An abstract derivation of the inequality related to Heisenberg uncertainty principle (Q1073308)

Heisenberg's uncertainty principle concerns a reciprocal relationship between the dispersions of certain incompatible magnitudes A and B as the statistical state varies between a dispersion-free state for A and a dispersion-free state for B. It is well-known that this principle can be derived in the Hilbert space formalism of quantum mechanics from some properties of inner products and operators. In the present paper it is shown that an abstract form of Heisenberg's principle can be derived in a more general mathematical framework without involving the Hilbert space. Namely, if m(A,\(\phi)\) denotes the statistical second moment of the observable A and we have \(m([A,B],\phi)=-m([A,-B],\phi)\) for all A, B and \(\phi\), then \(| m([A,B],\phi)|^ 2\leq m(A,\phi)m(B,\phi)\). If, moreover, there is a canonical pair of observables A, B such that \(m([A,B],[\phi,-\psi])=- m([A,B],[\phi,\psi])\) for all \(\phi\), \(\psi\), then the classical uncertainty principle follows. Here for a real function f on a vector space X f([x,y]) denotes \((f(x+y)-f(x)-f(y))\) for all x,y\(\in X\).