We show that for a system of two entangled particles, there is a dual description to the particle equations in terms of classical theory of conformally stretched spacetime. We also connect these entangled particle equations with Finsler geometry. We show that this duality translates strongly coupled quantum equations in the pilot-wave limit to weakly coupled geometric equations.

The classical and quantum dynamics of two particles constrained on $$S^1$$ S 1 is discussed via Dirac’s approach. We show that when state is maximally entangled between two subsystems, the product of dispersion in the measurement reduces. We also quantify the upper bound on the external field $$\vec {B}$$ B → such that $$\vec {B}\ge \vec {B}_{upper }$$ B → ≥ B → upper implies no reduction in the product of dispersion pertaining to one subsystem. Further, we report on the cut-off value of the external field $$\vec {B}_{cutoff }$$ B → cutoff , above which the bipartite entanglement is lost and there exists a direct relationship between uncertainty of the composite system and the external field. We note that, in this framework it is possible to tune the external field for entanglement/unentanglement of a bipartite system. Finally, we show that the additional terms arising in the quantum Hamiltonian, due to the requirement of Hermiticity of operators, produce a shift in the energy of the system.

For a system of n non-relativistic spinless bosons, we show by using a set of suitable matching conditions that the quantum equations in the pilot-wave limit can be translated into a geometric language for a Finslerian manifold. We further link these equations to Euclidean timelike relative Funk geometry and show that the two different metrics in both of these geometric frameworks lead to the same coupling.PACS numbers:

We show that for non-relativistic free particles, the (bosonic) many particle equations can be rewritten in geometric fashion in terms of a classical theory of conformally stretched spacetime. We further generalize the results for the particles subject to a potential.

The classical and quantum dynamics of particles constrained on a right helicoid is discussed via Dirac approach. We show how the uncertainty in measurement for observables of maximally entangled system is affected by the total number of constrained particles $$\sigma $$ σ , external magnetic field $$\vec {\mathcal {B}}$$ B → ; as well as by geometric parameters like the pitch$$\rho $$ ρ and the radial position of particles. In doing so we also highlight numeric bounds on the external field strengths to tune both intraparticle and bipartite entanglement and we remark that the bipartite entanglement is more robust to changes in the fields than the intraparticle entanglement, in this framework. We also highlight specific parameter regimes which lead the uncertainty (in measurement) to achieve respective parameter independence and, for a particular subset of commutation relations, the system remains confined in the quantum regime even in the limit $$\sigma \rightarrow \infty $$ σ → ∞ . It is observed that the uncertainties are strongly influenced by the geometric parameters e.g., $$\rho $$ ρ , and the strength of bipartite as well as intraparticle entanglement might be controllable through $$\rho $$ ρ . The energy equation for this setup is obtained and the additional terms are discussed which arise due to quantum correlations, orbit–orbit interaction and the normal Zeeman effect, which leads to the splitting of the energy level into 11 non-degenerate levels. Finally we comment that, a linkage of this phenomenology with Aharonov–Bohm like effect might be possible by strictly confining $$\vec {\mathcal {B}}$$ B → along the central axis of the helicoid.