Revisiting the Algebraic and Analytic Descriptions of Quantum Mechanics

Abstract: We study Heisenberg's matrix mechanics within an algebraic pre-Hilbert framework of arbitrary finite dimension. The commutator of the position and momentum matrices naturally generates a third Hermitian operator whose unbounded character originates from boundary contributions and whose structure induces a discrete analogue of the Cauchy-Hilbert kernel. Compared with the separable Hilbert-space completion, the algebraic framework reproduces the standard spectra, canonical commutation relations, and Heisenberg uncertainty relation for finite-energy states, while the discrete kernel is absorbed into its continuous integral counterpart under completion. The comparison shows that both formulations require restrictions on admissible states for effective calculations -- analytic domain restrictions in Hilbert space and finite-energy restrictions in the pre-Hilbert framework. Finally, we discuss to what extent quantum randomness arises from the algebraic structure of the pre-Hilbert framework.