Symmetry and observability

SEBASTIAN FORTIN; OLIMPIA LOMBARDI

Lovaina

Congreso; Summer Symposium 2012 of the International Society for the Philosophy of Chemistry; 2012

International Society for the Philosophy of Chemistry (ISPC), Catholic University of Leuven

Traditionally, symmetry has played a central role in the stereochemical study of the spatial arrangements of atoms within molecules, under the scientific assumption that molecular structure has definite consequences on the macroscopic properties of chemical substances. In this context, the interest was mainly focused on spatial symmetries, as those related with rotations or reflections, that is, transformations on physical space. However, when the concept of symmetry is expressed from an abstract and formal perspective, it includes symmetries that are not defined in spatial terms, but involve non-spatial transformations. In this presentation we will consider this formal notion of symmetry and its relationship with observability, in particular in the context of quantum chemistry. For this purpose, we will begin by recalling the abstract concept of symmetry as invariance under a transformation. Then, we will distinguish between symmetries of laws and symmetries of objects, and we will stress the difference between invariance and covariance. On this basis, we will apply these general considerations to the particular case of quantum mechanics, in order to distinguish the invariance and the covariance of the Schrödinger equation, and to consider the invariance of the Hamiltonian. In the particular quantum context, we will focus on the Hamiltonian, in order to recall that, when the Hamiltonian is invariant under a certain continuous transformation, the generator of that transformation is a constant of motion of the system: each symmetry of the Hamiltonian defines a conserved quantity. We will also stress the strong relationship between symmetry of the Hamiltonian and energy degeneracy. On this conceptual and theoretical basis, finally we will consider the relationship between symmetry and observability in the case of the Hamiltonian: we will propose a thesis according to which the observables whose eigenvalues break the symmetries of the Hamiltonian have no observable energetic manifestations in spite of the term ?observable? to designe them. We will illustrate the claim with the simple case of the hydrogen atom, in the free case and under the action of an external magnetic field (Zeeman effect and fine structure). As a future perspective of this work, we will venture a general hypothesis about measurement, according to which in the quantum realm measurement is a symmetry breaking process that renders a symmetry generator of the system?s Hamiltonian empirically accessible.