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arXiv:1401.7018 · International Journal of Modern Physics A

Locality and nonlinear quantum mechanics

Chiu Man Ho, Stephen D. H. Hsu

Published Feb 11, 2015• 5 citations
arXivHTMLPDF

Abstract

Nonlinear modifications of quantum mechanics generically lead to nonlocal effects which violate relativistic causality. We study these effects using the functional Schrödinger equation for quantum fields and identify a type of nonlocality which causes nearly instantaneous entanglement of spacelike separated systems. We describe a simple example involving widely separated wave packet (coherent) states, showing that nonlinearity in the Schrödinger evolution causes spacelike entanglement, even in free field theory.

DOI
10.1142/s0217751x15500293

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Nonlinear modifications of quantum mechanics generically lead to nonlocal effects which violate relativistic causality. We study these effects using the functional Schrödinger equation for quantum fields and identify a type of nonlocality which causes nearly instantaneous entanglement of spacelike separated systems. We describe a simple example involving widely separated wave packet (coherent) states, showing that nonlinearity in the Schrödinger evolution causes spacelike entanglement, even in free field theory.
This work presents a conceptual and semi-formal analysis of quantum entanglement, locality, and determinism, motivated by a thought experiment involving double-slit measurements performed on entangled particles at spacelike separation. While quantum correlations appear instantaneous, the paper demonstrates that no superluminal information transfer or causal influence is required to explain observed outcomes.Building upon the Einstein–Podolsky–Rosen framework and Bell’s theorem, the study distinguishes between local dynamical processes and nonlocal informational correlations encoded in the global quantum state. It argues that measurement outcomes are locally realized yet globally constrained, leading to an effectively deterministic universal description while preserving epistemic uncertainty for observers.The paper reframes quantum nonlocality as a manifestation of global consistency rather than physical influence, offering insights into the relationship between realism, free will, and relativistic causality. The discussion is interpretational in nature and does not introduce new experimental claims, aiming instead to clarify foundational assumptions underlying quantum theory.This work is intended for researchers and students interested in quantum foundations, the philosophy of physics, and the conceptual structure of modern physical theories.
We add nonlinear and state-dependent terms to quantum field theory. We show that the resulting low-energy theory, nonlinear quantum mechanics, is causal, preserves probability and permits a consistent description of the process of measurement. We explore the consequences of such terms and show that nonlinear quantum effects can be observed in macroscopic systems even in the presence of decoherence. We find that current experimental bounds on these nonlinearities are weak and propose several experimental methods to significantly probe these effects. The locally exploitable effects of these nonlinearities have enormous technological implications. For example, they would allow large-scale parallelization of computing (in fact, any other effort) and enable quantum sensing beyond the standard quantum limit. We also expose a fundamental vulnerability of any nonlinear modification of quantum mechanics---these modifications are highly sensitive to cosmic history and their locally exploitable effects can dynamically disappear if the observed universe has a tiny overlap with the overall quantum state of the universe, as is predicted in conventional inflationary cosmology. We identify observables that should persist in this case and discuss opportunities to detect them in cosmic ray experiments, tests of strong field general relativity and current probes of the equation of state of the universe. Nonlinear quantum mechanics also enables novel gravitational phenomena and may open new directions to solve the black hole information problem and to uncover the theory underlying quantum field theory and gravitation.
Principle of locality means that any local change (perturbation) of the stationary state wave function field propagates with finite speed, and therefore reaches distant regions of the field with time delay. If a one-particle or multi-particle non-relativistic quantum system is initially in a stationary state, and its wave function field is locally perturbed, then perturbed and non-perturbed sub-regions appear in the region. According to Schrödinger equation, borders of the perturbed sub-region propagate with infinite speed, and the perturbation instantaneously affects all infinite region. It means that Schrödinger equation predicts infinite speed of the wave function perturbations propagation. This feature of classical Schrödinger equation is traditionally interpreted as non-locality of quantum mechanics. From physical point of view, such mathematical behavior of Schrödinger equation solutions is questionable because it is difficult to accept that in real world infinite physical objects can instantaneously appear. We introduce a hypothesis that in reality speed of propagation of the perturbed sub-region borders is equal speed of light. On this basis we develop and analyze a finite propagation speed concept for non-relativistic quantum equations. It leads to local interpretation of non-relativistic quantum mechanics consistent with principle of locality and free of local hidden variables. The theory is applied to analysis of Einstein-Podolsky-Rosen (EPR) paradox, entanglement and properties of perturbed matter waves. We proved that formulated theory agrees with results of classical experiments on electron matter waves diffraction.
Utilizing the general theory of open quantum systems to investigate the exact dynamical evolution of simple bilinear systems, we discover a mechanism of the dynamical genesis of quantum entanglement. We focus in detail on the exact quantum evolution dynamics of two photonic modes (or any two bosonic modes) coupled to each other through a linear interaction, as the simplest system of open quantum systems that we have investigated in the past two decades. Such a linear coupling alone fails to produce two-mode entanglement. We also start with an initially separable pure state of the two modes. By solving exactly the quantum equation of motion without relying on the probabilistic interpretation, we find that when the initial state of one mode is different from a coherent state (a minimum uncertainty wave packet with equal variance in the conjugate quadratures that corresponds to a well-defined classically "particle"), the causality in the time-evolution of each mode is internally violated. It also leads to the emergence of quantum entanglement between the two modes. The lack of causality is the nature of statistics. We discover that it is the internal violation of causality in the reduced (subsystem) dynamical evolution that results in the emergence of entanglement and statistic probability in quantum mechanics, even though the dynamical evolution of the whole system completely obeys the deterministic Schrödinger equation. This conclusion is valid for the quantum dynamics of more complicated composite systems. It may provide the fundamental mechanism of the dynamical genesis for both the entanglement and the statistical probability within the deterministic framework of quantum mechanics, which is the longest-standing problem that has not been fully understood since the birth of quantum mechanics.
We use the Tomonaga-Schwinger (TS) formulation of quantum field theory to determine when state-dependent additions to the local Hamiltonian density (i.e., modifications to linear Schrodinger evolution) violate relativistic covariance. We derive new operator integrability conditions required for foliation independence, including the Frechet derivative terms that arise from state-dependence. Nonlinear modifications of quantum mechanics affect operator relations at spacelike separation, leading to violation of the integrability conditions.
Principle of locality means that any local change (perturbation) of the stationary state wave function field propagates with finite speed, and therefore reaches distant regions of the field with time delay. If a one-particle or multi-particle non-relativistic quantum system is initially in a stationary state, and its wave function field is locally perturbed, then perturbed and non-perturbed sub-regions appear in the region. According to Schrodinger equation, borders of the perturbed sub-region propagate with infinite speed, and the perturbation instantaneously affects all infinite region. It means that Schrodinger equation predicts infinite speed of the wave function perturbations propagation. This feature of classical Schrodinger equation is traditionally interpreted as non-locality of quantum mechanics. From physical point of view, such mathematical behavior of Schrodinger equation solutions is questionable because it is difficult to accept that in real world infinite physical objects can instantaneously appear. We introduce a hypothesis that in reality speed of propagation of the perturbed sub-region borders is equal speed of light. On this basis we develop and analyze a finite propagation speed concept for non-relativistic quantum equations. It leads to local interpretation of non-relativistic quantum mechanics consistent with principle of locality and free of local hidden variables. The theory is applied to analysis of Einstein-Podolsky-Rosen (EPR) paradox, entanglement and properties of perturbed matter waves. We proved that formulated theory agrees with results of classical experiments on electron matter waves diffraction.