Entropic Dynamics approach to Relational Quantum Mechanics (2506.07921v1)

Abstract: The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational quantum mechanics from well known inference principles -- probability, entropy and information geometry. Although only partially relational -- the absolute structures of simultaneity and Euclidean geometry are still retained -- these models provide a useful testing ground for ideas that will prove useful in the context of more realistic relativistic theories. The fact that in ED the positions of particles have definite values, just as in classical mechanics, has allowed us to adapt to the quantum case some intuitions from Barbour and Bertotti's classical framework. Here, however, we propose a new measure of the mismatch between successive states that is adapted to the information metric and the symplectic structures of the quantum phase space. We make explicit that ED is temporally relational and we construct non-relativistic quantum models that are spatially relational with respect to rigid translations and rotations. The ED approach settles the longstanding question of what form should the constraints of a classical theory take after quantization: the quantum constraints that express relationality are to be imposed on expectation values. To highlight the potential impact of these developments, the non-relativistic quantum model is parametrized into a generally covariant form and we show that the ED approach evades the analogue of what in quantum gravity has been called the problem of time.

• The paper introduces an Entropic Dynamics framework that reconstructs relational quantum mechanics using inference-based constraints instead of classical dynamics.
• The methodology employs information geometry and Hamilton-Killing flows to define best matching conditions and quantum state equivalence.
• Insights include circumventing operator ambiguities and addressing the 'problem of time' by formulating temporality as a dynamical variable.

Entropic Dynamics Approach to Relational Quantum Mechanics

In the paper titled "Entropic Dynamics approach to Relational Quantum Mechanics," authors Ariel Caticha and Hassaan Saleem explore the utilization of the Entropic Dynamics (ED) framework to construct non-relativistic models of relational quantum mechanics. The ED approach is leveraged to create a theoretical model where particle positions are ontic, and quantum mechanics is reconstructed on the basis of inference principles such as probability theory, entropy, and information geometry.

Key Contributions and Theoretical Implications

The paper investigates a partially relational approach to quantum mechanics, retaining absolute structures like simultaneity and Euclidean geometry. However, it establishes models in which spatial relationality operates with respect to rigid translations and rotations, markedly different from the classical precedents set by Newtonian mechanics. Notably, within these models, constraints such as those regarding the positions of particles are imposed on expectation values rather than ontic states, aligning with epistemic interpretations of quantum mechanics. This is a pivot from classical constraints and offers insights into how relational dynamics may be integrated into quantum mechanics.

The authors demonstrate that the conditions for best matching in quantum mechanics — finding conditions under which two states are considered equivalent — can be formulated utilizing information and symplectic geometric tools natural to the ED framework. By adjusting kinematics using these tools, the implications are profound for quantizing theories that traditionally rely on classical dynamics foundations. It eschews operator ambiguities and provides a direct path to formulating quantum theory which does not require supplementary classical dynamics.

Methodology

The methodology involves reinterpreting the dynamics of probabilities not through ontic particle configurations but through the evolution of wave functions modeled within epistemic phase space. This is accomplished without relying on predicated classical dynamics templates — a departure from traditional quantization methods. The authors introduce the notion of Hamilton-Killing flows, nonlinear transformations in the information geometry arena, preserving symplectic structures in the epistemic space.

The paper also addresses the challenge known in quantum gravity as the "problem of time," by making temporality relational. Through the parametrization technique borrowed from classical general relativity, time becomes a dynamical variable, demonstrating that ED can inherently circumvent the traditional pitfalls faced in canonical quantum gravity.

Future Developments

The relational ED framework developed here is instrumental for further explorations into quantum theories beyond particles, anticipating use in complex systems such as quantum fields and gravity. The insights into conditional constraints on relational dynamics may extend to better formalizations across various fundamental interactions, potentially redefining approaches in electromagnetism and Yang-Mills theories.

Conclusion

The approach laid out offers a novel viewpoint on integrating relational concepts within quantum mechanics. While restricted to non-relativistic considerations currently, its extension to relativistic cases appears promising. By manifestly disentangling epistemic from ontic components, this paper offers an innovative perspective on understanding dynamics in quantum mechanics through inference rather than classical paradigms. As such, it opens new pathways for thinking about relational dynamics in quantum theories.