On Classical and Quantum Objectivity

Abstract

Abstract. We propose a conceptual framework for understanding the relationship between observables and operators in mechanics. To do so, we introduce a postulate that establishes a correspondence between the objective properties permitting to identify physical states and the symmetry transformations that modify their gauge dependant properties. We show that the uncertainty principle results from a faithful -or equivariant -realization of this correspondence. It is a consequence of the proposed postulate that the quantum notion of objective physical states is not incomplete, but rather that the classical notion is overdetermined.

Quantum Ontology in the Light of Gauge Theories

“…[12]. By using the so-called momentum map formalism, we showed that the injectivity of the map between observables and operators would permit us to define physical observables as quantities that faithfully quantify the representation of abstract operators belonging to the corresponding Lie algebra g as operators acting on the space of states.…”

“…In Ref. [12], we introduced the following terminology: the fundamental vector field v ξ ∈ T P evaluated at a point x ∈ P is said to be the representation of the abstract operator ξ ∈ g on the particular state x. A symplectic G-action is said to be Hamiltonian if there exists a map µ : g → C ∞ (P ) (called co-momentum map) such that τ •μ = ι, where τ : C ∞ (P ) → HP .…”

“…In Refs. [11,12] we argued that Heisenberg uncertainty principle, far from resulting from an epistemic restriction to the amount of information an observer can have about an object, can be understood as a consequence of the circular imbrication between objective properties and non-objective phases established by the phase postulate. Indeed, the phase postulate entails that an objective property of an object must be invariant under the phase transformations induced by all the other objective properties of the same object.…”

“…While the phase postulate was introduced in Ref. [12], a first formulation of the quantum postulate was proposed in Ref. [11].…”

Can Classical Description of Physical Reality Be Considered Complete?

“…This article develops, in more conceptual terms, the interpretation of quantum mechanics begun in Catren [2008]. This interpretation is founded on an analysis of the symplectic formulation of mechanics (Abraham and Marsden [1978]; Libermann and Marle [1987]; Marsden and Ratiu [1999]; Souriau [1997]) and the geometric quantization formalism (Brylinski [1993]; Kostant [1970]; Souriau [1997]; Woodhouse [1992]).…”