Quantum Mechanics and Hidden Variable Theories

Quantum mechanics represents one of the most successful scientific theories of the twentieth century, describing the behaviour of matter and energy at atomic and subatomic scales. Central to its formalism is the wave function, a complex-valued object whose squared modulus yields probability densities for measurement outcomes. Despite its predictive power, foundational questions persist concerning the nature of quantum indeterminacy, wave-function collapse and the role of observers. Hidden-variable theories, most notably the de Broglie–Bohm pilot-wave interpretation, propose that particles possess definite positions guided by an underlying wave field, restoring determinism at the cost of intrinsic non-locality. Bell’s theorem and related experiments have demonstrated that any viable hidden-variable model must incorporate non-local correlations, as seen in the instantaneous influence of the pilot wave on spatially separated particles. The concept of quantum equilibrium, wherein the distribution of hidden variables matches Born’s rule, underpins the empirical success of the pilot-wave framework yet invites questions about relaxation timescales and the emergence of classicality. Recent research has extended these ideas to relativistic regimes, quantum field settings and non-normalisable states, and has explored practical implications for sensing, communication and quantum computing. Such investigations deepen our understanding of quantum foundations and guide the development of next-generation quantum technologies.

Research from Nature Portfolio

Recent studies have advanced the pilot-wave approach in several directions. A recent analysis of the harmonic oscillator has shown that non-normalisable eigenstates can be interpreted as bound configurations within a pilot-wave framework, leading to a generalised notion of quantum equilibrium governed by an H-function that obeys an analogue of Boltzmann’s H-theorem. Another work has deployed weak measurement techniques to reconstruct relativistic Bohmian trajectories of photons in interferometric setups, demonstrating operationally that these trajectories respect both quantum-mechanical continuity and relativistic velocity addition. A complementary computational development utilises ensembles of one-particle pilot waves to simulate strongly entangled many-body bosonic systems, revealing how entanglement and ground-state dynamics emerge naturally from interacting Bohmian trajectories and paving the way for efficient simulations of complex quantum systems.

Quantum Mechanics and Hidden Variable Theories publication trend

The graph below shows the total number of articles in quantum mechanics and hidden variable theories across all publications each year (not limited to Nature Index journals).

Technical terms

Wave function: A complex-valued function describing the complete quantum state, with its squared modulus giving probability densities for measurement outcomes.

Pilot-wave theory: A non-local hidden-variable interpretation positing that particles follow deterministic trajectories guided by a wave field, reproducing standard quantum predictions under quantum equilibrium.

Quantum equilibrium: The statistical distribution of hidden variables that yields the Born rule, ensuring empirical agreement between pilot-wave models and conventional quantum mechanics.

H-function: A measure of deviation from quantum equilibrium in pilot-wave theory, analogous to Boltzmann’s H-function in classical thermodynamics.

Casimir–Polder force: A quantum fluctuation-induced attraction between neutral atoms and surfaces, significant in cold-atom physics and quantum sensing.

Weak measurement: A technique that minimally disturbs the quantum system, allowing the extraction of average values of observables and reconstruction of trajectories in certain interpretations.

References

1. Physical interpretation of non-normalizable harmonic oscillator states and relaxation to pilot-wave equilibrium. Scientific Reports (2024).
2. Relativistic Bohmian trajectories of photons via weak measurements. Nature Communications (2022).
3. Entangled Quantum Dynamics of Many-Body Systems using Bohmian Trajectories. Scientific Reports (2018).
4. Countering a fundamental law of attraction with quantum wave-packet engineering. Physical Review Research (2023).
5. Justifying Born’s Rule Pα = |Ψα|2 Using Deterministic Chaos, Decoherence, and the de Broglie–Bohm Quantum Theory. Entropy (2021).
6. Time scales for dynamical relaxation to the Born rule. Proceedings of the Royal Society A (2011).