Fermionic quantum orthogonalizations. I (Q2360826)

The paper under review initiates an axiomatic study of the so-called fermionic quantum (FQ) operations in a unital real algebra \({\mathfrak A}\) endowed with a filtration consisting of finite-codimensional subalgebras satisfying the natural completeness and Hausdorff conditions. An FQ operation is a mapping from a suitable set of \(n\)-tuples of elements of \({\mathfrak A}\) to \(n\)-tuples of elements of \({\mathfrak A}\), which is equivariant with respect to homomorphisms of \({\mathfrak A}\) and is analytic in a suitable sense. Such an FQ operation is called an FQ orthogonalization procedure if its image consists of Clifford \(n\)-tuples, that is, \(n\)-tuples \((C_1,\dots,C_n)\) satisfying \(C_iC_j+C_jC_i=-2\delta_{ij}1\) for \(i,j=1,\dots,n\). The generalization of the Gram-Schmidt orthogonalization procedure to this abstract setting is a specific example that turns out to play an important role for the general investigation of FQ operations.