A Heisenberg inequality, involving a differentiation operator, its adjoint, and the second quantization operator of a unitary operator, is proved in the context of white noise analysis.
【The Heisenberg inequality associated with the uncertainty principle is extended to an infinite dimensional abstract Wiener space (H, B) with an abstract Wiener measure p 1 . For ϕ∈ L 2 (p 1 ) and T ∈ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) ϕ is the Fourier-Wiener transform of ϕ . The conditions when the equality holds also discussed.】
Abstract In this paper we discuss some aspects of the Heisenberg uncertainty relation, mostly from the point of view of non self-adjoint operators. Some equivalence results, and some refinements of the inequality, are deduced, and some relevant examples are discussed. We also begin a sort of dynamical analysis of the relation, in connection with what has been recently called γ-dynamics and γ-symmetries, and we discuss in some details the role of different scalar products in our analysis. The case of self-adjoint operators is recovered as a special case of our general settings.
ABSTRACT. The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f: R → C is a complex valued function of a random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the product of the second moment of the random real x for |f | 2 and the second moment of the random real ξ for ∣ f ∣ 2 is at least E 2 R,|f | 4π, where f is the Fourier transform of f, f (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R e2iπξx ˆ f (ξ) dξ, and E 2 R,|f | = ∫ R |f (x)|2 dx. This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.