, but still 840 existing, electron. 841 3) Dynamics: Let us now investigate the quaternionic 842 wavefunction dynamics. To begin, consider the three particles, 843 PR,PN and the neuronal group GT-generated PT.Considering 844 Item IV.b, neuronal group GN is connected to GR and GT.In 845 accord with Item V-E1.25, a quaternionic wavefunction ΨN,R846 is formed on the PIAs connecting the spacetime-outputting 847 neuronal sets of the neuronal groups GN and GR. Suppose the 848 travelling quaternionic wavefunction 849 ΨN,R=U(t−t0)exp ˜ iSN,RGSN,R; (V.11) where Uis a unit step function; 850 SN,R=WN ˆ vN,R·xN,R−ω(t−t0); Gis a normalized Gaussian function; WN is a wavenumber; 851 ωis the angular frequency; ˆ vN,Rand xN,Rare the quaternionic 852 wavefunction travel direction (or group velocity direction) and 853 the distance from particle PN to PR, respectively; and t0≤tis 854 the RT at which particle PN starts to randomly travel towards 855 PR according to the quaternionic wavefunction. Suppose fur856 ther that the PIAs between the spacetime-outputting neuronal 857 sets of the neuronal groups GN and GT do not vibrate thereby 858 the quaternionic wavefunction ΨN,T=0 over the PIAs, i.e., 859 particle PN has no probability of going towards particle PT.860 Now, suppose that at RT tr>t0the neuronal group GN 861 encounters GR where the quaternionic wavefunctions ΨN,R=0862 and 863 ΨN,T=U(t−tr)exp ˜ iSN,TGSN,T; (V.12) 864 SN,T=WN ˆ vR,T·xR,T−ω(t−tr); and ˆ vR,Tand xR,Tare the quaternionic wavefunction ΨN,T865 travel direction and the distance from particle PR to PT,866 respectively. Notice that the quaternionic wavefunction ΨN,T867 is not over the PIAs of the spacetime-outputting neuronal 868 sets connecting the neuronal groups GR and GT but rather 869 ostensibly running from PR towards PT.Inferring from Eqs. 870 V.11 and V.12, the quaternionic wavefunction that starts from 871 GN runs towards GR, reflects, and then ostensibly runs towards 872 GT.This exemplifies a quaternionic wavefunction reflection. 873 Following the same line of reasoning, it can be shown that 874 quaternionic wavefunctions can refract or diffract. 875 Considering Eqs. V.11 and V.12 again, the reflection did not 876 change the quaternionic wavefunction type — analogous QM 877 wavefunction dynamics had been experimentally observed. 878 Thus, particle PN has constant ensemble average energy of 879 any type, is not lost, and does not emit other particles, i.e., it 880 does not interact despite its quaternionic wavefunction group 881 velocity direction changes (as manifested in the equations) 882 contradicting Classical Physics. Thus, considering Item V-A.b 883 and the definition of quaternionic wavefunction type above: 884 (29) Any particle does not interact if and only if the type of its 885 propagating, reflecting, refracting, or diffracting quaternionic 886 wavefunction does not change. ■An incident particle may 887
16 1≤s≤NS (τ, t,u,p);σis referred to as the State Function1421 (SF); and NS (τ, t,u,p)is the number of possible charstates1422 particle phas. Again, for brevity,1423 στ, t,u, χ(t)(p)→σ(p) and1424 C→ |Ω(τ, t,u, χ, σ(p),p)⟩.(V.45) Let us explore the transition from one snapshot to the1425 next snapshot. (44) The snapshot with SUS St−1≡1426 |U(τ, t−1,u, χ)⟩,t>0, of the trunk or imagined universe1427 at IC (τ, t−1)with US Ut−1≡ |U(τ, t−1,u)⟩is only allowed1428 to transition to the snapshot with SUS St≡ |U(τ, t,u, χ)⟩of1429 the imagination product universe with US Utat the next level1430 IC (τ, t), i.e., only one climb step in the analogue tree. The1431 transition is expressed as1432 St=˚ T(τ, t,u, χ :τ, t−1,u, χ)St−1(V.46) where ˚ Tis the operation that:(a) searches the Eq. V.331433 solution |Ψ(l),l=(τ+t)δ⟩,using the initial condition US1434 Ut−1of the trunk or imagined universe having the SUS St−1;1435 (b) uniquely indexes all elements of the collection |Ψ(l)⟩(refer1436 to Eq. V.40); (c) picks a collection element Utsay with index1437 ut, i.e., the last element of u(τ, t);(d) takes all snapshots of1438 the universe with state Utand uniquely index them; (e) and1439 picks a snapshot of Ut, say with index ht(i.e., the last element1440 of χ(u(τ, t))), which then becomes St.■The transition steps1441 44c to 44e are applied to all USs in the collection and their1442 respective snapshots. Some SUS transitions are indicated as1443 circular disk-connecting dashed arrows in Figure 6.1444 I. History1445 As implied in Section V-H, the successive trunk or imagined1446 universe state transitions over IT during the imaginations for1447 RTI τfrom a given trunk universe with US |U(τ, 0,[1])⟩is1448 undertaken endlessly. From the ordered (symbolized by the1449 operator ⊔) set of SUSs produced by the transitions the entity1450 referred to as an history is defined and expressed as,1451 Hτ, ∞ u, χ=⊔∞ t=0|U(τ, t,u, χ)⟩(V.47) where ∞ u=u(τ, ∞)(in unabridged form) is the HIU till infinity.1452 The order of any SUS in a given history is the IT it corresponds1453 to, e.g., the zeroth SUS of His |U(τ, 0,[1], χ)⟩, a SUS of the1454 trunk universe with US |U(τ, 0,[1])⟩.1455 The non-zeroth SUSs of any history are found by the1456 successive application of Eq. V.46 starting with the zeroth1457 SUS of the history whose source is the initial condition of1458 the Schr¨ odinger Eq. V.33. However, their parameters HIU1459 and SHF cannot be given concrete expressions if no concrete1460 expression is also given to the Hamiltonian in Eq. V.33.1461 Providing the Hamiltonian of a trunk universe is impractical if1462 the universe has myriad of particles. Thus, from here onward,1463 the concrete expressions of the parameters HIU and SHF of1464 any history are not provided, but assumed appropriate for the1465 given qualitative descriptions of the history or part thereof.1466 At any given RTI τ:(45) Imaginator creates all types1467 of histories which have non-zeroth SUSs made through the1468 transition that abides by the 7.1MSchr¨ odinger equation as 1469 described in Item V-H4.44. ■The set of all types of histories 1470 created by Imaginator for RTI τusing the trunk universe with 1471 US |U(τ, 0,[1])⟩(as initial condition of the 7.1MSchr¨ odinger 1472 Eq. V.33) is defined as, 1473 HH =Hτ, ∞ u, χ ∞ u∈R+∞, χ ∈Υ =I(|U(τ, 0,[1])⟩) (V.48) where Iis the Imaginator function that produces the histories; 1474 R+∞=(R+)∞; and Υis the space of SHFs. 1475 And, the probability of an history to exist is defined as,1476 PHτ, ∞ u, χ≡ LM τ, ∞ u, χ 2 (V.49) where 1477 LM τ, ∞ u, χ=lim N→∞ΠN−1 t=0TR (τ, t:t+1,u, χ); (V.50) and 1478 TR (τ, t:t+1,u, χ)=⟨U(τ, t+1,u, χ)||U(τ, t,u, χ)⟩ is the transition probability from SUS |U(τ, t,u, χ)⟩to 1479 |U(τ, t+1,u, χ)⟩. Notice that the parameters u, χ of TR are 1480 just used as tokens and they do not abide by Convention V.43. 1481 1) Event: The history Hτ, ∞ u, χexpressed in Eq. V.47 1482 is defined whereby each of its series SUSs is of a snapshot 1483 containing a number of particles/clones assumed overwhelm1484 ing to account. Thus, in each of the snapshots, only the 1485 particles/clones in a given set Prelevant (defined in Item 1486 V-A.b) to a given context are considered. Consequently, the 1487 population-limited version of the SUS |U(τ, t,u, χ)⟩is denoted 1488 as, 1489 |L(τ, t,u, χ, P)⟩; (V.51) referred to as the State of Population-Limited (trunk or imag1490 ined) Universe Snapshot (SPLUS); is the charstate of the 1491 system comprised of particles in Pat IT t;and is a vector 1492 in the Hilbert space associated to the particles in Pat IT t.1493 The probability for the SPLUS |L⟩to occur is defined as, 1494 P(|L(τ, t,u, χ, P)⟩)= ⟨L(τ, t,u, χ, P)||L(τ, t,u, χ, P)⟩.(V.52) If the SPLUS |L⟩population is restrict to a single particle only 1495 (i.e., P={p}), it becomes the charstate 1496 |L(τ, t,u, χ, {p})⟩=|Ω(τ, t,u, χ, σ(p),p)⟩(V.53) based on the definition of SPLUS and on Convention V.45. 1497 As expressed in Eq. V.47, any history has an infinite 1498 sequence length. Thus, for a given imagined dynamics of 1499 particles in a set Pterminating at IT tf, the history section 1500 considered is limited only from its zeroth to tth fSUS. The 1501 sequence length and particle/clones population-limited history 1502 is referred to as the event, and expressed as, 1503 Vτ, u, χ, P,tf=⊔tf t=0|L(τ, t,u, χ, P)⟩,(V.54) i.e., it is imagined for UR τ. For being constituted by the 1504 SPLUSs corresponding to the lesser-ordered SUSs of the 1505 history H, event is also referred to as the population and 1506
17 duration-limited beginning section (PDLBS) of the history H.1507 Every SPLUS is also referred to as an instance of the event it1508 belongs to. The tth,0≤t≤tf,instance of the event for UR τis1509 defined to correspond to IC (τ, t).In any event expression (such1510 as Vin the LHS of Eq. V.54), the parameters HIU and SHF1511 are simply made as tokens but do not abide by Conventions1512 V.37 and V.43 as there is no IT parameter before the HIU1513 parameter, unlike in the expression of |L⟩in Eq. V.54. The1514 probability of event Vto happen is defined as,1515 PVτ, u, χ, P,tf≡ tf Y t=0 P(|L(τ, t,u, χ, P)⟩)(V.55) through the use of Eq. V.52.1516 All events are derived from histories (based on Item V-I.451517 and Eqs. V.47, V.48, V.51, and V.54) created by Imaginator.1518 Thus: (46) Imaginator creates all types of imagined events Vi 1519 that: have instance transitions abiding by the 7.1MSchr¨ odinger1520 equation (as described in Item V-H4.44);■pertain to a given1521 population Pof particles/clones; imagined for various URs τi;1522 and defined to comprise a set, such as,1523 V={Vi(τi,ui, χi,P,ti)|i∈I⊂R}(V.56) where the IT ti∈Z+is the terminal IT of the event Vi; and1524 the HIU uiand SHF χiparameters are appropriate to the event1525 Vibut not given concrete expressions as mentioned above.1526 As all types of 7.1MSchr¨ odinger equation-abiding events1527 are created, different events Viand Vj,j,i,i,j∈I, in Eq.1528 V.56 can possibly differ by a single feature only. For example,1529 suppose that τj=τi, and tj=ti, but the two clones of the1530 same particle p∈Pare in different locations xiand xjin the1531 events Viand Vj, respectively. This implies that clones of the1532 same particle can traverse different paths in different events.1533 Considering Eqs. V.47, V.48, V.51, and V.54, each event1534 Vi∈Vin Eq. V.56 can be the PDLBS of various histories1535 Hj, such that, one can define the set,1536 HT =Hjτj,∞ uj, χj j∈J⊃I. Thus, the set of indices J⊂Ris larger than Idefined in Eq.1537 V.56 for the event set V.1538 Using Eqs. V.53 and V.54, if P={p}is a singleton, the1539 event1540 Vτ, u, χ, P,tf=⊔tf t=0|Ω(τ, t,u, χ, σ(p),p)⟩.(V.57) From this event, the probability of particle pto be at space-IT1541 (x,t)is,1542 |Ψ(x,t)|2≡ ⟨Ω(τ, t,u, χ, σ(p),p)||Ω(τ, t,u, χ, σ(p),p)⟩ by using Eq. V.17 and Convention V.20. Thus, in imagination,1543 the quaternionic wavefunction Ψ(x,t)norm may be derived1544 from the imagined event. Further, particle pmay traverse a1545 stochastic path (according to its quaternionic wavefunction1546 norm square based on Eq. V.10) — in the imagination as1547 IT tprogresses — similar to that illustrated in Figure 5(a)1548 whose RT taxis replaced with IT t.1549 2) Path bunch: Let a non-interacting particle pclone tra1550 verses a continuous path pp.If the tangential direction at any 1551 given point on path ppis identical to particle pquaternionic 1552 wavefunction group velocity direction at that point, then path 1553 ppis referred to as the ray.The set of all paths, referred to 1554 as the path bunch, has elements beginning and terminating 1555 at coordinates, and having passage features, similar to that 1556 of a ray, and being traversed in one-to-one correspondence 1557 (as described in Item V-E6.30) by the clones of the same 1558 non-interacting particle. For example, a ray ruin a diffraction 1559 experiment starts at an emitter molecule, passes by the upper 1560 slit, and then terminates at a screen molecule. Thus, each 1561 element of a path bunch PU established from ray rustarts 1562 at the same emitter molecule, passes by the upper slit, and 1563 then terminates at the same screen molecule. Notice that the 1564 ray is an element of the path bunch. 1565 Now, there could be several path bunches in one experiment, 1566 e.g., the path bunches PU and PL have elements that pass by 1567 the upper and lower slits of an electron diffraction experiment, 1568 respectively. In case of several different path bunches PBi,j,1569 the set PB =SNBj,NP i,j=1PBi,jindicates that each of the of NP 1570 particles pjhas NBjdifferent path bunches, each has elements 1571 simultaneously traversed by the particle pjclones in one-to1572 one correspondence. 1573 In accord with the definition of ray above, the NB number 1574 of path bunches of the same particle palso have NB rays 1575 rk,k∈K={k|1≤k≤NB },and each traversed by a differ1576 ent particle pclone clk,k∈K,that has a corresponding ray 1577 rk-travelling quaternionic wavefunction Ψk.The wave nature 1578 of quaternionic wavefunction hints to it being superimposable. 1579 Suppose all the rays terminate at the same point. It is then 1580 defined that: (47) The probability amplitude Ψof particle p1581 at the rendezvous terminal coordinate is defined as the sum 1582 Ψ = Pk∈KΨk, thereby Ψ∗Ψis its existence probability at this 1583 coordinate. ■1584 J. Selector in 7.1M1585 Let us now investigate the implementation of 7.1MSelector. 1586 The latter randomly chooses, being the action of its function 1587 Sat every RTI τ, one history Hτ, ∞ u, χ=S(HS ),∃∞ u∈1588 (R+)∞, χ ∈Υ, among the Imaginator outputted histories in 1589 HS =I(|U(τ, 0,[1])⟩)(defined in Eq. V.48) based on their 1590 probability attribute (defined in Eq. V.49). In the chosen 1591 history H(defined in Eq. V.47), the first SUS is that of 1592 the snapshot at IC (τ, 1).Suppose its state is denoted as 1593 S1≡ |U(τ, 1,u, χ)⟩; it is indicated as the small square 1594 enclosing a circular disk and contained inside the large square 1595 at IC (τ, 1)level in Figure 6; and it has particles interacting 1596 or created at IC (τ, 1)that comprise the set CN. Thus, ap1597 plying Item V-D2.21, let us define that: (48) All particles 1598 in CN are consciously perceived at RTI τ+1 from being 1599 subconsciously perceived for being Imaginator outputs (based 1600 on Item V-H.42), i.e., analogous to the Heisenberg theory in 1601 [7]. Further, all particles not in CN but in the snapshot source 1602 are subconsciously perceived at RTI τ+1. ■1603 Now, as reality for us humans is that which we are conscious 1604 of, as being conscious of any mental concept (based on 1605
18 Item II.e) may be due to neuronal group firing, and as this1606 firing is analogous to the neuronal group-generated particle1607 interaction (as mentioned in Section V-D2), it is then defined1608 that: (49) All the particles (created or interacting) comprising1609 CN constitute the singular reality at RTI τ+1 (i.e., at the1610 RTI next to τ), or at RT (τ+1)δ.Further, all particles p1611 (i.e., consciously or subconsciously perceived) — at definite1612 charstate |Ω(τ, 1,u, χ, σ(p),p)⟩and in the source universe1613 of the first snapshot with state S1=|U(τ, 1,u, χ)⟩of the1614 chosen history Hτ, ∞ u, χ— are used by Imaginator to create1615 histories for RTI τ+1. ■Thus, they constitute the trunk1616 universe where each of them is considered as having several1617 possible charstates (based on Item V-G.40) and the universe1618 state, say Uτ+1≡ |U(τ+1,0,[1])⟩, is the initial condition of1619 7.1MSchr¨ odinger Eq. V.33 for the history creation at RTI1620 τ+1. Consequently, for computational purpose only, the trunk1621 universe is devised in such a manner despite each of its1622 constituting particles is actually at definite charstate at RTI1623 τ+1, i.e., it is not the physical universe (reality). By being1624 singular, the 7.1M-defined reality is simpler in description than1625 that in the many-world interpretation by Everett [54].1626 Implied in Item V-J.49, all particles/clones not found in1627 the snapshot at SUS S1but in the trunk universe with US1628 U(τ, 0,[1])and/or in the snapshots with SUSs belonging to all1629 the unselected histories are not considered in the Imaginator1630 actions for RTI later than τ.1631 Considering that the US |U(τ, 0,[1])⟩is used for RTI τby1632 Imaginator to output histories used by Selector to obtain the1633 US |U(τ+1,0,[1])⟩, one obtains the recursive equation,1634 |U(τ+1,0,[1])⟩=S(I(|U(τ, 0,[1])⟩)) ,0≤τ < ∞.(V.58) This implies that the step starting from Imaginator use of the1635 current trunk universe and ending at the formation of a new1636 trunk universe is repeated sequentially, a process referred to1637 as the trunk universe evolution.Further, there is no negative1638 UR, and that the 7.1Minterpretation does not allow time1639 reversal. Furthermore, the new trunk universe (with US Uτ+1)1640 is computed using the last trunk universe (with US Uτ)1641 through the stochastic imagination and selection by Imaginator1642 and Selector, respectively. Thus, its comprising particles have1643 non-essential attributes that cannot be predicted, due to the1644 randomness (which follows the statistical constraint implied1645 in Eq. V.46), using the attributes of the particles in the last1646 trunk universe. Thus, so to speak, one needs to roll the dice1647 first before the trunk universe at a given RTI can be computed1648 given the trunk universe at the last RTI. Therefore, it is not1649 possible to compute the US |U(τ+y,0,[1])⟩, for example,1650 from the US |U(τ+x,0,[1])⟩,x+1<y, if one does not1651 successively compute (roll the dice), beforehand, the USs1652 |U(τ+w,0,[1])⟩,x<w<y,∃w,x,y∈Z+. It follows1653 that, Eq. V.58 expresses an inexorably irreducible computation1654 in determining the trunk universes across RTI. As explained1655 by Wolfram, an inexorably irreducible series of computations1656 corresponds to the flow direction of time [55]. This is the 7.1M 1657 interpretation of time flow direction.1658 1) Wave-particle dilemma: The random pick of Selector1659 is now used to resolve the wave-particle dilemma in QM.1660 To start, it is proposed that: (50) At every IT, every free 1661 particle is substituted to only one of its clones randomly 1662 chosen according to its existence probability at the location of 1663 these clones in the history randomly chosen by Selector. At the 1664 moment of substitution, only the substituted clone is allowed to 1665 interact with, and transfer ensemble average total energy to or 1666 from, the other particles. And, if in the interaction the particle 1667 banishes, then all of its clones inside the chosen history banish 1668 as well (clones in unselected history are discarded after the 1669 selection as mentioned above). ■Thus, if the substituted 1670 clone interacts with a single particle only, its energy can be 1671 shared only with that particle and not to other particles near 1672 the interaction point. This interaction mode is that of particles 1673 and not of any wave whose energy is distributed to the objects 1674 its front encounters. Thus, any free massive particle randomly 1675 moves according to its wave-like quaternionic wavefunction 1676 (as explained at the beginning of Section V-E), but has particle 1677 nature when interacting, i.e., there is no wave-particle dilemma 1678 in 7.1M.1679 According to Item V-J1.50, each of the infinite number of 1680 clones has ensemble average total energy equal to that of their 1681 singular parent. If all the clones interact to particles not their 1682 parent, then there is infinite energy available. However, only 1683 one of the clones (e.g., the one that reaches the screen particle 1684 s3illustrated in Figure 5(a)) is substituted by its parent particle 1685 and would be able to interact. Thereby, if interacting with 1686 another particle, the energy of the non-substituted clones (e.g., 1687 the clones that reached the screen particles other than s3) are 1688 unused during the interaction, i.e., the infinite energy is an 1689 illusion. 1690 2) State series over RTI: Let us now investigate the free 1691 particle dynamics over RTI (or UR). To begin, suppose Eq. 1692 V.57 pertains to a free particle dynamics over IT. Further, 1693 suppose Imaginator creates histories for every RTI τ+j,0≤1694 j≤te,∃τ, te∈Z+, containing the dynamics and comprising 1695 the set, 1696 HRj=Rijτ+j,∞ uij, χij ij∈Ij⊂R(V.59) where HIU ∞ uijand SHF χijare appropriate to the histories but 1697 not given concrete expressions as cited in Section V-I. 1698 Suppose the history Ri0∈HR0,∃i0∈I0,has the creation 1699 of a particle pat the snapshot with SUS being its first, say 1700 with charstate 1701 |Ωτ+0,1,ui0, χi0, σ(p),p⟩, and this particle then randomly moves as free at the snapshots 1702 with SUSs being its succeeding SUSs (over IT). Suppose 1703 further that it is chosen by Selector at UR τ.Thus, particle pis 1704 created (i.e., consciously perceived according to Item V-D2.21) 1705 and, at UR τ+1, present in reality and in the trunk universe 1706 (based on Item V-J.49) with US, say |U(τ+1,0,[1])⟩. Using 1707 the latter in Eqs. V.33 and V.46, all the histories (constituting 1708 HR1) created by Imaginator for RTI τ+1 are restricted to have 1709 first SUSs which are of snapshots, each has particle/clone p, as 1710 the latter is a free particle (based on Item V-G.40). Inevitably, 1711
19 the chosen history by Selector at RTI τ+1 has first SUS that is1712 of the snapshot containing particle/clone p, say with charstate1713 |Ωτ+1,1,ui1, χi1, σ(p),p⟩,∃i1∈I1. It follows that this particle is in the trunk universe, say with US1714 |U(τ+2,0,[1])⟩, at UR τ+2 and subconsciously perceived as1715 it did not interact (according to Item V-D2.21) at IC (τ+1,1) 1716 for being free. This illustrates the trunk universe evolution1717 (defined in relation to Eq. V.58) from RTI τto τ+1.1718 For the trunk universe evolution from RTI τ+2 until1719 τ+te, the free particle pis always subconsciously perceived1720 (according to Item V-J.49) from RTI τ+3 until τ+te+1, i.e.,1721 it is only consciously perceived at RTI τ+1. The evolutions1722 yield a series over RTI of particle pcharstates as1723 SR =⊔te j=0|Ωτ+j,1,uij, χij, σ(p),p⟩,ij∈Ij.(V.60) As they involve stochastic actions (by Imaginator and Selec-1724 tor), the HIU uijand SHF χijof Eq. V.60 are viewed as the1725 sample values at RTI τ+jof the random variables HIU ujand1726 SHF χj, respectively. Eq. V.60 implies that the free particle p1727 can be at stochastic location at RTI τ+jaccording to the1728 PDF ⟨Ωτ+j,1,uij, χij, σ(p),p||Ωτ+j,1,uij, χij, σ(p),p⟩1729 which by Convention V.20 can vary in space. Thus, the1730 stochastic path traversed by the free particle pcan be the one1731 illustrated in Figure 5(a) having RTI as the domain.1732 K. Fire fly interact event1733 Let us consider another free particle dynamics. To begin,1734 consider the event Ni0,∃i0∈I0⊂R,created by Imaginator1735 for RTI τ∈Z+and has the following instances (defined in1736 Section V-I1) occurring at increasing ITs t(i.e., at IC (τ, t)):1737 [1) A particle αis just emitted, located at aij, and then1738 move as free particle.1739 [mij) It reaches the location bijto meet another particle β.1740 [tij)The particles αand βinteract (e.g., αabsorbed by1741 β).1742 The series of instances similar — in IT order but may differ in1743 IT values, and particle names, locations, and flight pattern —1744 to the last-enumerated dynamics from IT one till tijinclusive1745 defines the event referred to as the fire-fly-interact (FFI) event.1746 The enumerated FFI instances are either about an interaction1747 or particle arrival, excluding the movements of free particle α.1748 The instances wherein at least one particle interacts, reflects,1749 refracts, diffracts, or arrives at other particles are referred to1750 as the remarkable instances. The instances exclusive between1751 any two successive remarkable instances are referred to as1752 mediocre instances. If a remarkable instance is found at the1753 same IC of every event set element, it is referred to as the1754 common remarkable instance (CRI) of the event set.1755 Event Ni0is about the particles comprising the set1756 P={α, β}. It is an element of the event set,1757 ENj=Nijτ+j,vij, φij,P,tijij∈Ij⊂R(V.61) with j=0, and HIU vijand SHF φijappropriate to the events1758 in the set but not given concrete expressions (as mentioned1759 in Section V-I). The events in the set are the PDLBSs of the 1760 histories comprising the set 1761 HYj=Yejτ+j,∞ vej, φej ej∈Ej⊃Ij.(V.62) 1) Reenactment: Suppose the FFI event Ni0describe above 1762 is the PDLBS of the history Ye0∈HY0. Thus, history Ye0first 1763 SUS is of the snapshot where the αparticle firing occurs, 1764 i.e., at IC (τ, 1).Suppose further that it is chosen by Selector 1765 at RTI τ. Thus, particle αis in the trunk universe, say with 1766 US Uτ+1≡ |U(τ+1,0,[1])⟩, and at space-RTI ai0, τ +1as 1767 implied by Item V-J.49 and indicated in Figure 7. The US Uτ+11768 is used as the seed of the trunk universe evolution wherein 1769 particle αis a free particle, as implied in Item V-K.[1). It is 1770 straightforward to show that there exist an evolutionary stage 1771 RTI τ+x,x=mix−1>0,when Imaginator creates the 1772 event Nix∈ENx,∃ix∈Ix,wherein αand βmeet at space1773 IT bix,1or at the Additional Imagination Coordinate (AIC) 1774 {x,1}≡the IC (τ+x,1)indicated in Figure 7. Suppose at this 1775 stage Selector chooses the history Yix∈HYx(defined in Eq. 1776 V.62) with PDLBS being the event Nixwhich exists based on 1777 Item V-I1.46. It follows that the meeting happens at space-RTI 1778 bix, τ +mixbased on Item V-J.49. Suppose further that in the 1779 next trunk evolutionary step RTI τ+y,y=x+1=tiy−1,1780 Imaginator creates the event Niy∈ENy,∃iy∈Iy,wherein the 1781 particles αand βinteracts at AIC {y,1}indicated in Figure 1782 7. Suppose furthermore that at this step Selector chooses the 1783 history Yiy∈HYywith PDLBS being the event Niywhich 1784 exists based on Item V-I1.46. Thus, the interaction happens, 1785 and the particles αand βare in reality (based on Item V-J.49), 1786 at RTI τ+y+1=τ+tiyindicated in Figure 7. 1787 RTI τ + 1 τ + t - 2 τ + t - 1 τ + t ● ● ● ● ▪ ▪ ▪ {1, m > 1} {m – 1, 1} {t – 1, 1} ▪ ▪ ▪ meet interact ix fire ix iy iyiyiy Fig. 7. Series of AICs corresponding to the snapshots in the Selector chosen episodes created by Imaginator for the RTI indicated at the horizontal axis. Some AICs correspond to the CRIs such as the particle αand βmeeting and interaction. To recap, the defined sequence of remarkable instances of 1788 the FFI event Ni0that transpire at IT 1 <mi0<ti0also 1789 transpire at RTI τ+1< τ +mix< τ +tiy, respectively, i.e., 1790 the remarkable instances of Ni0over IT can be reenacted at 1791 the same order but over RTI. 1792 2) Shortening: In Eq. V.61, let us set tij= Ξ −j,Ξ∈1793 Z+,0≤j<Ξ.And, suppose that the CRIs of the events in 1794 ENjthat occur at IT IL −j>0 are also the CRIs of the events 1795 in EN0that occur at IT IL,0<IL ≤Ξ. For example, in the 1796
20 event Nij,∃ij∈Ij, the αand βinteraction occurs at IT ti0−j1797 which in the event Ni0occurs at IT ti0.Due to the restriction1798 IL −j>0, the CRIs in event Ni0at IT zero till IL −1, are not1799 found in the event Nij,j≥IL, i.e., the latter event is shorter1800 than the former. Thus, the set1801 SE ≡nENj0≤j<Ξo(V.63) of the event sets ENjwhere the shortening occurs is referred1802 to as the Set of Shortening Events (SSE).1803 3) More definitions: Before proceeding, consider more1804 definitions: (a) The general scenario of particles is defined1805 as the sequence of interactions, and/or propagations of the1806 particles whereby they are generally defined. For example,1807 a general scenario with an electron is the sequence starting1808 from its emission, free flight, slit passage, free flight, and1809 then absorption, where its location across the sequence and1810 the features of the particles it interacts are unspecified. (b) If1811 a general scenario SA (or SB) cannot alter another general1812 scenario SB (or SA), it and the particles in it are referred to1813 as irrelevant to SB (or SA). If the particles in SA (or SB)1814 interact with the particles in SB (or SA), their interaction is also1815 referred as irrelevant to SB (or SA). For example, in an elec-1816 tron double-slit interference experiment, the non-experiment1817 interfering beaker displacement, the ambient electromagnetic1818 radiation, the experiment-immersing atmospheric gas, and the1819 constant thermal agitation on the slits and screen molecules,1820 are irrelevant to the general scenario of electron emission,1821 flight, slit passage, flight, and then absorption to a screen1822 particle.1823 VI. Electron Double-slit Diffraction1824 Let us apply the provided 7.1Mattributes to the two-1825 dimensional electron double-slit diffraction experiment whose1826 schematic diagram is illustrated in Figures 8(a)–(f). In the1827 figures: (a) All the experiment-relevant particles are depicted1828 as static and situated at their ensemble average location using1829 their respective PDF, such as the one defined in Eq. V.20. This1830 depiction is advocated by their stochastic locations (as sup-1831 ported in Item V-D.18) which cannot be depicted in any still1832 illustration such as the figures. (b) The emitter is represented1833 as a shaded square and constituted of the particles comprising1834 the set EM including the molecule em.The latter exists, and1835 made as the RP, from its emission, until the absorption by the1836 screen particle z,ofelectron e.(c) The electron eclones pass1837 through the two equal-sized slits suitable for its quaternionic1838 wavefunction to diffract. The slits are formed by the non-1839 illustrated but existing upper and lower blocks and the middle1840 block containing particles uand das shown. (d) All the blocks1841 are comprised of particles constituting the set BP and have1842 dimensions whereby any emitted electron clone not passing1843 through the slits strikes, and then absorbed by, them. They lie1844 in an imaginary vertical plain and are made of materials that1845 do not interact with the electron clones passing through the1846 slits, and prevent these clones from tunnelling through them.1847 (e) They are parallel to the non-illustrated but existing flat1848 black screen. The latter is constituted of particles comprising1849 the set SP including particle zand has dimensions whereby all1850 the slit-passing electron clones strike, and then are absorbed 1851 by, it. (f) The ensemble average particle zlocation ¯ z(i.e., 1852 relative to the RP) is at one end of an imaginary horizontal line 1853 whose other end is at the RP, i.e., at 0.The line perpendicularly 1854 strikes the screen at ¯ zand the ensemble average middle block 1855 vertical center ¯ m(relative to the RP). Let the distances from ¯ m1856 to the RP, ¯ z, and the ensemble average center of any slit be a,1857 b, and c, respectively. (g) The particles z,u,dhave RP-relative 1858 instantaneous locations denoted as the subscripted vectors 1859 z,u,d, respectively. Any of them or electron e(or clone) 1860 has instantaneous location being the center of the circular 1861 disk enclosing its label (e.g., cu0,3indicated in Figure 8(a)). 1862 The continuous (or dashed) circles represent electrons/clones 1863 belonging to the trunk (or imagined) universes. 1864 A. Events 1865 Consider first a particular diffraction experiment event Ea01866 having two rays pfuand pfdrespectively elements of the 1867 path bunches PFuand PFdof electron e, and illustrated 1868 approximately as the upper and lower connected arrows in 1869 Figures 8(a)–(f), respectively. Let it be an element of the event 1870 set EE0whose CRIs are at the following ITs: 1871 (1) Electron eis emitted by particle em ∈EM.1872 (fd)One of its clones cd passes by the lower slit as this 1873 clone randomly situates along ray pfd.1874 (fu) Another of its clone cu passes by the upper slit as 1875 this clone randomly situates along ray pfu.1876 (t0)It is substituted to only one of its clones cd and cu 1877 that arrives at screen particle z∈PS in accord with 1878 Item V-J1.50. 1879 (ns) It is absorbed by particle zwhich then creates a white 1880 spot on the black screen. 1881 In accord with the enumerated CRIs, the set of experiment1882 relevant (defined in Item V-A.b) particles is defined as,1883 DP ≡{EM,BP,SP,e}.(VI.1) Further, although 1 <fd,fu<t0<ns but the order of the ITs 1884 fdand fudepends on the stochastic clone space-IT coordinate 1885 on the rays pfdand pfu.1886 Now, with regards to electron e:(A) Right after its emission 1887 by molecule em and right before its absorption by particle z,1888 it is an approximated-free (defined in Item V-A.d) particle. 1889 Without loss of generality, it is assume as free. (B) In its 1890 slit passage, it does not interact with the block particles as 1891 its associated quaternionic wavefunction type is maintained in 1892 the passage (base on Item V-E3.29). (C) However, its gen1893 erating neuronal group encounters with the neuronal groups 1894 generating the particles in BP that causes its associated quater1895 nionic wavefunction to diffract, as supported in Section V-E3. 1896 (D) Its quaternionic wavefunctions are approximated as non1897 dispersive 7.1MGaussian wave packets Γuand Γdtravelling 1898 along the rays pfuand pfd, respectively, with constant group 1899 speed GS and magnitude peaking at the spatio-IT (0,0), where 1900 0is the RP location as defined in Item VI.b. 1901
21 e u 0 d 0 z 0 (a) (b) (c) (d) (e) cu 0,3 cu 0,5 cd 0,2 cd 0,4 e u 1 d 1 z 1 cu 1,2 cu 1,4 cd 1,1 cd 1,3 e u 2 d 2 z 2 cu 2,1 cu 2,3 cd 1,1 cd 2,2 e u 3 d 3 z 3 cu 2,1 cu 3,2 cd 1,1 cd 3,1 (f) e u 4 d 4 z 4 cu 2,1 cu 4,1 cd 1,1 cd 3,1 e u 5 d 5 z 5 cu 2,1 cu 4,1 cd 1,1 cd 3,1 c u c d f u f d b l o c k ^ ^ ^ ^ m i d d l e b l o c k m i d d l e b l o c k m i d d l e b l o c k m i d d l e b l o c k m i d d l e b l o c k m i d d l e Fig. 8. Figures (a) to (f) illustrate the events corresponding to RTI τ+j,j=0 to 5, respectively, in an electron double slit experiment. In them, the shaded, unshaded continuous-perimeter, and unshaded dashed-perimeter circular disks represent the original, the trunk universe contained, and the imagined clone particles, respectively. The encircled bold letters indicate the vector location of the particles represented by the circle. Each encircled italic letter and its subscript indicate the average location and AIC, respectively, of the particle represented by the circle. The dashed arrows, some indicated by cu,cd,fu,fd, represent the peak quaternionic wavefunction displacement for the duration of Θ. The shaded square represents the electron emitter. The above descriptions prepare for the definition of the1902 diffraction experiment events which start at IC (τ+j,0), end1903 at τ+j,tj,∃τ∈Z+,0≤j≤4, and comprise the set1904 EE j=Eijτ+j,wij, ϱij,DP,tjij∈Ij(VI.2) where Ijis a set of uncountably infinite alphanumeric single-1905 tons and composites;1906 tj=(5−j)(VI.3) is the IT when electron earrives at screen particle zat a given1907 j;and HIU wijand SHF ϱijare appropriate for the event Eij 1908 but not given concrete expressions as mentioned in Section1909 V-I.1910 The only event in EE jdepicted in one of Figures 8(a)–(f) is 1911 the one having electron eclone paths being the rays pfuand 1912 pfd.Any event in EEjhas at most six SPLUSs, such as, 1913 Eij=⊔tj t=0|Mτ+j,t,wij, ϱij,DP⟩ ∃ ij∈Ij.(VI.4) And its RTs have interval approximated as 1914 Φ = L 4δGS δ(VI.5) instead of δ(as indicated in Item V-C1.16), to simplify the 1915 foregoing discussions, where Lis the length of every ray pfu1916 and pfd.1917 The events in EE jalso belong to the SSE 1918 E≡nEE j0≤j≤4o,(VI.6)
22 Thereby, the depicted events in Figures 8(a)–(e) are illustrated1919 as shortening as jincreases (as explained in Section V-K2),1920 such that, the trunk universe electron erepresented in these1921 figures as continuous circles do not belong to them.1922 For every event in EEj, Imaginator has created for RTI1923 τ+jall types of 7.1MSchr¨ odinger equation-abiding histories1924 Ikj(based on Item V-I.45), whereby each has this event as1925 its PDLBS. Thereby, the creation process yields the histories1926 defined as comprising the set,1927 HIj=Ikjτ+j,w∞ kj, ϱkjkj∈Kj⊃Ij.(VI.7) It is emphasized that the histories in HIj,0≤j≤4,may not be1928 the only ones created by Imaginator from the a trunk universe1929 (according to Item V-I.45), say with US |U(τ+j,0,[1])⟩, at1930 RTI τ+j.1931 B. Imagination across ITs1932 Let us now investigate an imagined electron edynamics1933 over IT in the diffraction experiment. To begin, consider1934 the following suppositions for RTI τ, ∃τ∈Z+:There is1935 a trunk universe with US U0≡ |U(τ, 0,[1])⟩that contains1936 the particle set DP −e(the set DP defined in Eq. VI.1 less1937 electron e), and used by Imaginator to create various imagined1938 universes (following the scheme explained in Section V-H3).1939 One of the imagined universes is created for IT t=1 and1940 has US U0,1≡ |Uτ, 1,wa0⟩,∃a0∈I0.Further, it has a1941 snapshot in SUS S0,1≡ |Sτ, 1,wa0, ϱa0⟩(wa0, ϱa0and I0 1942 are defined via Eq. VI.2) where the electron eis fired from1943 the emitter molecule em.In turn, the SUS S0,1has SPLUS1944 M0,1≡ |Mτ, 1,wa0, ϱa0,DP⟩(defined in Eq. VI.4) being the1945 first in the event Ea0, which say is the PDLBS of the history1946 Ia0(expressed via Eq. VI.7). Its electron ehas charstate1947 Ω0,1≡ |Ωτ, 1,wa0, ϱa0, σ(e),e⟩ and located at xa0(with probability ⟨Ω||Ω⟩=1948 Ψ(xa0,(τ+1)Φ) 2in accordance with Eq. V.20) being1949 the center of the circular disk inside the square disk in Figure1950 8(a).1951 Now, in event Ea0and according to Item VI-A.A, electron1952 eis a free particle starting at IT two (right after being emitted1953 by molecule em) until IT t0(when it arrives at screen particle1954 z). Thus, applying the approach provided in Section V-J2, one1955 obtains its series of charstates,1956 SD =⊔t0 t=2|Ωτ, t,wat, ϱat, σ (e(t)),e(t)⟩(VI.8) where e(t) is its clone label being the tth element of the vector1957 [#,cd,cu,cd,cu](the symbol “#” denotes a dummy); and |Ω⟩1958 determines its probability (using Eq. V.20) of being randomly1959 located (as explained in Section V-D1) at cd0,2,cu0,3,cd0,4,1960 cu0,5on ITs two to five, respectively, when it substituted its1961 clones cu and cd (in accord with Item V-J1.50). Evidently from1962 Figure 8(a), it is transferring randomly over the mentioned ITs.1963 However, according to Items V-E3.29 and VI-A.A, it does not1964 emit nor absorb any particle despite changing transfer/velocity1965 direction for being a free particle.1966 In event Ea0and in accord with Item V-G.40, the transfer1967 of the electron e 7.1Mwave packet peak from the beginning1968 to the end of any arrow illustrated in Figures 8(a)-(f) is for a 1969 duration of one Φ(defined in Eq. VI.5) and by a distance of 1970 AL ≡ΦGS ≈L/4 assumed to be the arrow length also. Further, as illustrated in 1971 Figure 8(a), the electron equaternionic wavefunction group 1972 velocity and arrow direction cdis towards the location cd0,21973 (not necessarily inside the arrow length) at IT t=2 of the 1974 electron clone cd transferring along the lower ray pfd.1975 Notice in the event Ea0depicted in Figure 8(a) that clone cu 1976 is not shown for IT t=2 as electron esubstitutes only clone 1977 cd at this IT in accord with Item V-J1.50. At IT t=4, the 1978 wavelike electron equaternionic wavefunction diffracts at the 1979 slits (aligned with Item V-G.40) thereby the clones cu and cd 1980 (do not interact with the slit block particles as implied in Item 1981 VI-A.A) change directions (without emitting nor absorbing 1982 particles based on Item V-E3.29) toward z0from cuand cd1983 to fuand fd, respectively. At IT t=5, electron eby chance 1984 substitutes the clone cu (in accord with Item V-J1.50) which 1985 appears at cu0,5that coincides with the location z0of the screen 1986 particle z.The coincidence is expressed in Figure 8(a) as the 1987 overlapping of circles corresponding to clone cu and particle 1988 zto distinguish them. At the coincidence space-IT coordinate 1989 (z0,5), the sum WF = Γu+ Γw(justified in Item V-I2.47) of 1990 the ray-traversing wave packets Γuand Γd(defined in Item 1991 VI-A.D) yields the electron eprobability WF∗WF to exist at 1992 this coordinate. At the coordinate, the electron eexistence 1993 does not imply that it and particle zinteract as implied by 1994 Item VI-A.(t0). However, they interact at IT t=6 thereby 1995 creating a white spot on the black screen at particle zlocation 1996 (as cited in Item VI-A.(ns)). 1997 As just presented, the electron eclones in event Ea0transfers 1998 from molecule em to particle zvia the slits. Now, consider a 1999 fictitious diffraction experiment event Zwherein the clones 2000 from em do not pass by the slits but rather tunnels through 2001 the blocks (possible based on Item V-D1.B) going to z.2002 However, the neuronal groups generating electron eand the 2003 block particles encounter to form a quaternionic wavefunction 2004 (supported in Item V-E1.25) which travels from molecule em 2005 towards the blocks but does not penetrate the blocks due to the 2006 block material type according to Items V-G.40 and VI.d. Thus, 2007 there is no probability for the electron eclones to travel from 2008 em, tunnel over the blocks, and then proceed to the screen. 2009 Therefore, the fictitious event Z<EE0.2010 C. Imagination across RTIs 2011 The investigation in Section VI-B is only for a single RTI 2012 τ.Let us now explore the diffraction dynamics at later RTIs. 2013 To begin, one of the imagined histories created by Imaginator 2014 for RTI τis randomly chosen by Selector according to their 2015 probabilities (defined through Eq. V.49). Suppose the history 2016 Ia0(defined through Eq. VI.7) is chosen by Selector at RTI 2017 τ, has first SUS S0,1, and PDBLS event Ea0.The latter has 2018 first SPLUS M0,1containing the set DP of relevant particles 2019 including electron ejust fired by molecule em at RTI τ.2020 Consequently, eand em are consciously perceived (i.e., they 2021 are in reality at RTI τ+1 based on Item V-J.49) and in a 2022
23 new trunk universe, say with US |U(τ+1,0,[1])⟩, constituted2023 by all the particles in the snapshot in SUS S0,1.Being in the2024 snapshot, the particles e,u,d,z∈DP are indicated in Figure2025 8(b) as circles with continuous perimeter.2026 Next, using |U(τ+1,0,[1])⟩as trunk universe US, Imag-2027 inator creates events for RTI τ+1; One of them is event2028 Eb1∈EE1,∃b1∈I1(defined through Eq. VI.2), terminates2029 at IT t1=5−1 (related in Eq. VI.3) and is the PDLBS of the2030 history Ib1which has first SUS S1,1.In event Eb1, electron eis2031 located by chance at cd1,1,cu1,2,cd1,3,cu1,4when it substitutes2032 at ITs one to four, respectively, the clones cu and cd illustrated2033 in Figure 8(b) as dashed circles for being in imagination. In2034 the figure, electron eis indicated as a continuous circle for2035 being absent at the emitter in event Eb1∈EE1⊂Ewhich is2036 an SSE.2037 Suppose at RTI τ+1, Selector chooses the history Ib1.Thus,2038 following the argument above, electron esubstitutes its clone2039 cd at space-IT coordinate cd1,1,1, and is in the latest trunk2040 universe say with US |U(τ+2,0,[1])⟩. However, it does not2041 interact, such that, it is subconsciously perceived at RTI τ+22042 according to Item V-J.49. Being in the snapshot at SUS S1,1,2043 the particles cd1,1,u,d,z∈DP are illustrated in Figure 8(c) as2044 circles with continuous perimeter.2045 According to Item VI-A.A, electron eis assumed as free in2046 the imaginations by Imaginator for each RTI τ+j2≤j≤4,2047 to create the events in EE j(defined in Eq. VI.2) terminating at2048 IT tj(defined in Eq. VI.3), such that, the approach in Section2049 V-J2 can be applied to the events. The application outcomes2050 for RTI τ+2 till τ+4 are illustrated in Figures 8(c) to (e),2051 respectively. Over these RTIs, the non-interacting electron eis2052 subconsciously perceived.2053 At RTI τ+4, the by-chance clone cu location cu4,1(i.e.,2054 at AIC {4,1}) indicated in Figure 8(e) coincides with the by-2055 chance screen particle zlocation z4, such that they interact2056 at AIC {4,2}, according to Item VI-A.(ns). Now, suppose2057 Selector chooses at RTI τ+4 the history Ie4,∃e4∈I4,2058 having first SUS S4,1and PDLBS event Ee4depicted in Figure2059 8(e). In the figure, electron eat the emitter and the clones2060 cd1,1,cu2,1,cd3,1it substituted are indicated as continuous2061 circles as they are absent in event Ee4∈EE4⊂Ewhich is2062 an SSE. The snapshot in SUS S4,1contains the free electron2063 e(after substituting its clone cu) which does not interact at2064 AIC {4,1}but rather at AIC {4,2}and hence not in reality2065 at RTI τ+5 (according to Item V-J.49). For being in the2066 snapshot in SUS S4,1, the electron at cu4,1is represented with2067 a continuous-perimeter circle in Figure 8(f) which serves as a2068 summary of the electron journey over RTI.2069 D. Experiment implications2070 Let us now explore the few implications on the investigated2071 diffraction experiment:2072 [a] To begin, suppose the electron echarstate at AIC {4,1} 2073 is,2074 Ω4,1≡ |Ωτ+4,1,we4, ϱe4, σ(cu),cu⟩,∃e4∈I4, when it substituted its clone cu. Suppose further that it has the2075 corresponding wavefunction (defined in Item VI.D)2076 WF = Γu(dscu,t4)+ Γd(dscd,t4) where the Gaussian wave packet (in accord with Eq. V.10), 2077 Γcdspn,tx≡exp ˜ iWNdspn −ωtxΦqPDSpn dspn; (VI.9) tx=t4is defined in Eq. VI.3; c=uor d;pn =cu or cd;2078 dspn is the distance travelled by clone cu (cd) along the ray 2079 pfu(pfd) starting from the emitter, passing through the upper 2080 (lower) slit and ending at a screen particle z;PDSpn dspnis the 2081 non-wavelike electron eexistence PDF at dspn;ωis a given 2082 angular speed; WN =ω/GS ; and GS is a given wave function 2083 group speed defined in Item VI.D. 2084 To illustrate the wavelike electron eexistence probability, 2085 suppose that instead of rays pfuand pfdterminating at the 2086 screen particle zlocation one has rays sfuand sfdterminating 2087 at the screen saway from the ensemble average zlocation 2088 ¯ z, starting from the RP, and passing by the upper and lower 2089 slits, respectively. Suppose further that the electron clones su 2090 and sd travel along their respective rays sfuand sfdby the 2091 distances dssu and dssd =dssu +dl,∃dl ∈R.Conforming 2092 with Section V-E4, either or both of the electron clones can 2093 arrive on RT tsΦ,ts≥t4,at screen location s=lsu ˆ rsu +2094 msuˆ ssu =lsu ˆ rsd +(msu +dl)ˆ ssd,dssu =msu +lsu. The clone su 2095 (sd) travelling on ray sfu(sfd) is displaced from emitter to the 2096 upper (lower) slit center approximately by lsu ˆ rsu (lsu ˆ rsd), and 2097 then displaced approximately by msuˆ ssu ((msu +dl)ˆ ssd) to the 2098 screen location s.Thus, given s, and a,b,cdefined in Item 2099 VI.f, one obtains msu,lsu,dl,ˆ rsu,ˆ ssu,ˆ rsd,ˆ ssd. Consequently, at 2100 space-RTI coordinate (s,ts), the probability for electron eto 2101 exist is SF∗SF where SF = Γu(dssu,ts)+ Γd(dssd,ts), i.e., it 2102 has a wavelike probability to exist along the vertical screen. 2103 By Item VI-A.(ns), electron einteracts with the black screen 2104 molecule at location sthereby creating a white screen spot 2105 on this location. Applying the above logic for every screen 2106 location whereby the electrons fired by the emitter exist proves 2107 that an interference pattern is created on the screen. 2108 The electron particle behaviour can be shown by blocking 2109 one of the slits, say the lower slit, that yields its existence 2110 probability at the screen as Γ∗ u(dssu,ts)Γu(dssu,ts)(based on 2111 Item V-G.40). Thus, no interference pattern is formed on 2112 the screen according to Eq. VI.9 but rather the non-wavelike 2113 pattern due to the PDF PDSsu (dssu).2114 The explanation just provided implies that the existence (or 2115 non-existence) of the screen interference pattern is not because 2116 7.1M, being brain-like, does (or does not) give the interference 2117 if the type of the path traversed by the electrons cannot (or 2118 can) be determined (i.e., akin to the which-way information), 2119 but rather because the experimental set up allows (or prevents) 2120 the clone quaternionic wavefunctions of the same electron to 2121 interfere (or not interfere). 2122 [b] Figure 8(f) illustrates that electron ecan suddenly 2123 change path/ray (or location) from one RTI to the next despite 2124 not interacting. Base on the above discussion, this sudden 2125 change is primarily due to the electron estochastic substitution 2126 of one of its clones (abiding by Item V-J1.50), thereby this 2127 electron is at different path/ray (or location) at AIC {j,1}2128 than at {j−1,1},∀j,0<j≤4. As cited in Item VI-A.A 2129 electron eis free, implying that it: is subconsciously perceived 2130 (according to Item V-D2.22) over the RTIs τ+j,0<j≤4, as 2131
24 explained in Section VI-C; does not require energy to suddenly2132 change location as supported by Section V-D2; and does not2133 interact (according to Item V-A.c), i.e., it does not emit any2134 particle even if charged based on Item V-A.b. Thus, 7.1M 2135 defies Classical Physics in this respect.2136 [c] Figure 8(f) indicates electron eas located at cu2,1and2137 cd3,1(on RTIs τ+2 and τ+3) that, relative to the RP, is2138 prior and pass the upper and lower slits, respectively. Thus, it2139 can be absent at the slits despite its quaternionic wavefunction2140 diffracts on them.2141 [d] Suppose electron eis absorbed by a screen particle zat2142 space-RTI coordinate (z5, τ +5).By the absorption, it is now2143 in a region Rbwhere it is bound to the particle znucleus and2144 hence has no probability to exist outside region Rbwithout2145 interaction. Thus, outside region Rb, its prior to absorption2146 quaternionic wavefunction (when it is still a free particle)2147 is now zero and considered as collapsed. This is the 7.1M 2148 interpretation of the wavefunction collapse in QM. However,2149 inside region Rb, electron eexists and hence has non-zero2150 existence probability. Therefore, its quaternionic wavefunction2151 (which is its charstate as a bound particle) does not collapse2152 inside region Rbdespite its interaction, or measurement if the2153 experiment is treated as such.2154 [e] The investigation and explanation of the diffraction2155 experiment, using the observer-independent 7.1Mattributes,2156 account the particles of the experimental measuring devices.2157 Thus, the attributes do not yield the measurement problem2158 which is the product of the Copenhagen interpretation of2159 QM [56]. Further, they are applicable to any object size, i.e.,2160 they are applicable to investigate the atomic as well as the2161 macroscopic scale experimental objects.2162 [f] As cited in Item V-I.45, Imaginator creates all 7.1M 2163 Schr¨ odinger equation-abiding histories at every given RTI τ.2164 Only one of the created is chosen by Selector, thereby the2165 interacting particles in the snapshot in SUS being its first2166 (i.e., at IT one) are the only ones consciously perceived and2167 constituting a single reality at RTI τ+1 according to Items2168 V-J.48 and 49. Thus, the Schr¨ odinger’s cat is in reality either2169 dead or alive, but not simultaneously, before being observed2170 or not by any conscious being.2171 [g] Some 7.1Mattributes are related to: [A] the Copenhagen2172 interpretation [2] in their definition of non-essential particle2173 attributes as stochastic — supported in Section V-D1; [B] the2174 proposal of Heisenberg [7] explained in Section I in their2175 definition of any particle to exist at any point where its2176 quaternionic wavefunction absolute square is non-zero, and to2177 emerge in reality via interaction (according to Item V-J.49);2178 [C] the Quantum information theory [8] in their definition of2179 particles as information as cited in Item IV.c; [D] the Everett2180 interpretation [54] in their use of a universal wavefunction2181 analogue cited in Eq. V.39; [E] and the theory of Eugene2182 Wigner [16] in their use of the concept of consciousness as2183 cited in Item IV.e.2184 VII. Future work2185 The 7.1Mattributes applied in this paper are use to inter-2186 pret only few mechanics of the non-relativistic Schr¨ odinger2187 equation-abiding particles. Thus, the successors of this paper 2188 to be released in the future will deal on the 7.1Minterpretation 2189 of: (i) the interference pattern formed by massive particles 2190 from independent sources, i.e., analogous to that investigated 2191 in [57]. (ii) the delayed choice between the wave and particle 2192 nature, analogous to that presented in [58]. (iii) STR and 2193 particle spin. Section V-C explains the absence of space based 2194 on the analogy regarding the blind man having an illusion 2195 on the existence of an object at a spatial location based 2196 on the sound he hears ostensibly from the object. The spin 2197 interpretation will also be using the blind man analogy but, in 2198 this case, he has illusion on the existence of a spinning and 2199 running object (say a bullhorn) based on the sound he hears. 2200 (iv) delayed choice quantum eraser [59], entanglement swap2201 ping [60], and entanglement of non-coexisting particles [61]. 2202 In their interpretations, particle entanglement will be defined 2203 as the state of several information/particles being generated 2204 by many neuronal groups with at least one common and one 2205 different neurons (supported by Item II.C). Having at least 2206 one common neuron, the neuronal groups constitute a single 2207 inseparable system, and thus the effect on one of them affects 2208 the rest instantaneously. Thus, the interaction of one particle 2209 instantaneously affects its entangled particles (e.g., if particle 2210 Al interacts with spin up then its entangle particle Bb will have 2211 spin down instantaneously on the interaction/measurement) 2212 where prior to the interaction they have stochastic but related 2213 non-essential attributes (e.g., Al and Bb have spin down and 2214 up, respectively) in conformity with Item V-D.18. (v) Quantum 2215 Information Theory based on its treatment of particles as 2216 information as in 7.1M.(vi) vacuum fluctuation mentioned in 2217 Section V-D and QFT using field of neurons similar to the 2218 methodology in [62]; (vii) GTR based on the 7.1Mprinciple 2219 that particle mass is an element of the particle-generating 2220 neuronal group attribute INGI which influences the particle 2221 spacetime coordinate relative to another particle (according to 2222 Item V-B.7). Thus, mass is envisioned to also influence the 2223 coordinate. (viii) any type of force and dark matter where the 2224 Contrector-like Imaginator action is deemed to play a central 2225 role. (ix) and the time interference experiment similar to that 2226 explored in [63]. 2227 Additional future publications will also provide: (x) a pro2228 posal to experimentally justify 7.1Mprinciples, (xi) and a new 2229 paradigm that uses human neurological dynamics as one of the 2230 foundations (along with physical law symmetry for example) 2231 for formulating physical theories. 2232 VIII. Acknowledgements 2233 This work was inspired by the Blessed Trinity, i.e., God the 2234 Father, God the Son, and the Holy Spirit. It was made through 2235 the intercession of St. Joseph, the Blessed Virgin Mary, St. 2236 Anthony, St. Pancratius, St. Benedict, and St. Michael the 2237 Archangel. Further, it is indirectly supported by my late 2238 parents and other people who had helped me in my life. 2239 The author is also highly indebted to the assistance of the 2240 following kind people that leads to the making of this paper. 2241 They are: Prof. Lev Vaidman, Dr. Julian Barbour, and Dr. 2242 Matthias Egg. 2243
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