# Local Quantum Theory with Fluids in Space-Time

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## Abstract

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## 1. Introduction

**only**in the sense that there are many world-lines for the many such particles in space-time, and each particle experiences a unique perspective from its location in space-time. According to relativity theory, all empirical experiences necessarily follow from these unique local perspectives, and are fully restricted to an observer’s past light cone. That is, for each observer, the ‘world’ is the image of the surface of that observer’s past light cone. This definition makes the set of events in a ’world’ Lorentz invariant. There are no global ‘worlds’ in this theory - there is only the one global space-time, containing many particles on world-lines, each with its own past light cone and ’world’, as shown in Figure 1. To be very explicit, even though their resolutions to the measurement problem are similar, the local space-time model presented here is fundamentally different from the many-worlds theory of Everett [23,24], which is delocalized in configuration space, and describes global worlds in a particular Lorentz frame. There is no space-like hypersurface that is observed by even one, let alone many observers, and it is a mistake to define global worlds on these hypersurfaces. This is corroborated by the fact that one cannot Lorentz transform the delocalized wavefunctions defined for these surfaces between different inertial frames (a Lorentz transform is a mapping that can act only on a 4-vector or field tensor at a single event, and the descriptions of these 4-vectors and fields are inherently separable event-by-event). All empirical data pertains to events within the observer’s past light cone, and that data defines the world for that observer. Importantly, no observer’s world contains the results of two space-like separated measurement events until signals have arrived to the observer from both of those events (i.e., once the observer has seen the empirical data).

## 2. Internal and External Memory

## 3. Macroscopic Preferred Bases and Relative Collapse

## 4. Spins

#### 4.1. Two Spins

#### 4.2. Three or More Spins

#### 4.3. Local Entanglement

#### 4.4. The Interaction Boundary

#### 4.5. Boundary Conditions

#### 4.6. Von Neumann Measurement and the Born Rule

#### 4.7. Synchronization and General Transfer Matrices

## 5. Bell Test

#### 5.1. Case 1

#### 5.2. Case 2

#### 5.3. Demonstrating the Local Hidden Variables

## 6. Single-Particle Unitaries and Spatial Superpositions

#### 6.1. The Beam Splitter and Einstein’s Objection to Nonlocal Collapse

#### 6.2. Stern-Gerlach Devices

## 7. Conclusions

## Supplementary Materials

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The distinct perspectives of observers on different world-lines in special relativity already provide a clear notion of many worlds. Each fluid particle has its own world in exactly this sense, and the worlds of two particles can only coincide if and when they are both present at the same event. No one observes a space-like hypersurface, so it is a mistake to define global worlds on those surfaces.

**Figure 2.**A local ballistic model is an unambiguous local hidden variable theory wherein all causal information is carried along world-lines in point-like packets. When two or more such packets meet at an event in space-time, their information undergoes a joint evolution resulting in new information that all of the packets may carry away (e.g., ${\lambda}_{4}$ results from the joint evolution of ${\lambda}_{1}$ and ${\lambda}_{2}$, etc.).

**Figure 3.**An illustration of the local hidden variables of the present model, showing three systems undergoing a series of local spin-spin interactions in space-time. The memories of systems 1 and 2 synchronize when they meet, and the interaction unitary ${U}^{12}$ is added to both. Then the memories of systems 1 and 3 synchronize when they meet, and the interaction unitary ${V}^{13}$ is added to both. The memory of system 2 is unaffected by this space-like separated interaction. Finally, the memories of systems 2 and 3 are synchronized when they meet, the interaction unitary ${W}^{23}$ is added to both, and the expected entanglement correlations between those systems are obeyed. The internal states can be expanded in any product basis to give the local set of single-particle wavefunctions in space-time indexed by the external memories in that basis, where the interaction unitaries define boundary conditions connecting the pre-interaction fluids to the post-interaction fluids. If there is a macroscopic preferred basis, then expanding the internal memory in that basis gives the set of different external memories experienced by different copies of macroscopic observers in space-time. If one considers a Schwinger state on any space-like hypersurface that cuts across this diagram, it is easy to see what information is encoded at each event on that surface, and to verify that this information only pertains to that event’s past light cone. It is also easy to see that the Schwinger state on the ‘present’ surface contains too much information to be reconstructed from a standard delocalized quantum state.

**Figure 4.**Three frames showing the local interaction process as two particles in one dimension pass through each other, with only their spins interacting. The spatial density ${|\psi (x)}_{i}{|}^{2}$ of each fluid pulse is shown, each indexed by past interactions, for the particular case that $|{a}_{1}{|}^{2}=|{b}_{1}{|}^{2}=|{a}_{2}{|}^{2}={|{b}_{2}|}^{2}=1/2$ and ${|c|}^{2}={|d|}^{2}={|e|}^{2}={|f|}^{2}=1/4$. The piece-wise wave-field ${|\mathsf{\Psi}(x,t)\rangle}^{s}$ of each system formally includes all six wavefunctions as separated at the dynamic boundary ${x}_{12}(t)$ (stationary in this example) which all occupy the same space-time. Also consider this example in a boosted Lorentz frame, where the boundary is moving such that the fluid fluxes of the two systems are equal and opposite.

**Figure 5.**Three frames showing the local interaction process as two particles in one dimension pass through each other, with only their spins interacting. The interaction is a Von Neumann measurement of the binary basis, where system 2 is the pointer, which starts in ‘ready’ state ${|0\rangle}^{2}$. The spatial density ${|\psi (x)}_{i}{|}^{2}$ of each fluid pulse is shown, for the particular case that $|{a}_{1}{|}^{2}={|{b}_{1}|}^{2}$, each indexed by past interactions.

**Figure 6.**Three frames showing the steps of the Mermin–Wigner Bell test, for the case that Alice and Bob measure the same setting. The

**top**frame shows the Bell state being sent to Alice and Bob, the

**middle**frame is after Alice’s and Bob’s measurements are completed, and the

**bottom**frame is after Alice and Bob meet to share their results. In the experience of each Alice and Bob, the proper entanglement correlations have been obeyed.

**Figure 7.**Three frames showing the steps of the Mermin-Wigner Bell test, for the case that Alice and Bob measure the different settings. The

**top**frame shows the Bell state being sent to Alice and Bob, the

**middle**frame is after Alice’s and Bob’s measurements are completed, and the

**bottom**frame is after Alice and Bob meet to share their results. In the experience of each Alice and Bob, the proper entanglement correlations have been obeyed.

**Figure 8.**Three frames showing the local entanglement of the spin and path degrees of freedom as particle s passes through a Stern-Gerlach device, approximated as a point (vertical dotted line), which either transmits or reflects the particle. The incoming wavefunction ${\psi}^{{\mathrm{I}}_{s}}(x,t)$ is identical for both spin states prior to this entanglement (as in all previous examples in this article). After the interaction, there are two different wavefunctions, $a{\psi}_{{0,|1\rangle}^{\mathrm{I}}{|0\rangle}^{\mathrm{II}}}^{{\mathrm{I}}_{s}}(x,t)$ continuing in the same direction and $b{\psi}_{{1,|0\rangle}^{\mathrm{I}}{|1\rangle}^{\mathrm{II}}}^{{\mathrm{II}}_{s}}(x,t)$ along a different direction. The interference between the incoming and reflected waves is not shown. The vacuum modes are also omitted for clarity.

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**MDPI and ACS Style**

Waegell, M.
Local Quantum Theory with Fluids in Space-Time. *Quantum Rep.* **2023**, *5*, 156-185.
https://doi.org/10.3390/quantum5010011

**AMA Style**

Waegell M.
Local Quantum Theory with Fluids in Space-Time. *Quantum Reports*. 2023; 5(1):156-185.
https://doi.org/10.3390/quantum5010011

**Chicago/Turabian Style**

Waegell, Mordecai.
2023. "Local Quantum Theory with Fluids in Space-Time" *Quantum Reports* 5, no. 1: 156-185.
https://doi.org/10.3390/quantum5010011