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arXiv:2107.06575 · Quantum Reports

Local Quantum Theory with Fluids in Space-Time

Mordecai Waegell

Published Feb 21, 2023• 10 citations

Abstract

In 1948, Schwinger developed a local Lorentz-covariant formulation of relativistic quantum electrodynamics in space-time which is fundamentally inconsistent with any delocalized interpretation of quantum mechanics. An interpretation compatible with Schwinger’s theory is presented, which reproduces all of the standard empirical predictions of conventional delocalized quantum theory in configuration space. This is an explicit, unambiguous, and Lorentz-covariant “local hidden variable theory” in space-time, whose existence proves definitively that such theories are possible. This does not conflict with Bell’s theorem because it is a local many-worlds theory. Each physical system is characterized by a wave-field, which is a set of indexed piece-wise single-particle wavefunctions in space-time, each with its own coefficient, along with a memory which contains the separate local Hilbert-space quantum state at each event in space-time. Each single-particle wavefunction of a fundamental system describes the motion of a portion of a conserved fluid in space-time, with the fluid decomposing into many classical point particles, each following a world-line and recording a local memory. Local interactions between two systems take the form of local boundary conditions between the differently indexed pieces of those systems’ wave-fields, with new indexes encoding each orthogonal outcome of the interaction. The general machinery is introduced, including the local mechanisms for entanglement and interference. The experience of collapse, Born rule probability, and environmental decoherence are discussed, and a number of illustrative examples are given.

DOI: 10.3390/quantum5010011

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In 1948, Schwinger developed a local Lorentz-covariant formulation of relativistic quantum electrodynamics in space-time which is fundamentally inconsistent with any delocalized interpretation of quantum mechanics. An interpretation compatible with Schwinger’s theory is presented, which reproduces all of the standard empirical predictions of conventional delocalized quantum theory in configuration space. This is an explicit, unambiguous, and Lorentz-covariant “local hidden variable theory” in space-time, whose existence proves definitively that such theories are possible. This does not conflict with Bell’s theorem because it is a local many-worlds theory. Each physical system is characterized by a wave-field, which is a set of indexed piece-wise single-particle wavefunctions in space-time, each with its own coefficient, along with a memory which contains the separate local Hilbert-space quantum state at each event in space-time. Each single-particle wavefunction of a fundamental system describes the motion of a portion of a conserved fluid in space-time, with the fluid decomposing into many classical point particles, each following a world-line and recording a local memory. Local interactions between two systems take the form of local boundary conditions between the differently indexed pieces of those systems’ wave-fields, with new indexes encoding each orthogonal outcome of the interaction. The general machinery is introduced, including the local mechanisms for entanglement and interference. The experience of collapse, Born rule probability, and environmental decoherence are discussed, and a number of illustrative examples are given.

Bell's theorem establishes that a local and spatially separable physical theory with no superdeterminism or retrocausality must be a theory of many local worlds. The local many worlds formalism of quantum mechanics is a transparent separable physical theory in spacetime with a Lorentz covariant causal structure. Unlike the Everett many worlds interpretation, there is no nonseparable universal wavefunction in configuration space, but like Everett, there is only local unitary evolution which conserves probability (fluid) current, and no wavefunction collapse. Each event in spacetime has its own local wavefunction which is built up using only the local wavefunctions and local unitary evolution operators within its past light cone, and which is entirely disconnected from spacelike separated local wavefunctions. We call this the local Schrödinger picture. Local many worlds can also be described in the Heisenberg picture by ascribing Deutsch-Hayden (DH) descriptors to each individual system, each of which evolves under local unitary operations. Both the local wavefunctions and DH descriptors contain records of past interactions, and while the local wavefunctions provide a clear and intuitive physical narrative, the descriptors contain all of the same information in a much more compact form. Unfortunately, the compact form is also somewhat cryptic, which has made the local Heisenberg picture relatively inaccessible. The purpose of this article is to show that the local wavefunctions for any

N

-qubit system can always be constructed directly from the DH descriptors, allowing us to combine the benefits of both pictures.

We demonstrate how to construct a lorentz-invariant, hidden-variable interpretation of relativistic quantum mechanics based on particle trajectories. The covariant theory that we propose employs a multi-time formalism and a lorentz-invariant rule for the coordination of the space-time points on the individual particle trajectories. In this way we show that there is no contradiction between nonlocality and lorentz invariance in quantum mechanics. The approach is illustrated for relativistic bosons, using a simple model to discuss the individual non-locally correlated particle motion which ensues when the wavefunction is entangled. A simple example of measurement is described.

We demonstrate how to construct a Lorentz-invariant, hidden-variable interpretation of relativistic quantum mechanics based on particle trajectories. The covariant theory that we propose employs a multi-time formalism and a Lorentz-invariant rule for the coordination of the spacetime points on the individual particle trajectories. In this way we show that there is no contradiction between non-locality and Lorentz-invariance in quantum mechanics. The approach is illustrated for relativistic bosons, using a simple model to discuss the individual non-locally correlated particle motion which ensues when the wavefunction is entangled. A simple example of measurement is described.

A local interpretation of quantum mechanics is presented. Its main ingredients are: first, a label attached to one of the virtual paths in the path integral formalism, determining the output for measurement of position or momentum; second, a mathematical model for spin states, equivalent to the path integral formalism for point particles in space time, with the corresponding label. The mathematical machinery of orthodox quantum mechanics is maintained, in particular amplitudes of probability and Born's rule; therefore, Bell's type inequalities theorems do not apply. It is shown that statistical correlations for pairs of particles with entangled spins have a description completely equivalent to the two slit experiment, that is, interference (wave like behaviour) instead of non locality gives account of the process. The interpretation is grounded in the experimental evidence of a point like character of electrons, and in the hypothetical existence of a wave like, the de Broglie, companion system. A correspondence between the extended Hilbert spaces of hidden physical states and the orthodox quantum mechanical Hilbert space shows the mathematical equivalence of both theories. Paradoxical behaviour with respect to the action reaction principle is analysed, and an experimental set up, modified two slit experiment, proposed to look for the companion system.

I propose a local, realist interpretation of quantum mechanics based on a physical scalar wave field in ordinary three-dimensional space. Each fundamental particle (electron, photon, proton, etc.) possesses its own wave field. The field is fundamental — it is not emitted by the particle, nor does it draw energy from the particle’s motion. The particle is guided by the phase gradient of its own wave via a Bohmian guidance law. The wave evolves according to a linear Klein-Gordon equation. For a single particle this reproduces the full empirical success of quantum mechanics, including the hydrogen spectrum and the double-slit interference pattern. For multiple particles, each particle retains its own local 3D wave; no configuration space is needed. Entanglement is rejected as a fundamental physical phenomenon: the correlations observed in Bell-type experiments are interpreted as ordinary statistical correlations arising from common past interactions, not as evidence of non-locality or of a non-separable quantum state. The quantum mechanical wave function in configuration space is regarded as a mathematical tool, not a physical field. The theory is local, realist, and fully three-dimensional.

In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.