## Final Activity and Management Report Summary - FPQT (From Fermat's principle to quantum teleportation)

My research on the foundations of quantum mechanics in this period has produced three papers. In 'can nothing cause quantum jumps,' written with Gino Tarozzi, a paradox is discussed in which a photon can occupy one of two positions, 'left' or 'right.'

Quantum mechanics allows the two possibilities-'nothing left, photon right' and 'photon left, nothing right'-to be combined in a (coherent) superposition; or alternatively in an (incoherent) mixture, with similar terms, but no phase relation between them.

As the phase relation is statistically significant, and can in principle be revealed experimentally, the two superposed possibilities must both be present in nature, together, for they somehow 'communicate' with one another through that relation. The eigenvalue +1 can be assigned to the presence of the photon on the right, -1 to its absence, to construct a measurable physical quantity, 'photon-right.' Its expectation vanishes for the aforementioned superposition. Now, suppose we look for the photon on the left and do not find it. So it must be on the right. The superposition accordingly collapses to the term 'nothing left, photon right,' whose expectation for photon-right jumps right away from zero to one. Only with a mixture could one ascribe the initial (vanishing) expectation to an ignorance which is then overcome once the photon is not found on the left.

The issue is: what exactly happens on the left to cause the ontic (and not merely epistemic) jump on the right? Is it some mental event? Or is it nothing at all? In 'If Bertlmann had three feet' I argue that perfect quantum correlations cannot be due to additive conservation. In 'Duhem, Quine and the other dogma', I make a more philosophical point, illustrated by a quantum-mechanical example. By linking (verificationist) meanings and analyticity, Quine does away with both 'dogmas of empiricism' together, as 'two sides of a single dubious coin.' His denial of the second ('reductionism') has been associated with Duhem's rejection of crucial experiments - which relies on the 'cleavage,' repeatedly invoked, between mathematics and physics. The other dogma repudiated by Quine is the 'cleavage between analytic and synthetic truths'; but aren't the truths of mathematics analytic, those of physics synthetic? Exploiting Quine's association of essences, meaning, synonymy and analyticity, and appealing to a 'model-theoretical' notion of abstract test derived from Duhem and Quine themselves - which can be used to overcome their holism by separating essences from accidents - I reconsider the crucial experiment, as well as both 'cleavages'; and propose a characterisation of the meaning and reference of sentences, which naturally extends the distinction first applied to words.

My research has also dealt with the classical roots (Fermat's principle etc.) of quantum mechanics, in two papers. In 'optico-mechanical analogy: an axiomatic approach,' an axiomatic characterisation of a 'two-level Hamiltonian structure' is proposed, which expresses the optico-mechanical analogy by representing optics and mechanics as (disjoint) classes of models satisfying the axioms.

There is the 'Hamilton-Jacobi level,' which involves a differential manifold on which the characteristic function satisfying the Hamilton-Jacobi equation is defined; and the 'symplectic level,' involving the Hamiltonian, defined on the cotangent bundle of the manifold. The two levels, with the (analogous) structures on them, concern both optics and mechanics. In 'Cartesian and Lagrangian momentum,' historical, physical and geometrical relations between two different momenta, characterised here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical.

Quantum mechanics allows the two possibilities-'nothing left, photon right' and 'photon left, nothing right'-to be combined in a (coherent) superposition; or alternatively in an (incoherent) mixture, with similar terms, but no phase relation between them.

As the phase relation is statistically significant, and can in principle be revealed experimentally, the two superposed possibilities must both be present in nature, together, for they somehow 'communicate' with one another through that relation. The eigenvalue +1 can be assigned to the presence of the photon on the right, -1 to its absence, to construct a measurable physical quantity, 'photon-right.' Its expectation vanishes for the aforementioned superposition. Now, suppose we look for the photon on the left and do not find it. So it must be on the right. The superposition accordingly collapses to the term 'nothing left, photon right,' whose expectation for photon-right jumps right away from zero to one. Only with a mixture could one ascribe the initial (vanishing) expectation to an ignorance which is then overcome once the photon is not found on the left.

The issue is: what exactly happens on the left to cause the ontic (and not merely epistemic) jump on the right? Is it some mental event? Or is it nothing at all? In 'If Bertlmann had three feet' I argue that perfect quantum correlations cannot be due to additive conservation. In 'Duhem, Quine and the other dogma', I make a more philosophical point, illustrated by a quantum-mechanical example. By linking (verificationist) meanings and analyticity, Quine does away with both 'dogmas of empiricism' together, as 'two sides of a single dubious coin.' His denial of the second ('reductionism') has been associated with Duhem's rejection of crucial experiments - which relies on the 'cleavage,' repeatedly invoked, between mathematics and physics. The other dogma repudiated by Quine is the 'cleavage between analytic and synthetic truths'; but aren't the truths of mathematics analytic, those of physics synthetic? Exploiting Quine's association of essences, meaning, synonymy and analyticity, and appealing to a 'model-theoretical' notion of abstract test derived from Duhem and Quine themselves - which can be used to overcome their holism by separating essences from accidents - I reconsider the crucial experiment, as well as both 'cleavages'; and propose a characterisation of the meaning and reference of sentences, which naturally extends the distinction first applied to words.

My research has also dealt with the classical roots (Fermat's principle etc.) of quantum mechanics, in two papers. In 'optico-mechanical analogy: an axiomatic approach,' an axiomatic characterisation of a 'two-level Hamiltonian structure' is proposed, which expresses the optico-mechanical analogy by representing optics and mechanics as (disjoint) classes of models satisfying the axioms.

There is the 'Hamilton-Jacobi level,' which involves a differential manifold on which the characteristic function satisfying the Hamilton-Jacobi equation is defined; and the 'symplectic level,' involving the Hamiltonian, defined on the cotangent bundle of the manifold. The two levels, with the (analogous) structures on them, concern both optics and mechanics. In 'Cartesian and Lagrangian momentum,' historical, physical and geometrical relations between two different momenta, characterised here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical.