Conservation Laws and Deterministic Evolutions

Jun Liu

Bassano Vacchini

Marriam Naeem

Deterministic microscopic laws constrain physical systems through conservation laws and boundary conditions, yet macroscopic evolution routinely exhibits irreversible collapse, destruction, and path dependence. We analyze deterministic classical and quantum systems admitting multiple metastable macroscopic trajectories under fixed global constraints and show that deterministic dynamics generally underdetermine macroscopic history. We introduce a distinction between constraint-defined macroscopic equivalence classes, fixed by invariant quantities, and metastable macroscopic trajectories, defined by dynamical stability. Using stability and basin-of-attraction arguments, we characterize collapse as a deterministic loss of metastability in which an entire macroscopic trajectory becomes dynamically inaccessible without violation of microscopic reversibility or conservation laws. Irreversible destruction corresponds to elimination of physically realizable paths rather than loss of information or coarse-grained inaccessibility. We show that explanations based solely on coarse-graining or decoherence are insufficient to account for such trajectory elimination, as they preserve admissibility of erased histories. In contrast, instability-driven trajectory destruction constitutes an ontic dynamical process, directly realized in many-body and condensed matter systems. The framework provides a unified account of irreversible macroscopic change and causal differentiation within deterministic dynamics, without invoking stochasticity, modified laws, or observer-dependent mechanisms. Deterministic laws constrain admissible macroscopic states, while instability and metastability govern the physically realizable histories connecting them.

Gerard ’t Hooft

A sharper formulation is presented for an interpretation of quantum mechanics advocated by author. As an essential element we put forward conservation laws concerning the ontological nature of a variable, and the uncertainties concerning the realisation of states. Quantum mechanics can then be treated as a device that combines statistics with mechanical, deterministic laws, such that uncertainties are passed on from initial states to final states.

Dennis Dieks

James L Park, William Band

Ordinary quantal conservation laws are associated with null operators for time rate of change and are valid for causal evolution of the system through either pure or mixed states. There is, however, a larger class of quantal constants of the motion, some of which are conserved only for pure state evolution. After an analysis of the theoretical origins of these exotic conserved quantities, several illustrations are presented with empirical interpretations based on the quorum theory methods used in Paper I.