The Proposal and Formulation of Matrix Mechanics

Olivier Darrigol

This chapter is devoted to several attempts at showing the necessity of quantum mechanics. It first examines whether necessity arguments can be extracted from the history of quantum mechanics, both on the matrix-mechanics side and on the wave-mechanics side. It then recounts the stunning mathematical discovery (1979) that the only one-parameter deformation of the Lie algebra of classical Hamiltonian mechanics is the Moyal algebra, which is isomorphic to the algebra of infinitesimal evolutions of quantum mechanics. The chapter goes on with an account of “quantum logic” as an attempt to justify the matrix-density representation of quantum states and the unitary evolution of these states by a natural logic of Yes–No experimental questions. The last section is a discussion of Lucien Hardy’s (2001) and more recent proposals of “natural axioms” for statistical measurement correlations, leading again to the matrix-density formulation of quantum mechanics.

Roland L. Omnes

Pavel Bóna

Proposals for nonlinear extenstions of quantum mechanics are discussed. Two different concepts of "mixed state" for any nonlinear version of quantum theory are introduced: (i) >genuine mixture< corresponds to operational "mixing" of different ensembles, and (ii) a mixture described by single density matrix without having a canonical operational possibility to pick out its specific convex decomposition is called here an >elementary mixture

Gary E. Bowman

Christian DE RONDE, Raimundo Fernández Mouján, César Massri

In this work we discuss the establishment of Standard Quantum Mechanics (SQM) developed through Schrödinger's and Dirac's wave-vectorial reformulations of Heisenberg's original matrix mechanics. We will argue that while Heisenberg's approach was consistently developed -- taking as a standpoint the intensive patterns that were observed in the lab -- as an invariant-operational formalism, Dirac's axiomatic re-formulation was, instead, developed -- taking as a standpoint Schrödinger's wave mechanics and the methodological guide of Bohr and logical positivists -- as an essentially inconsistent "recipe" intended (but unable) to predict (binary) measurement outcomes. Leaving SQM behind and attempting to restore the consistent and coherent account of a real state of affairs, we will present a new tensorial proposal which -- taking as a standpoint Heisenberg's original approach -- will prove capable not only to extend the matrix formalism to a tensorial representation but also to account for new experimental phenomena.