Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator

Abstract

We introduce a bi-oriented interlaced matrix product on $2\times 2$ matrices over a unital, possibly non-associative algebra, extending a matrix mnemonic for Cayley--Dickson doubling. The opposite orientations $X_R$ and $X_L$ coincide precisely when the underlying algebra is commutative. For the canonical position and momentum operators $Q,P$ on $L^2(\mathbb{R})$, a matrix $D$ and its companion $\widetilde{D}$ satisfy, on the common Schwartz domain, $\widetilde{D} X_R D =\widetilde{D} X_L D =2H Id_2$, where $H=(P^2+Q^2)/2$. We study the associated interlaced eigenvalue equations for operator-valued columns. The combinations $Φ=A+iB$ and $Ψ=A-iB$ decouple the equations into two-sided Sylvester equations. We construct explicit rank-one generalized eigenoperators from position and momentum eigenstates. Their distributional kernels belong to the doubled rigged Hilbert space $S(\mathbb{R}^2)\subset L^2(\mathbb{R}^2) \subset S'(\mathbb{R}^2)$; the corresponding position and momentum operators act on different tensor factors and therefore commute. The rank-one families do not exhaust the distributional solutions: higher-order generalized solutions are also exhibited. We distinguish the algebraic identities on the Schwartz core from questions concerning closed-operator realizations.