Local Quantum Theory with Fluids in Space-Time

Mordecai Waegell1,2

1Institute for Quantum Studies, Chapman University, Orange, CA 92866, USA 2Schmid College of Science and Technology, Chapman University, Orange, CA 92866, USA July 14, 2021

A local theory of relativistic quantum article we lay out the general framework for lo- physics in space-time, which makes all of cal realist collapse-free theory quantum mechan- the same empirical predictions as the con- ics, and work through the simplest examples, ventional delocalized theory in configura- with all dynamics occurring explicitly in space- tion space, is presented and interpreted. time. This realizes an unachieved goal of Ein- Each physical system is characterized by stein, Schrödinger,and Lorentz, who were never a local memory and a set of indexed satisfied with the configuration space treatment, piece-wise single-particle wavefunctions in precisely because it introduced fundamental non- space-time, each with with its own coeffi- locality [2]. The new model makes identical em- cient, and these wave-fields replace entan- pirical predictions to standard quantum theory, gled states in higher-dimensional spaces. and can serve as a full replacement. This model is Each wavefunction of a fundamental sys- consistent with the Lorentz covariant Heisenberg- tem describes the motion of a portion Schrödinger model proposed by Schwinger in of a conserved fluid in space-time, with 1948 [3], and restores the equivalence between the fluid decomposing into many classical the local Heisenberg and Schrödingerpictures. point particles, each following a world-line However, we now know from Bell’s theorem and recording a local memory. Local in- [4,5,6,7,8] that if we wish to maintain inde- teractions between two systems take the pendence of measurement settings, then this is form of local boundary conditions between unavoidably a theory of many local worlds [9]. the differently indexed pieces of those sys- It it important to emphasize here the breadth tems’ wave-fields, with new indexes encod- of Schwinger’s accomplishment. In deriving ing each orthogonal outcome of the inter- quantum electrodynamics in parallel to Fey- action. The general machinery is intro- namn, he obtained a new Lorentz covariant state duced, including the local mechanisms for vector, defined on a single space-like hypersur- entanglement and interference. The expe- face, with information at each point in that sur- rience of collapse, Born rule probability, face restricted to that point’s past light cone. and environmental decoherence are dis- He also obtained the localized Schrödinger-like cussed. A number of illustrative examples dynamics that shows how this state evolves lo- are given, including a Von Neumann mea- cally to the next parallel space-like hypersurface, surement, and a test of Bell’s theorem. and obtained a space-time invariant local interaction unitary for QED. This treatment is at arXiv:2107.06575v2 [quant-ph] 22 Sep 2021 1 Introduction the heart of modern particle physics, but these state vectors are completely inconsistent with Despite insubstantial but influential claims from the configuration-space wavefunctions in preva- the early days of quantum theory, Bohm proved lent use throughout modern quantum founda- in 1952 [1] that it is possible to give a straightfor- tions and information theory. ward realist interpretation of quantum mechan- The present model is an attempt to interpret ics with particles in space-time. However, in that the empirical data from table-top quantum ex- theory the underlying dynamics occurs in higher- periments, rather than high energy particle colli- dimensional configuration space, resulting in ex- sions, given Schwinger’s theory, by deconstruct- plicitly nonlocal dynamics in space-time. In this ing his new state vector into more familiar single-

1 particle spatial wavefunctions. This turns out theory. The empirical experience of collapse and to be the natural theoretical framework for re- many of its consequences are explained later, but fining the local Schrödingerpicture of the Par- for now, the right intuition is that each funda- allel Lives interpretation of quantum mechanics mental quantum system comprises a conserved [10, 11], and should also be consistent with the nonclassical fluid in space-time - and it helps to local Heisenberg picture frameworks that have keep in mind that the fluid is composed of parti- been developed elsewhere [12, 13, 14, 15, 16, 17, cles on world-lines. 18, 19, 20, 21, 22]. This is a local ballistic model of the uni- In the present model, all (quantum) systems verse, meaning all interactions are local scatter- are comprised by pseudo-classical fluids in a sin- ing events between ballistic classical particles, gle objective locally-Minkowski space-time and and there are no nonlocal or long-range interac- the classical particles in these fluids follow world- tions or objects of any kind (i.e., all long-range lines through space-time. There are many worlds effects are mediated by force-carrying particles on only in the sense that there are many world- world-lines which undergo local collisions). In the lines for the many particles in space-time, and most general local ballistic model, classical par- each particle experiences a unique perspective ticles can carry an internal memory tape with an from its location in space-time. According to arbitrary amount of information, and when two relativity theory, all empirical experiences nec- particles interact locally, nature performs a com- essarily follow from these unique local perspec- putation using those two memory tapes, and then tives, and are fully restricted to an observer’s updates both of them. In the coarse-grained fluid past light cone. There are no global ‘worlds’ in picture, the set of scattering rules for such local this theory - there is only the one global space- collisions should ultimately come from the Stan- time, containing many particles on world-lines. dard Model Lagrangian, and these take the form To be very explicit, even though their resolu- of boundary conditions between different packets tions to the measurement problem are similar, of fluid, while the memories become local prop- the local space-time model presented here is fun- erties of the continuum fluid packets. damentally different from the many-worlds the- A single quantum system may comprise a ory of Everett [23, 24], which is delocalized in superposition of many different indexed single- configuration space. particle wavefunctions, each evolving indepen- There are some similarities between the dently of the others in space-time, in the ab- present model and the work of Madelung [25], sence of an interaction with another system. We and also various works on many-interacting- can think of the indexes that delineate the differ- worlds [26, 27, 28, 29, 30, 31] for single quantum ent wavefunctions of a given system as belonging particles. to its internal memory tape. For each system, We will not be working with the individual it is the local scattering interactions with other trajectories of the classical particles in the fluids fluid particles (of the same system) that produces here, since we do not yet know how to choose a the collective Schrödinger/Diracwave evolution unique solution. The decomposition of the single- in the fluid. particle quantum probability current into fluid We call the collective description of all in- streamlines is always possible, and serves as the dexed packets of a quantum systems in space- simplest example of a viable set of trajectories. time a wave-field. As we will show later, the Here we consider the continuum fluid equa- wave-field for a single fundamental system is ex- tions obtained by coarse-graining over the bal- pressed as a piece-wise multi-valued wavefunc- listic trajectories of the individual particles com- tion in space-time, where each indexed value prising the fluid - and we take these to be the evolves independently according to the single- single-particle Schrödinger/Diracequations. The particle Schrödinger/Dirac equation. The pieces behavior for multiple quantum particles is com- are separated in space-time by interaction-based pletely different than in the standard treatment, boundary conditions, which are also where the which is the main focus of this article. The fluid number of indexes changes. The wave-field of a is conserved, which corresponds to conservation system is a separable mathematical description of probability current in collapse-free quantum for that system alone - even if it is entangled

2 with other systems. The set of all wave-fields on Quantum tunneling through a finite barrier a given space-like hypersurface corresponds to co- highlights the nonclassicality of the fluid. As a variant state introduced by Schwinger. pulse is incident upon a barrier, the interference In the non-relativistic limit, we can use Bohm’s with the reflected wave may cause temporary ze- eikonal form of a single-particle wavefunction ros to form in front of the barrier, and the fluid to iS form a series of compressed and rarefied regions, ψi = Rie i to elucidate the fluid picture, where 2 which quickly vanish as the reflected pulse moves Ri is the fluid density, Si is Hamilton’s principal away. Part of the packet also penetrates inside function, and ∇~ Si/m is the local average veloc- the barrier, and the probability current there is ity field of the particles in the fluid. Then Ri nonzero, so the fluid particles’ world-lines are lit- and Si evolve according to the coupled continu- ity equation and Hamilton-Jacobi equation, as in erally passing through the barrier and continuing single-particle Bohmian mechanics. The coeffi- on the other side - and clearly with a nonzero iφa tunneling time. cients ai = |ai|e i give the total quantity and global phase of the packet of fluid with index i. As we will see, the complexity of the picture These are also the relative phases and propor- grows with the dimension of the usual Hilbert tions of the total fluid in the total wave-field, space, which only obscures some of the relevant which are relevant for interference. As we will features, and this is why we focus most of our see, it is still essential that each ballistic particle attention on examples with 2-level systems. This in the fluid carries its own copy of all of the lo- article begins with analysis of states of one, two, cal state in its memory, in order to properly de- and three spins in space-time, including Von Neu- fine the local transfer matrices for interactions. mann measurement and Born’s rule. We then The relativistic treatment is conceptually identi- give a full demonstration of the local treatment cal, with the fluid particles moving along world- of an experimental test of Bell’s theorem, treat- lines. ing Alice and Bob as spins. Spatial entanglement and the Stern-Gerlach are also discussed. In general, the fluid particles in different wave- Next, we present a possible experimental imple- functions of different systems can interact via lo- mentation of spin-spin entanglement that would cal scattering, and when this results in entangle- closely approximate the proper treatment given ment, this actually means that the scattering has here. We conclude with some discussion of the increased the number of wavefunctions of each historical context of this model as well as its po- system, with correlations encoded in their local tential future applications. memories, and the particles in the fluid divided up to occupy these new wavefunctions. These points of division form the spatial boundaries 2 Spins that separate the different pieces of the wave- field. In this model, the wave-field of an isolated Finally, to get some physical intuition for Pauli spinor comprises a superposition of two this model, it helps to think of each indexed indexed single-particle wavefunctions in space- 2 wave packet as an isolated drop of fluid float- time, which is to say a fluid density Ri (x, t) and ing through space. This is a very nonclassical a principal function Si(x, t) with velocity field fluid, which behaves more like a gas than a liquid, ∇~ Si(x, t)/m. If two spins are entangled, then allowing significant compression and rarefaction each spin comprises up to eight wavefunctions as it moves. This facilitates longitudinal waves on space-time, with four different interaction in- passing through the drop, which produces famil- dexes. For three entangled spins, each comprises iar wave behavior. Unlike a classical gas, these up to 32 wavefunctions space-time, with 16 in- waves can create zeros in the fluid density - for teraction different indexes. We can work in any example the nodes of a stationary state - so the basis without loss of generality, so we use the bi- local ballistic scattering rules for the particles in nary basis for all Pauli spinors, meaning i = 0, 1, the fluid must also be quite nonclassical. Despite corresponding to spin states |0i and |1i. this, fluid particles never cross these zeros, and For a single spin (system 1, denoted by super- 1 1 the entire motion can always be decomposed into script), the two wavefunctions aψ0(x) and bψ1(x) their world-lines. correspond to the spin states |0i1 and |1i1, re-

3 spectively, and it is the sum of these two proba- equation. Local interactions take the form of spa- bility densities that is normalized in space (|a|2 + tial boundary conditions that connect the differ- 2 R ∞ 1 2 |b| = 1, −∞ |ψi (x)| dx = 1). The point is, if ent spatial wavefunctions. 1 1 1 ψ0(x) = ψ1(x) = ψ (x), then the spin-position After the local interaction, each of them carry a 1 1 1 12 1 2 Hilbert space product state (a|0i + b|1i )ψ (x1) copy of the information U |0i |0i in their local in standard quantum theory is replaced in the memories. This process of collecting local inter- new theory by the pair of fluid packets in 3-space action unitaries and initial states along a single 1 1 {aψ0(x), bψ1(x)}. Note that if the spin and po- world-line is the essence of the local Heisenberg sition were entangled the description would be treatment. It is important to emphasize that more complicated. We discuss the Stern-Gerlach each of these copies is separate and independent later, but the general case goes beyond the scope from the others, and each copy encodes local in- of this article. formation at a single point in space (a single fluid We can change the basis used for the repre- particle). Whenever two systems locally interact, sentation, which results in new coefficients, and before the new interaction unitary is applied, the a new division into two different fluids. These memories of the two systems first synchronize, so still evolve independently, and this still produces that they now share all unitary operations and the correct evolution of the state. From this it initial states from both of their past local interac- should be clear that the division into two fluids is tions. Then the new interaction unitary is added arbitrary, and thus each fluid particle must carry to both of them, resulting in an equal number of both coefficients a and b in its memory. indexed spatial wavefunctions for each system, with matching coefficients. 2.1 Two Spins Taking the simple case that the spin and spatial wavefunctions are not entangled, we have If two spins (initially in state |0i) have inter- 1 1 1 1 ψ(x) 2 = ψ(x) 2 = ψ(x) 2 = ψ(x) 2 = acted via a unitary U 12, then to find the spa- 0,|0i 0,|1i 1,|0i 1,|1i ψ(x)1 and ψ(x)2 = ψ(x)2 = ψ(x)2 = tial wavefunctions of each system, we construct 0,|0i1 1,|0i1 0,|1i1 2 2 the 2-spin Hilbert space state U 12|0i1|0i2 = ψ(x)1,|1i1 = ψ(x) , the Hilbert space state of two P1 1 2 entangled spins with separable position states of i,j=0 aij|ii |ji , and then treat the states of all other systems as indexes that delineate separate standard quantum theory, wavefunctions of the present system. For exam- 1 1 2 X 1 2 ple, the four local states of system 1 are, ψ(x1) ⊗ ψ(x2) ⊗ aij|ii |ji , (5) i,j=0 1 1 1 1 a00|0i 2 , a01|0i 2 , a10|1i 2 , a11|1i 2 , |0i |1i |0i |1i is replaced in the local theory by the above set of (1) eight fluid packets in 3-space, and the many lo- and the four corresponding wavefunctions are cal copies of the Hilbert-space state they carry 1 1 a00ψ(x)0,|0i2 , a01ψ(x)0,|1i2 , in memory. It should be clear that a change 1 1 (2) of (product) basis still results in new coefficients a10ψ(x)1,|0i2 , a11ψ(x)1,|1i2 , and a new set of indexed spatial wavefunctions, and it is the sum of these four probability densi- but the underlying memory state is unchanged. 2 2 ties that is normalized in space (|a00| + |a01| + Note that the local memories of each spin 2 2 |a10| + |a11| = 1). Likewise for system 2 the carry all of the same information as the entan- local states are, gled Hilbert space state of conventional quantum theory, and at the fine-grained scale, every a |0i2 , a |1i2 , a |0i2 , a |1i2 , 00 |0i1 01 |0i1 10 |1i1 11 |1i1 fluid particle of each system carries this informa- (3) tion as well. The main point here is that these and the four corresponding wavefunctions are eight fluid packets in space-time contain all of 2 2 the information needed to produce the correct a00ψ(x)0,|0i1 , a01ψ(x)1,|0i1 , 2 2 (4) empirical probabilities and entanglement corre- a10ψ(x) 1 , a11ψ(x) 1 . 0,|1i 1,|1i lations for these systems. As we will show, be- In the absence of an interaction, each of these cause all interactions are local, we can completely spatial wavefunctions evolves independently ac- discard the delocalized wavefunction in a higher- cording to the single-particle Schrödinger/Dirac dimensional space, and instead obtain all of the

4 correct empirical predictions from this set of flu- wavefunctions of system 2 into eight, such that ids in space-time. the entanglement correlations between systems 2 By way of notation in the present formalism, and 3 become physically manifest. Note that if we will use superscripts to indicate which system two interacting systems already share some uni- a spatial wavefunction belongs to, rather than taries or initial states in memory, they will nec- subcripts on the coordinates in the one configu- essarily match, and so the two memories can be ration space wavefunction. As shown above, all simply be merged, as in this case (before the in- internal degrees of freedom (like spin) now cor- teraction, system 2 had U 12|0i1|0i2). respond to additional indexed spatial wavefunc- Now the new interaction unitary is added to tions of a given system, and entanglement with both memories, resulting in other systems also results in additional spatial 1 wavefunctions for both systems. 23 13 12 1 2 3 X 1 3 2 W V U |0i |0i |0i = djkl|ki |li |ji , j,k,l=0 2.2 Three Or More Spins (9) and eight new wavefunctions for systems 2 and Now, suppose that system 1 interacts locally with 3. The eight wavefunctions of system 1 are not system 3, while system 2 is not involved, and involved in this interaction, and are unchanged. does not change in any way. The interaction uni- Hopefully the general picture for larger numbers tary is V 13 and the initial state of system 3 is of spins is clear at this point. |0i3, and system 3 carries no other relevant mem- A final issue is single-spin unitaries that act ory. First, the two systems synchronize memory, on any system in isolation. Those are added to so system 3 acquires a copy of the U 12|0i1|0i2, memory, and participate in synchronization as which splits system 3 into four indexed wavefunc- normal, and thus they seem like a simple case tion, whose coefficients match those of systems 1. to discuss. However, in actuality, all interactions Then V 13|0i3 is added to both memories, result- are of the pairwise type described above, and a ing in state, single-spin unitary is truly an approximation of 1 a case where entanglement is extremely weak (as 13 12 1 2 3 X 1 3 2 V U |0i |0i |0i = aijbikl|ki |li |ji with the momentum exchange between a photon i,j,k,l=0 and a beam splitter), which is a bit more complicated. 1 X 1 3 2 = cjkl|ki |li |ji . (6) j,k,l=0 2.3 Local Entanglement From this, we see that there are eight local spin What remains is to show how two systems in- states for each system, teract locally and become entangled in this way. We will begin with the simplest possible exam- n 1 o n 3 o cjkl|ki|li3|ji2 , cjkl|li|ki1|ji2 , (7) ple, which is also quite illustrative. We expect the general theory to contain only one type where in either case the other two systems are of coupling potential, and this is of the form treated as indexes, and thus eight spatial wave- δ(x1 − x2)V , where V is a general space-time- functions for each system as well, independent potential. This says that when two systems meet at an event in space-time, the po- n 1 o n 3 o cjklψ(x)k,|li3|ji2 , cjklψ(x)l,|ki1|ji2 . (8) tential V produces the local scattering between them, via the unitary U = e−iV/~, and the new Now, systems 2 and 3 have not interacted, but states are written into the memories of the fluid there are now entanglement correlations between particles as this happens, causing them to sepa- them, which will effect what happens if they in- rate into more distinct fluids than before. This teract in the future. Let us consider that case general potential should be uniform throughout next. space-time, and encompass all possible scattering Systems 2 and 3 now interact via unitary events between all types of quantum systems. In W 23. First, their memories are synchronized to other words, all Standard Model particle interac- V 13U 12|0i1|0i2|0i3, which splits the four indexed tions should be encoded in V .

5 Because this model does not support long- and the fully normalized piece-wise wave-field range interactions it is relatively complicated to |Ψ(x, t)is of each system includes contributions recover Coulomb-potential based interactions be- from all six, and all twelve for both systems. The tween charges, which are mediated by massless situation is shown in Fig.1. force carriers. To demonstrate the general mechanism, we consider a gedanken experiment with 2.4 The Interaction Boundary just two quantum systems in space-time, where the only coupling potential is a spin-spin interac- The next important detail we need to examine tion - thus U = e−iV/~ is some 4 × 4 2-spin ma- is the actual location x12 between the systems, trix. With this potential, the spatial wavefunc- which is not a fixed boundary at all, but rather a tions never change or become entangled with the dynamic one that moves in time depending on the spins. Thus, if the two systems are incident upon shapes of the two incident systems’ wavefunctions one another, the fluid packets pass through the (in 3D this is a dynamic boundary surface). The interaction undeflected and undeformed, but the boundary is defined by a special rule that applies spin states interact locally as this occurs, caus- to all entanglement couplings in this model - the ing the fluid to acquire new indexes that separate fluid flux of the two systems across the boundary it into more distinct packets than before - each must be equal and opposite. For any two normal- moving independently but identically. These lo- ized wavefunctions ψ(x) and φ(x), there is always cal interactions are the origin of empirical entan- a boundary point where, glement correlations, so it is still appropriate to Z x12 Z ∞ 1 2 2 2 say that two systems became entangled during |ψ (x)| dx = |ψ (x)| dx, (10) −∞ x this interaction, even if there is no unfactorizable 12 state in configuration space. and

Z x12 Z ∞ We now consider such an entangling inter- |ψ2(x)|2dx = |ψ1(x)|2dx. (11) action for two spins that begin in a separable −∞ x12 state, each with two wavefunctions in space- The initial value of the boundary can be found in 1 1 2 time, a1ψ0(x, t) and b1ψ1(x, t), and a2ψ0(x, t) this way (ideally when the two packets are well- 2 and b2ψ1(x, t), respectively. They can only inter- separated), and then it moves according to, act locally, and thus the only reason they have 1 2 not interacted is that they have no overlapping j (x12, t) + j (x12, t) x˙ 12(t) = 1 2 2 2 , (12) support. In fact, there must be a boundary point |ψ (x12, t)| + |ψ (x12, t)| x12 that separates their supports. For this simple one-dimensional example, we will begin with where js is the current density of each fluid. spin 1 located fully to the left of x12 and propa- This equal-and-opposite flux condition guaran- gating towards spin 2, which is fully to the right tees that the an equal amount of fluid from each of x12. In this example, once their supports begin system is always crossing the boundary in a given to overlap, the spins will directly interact via a 4- time. dimensional unitary U (strictly speaking, it need This condition is required to guarantee that 1 2 the a00 in a00ψ is the same as in a00ψ 1 2 , the not be unitary so long as it is norm-preserving), 00102 0 0 1 2 which maps the two pre-interaction wavefunc- a01 in a01ψ0,|1i2 is the same as the a01 in a01ψ|0i1 , tions of each system into its four post-interaction etc. As they cross the interaction boundary, the wavefunctions. Note that for spin 1 the two pre- fluid particles of both systems acquire all of the interaction wavefunctions are only supported at 2-spin entanglement information. Importantly, x ≤ x12, while the four post-interaction wave- these are independent copies of the coefficients, functions are only supported at x ≥ x12 (since in different memory records, and local interven- the wave-packets continue to propagate with the tions on one copy have no nonlocal effect on other same momentum). From here, it is clear that U copies. simply defines the boundary conditions that con- In 3D, there is no longer a unique boundary nect these six wavefunctions at x12 (four post- surface for two normalized functions, so an ini- interactions wavefunctions on one side, and two tial boundary must be assumed. The motion of pre-interaction wavefunctions on the other side), this boundary is then defined locally such that as

6 Figure 1: Three frames showing the local interaction ferent interaction boundaries for different wave- process as two particles in one dimension pass through functions of a given pair of systems. each other, with only their spins interacting. The spa- 2 Finally, the hard boundary presented here may tial density |ψ(x)i| of each fluid pulse is shown, each indexed by past interactions, for the particular case that be a proof of concept, rather than a physically 2 2 2 2 2 2 correct rule. A possible generalization is that |a1| = |b1| = |a2| = |b2| = 1/2 and |c| = |d| = |e|2 = |f|2 = 1/4. The piece-wise wave-field |Ψ(x, t)is when two fluids meet, only some fraction of them of each system formally includes all six wavefunctions as interacts, and the remainders simply continue in separated at the dynamic boundary x12(t) (stationary their pre-interaction states. This would effec- in this example) which all occupy the same space-time. tively smear the boundary between the pre- and Also consider this example in a boosted Lorentz frame, post-interaction wavefunctions, but it would also where the boundary is moving such that the fluid fluxes mean that two fluids can never fully switch into of the two systems are equal and opposite. their post-interaction states. In any event, for the purpose of this article, we will use the hard boundary. x 2.5 Boundary Conditions

x To obtain the boundary condition at x12, we consider the action of a general norm-preserving transformation matrix on a general product state of two spins. The matrix can be expanded as

x 12 X 1 2 1 2 U = uijkl|ii |ji hk| hl| (13) i,j,k,l∈[0,1] x

x 12 1 2 1 2 1 2 1 2 U a1a2|0i |0i +a1b2|0i |1i +b1a2|1i |0i +b1b2|1i |1i x 1 2 = u0000a1a2+u0001a1b2+u0010b1a2+u0011b1b2 |0i |0i x 1 2 + u0100a1a2+u0101a1b2+u0110b1a2+u0111b1b2 |0i |1i x 1 2 + u1000a1a2+u1001a1b2+u1010b1a2+u1011b1b2 |1i |0i 1 2 + u1100a1a2+u1101a1b2+u1110b1a2+u1111b1b2 |1i |1i x = c|0i1|0i2 + d|0i1|1i2 + e|1i1|0i2 + f|1i1|1i2 This allows us to deﬁne the two 4×2 transfer ma- x trices T1 and T2 that map the two pre-interaction wavefunctions of each system onto its four post- x interaction wavefunctions,

x U 12 2 2 T1 = U a2|0i + b2|1i

= X (u a + u b )|ii1|ji2hk|1 two systems move together, an equal amount of ijk0 2 ijk1 2 i,j,k∈[0,1] fluid from each crosses per unit time. At the fine-   grained scale, the boundary corresponds to the u0000a2 + u0001b2 u0010a2 + u0011b2   locations where fluid particles of the two systems     are locally scattering and synchronizing memory.  u0100a2 + u0101b2 u0110a2 + u0111b2    A boundary like this exists between every pair =   ,   of systems. It goes beyond the scope of this arti-  u a + u b u a + u b   1000 2 1001 2 1010 2 1011 2    cle, but once the spatial degrees of freedom of a   system are entangled, there can generally be dif- u1100a2 + u1101b2 u1110a2 + u1111b2

7 and and U 12 1 1 T2 = U a1|0i + b1|1i  ˜2  ψ0,|0i1 x12(t), t " # 2 X 1 2 2 ˜2  ˜  ψ0 x12(t), t U † ψ0,|1i1 x12(t), t  = (uij0la1 + uij1lb1)|ii |ji hl| = T   , ψ˜2x (t), t 2 ψ˜2 x (t), t i,j,l∈[0,1] 1 12  1,|0i1 12  ψ˜2 x (t), t   1,|1i1 12 u0000a1 + u0010b1 u0001a1 + u0011b1 (18)     where the ψ˜ are general un-normalized individual    u0100a1 + u0110b1 u0101a1 + u0111b1    wavefunctions for each index. =   .   These reduce back to simple mappings between  u a + u b u a + u b   1000 1 1010 1 1001 1 1011 1  the spin coefficients, since all of the normalized     packets are identical, so the transfer matrices re- u1100a1 + u1110b1 u1101a1 + u1111b1 ally only produce the coefficients c, d, e, and f, and show how as and bs define them. The piece-wise multivalued wave-fields of each † −1 Because U = U , we also have system are,

U † U ˆ  1 Ts Ts = Is, (14)  a1ψ0(x, t)  1 x ≤ x12(t)  b1ψ (x, t)  1 ˆ  where Is is the identity for system s alone. Fi-  1  1 nally, if U is expanded into outer products then |Ψ(x, t)i = cψ0,|0i2 (x, t)  1 the T s can be expressed using the subscripts and  dψ0,|1i2 (x, t)  1 x > x12(t) without using matrices (see the Bell Test exam-  eψ 2 (x, t)  1,|0i ple below).  1  fψ1,|1i2 (x, t) It is clear that the local state of the other spin (19) appears in each transfer matrix, which makes and perfect sense given that this is a local interaction  2 between the two spins.  a2ψ0(x, t)  2 x > x12(t)  b2ψ (x, t) We can read off the coupled boundary condi-  1  tions for the four post-interaction wavefunctions  2  2 of each system as, |Ψ(x, t)i = cψ0,|0i1 (x, t)  2  dψ0,|0i1 (x, t)  ˜1   2 x ≤ x12(t) ψ 2 x12(t), t  eψ 1 (x, t) 0,|0i  1,|1i  ˜1  " ˜1 #  2 ψ0,|1i2 x12(t), t  U ψ x12(t), t  fψ 1 (x, t)   = T 0 , (15) 1,|0i ψ˜1 x (t), t 1 ψ˜1x (t), t (20)  1,|0i2 12  1 12 ˜1 Since all six spatial wavefunctions of each sys- ψ1,|1i2 x12(t), t tem are identical and normalized, as are the co- 2 2 and efficients in each region (|as| + |bs| = 1 and |c|2 + |d|2 + |e|2 + |f|2 = 1), we can verify that  ˜2  ψ0,|0i1 x12(t), t the wave-field of each system is a normalized fluid  ˜2  " ˜2 # ψ0,|1i1 x12(t), t  U ψ x12(t), t distribution in space-time.   = T 0 , (16) ψ˜2 x (t), t 2 ψ˜2x (t), t  1,|0i1 12  1 12 ˜2 2.6 Von Neumann Measurement and the Born ψ1,|0i1 x12(t), t Rule and for the pre-interaction wavefunctions as, We can simplify this example to illustrates the role of local entanglement during the measure- ψ˜1 x (t), t 0,|0i2 12 ment process and the experience of collapse with " ˜1 #  ˜1  ψ x12(t), t U † ψ0,|1i2 x12(t), t  0 = (T )   , Born rule probability [32] in the new fluid pic- ψ˜1x (t), t 1 ψ˜1 x (t), t 1 12  1,|0i2 12  ture. ˜1 ψ1,|1i2 x12(t), t For the Von Neumman measurement [33], we (17) keep the initial state of spin 1, expressed as

8 Figure 2: Three frames showing the local interaction pro- tions cess as two particles in one dimension pass through each other, with only their spins interacting. The interaction 1 1 is Von Neumann measurement of the binary basis, where a1ψ0,|0i2 (x), b1ψ1,|1i2 (x), system 2 is the pointer, which starts in ‘ready’ state |0i2. (22) 2 2 2 The spatial density |ψ(x)i| of each fluid pulse is shown, a1ψ0,|0i1 (x), b1ψ1,|1i1 (x), 2 2 for the particular case that |a1| = |b1| , each indexed by past interactions. 2 with a fraction |a1| of the particles in the spin 2 fluid recording the interaction |0i1 into their in- 2 1 x dices, and fraction |b2| recording |1i (see Fig. 2). These index records are part of the memory tape of each particle in the fluid, and they also x define the experience of that system, and thus from the perspective of each particle in the fluid of spin 2, spin 1 seems to collapse into one of its eigenstates or the other. Furthermore, in a large ensemble of identically prepared runs, spin 2 will 1 2 experience |0i with relative probability |a1| and x 1 2 |1i with relative probability |b1| , which is exactly the Born rule. x To round out the example, the two transfer matrices are x

1 0   U 12 2 0 0 T1 = U |0i =   , x 0 0 0 1 (23)   x a1 0   U 12 1 1  0 a1 T2 = U a1|0i + b1|1i =   ,  0 b1  1 1 1 b1 0 (a1|0i + b1|1i )ψ (x1) in the conventional the- 1 1 ory and {a1ψ0(x), b1ψ1(x)} in the present theory, but set the initial state of spin 2 to the ‘ready’ 2 2 and we have, state |0i ψ (x2), meaning the we have only one 2 spatial wavefunction ψ0(x) for spin 2 in this basis (a2 = 1, b2 = 0). U 1 1 1 1 T1 a1|0i +b1|1i = a1|0i 2 +b1|1i 2 , (24) For a projective measurement, the unitary is |0i |1i then U 12 = CNOT [34], with spin 1 as the control qubit, which produces the standard entan- and gled state,

U 2 2 2 T2 |0i = a1|0i 1 + b1|1i 1 , (25) 1 2 1 2 1 2 |0i |1i a1|0i |0i + b1|1i |1i ψ (x1)ψ (x2) (21)

which shows why we only have four nonzero in the conventional theory. In the present theory, wavefunctions. this corresponds to the each system carrying the 12 1 1 2 local memory state U (a1|0i + b1|1i )|0i and We can use these relations to deﬁne the bound- the corresponding set of four spatial wavefunc- ary conditions at x12 as the packets pass through

9 one another, As an example of synchronization, let us re- turn to the interaction between systems 1 and ˜1 ˜1 ψ0,|0i2 (x12(t), t) = ψ0(x12(t), t), 3 in Sec.2.1. The unitary synchronization operation S, when system 1 interacts interacts with ˜1 ˜1 ψ1,|1i2 (x12(t), t) = ψ1(x12(t), t), system 3 results updating the memory of system 3 from |0i3 to U 12|0i1|0i2|0i3, and thus the syn- ˜1 ˜1 chronization matrix is S = U 12I3|0i1|0i2, where ψ1,|1i2 (x12(t), t) = ψ0,|0i2 (x12(t), t) = 0, 3 the identity I3 has been added to emphasize that ˜2 ˜2 this is an operation on system 3. In this case, ψ 1 (x12(t), t) = a1ψ0(x12(t), t), 0,|0i all the operation does is introduce additional in- ˜2 ˜2 dexes that separate system 3 into four identical ψ1,|1i1 (x12(t), t) = b1ψ0(x12(t), t), 12 3 spatial wavefunctions. Applying S1 = I |0i to the memory of system 1 does not change number ψ˜2 (x (t), t) = ψ˜2 (x (t), t) = 0, 1,|0i1 12 0,|1i1 12 of distinct indexes, so there are still four spatial wavefunctions. ψ˜2(x (t), t) = a∗ψ˜2 (x (t), t) + b∗ψ˜2 (x (t), t), 0 12 1 0,|0i1 12 1 1,|1i1 12 After the synchronization, the interaction V 13 is added to the memory of both systems, result- ˜2 ∗ ˜2 ∗ ˜2 ψ1(x12(t), t) = a1ψ 1 (x12(t), t) + b1ψ 1 (x12(t), t), VS 13 0,|1i 1,|0i ing in the 8 × 2 transfer matrix T3 = V S3 = (26) V 13U 12|0i1|0i2 for system 3. This defines the along with the spatial derivative of these expres- boundary condition between the two initial wave- sions evaluated at the boundary. functions (in most bases), and the eight resulting Each of the spatial wavefunctions ψ(x, t) wavefunctions of system 3. For system 1, the 8×4 evolves under its own single-particle VS 13 13 2 3 transfer matrix is T1 = V S1 = V I |0i , Schrödinger/Diracequation, and all interactions which defines the boundary conditions between occur via the boundary conditions, which auto- the four initial and eight resulting wavefunctions matically produce the correct splitting into more of system 1. wave packets. When using the single-particle unitary approx- In summary, by enforcing all spin-spin interac- imation, the transfer matrix is simply identical to tion unitaries through the local boundary con- the unitary, with identity matrices tacked on for ditions, we obtain the complete multiparticle other systems in the local memory. quantum dynamics, using an ensemble of single- As we can see, branching is never global in the particle evolution equations in space-time (with- new model. New branchings arise from the cre- out any delocalized Hilbert/configuration space ation of new indexes during a local interaction, evolution). and each branching spreads via synchronization from the systems where it originates to any other 2.7 Synchronization and General Transfer Ma- systems they interact with, and then to other trices systems that those interact with, and so on, in We have considered an interaction between two a chain that eventually applies that branching to systems which were not already entangled with the entire environment. This is the mechanism of any other systems, and so no synchronization was decoherence in the new model, and explains the necessary prior to the applying the interaction emergence of classical macroscopic experiences in unitary. However, in general, the synchroniza- dense and frequently-interacting systems. tion process will extend the unitary operation This also explains how thought experiments being applied to each system. The transfer ma- like Schrödinger’scat [35], and Wigner’s friend trices come from the overall unitary operation, [36, 37] are resolved as sequences of local interac- and so they are not generally 4 × 2. Note that tions where branching is spread from one system the synchronization unitary is applied even if the to the next. interaction unitary is identity. Nevertheless, the It is also worth noting that this mechanism basic construction above gives the right idea for produces the correct empirical experience of col- how to construct the transfer matrices and the lapse after a measurement, because if the same corresponding boundary conditions. two systems interact again via identity, or they

10 measurement is repeated using a newly prepared Figure 3: Three frames showing the steps of the Mermin- device, the indexes all stay unchanged, meaning Wigner Bell test, for the case that Alice and Bob measure the same setting. The top frame shows the Bell each observer sees the same outcome as before. state being sent to Alice and Bob, the middle frame If this were not the case, then the observer would is after Alice’s and Bob’s measurements are completed, in general experience violation of natural conser- and the bottom frame is after Alice and Bob meet to vation laws of the interacting properties (energy, share their results. In the experience of each Alice and momentum, etc.). Even if the other system has Bob, the proper entanglement correlations have been participated in another interaction before being obeyed. measured again, the results will still be consistent with the collapsed state have undergone that operation, as when preparing a state, and then applying a unitary to put it in superposition. x

3 Bell Test x As a ﬁnal example of the model, and also to demonstrate the full local treatment of entanglement, we go through a simple gedanken example of a test of Bell’s theorem. This is the x Mermin-Wigner test [38, 39], where Alice and Bob each choose to measure their spin in one of three equally spaced directions in the zx-plane of the Bloch sphere. x We will begin with systems 1 and 2 having locally interacted to the anticorrelated Bell state of two spins, √1 |0i1|1i2 − |1i1|0i2ψ1(x )ψ2(x ) 2 1 2 in the old quantum theory, and in the present x theory, both systems carry the local states U 12|0i1|0i2 = √1 |0i1|1i2 − |1i1|0i2 in their 2 memory, and the four corresponding spatial the x four spatial wavefunctions:

√1 ψ1 (x), − √1 ψ1 (x), 2 0,|1i2 2 1,|0i2 right point across. To complete the experiment (27) and obtain the entanglement correlations, Alice − √1 ψ2 (x), √1 ψ2 (x), and Bob meet and interact via the identity. 2 0,|1i1 2 1,|0i1 in the new theory. 3.1 Case 1 By symmetry, there are only two distinct types of measurement - those with parallel settings and Alice begins in state |0iA with a single spatial A those with nonparallel settings, so we only need wavefunction ψ0 (x) and her measurement is a to consider one example of each type. In both V 1A = CNOT gate with system 1 as the control cases, Alice measures system 1 in the binary ba- and Alice as the target. The local state carried sis, while in Case 1, Bob measures system 2 in in memory synchronizes and update to . the binary basis, and in Case 2, Bob measures in 1 the basis, V 1AU 12|0i1|0i2|0iA = √ |0i1|1i2|0iA−|1i1|0i2|1iA, 2 √ |φ+i2 = 1 |0i2 + 3|1i2, (29) 2 resulting in the four spatial wavefunctions, √ (28) |φ−i2 = 1 3|0i2 − |1i2. 2 √1 ψ1 (x), − √1 ψ1 (x), 2 0,|0iA1i2 2 1,|1iA|0i2 We treat Alice and Bob as 2-level systems for (30) √1 ψA (x), − √1 ψA (x). simplicity in this analysis, and because it gets the 2 0,|0i1|1i2 2 1,|1i1|0i2

11 The situation is symmetric for Bob and system resulting in the eight wavefunctions, 2, with local state, q q 3 ψB (x), − 1 ψB (x), 1 8 0,|0i1|φ+i2 8 0,|0i1|φ+i2 W 2BU 12|0i1|0i2|0iB = √ |0i1|1i2|1iB−|1i1|0i2|0iB, 2 q 1 B q 3 B (31) − 8 ψ1,|0i1|φ−i2 (x), − 8 ψ1,|0i1|φ−i2 (x), resulting in the four wavefunctions, q q 3 ψ2 (x), − 1 ψ2 (x), − √1 ψ2 (x), √1 ψ2 (x), 8 φ+,|0i1|0iB 8 φ+,|0i1|0iB 2 0,|0iB |1i1 2 1,|1iB |0i1 (32) q q 1 B 1 B 1 2 3 2 − √ ψ (x), √ ψ (x). − 8 ψφ−,|0i1|1iB (x), − 8 ψφ−,|0i1|1iB (x). 2 0,|1i1|0i2 2 1,|0i1|1i2 Now, Alice and Bob meet (the identity transfor- (37) mation), and their local memories synchronize to Now, Alice and Bob meet, and their memories synchronize to the local state, V 1AW 2BU 12|0i1|0i2|0iA|0iB (33) V 12W 2BU 12|0i1|0i2|0iA|0iB (38) 1 = √ |0i1|1i2|0iA|1iB − |1i1|0i2|1iA|0iB, 2 r3 r1 = |0i1|φ+i2|1iA|0iB − |1i1|φ+i2|1iA|0iB which results in the four wavefunctions, 8 8 r r √1 A √1 A 1 1 − 2 A B 3 1 − 2 A B ψ 1 2 B (x), − ψ 1 2 B (x), − |0i |φ i |0i |1i − |1i |φ i |0i |1i , 2 0,|0i |1i |1i 2 1,|1i |0i |0i 8 8 − √1 ψB (x), √1 ψB (x). resulting in the eight spatial wavefunctions, 2 0,|1i1|0i2|1iA 2 1,|0i1|1i2|0iA (34) q q 3 ψA (x), − 1 ψA (x), We can now see that any fluid particle of Al- 8 0,|0i1|φ+i2|0iB 8 0,|0i1|φ+i2|0iB ice that experienced system 1 up |0i1 (|1i1) also q 1 A q 3 A meets a fluid particle of Bob that experienced − 8 ψ1,|0i1|φ−i2|1iB (x), − 8 ψ1,|0i1|φ−i2|1iB (x). system 2 in state |1i2 (|0i2), and thus from the q q perspectives of all Alice’s and Bob’s the correct 3 ψB (x), − 1 ψB (x), entanglement correlations for the Bell state have 8 0,|0i1|φ+i2|0iA 8 0,|0i1|φ+i2|0iA been obeyed. The steps are shown in Fig.3. q 1 B q 3 B − 8 ψ1,|0i1|φ−i2|1iA (x), − 8 ψ1,|0i1|φ−i2|1iA (x). 3.2 Case 2 (39) The situation for Alice’s measurement of system It is again clear from these final wavefunctions 1 is the same as in Case 1. For Bob’s measure- that the entanglement correlations for the Bell B ment, with the same ready state |0i , a viable state have been correctly obeyed for the case that unitary is, the measurement settings were not aligned. The steps are shown in Fig.4. W 2B = |φ+i2|0iBhφ+|2h0|B +|φ−i2|1iBhφ−|2h0|B The two cases together show that the local + |φ+i2|1iBhφ+|2h1|B + |φ−i2|0iBhφ−|2h1|B. fluid model exactly reproduces all of the empiri- (35) cal predictions of standard nonlocal quantum me- Thus, when Bob measures system 2, the local chanics for this Bell test. state carried in the memory of the two systems It is important to note that the entanglement synchronizes and updates to, correlations are not obeyed in any meaningful sense until Alice and Bob meet, their memories W 2BU 12|0i1|0i2|0iB (36) synchronize, and their wavefunctions are paired by their indexes. Prior to that, there were Al- r r 3 1 ices in space-time who had experienced either = |0i1|φ+i2|0iB − |1i1|φ+i2|0iB 8 8 outcome, and also Bobs who had experienced ei- r1 r3 ther outcome, but there is no correlation among − |0i1|φ−i2|1iB − |1i1|φ−i2|1iB, 8 8 them, which is clear because their distributions

12 Figure 4: Three frames showing the steps of the Mermin- include spontaneous parametric down conversion Wigner Bell test, for the case that Alice and Bob mea- of entangled states [40, 41], delayed-choice quan- sure the different settings. The top frame shows the Bell tum erasure [42], measurements of weak values state being sent to Alice and Bob, the middle frame is after Alice’s and Bob’s measurements are completed, and [43], and the Delft Bell experiment [44]. the bottom frame is after Alice and Bob meet to share their results. In the experience of each Alice and Bob, 3.3 Local Hidden Variables the proper entanglement correlations have been obeyed. As mentioned in the introduction, the empirical facts of this theory lead to a picture with many- worlds. It is worth emphasizing that the treat- x ment we have just given is an explicit local hidden variable model of the Bell experiments, that successfully reproduces the entanglement correla- x tions (one-world is an often-unstated assumption of Bell’s theorem, which is violated here). To make this is completely unambiguous, we consider a quick demonstration using students which makes it clear that all entanglement corre- x lations and Born rule statistics are obeyed, and everything happens on world-lines in a single x space-time. We will have 8 students in one room x who play different copies of Alice, and 8 more in another room who play different copies of Bob, and we are using the same singlet state. Stu- x dents playing the same person ignore each other. The students in each room choose one of the three settings, and measure. The results is that a random 4 of the students in that room get ‘up’ x and the other four get ‘down,’ consistent with the 50% Born rule probability for the reduced x density matrix of the Bell state available in each room. Each students writes their chosen setting x and their result on a sign they then carry, but they are still completely separated, and have not x communicated in any way. The students then gather in a single room with absolutely no light, so they cannot see the other students’ signs. A referee with night-vision gog- in space-time always match their respective re- gles then pairs the students up (one Alice and one duced density matrices. Bob), and then sends these pairs out of the room. There is also a general lesson here about post- In both cases, the students will always find that selected ensembles of quantum measurements. the Born rule probabilities from the singlet state All of the measurement outcomes exist as wave- were obeyed, regardless of their settings. functions with different indexes, with an over- In Case 1, the referee does this by pairing the all distribution still given by the reduced density four ‘up’ Alices with the four ‘down’ Bobs, and matrix of that system. When a single observer lo- vice versa, so all eight Alices meet a Bob with the cally collects data from the post-selected system opposite spin, as expected for the singlet state. and the other systems of interest, this is where In Case 2, the referee pairs one ‘up’ Alice with the entanglement correlations associated with the an ‘up’ Bob, one ‘down’ Alice with a ‘down’ Bob, post-selection are realized. The observers who three ‘up’ Alices with three ‘down’ Bobs, and saw the desired post-selection will also see the three ‘down’ Alices with three ‘up’ Bobs. The anticipated entanglement correlations. Examples fraction of students with each outcome thus re-

13 produce the Born rule statistics, so in a large proximation, only the memory of the target sys- ensemble of identical trials, the students will ex- tem is updated from |0it to U t|0it = a|0it +b|1it. perience them as frequentist probabilities. This simple treatment for spin systems can be easily generalized to any pair of systems who in- 4 Single-Particle Unitaries and Spatial teraction results in a weakly entangled state. Superpositions 4.1 The Beam Splitter and Einstein’s Objec- tion to Nonlocal Collapse As discussed above, all single-particle unitary operations on a system really correspond to weak Although the full treatment of entangled infinite- entanglement in a standard two-system interac- dimensional systems in space-time is quite com- tion. This is made explicit here. The control plicated, we can still get a good idea what is going system may be macroscopic, but we treat it as on for cases where only a finite number of spatial a single quantum system in an environmentally modes need to be considered. decohered basis. After the interaction, the en- The simplest example is a particle incident on tangled state in local memory contains orthogo- a beam splitter, where we use the single-system nal terms for the target system and terms that unitary approximation. After the beam splitter are nearly indistinguishable for the control sys- the superposed particle state on paths I and II is tem. To get the single-particle approximation, √1 |1iI|0iII + |0iI|1iII in the Fock basis. Treat- 2 the observer (environment) measures the con- ing the vacuum mode as a local state, this corre- trol system in the same decoherent basis it be- sponds to the four wavefunctions, gan in, which results in multiple nearly identical √1 ψ(x, t)I , √1 ψ(x, t)I , wavefunctions, each having undergone approxi- 2 1,|0iII 2 0,|1iII mately the intended single-particle unitary. The (42) √1 ψ(x, t)II , √1 ψ(x, t)II , single-particle unitary approximation is to ignore 2 1,|0iI 2 0,|1iI the differences between these wavefunctions and treat them as one (dropping the indexes corre- where it is implicit that ψ(x, t)I and ψ(x, t)II are sponding to the control system). two different wavefunction, evolving along two For two spins, the initial state of the target different paths. system is |0it, and for the control system it is |0ic. As an aside, we can see that for a single-system After the interaction, the local state carried in unitary situation with a continuous path degree each system’s memory is a|0ic|0it + b cos |0ic + of freedom, like a single-slit diffraction, the local sin |1ic|1it, for || << 1. state in memory becomes, The six wavefunctions of the two systems are Z {x}−x x O j t c φ(x)|1i |0i dx (43) aψ c (x), aψ t (x), 0,|0i 0,|0i {x} j

t c b cos ψ1,|0i1 (x), b cos ψ0,|1it (x), (40) where {x} is the set of all paths, and the interaction produces some normalized distribution t c R |φ(x)|2dx = 1 over all of the paths. We then b sin ψ1,|1i1 (x), b sin ψ1,|0it (x), {x} have an inﬁnite number of spatial wavefunctions e The experimenter begins in state |0i . The ex- for each speciﬁc path x0, perimenter now measures the control system in x0 the binary basis, resulting in local state ψ(x, t) φ(x0), 1,N{x}−x0 |0ij j a|0it|0ic|0ie + b|1it cos |0ic|0ie + sin |1ic|1ie (44) x0 ψ(x, t) φ(x1), (41) {x}−x −x t t c e t t c e 0,|1ix1 N 0 1 |0ij ≈ a|0i + b|1i |0i |0i = U |0i |0i |0i j in both systems’ memories. The state of the where x1 is any path other than x0, where each x0 target system has undergone single-system uni- ψ(x, t)i is a distinct wavefunction that evolves t tary U , and the states of the control and exper- on path x0, which break down into one case where x imenter systems are unchanged. Under this ap- the particle is on path x0 (|1i 0 , upper Eq. 44),

14 and infinitely many others where there is vac- |0iI|1iII|1iB, and his two wavefunctions are, x uum on path x0 (|0i 0 , lower Eq. 44), because 1 B 1 B the particle is on path x1. We won’t spend any √ ψ(x, t) , √ ψ(x, t) . 2 0,|1iI|0iII 2 1,|0iI|1iII (46) more time on continuous degrees of freedom here, but this discussion is included to emphasize the Finally, when Alice and Bob meet, the local state completeness of the present theory. carried in both their memories synchronizes to Now, returning to the beam splitter, we have 1 a simple tool to demonstrate how the present √ |1iI|0iII|1iA|0iB + |0iI|1iII|0iA|1iB, (47) model resolves Einstein’s objection at the 1927 2 Solvay conference to the instantaneous and nonlocal nature of wavefunction collapse in the and we have the four expected wavefunctions, emerging quantum theory. We have already ex- where in each case, only one of only one case, plained how the experience of collapse and Born only one of Alice or Bob detected the particle, rule probabilities arise for the individual fluid √1 ψ(x, t)A , √1 ψ(x, t)A particles along their world-lines, so this is just 2 1,|1iI|0iII|0iB 2 0,|0iI|1iII|1iB a matter of applying these principles. The situa- √1 ψ(x, t)B , √1 ψ(x, t)B . tion is analogous to Case 1 in the Bell test. 2 0,|1iI|0iII|1iA 2 1,|0iI|1iII|0iA Suppose we send a particle through a beam (48) splitter, and then path I leads to Alice’s detector, and path II to Bob’s space-like separated 4.2 Stern-Gerlach Devices detector. After the particle is detected, Alice and Bob meet to compare results, and they al- The true function of a Stern-Gerlach device [45] ways find that only one of them has detected in the local ballistic model involves many force- the particle. Roughly speaking, Einstein’s objec- carrying particles being emitted locally from the tion was that in a single objective world where a magnet and then propagating to the spin and spatially superposed wavefunction causally medi- interacting locally with it. Here we approximate ates between the source and detectors, once Al- that entire process by a single local interaction ice detects the particle, something must instanta- unitary and boundary condition, using the single- neously prevent the wavefunction from also trig- system unitary approximation, and treating just gering Bob’s detector, which violates local causal- two output modes. ity. In conventional quantum theory, the incoming In the present local model, half of the fluid goes state will be a|0is + (b|1is|1iI|0iII, and the ac- to Alice and the other half to Bob. Alice branches tion of the magnetic field will be to transmit the into two subgroups of fluid; those that detected |0is state and reflect the |1is state, causing the the particle and those that did not. Bob likewise path and spin to become entangled, and the state branches into two subgroups, with the indexes to become, a|0is|1iI|0iII + (b|1is|0iI|1iII. reversed from Alice. When they meet, the Alices In the present quantum theory, there are who detected the particle have matched indexes initially two identical spatial wavefunctions, s s with the Bobs who did not, and vice versa, and aψ0(x, t) and bψ1(x, t) which move on path I, and so they always find that only one of them has after the interaction they have evolved to, detected the particle, as expected. s s Once Alice’s detector on path I has either fired aψ0,|1iI|0iII (x, t), bψ1,|0iI|1iII (x, t), or not, the local state in her memory updates to √1 |1iI|0iII|1iA + |0iI|1iII|0iA, where |1iA indi- aψI (x, t), bψI (x, t), 2 1,|0is|0iII 0,|1is|1iII (49) cates her detector has fired, and she has the two spatial wavefunctions, II II aψ0,|1is|1iI (x, t), bψ1,|0is|0iI (x, t)

1 A 1 A √ ψ(x, t) I II , √ ψ(x, t) I II . (45) Is 2 1,|1i |0i 2 0,|0i |1i where ψi (x, t) is a wavefunction that propagates IIs along path I as it evolves, and ψi (x, t) is a dif- Likewise for Bob’s detector on path II, the local ferent wavefunction that propagates along path state in memory updates to √1 |1iI|0iII|0iB + 2 II.

15 Figure 5: Three frames showing the local entangle- local ballistic interaction between the spins. The ment of the spin and path degrees of freedom as packets are moving slowly enough that if the two particle s passes through a Stern-Gerlach device, ap- spins are facing each other then the repulsion will proximated as a point (vertical dotted line), which either transmits or reflects the particle. The incom- cause them to bounce back along their respec- ing wavefunction ψIs (x, t) is identical for both spin tive waveguides, and if they are parallel they will states prior to this entanglement (as in all previous both continue in their original direction. Then, examples in this article). After the interaction, there by preparing a superposition of parallel and an- Is are two different wavefunctions, aψ (x, t)0,|1iI|0iII (x, t) tiparallel spin in each wave-guide, this will re- IIs and bψ (x, t)1,|0iI|1iII (x, t), traveling in opposite di- sult in an entangled state of the spins and paths. rections. The interference between the incoming and This is not a perfect analog of the original exam- reflected waves is neglected here, to simulate the 3D ple with the dynamic boundary point, since here cases where the reflection is in a different direction. The vacuum modes are also omitted for clarity. the two spins can only interact in the crossing region. However, if we send in symmetric pulses, Stern-Gerlach then the densities at the interaction point will always match, and there is zero amplitude for the x case that the two particles miss each other at the fixed boundary point. We will denote the incoming packet in each x wave-guide as |T is ≡ |1iIs |0iIIs , and the reflected packet as |Ris ≡ |0iIs |1iIIs , where s = {1, 2} in the system index. In conventional quantum the- x ory, the initial state of the spins and paths is then,

x |T i1||0i1 + |1i1|T i2|0i2 + |1i2/2 (50)

and after the local interaction it has evolved to, x 1h |T i1|T i2||0i1|1i2 + |1i1|0i2 (51) 2 i + |Ri1|Ri2||0i1|0i2 + |1i1|1i2 x Then if we post-select on the cases that bounced back, we have a Bell state of the spins |0i1|0i2 + 5 Physical Spin-Spin Interaction |1i1|1i2, and likewise if we focus on the cases that did not bounce. Since we know of no physical situation where we The full local scattering matrix for this case is can get the spins to interact ballistically with- 16 × 16, but we can restrict our attention to the out becoming entangled with the spatial wave- sector that acts on this input state, functions, we now consider a physical situation 1 2 1 2 1 2 1 2 which simulates many of the desired features of U = |Ri |Ri |0i |0i hT | hT | h0| h0| (52) the gedanken experiment. Consider two spins + |T i1|T i2|0i1|1i2hT |1hT |2h01|h1|2 which move down long wave-guides which are electromagnetically shielded, with identical wave + |T i1|T i2|1i1|0i2hT |1hT |2h1|1h0|2 packet proﬁles. The two wave-guides are stacked + |Ri1|Ri2|1i1|1i2hT |1hT |2h1|1h1|2 + ... in an ‘X’ shape, and in a small region near the point where they cross, the shielding is absent, In the present quantum theory, we begin with the so that the spins can interact in that region via set of four spin wavefunctions, the exchange of photons. The wave-guides are far √1 ψ1(x, t), √1 ψ1(x, t), enough apart that the two spins cannot jump be- 2 0 2 1 tween them at the intersection. We will approx- (53) √1 ψ2(x, t), √1 ψ2(x, t), imate this entire interaction region as a single 2 0 2 1

16 and after the interaction, and the path post- a delocalized approximation of the proper local selection |Ri1|Ri2, the local states carried in physics. This calls into question every develop- the memory of each spin is the Bell state ment in the foundations of quantum mechanics √1 |0i1|0i2 +|1i1|1i2, and we have our four wave- based on this delocalized treatment of entangle- 2 functions for the two entangled spins, ment. The fact that this was not better under- stood in the 1950s seems baffling at first, but √1 ψ1 (x, t), √1 ψ1 (x, t), 2 0,|0i2 2 1,|1i2 when one considers the historical context, the (54) picture starts to becomes clear. √1 ψ2 (x, t), √1 ψ2 (x, t), 2 0,|0i1 2 1,|1i1 First, Bohr and Heisenberg’s complementarity This protocol can be repeated to entangle addi- had made the pursuit of any realist interpretation tional spins, allowing us to perform a full Bell test of quantum mechanics with a clear narrative un- in this way, and the correlations are explained us- popular. Ideas like this started to be denigrated ing explicitly local variables. as ‘philosophy’ rather than ‘physics,’ and students were taught to ‘shut up and calculate.’ It was no longer encouraged for physicists to know 6 Conclusions what they were talking about, so long as their mathematics led to accurate predictions. The local space-time quantum fluid model pre- Second, the very successful formalism of quan- sented here fully supplants configuration-space tum field theory that developed in that environ- Schrödinger-picturequantum mechanics for mul- ment makes use of both past and future boundary tiple particles, and produces all of the same pre- conditions, and the mathematics can be inter- dictions. The relativistic generalization of this preted as retrocausal effects propagating from fu- model, following Schwinger’s covariant formula- ture to past. Notions of propagation from past to tion, should only contain world-lines as particle future, or even descriptions of what is happening trajectories, which results in a local Lorentz in- between distant past and future boundary condi- variant causal structure. tions, are heavily obscured in the mathematical We expect this will require a correction to the and conceptual machinery of these theories - par- single-particle Schrödingerequation even for non- ticularly in the path integral formalism. Further- relativistic energies, to prevent superluminal sig- more, the plane wave solutions of the Dirac and nalling [46, 47]. It may also require a correction Klein-Gordon equations are delocalized, filling all of the Dirac equation for the same reason, but of space, and local packets are treated as emerg- this is less clear. Either way, these equations are ing from their interference. The mathematics of still quite close to the true (unknown) equations the theory are delocalized in both space and time, of motion that should underlie this model, and and there is simply no clear physical narrative of if we use those equations for all of our single- what is going on. particle evolution, and the same coupling unitaries for our local boundary conditions, then the Third, Bell’s theorem has had a much more re- new model makes identical predictions. Under- cent impact on the community, and has created standing the local dynamics of the particles in the widespread and mistaken impression that any the fluid could also constrain the weight factors local realist interpretation of quantum mechan- for single-particle operations. ics is impossible. What Bell’s theorem actually These details notwithstanding, we now have proves is that either such a theory must be su- a quantum theory in the local Heisenberg- perdeterministic, or it must have multiple copies Schrödingerpicture that Schwinger called the ‘in- of each observer, who may experience different teraction representation’. This theory is consis- outcomes from the measurement, but who each tent with the local Heisenberg treatment used in experience just one - exactly like the multiple per- relativistic quantum field theory and the Stan- spectives of quantum fluid particles on different dard Model, while the configuration space treat- world-lines in space-time. ment is not. All told, it is easy to see why the pursuit of a That said, there are clearly many situations local realist narrative in space-time has not been where the configuration space wavefunction is a a high priority in the foundations community, but useful tool for calculations, but it is truly only this appears to have been a colossal mistake. In

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