An abstract derivation of the inequality related to heisenberg's uncertainty principle

Abstract

It is shown that Heisenberg's uncertainty principle can be derived from algebraic properties of observables, without involving Hilbert space formalism of quantum mechanics. Namely, if m( A,ϕ ) denotes the statisctical second moment of an observable A measured in the state ϕ and we define m([A,B]),ϕ)= 1 2 (m(A+B,ϕ)−m(A,ϕ) −(B,ϕ)) , then the property of oddness with respect to observables m([ A,− B], ϕ) =− m([ A, B), ϕ) implies an abstract from of Heisenberg's inequality. If, in addition, there is a canonical pair of observables A, B such that m([ A, B],[ ϕ,− ψ]) =− m([ A, B],[ ϕ, ψ]), then the classical uncertainty principle of Heisenberg follows. These results allow us to formulate and derive Heisenberg's principle in the framework of axiomatic quantum mechanics from an equational assumption about the profitability function of the system.