What do the basic laws of physics describe? Why is the state of a classical particle identified by position and momentum (i.e. a point on a cotangent bundle) while the state of a quantum system is identified by a wave function (i.e. a vector in a Hilbert space)? Could we have had different laws?

In this project we explore these questions by identifying a handful of physical principles from which the basic laws can be rigorously derived. The aim is to fully clarify under what physical assumptions each physical theory is valid, providing a clear understanding of their foundation that requires no interpretation because the physical meaning is made precise right from the start. Uncovering hidden assumptions forces us to understand their limitations, possibly leading to new insights and ideas.

Overview of the framework

In this table we list the core ideas of our work and how they translate into the mathematical framework.

Concept Consequence
Principle of scientific objectivity. Science is universal, non-contradictory and evidence based. Our general mathematical theory of experimental science. A scientific theory is a collection of statements that can be experimentally tested, which ultimately determine the possible cases that can be experimentally distinguished. It must keep track of what can be verified (a topology), what can be predicted (a sigma-algebra) and a way to establish the precision of different statements (a measure).
I Assumption of determinism and reversibility. Given the present state of the system under study, all future and past states are uniquely identified. Dynamical system. The state of the system is the finest description that can be determined experimentally. Deterministic and reversible evolution preserves what can be tested experimentally (the topology), what can be predicted (the sigma algebra), the precision of statements (the measure) and the nature of the system (e.g. vector space, metric, ...). That is: it is a self-homeomorphism in the category (i.e. a continuous transformation for the topology, a measure preserving map for the measure, a linear transformation for a vector space, ...).
II.a Classical assumption of infinitesimal reducibility. The system under study is made of a homogeneous material that can be decomposed into infinitesimal parts. Giving the state of the whole system is equivalent to giving the state of each infinitesimal part. The evolution of each part is deterministic and reversible. Classical Hamiltonian particle mechanics. The state of the composite system is identified by a distribution of the material over the states of the infinitesimal parts. The state space of the infinitesimal part must allow to define the distribution independently from what coordinate system is used (a symplectic manifold). Deterministic and reversible evolution can only move the density from one state to another without changing it and must conserve the number of states as they evolve. That is: each infinitesimal part evolves according to Hamilton's equations (a symplectomorphism).
II.b Quantum assumption of irreducibility. The system under study is made of a homogeneous material that can be decomposed into infinitesimal parts. Giving the state of the whole system does not provide any description for the infinitesimal parts. The evolution of the system as a whole is deterministic and reversible but not at the level of each infinitesimal part. Quantum Hamiltonian particle mechanics. The state space of the homogeneous material is identified by a distribution within which each infinitesimal part moves chaotically. This random motion can be described in terms two independent stochastic variables (a complex distribution). While the overall system can still be assembled/disassembled into parts (a vector space), the correlation between the random variables will result into constructive and descructive interference. Deterministic and reversible evolution will preserve the total amount of material (unitary evolution) and therefore evolves according to the Schroedinger equation.
III Kinematic equivalence. For the system under study, the kinematics (i.e. trajectories in physical space-time) and the dynamics (i.e. trajectories in state space) are equivalent. Giving a set of initial conditions, such as position and velocity, is equivalent to giving the initial state. Lagrangian particle mechanics with scalar/vector potential forces. As we must be able to re-express the distributions defined over state variables as distributions over kinematic variables, ranges of initial states have to map linearly to ranges of initial conditions (symplectic form induces a metric). This constrains the possible relationships between velocities and momentum and limits the system to massive particles moving under forces described by scalar or vector potentials.