The bicomplex algebra as a minimal extension of the scalar algebra of quantum mechanics

Quantum mechanics is formulated on complex Hilbert spaces, where the complex numbers constitute the underlying scalar algebra. In this work, the structural question is investigated whether this scalar structure is algebraically minimal or whether a consistent extension exists that preserves the analytical infrastructure of the theory. We formulate this problem as a minimality question for finite-dimensional real, commutative and associative algebras that contain the complex numbers as a subalgebra and at the same time possess a nontrivial ideal that is isomorphic to C as a real algebra and carries a complex-like exponential structure. It is shown that such structures necessarily require at least four real dimensions. The bicomplex algebra B=C ⊗R C realizes these requirements in minimal dimension and is unique up to isomorphism under the stated conditions. For the extended state space M=H ⊗C B one obtains a canonical decomposition M≅H⊕H, so that the bicomplex scalar extension structurally corresponds to two parallel complex quantum sectors. The Born rule remains unchanged. The present work provides a structural analysis of possible extensions of the scalar algebra underlying quantum mechanics.