We propose a group-theoretical interpretation of the fact that the transition from classical to quantum mechanics entails a reduction in the number of observables needed to define a physical state and \ to \ or \ in the simplest case). We argue that, in analogy to gauge theories, such a reduction results from the action of a symmetry group. To do so, we propose a conceptual analysis of formal tools coming from symplectic geometry and group representation theory, notably Souriau’s moment (...) map, the Mardsen–Weinstein symplectic reduction, the symplectic “category” introduced by Weinstein, and the conjecture according to which “quantization commutes with reduction”. By using the generalization of this conjecture to the non-zero coadjoint orbits of an abelian Hamiltonian action, we argue that phase invariance in quantum mechanics and gauge invariance have a common geometric underpinning, namely the symplectic reduction formalism. This stance points towards a gauge-theoretical interpretation of Heisenberg indeterminacy principle. We revisit this principle in the light of the difference between the set-theoretic points of a phase space and its category-theoretic symplectic points. (shrink)

We revisit Heisenberg indeterminacy principle in the light of the Galois–Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois–Grothendieck duality between finite K-algebras split by a Galois extension \ and finite \\) -sets can be reformulated as a Pontryagin duality between two abelian groups. We define a Galoisian quantum model in which the Heisenberg indeterminacy principle can be understood as a manifestation of a Galoisian duality: the larger the group of automorphisms \ of (...) the states in a G-set \ , the smaller the “conjugate” algebra of observables that can be consistently evaluated on such states. Finally, we argue that states endowed with a group of automorphisms \ can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations. (shrink)

En este artículo, se introducirá el formalismo de cuantificación canónica denominado "cuantificación geométrica". Dado que dicho formalismo permite entender la mecánica cuántica como una extensión geométrica de la mecánica clásica, se identificarán las insuficiencias de esta última resueltas por dicha extensión. Se mostrará luego como la cuantificación geométrica permite explicar algunos de los rasgos distintivos de la mecánica cuántica, como, por ejemplo, la noconmutatividad de los operadores cuánticos y el carácter discreto de los espectros de ciertos operadores. In this article, (...) We shall introduce the formalism of canonical quantization called "geometric quantization". Since this formalism let us understand quantum mechanics as a geometric extension of classical mechanics, we shall identify the insufficiencies of the latter that are resolved by such an extension. We shall show that geometric quantization permits us to explain some fundamental features of quantum mechanics, such as the non-commutativity of quantum operators and the discrete spectrum of some operators describing physical quantities. (shrink)

Observables have a dual nature in both classical and quantum kinematics: they are at the same time quantities, allowing to separate states by means of their numerical values, and generators of transformations, establishing relations between different states. In this work, we show how this twofold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look (...) at the algebraic description of both classical and quantum observables in terms of Jordan–Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the twofold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of Kähler manifolds. A key achievement of this reformulation is to show that the twofold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman’s general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other. (shrink)

Observables have a dual nature in both classical and quantum kinematics: they are at the same time quantities, allowing to separate states by means of their numerical values, and generators of transformations, establishing relations between different states. In this work, we show how this twofold role of observables constitutes a key feature in the conceptual analysis of classical and quantum kinematics, shedding a new light on the distinguishing feature of the quantum at the kinematical level. We first take a look (...) at the algebraic description of both classical and quantum observables in terms of Jordan–Lie algebras and show how the two algebraic structures are the precise mathematical manifestation of the twofold role of observables. Then, we turn to the geometric reformulation of quantum kinematics in terms of Kähler manifolds. A key achievement of this reformulation is to show that the twofold role of observables is the constitutive ingredient defining what an observable is. Moreover, it points to the fact that, from the restricted point of view of the transformational role of observables, classical and quantum kinematics behave in exactly the same way. Finally, we present Landsman’s general framework of Poisson spaces with transition probability, which highlights with unmatched clarity that the crucial difference between the two kinematics lies in the way the two roles of observables are related to each other. (shrink)

We revisit Heisenberg indeterminacy principle in the light of the Galois-Grothendieck theory for the case of finite abelian Galois extensions. In this restricted framework, the Galois-Grothendieck duality between finite K-algebras split by a Galois extension L and finite Gal-sets can be reformulated as a Pontryagin-like duality between two abelian groups. We then define a Galoisian quantum theory in which the Heisenberg indeterminacy principle between conjugate canonical variables can be understood as a form of Galoisian duality: the larger the group of (...) automorphisms H of the states in a G-set O = G/H, the smaller the ``conjugate'' observable algebra that can be consistently valuated on such states. We then argue that this Galois indeterminacy principle can be understood as a particular case of the Heisenberg indeterminacy principle formulated in terms of the notion of entropic indeterminacy. Finally, we argue that states endowed with a group of automorphisms H can be interpreted as squeezed coherent states, i.e. as states that minimize the Heisenberg indeterminacy relations. (shrink)

We propose the conjecture according to which the fact that quantum mechanics does not admit sharp value attributions to both members of a complementary pair of observables can be understood in the light of the symplectic reduction of phase space in constrained Hamiltonian systems. In order to unpack this claim, we propose a quantum ontology based on two independent postulates, namely the phase postulate and the quantum postulate. The phase postulate generalizes the gauge correspondence between first-class constraints and gauge transformations (...) to the observables of unconstrained Hamiltonian systems. The quantum postulate specifies the relationship between the numerical values of the observables that permit us to individualize a physical system and the symmetry transformations generated by the operators associated to these observables. We argue that the quantum postulate and the phase postulate are formally implemented by the two independent stages of the geometric quantization of a symplectic manifold, namely the prequantization formalism and the election of a polarization of pre-quantum states respectively. (shrink)

We propose a definition of physical objects that aims to clarify some interpretational issues in quantum mechanics. We claim that the transformations generated by the objective properties of a physical system must be strictly interpreted as gauge transformations. We will argue that the uncertainty principle is a consequence of the mutual intertwining between objective properties and gauge-dependant properties. The proposed definition implies that in classical mechanics gauge-dependant properties are wrongly considered objective. We will conclude that, unlike classical mechanics, quantum mechanics (...) provides a complete objective description of physical systems. (shrink)