Editing Walsh function (section)

===Fermion Walsh system===

The '''Fermion Walsh system''' is a non-commutative, or "quantum" analog of the classical Walsh system. Unlike the latter, it consists of operators, not functions. Nevertheless, both systems share many important properties, e.g., both form an orthonormal basis in corresponding Hilbert space, or [[Schauder basis]] in corresponding symmetric spaces. Elements of the Fermion Walsh system are called ''Walsh operators''.

The term ''Fermion'' in the name of the system is explained by the fact that the enveloping operator space, the so-called [[hyperfinite type II factor]] <math> \mathcal R</math>, may be viewed as the space of ''observables'' of the system of countably infinite number of distinct [[Spin (physics)|spin]] <math>1/2</math> [[fermion]]s. Each [[Rademacher function|Rademacher]] operator acts on one particular fermion coordinate only, and there it is a [[Pauli matrices|Pauli matrix]]. It may be identified with the observable measuring spin component of that fermion along one of the axes <math> \{x,y,z\}</math> in spin space. Thus, a Walsh operator measures the spin of a subset of fermions, each along its own axis.