| de Broglie–Bohm theory | |
|---|---|
| Name | de Broglie–Bohm theory |
| Author | Louis de Broglie; developed by David Bohm |
| Introduced | 1927 (pilot wave); 1952 (Bohmian formulation) |
| Discipline | Quantum mechanics |
| Notable concepts | Pilot wave, Hidden variable theory, Quantum potential |
de Broglie–Bohm theory
de Broglie–Bohm theory is a non-standard interpretation of Quantum mechanics that postulates determinate particle trajectories guided by a wavefunction. It provides a causal, realist account of quantum phenomena via a pilot wave (or "guiding equation") and an accompanying quantum potential, offering an alternative to the Copenhagen interpretation. The theory matters because it exposes the role of nonlocality and hidden-variable theorys in quantum foundations and suggests distinct approaches to quantum measurement and ontology.
The theory traces to Louis de Broglie's 1927 pilot-wave proposal presented at the Solvay Conference and was later revived and extended in 1952 by David Bohm in two papers that formulated a complete non-relativistic account. De Broglie originally sought to reconcile wave–particle duality for single particles such as the electron, building on his 1924 PhD thesis introducing matter waves. Bohm's work built on the Schrödinger equation and introduced the quantum potential and explicit trajectories to reproduce the predictions of standard quantum mechanics. Key historical interlocutors include Max Born (probabilistic interpretation), Niels Bohr and proponents of the Copenhagen school, and later critics and supporters such as John Bell, who highlighted the theory's clarity about nonlocality via Bell's theorem. Development has continued among researchers at institutions like Princeton University, University of London and University of Oxford and in works by Detlef Dürr, Sheldon Goldstein, and Antony Valentini.
In the de Broglie–Bohm formalism a system is described by a configuration (e.g., particle positions) and a universally evolving wavefunction satisfying the Schrödinger equation. The wavefunction ψ yields a guiding velocity field v = (ħ/m) Im(∇ψ/ψ) for a single nonrelativistic particle, producing deterministic trajectories via the guiding equation. The dynamics can be recast using the polar decomposition ψ = R e^{iS/ħ}; substitution into the Schrödinger equation yields a modified Hamilton–Jacobi equation with an extra term, the quantum potential Q = −(ħ^2/2m)(∇^2R)/R, which affects particle motions. For multiple particles the guiding equation couples all particle positions through the multi-particle wavefunction, manifesting explicit configuration-space interactions. The formalism respects the mathematical structure of Hilbert space and reproduces Born-rule statistics under appropriate conditions known as quantum equilibrium.
De Broglie–Bohm theory accounts for statistical predictions of quantum experiments via the quantum equilibrium hypothesis, which posits that an ensemble of systems has particle configurations distributed according to |ψ|^2. Measurement outcomes arise from deterministic evolution of system-plus-apparatus configurations without invoking wavefunction collapse: apparent collapse is effective, resulting from entanglement and conditional wavefunctions. The theory is manifestly nonlocal for entangled systems because the guiding equation for one particle depends on the instantaneous configuration of distant particles, a feature made precise by Bell's theorem and emphasized in discussions of EPR paradox thought experiments. Philosophically, the theory is realist and deterministic, contrasting with indeterministic readings of Quantum mechanics; it raises questions about the role of configuration space vs. three-dimensional space in ontology and about the status of the wavefunction as physical or nomological.
Extending the pilot-wave approach to relativistic regimes and quantum field theory has been an active research area. Relativistic particle models face challenges with Lorentz invariance because of the theory's nonlocal guidance in a preferred foliation of spacetime; proposals include adopting a dynamically preferred foliation or a covariant multi-time formalism. For quantum fields, Bohmian approaches assign beables such as field configurations (e.g., scalar field values) or particle creation/annihilation events; important contributions include works by P. R. Holland, Detlef Dürr, and Sheldon Goldstein adapting the scheme to Quantum field theory and Quantum electrodynamics. Attempts at a Bohmian approach to Quantum gravity and Dirac equation systems remain exploratory, with models often invoking additional structure (foliations, pilot-wave fields) to reconcile nonlocal guidance with relativistic causality.
Empirically, de Broglie–Bohm theory reproduces the statistical predictions of nonrelativistic quantum mechanics when quantum equilibrium holds, so it is largely empirically equivalent to standard formulations for typical quantum experiments such as double-slit experiments, Stern–Gerlach experiments, and Bell test experiments. Proposed empirical differences hinge on nonequilibrium distributions or extensions beyond standard quantum theory; Antony Valentini and collaborators have suggested that early-universe quantum nonequilibrium could leave imprints in the cosmic microwave background or permit superluminal signalling in principle. Laboratory tests exploring subtle trajectory predictions or weak measurement reconstructions of Bohmian trajectories have been performed (e.g., weak-value experiments), but these do not decisively discriminate interpretations because they depend on modelling choices and assumptions about equilibrium.
Critics argue that the theory's introduction of additional ontology (exact particle positions and a universal wavefunction) is ontologically extravagant and that its nonlocality conflicts with relativistic principles unless supplemented by extra structure. Supporters counter that it restores clarity about measurement and provides a coherent realist account. Debates intersect with discussions of the Copenhagen interpretation, many-worlds interpretation (Everettian approaches), and modal or spontaneous-collapse models such as Ghirardi–Rimini–Weber theory. Prominent defenders and analysts include John Bell, P. R. Holland, Detlef Dürr, and Sheldon Goldstein; prominent skeptics include proponents of operationalist frameworks. The ongoing research agenda includes developing relativistic, field-theoretic, and cosmological models, assessing potential nonequilibrium signatures, and clarifying the ontological status of the wavefunction within philosophy of physics.
Category:Quantum mechanics interpretations Category:Hidden variable theories