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, Schr¨odinger,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 Schr¨odinger 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 Schr¨odingerpictures. 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 Schr¨odinger-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 inter- action 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 Schr¨odingerpicture 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 Schr¨odinger/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 Schr¨odinger/Diracequations. The particle Schr¨odinger/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.