Heisenberg inequality and the complex field in quantum mechanics

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Observables A and B satisfy the Heisenberg inequality if the product of their variances has a positive lower bound independent of the state of the system. In the Hilbert space formulation of quantum mechanics it is a consequence of the Schwarz inequality that the Heisenberg-type inequality Var(A,φ)⋅Var(B,φ)≥ (1)/(4) ‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖2 holds for any pair of observables A and B (represented as self-adjoint operators) and for any (vector) state (represented as a unit vector). If inf{‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖ ‖φ∈dom(A) ∩dom(B)}≠0 then A and B satisfy the Heisenberg inequality. In the present paper the derivability of the Heisenberg-type inequality is analyzed within the general theoretical frame of a sum logic. It is shown that any real-valued non-negative function (A,α)→f(A,α) of observables A and of states α, which has a symmetry property f(A+B,α)+f(A−B,α)=2f(A,α)+2f(B,α) with respect to observables, satisfies the Heisenberg-type inequality f(A,α)⋅f(B,α)≥ (1)/(4) ‖ f(A+B,α)−f(A,α)−f(B,α)‖2 for all observables A and B and for all states α. The natural probabilistic realizations f1(A,α)=Exp(A2,α) and f2(A,α)=Var(A,α) of such functions are then analyzed. It turns out that only with the complex extension of the theory can the Heisenberg inequality be attained in the Hilbert space realization of the theory. This is used as an argument in favor of the complex field as the scalar field of quantum mechanics.