Publications

  • Rotor-Valued Orientation as Generalized Clifford Geometry

    Kagwe A. Muchane · 2026

    Grassmann's exterior orientation, encoded by the antisymmetry of the wedge product and inherited in Clifford algebra through the anticommutation of orthogonal generators, is shown to be an order-two rotor acting in the oriented plane. Clifford's geometric algebra therefore appears as the binary limit of a broader orientation geometry in which reversion carries rotor phase rather than merely sign reversal. In finite cyclic cases, the exchange symmetry induced by reversion admits a harmonic decomposition, and Clifford's geometric product is shown to be the Z2\mathbb{Z}_2 Fourier decomposition of transposition. At the root lie the generalized geometric product and phase-valued orientation invariants generalizing signed area and signed volume. We use these developments to derive and define generalized Clifford algebra (GCA) canonically from Clifford geometry rather than axiomatically through deformation.

    clifford algebramathematical physics

  • The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information

    Kagwe A. Muchane · 2025

    We revisit the Pauli--Clifford connection to introduce a real, grade-preserving algebraic framework for nn-qubit quantum computation based on the tensor product Cℓ2,0(R)⊗n\mathcal{C}\ell_{2,0}(\mathbb{R})^{\otimes n}. In this setting, the bivector J=e12J = e_{12} satisfies J2=−1J^{2} = -1 and supplies the complex structure on the JJ-closure of a minimal left ideal via right multiplication, while Pauli operations arise as left actions of Clifford elements. The Peirce decomposition organizes the algebra into sector blocks determined by primitive idempotents, with nilpotent elements generating transitions between sectors. Quantum states are represented as equivalence classes modulo the left annihilator, exhibiting the quotient description underlying the minimal left ideal. Adopting a canonical stabilizer mapping, the nn-qubit computational basis state ∣0⋯0⟩\ket{0\cdots 0} is given natively by a tensor product of these idempotents. This structural choice leads to a compatibility law that is stable under the geometric product for nn qubits and aligns symbolic Clifford multiplication with unitary evolution on the Hilbert space.

    PDFDOIKagwe A. Muchane. The State-Operator Clifford Compatibility: A Real Algebraic Framework for Quantum Information. arXiv:2512.07902 [quant-ph] (2025)
    quantum physicsclifford algebramathematical physics