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Journal of the Korean Mathematical Society

THE HEISENBERG INEQUALITY ON ABSTRACT WIENER SPACES

Yuh-Jia Lee

Published Jan 1, 2001• 1 citation
OpenAlex

Abstract

【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 $\phi$ $\in$ L $^2$ (p $_1$ ) and T $\in$ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) $\phi$ is the Fourier-Wiener transform of $\phi$ . The conditions when the equality holds also discussed.】

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A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 $$ \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,$$ where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3

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【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_11​ . For ϕ\phiϕ ∈\in∈ L 2^22 (p 1_11​ ) and T ∈\in∈ L(B, H), it is shown that (※Equations, See Full-text), where F(sub) ϕ\phiϕ is the Fourier-Wiener transform of ϕ\phiϕ . The conditions when the equality holds also discussed.】
A mathematical uncertainty principle is an inequality or uniqueness theorem concerning the joint localization of a function or system and its spectrum. A Fourier uncertainty principle is thus a statement that a function and its Fourier transform cannot both decay too rapidly. The most familiar form is the Wiener-Heisenberg inequality, 5.1 $$ \left\| {(x - x_0 )f(x)} \right\|_2 \left\| {(\xi - \xi _0 )\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f} (\xi )} \right\|_2 \geqslant \frac{{\mathop {\left\| f \right\|}\nolimits_2^2 }} {{4\pi }},$$ which becomes an equality only for multiples of Gaussians of the form exp(−πα(x − x0)2 + 2πiξ0x). In this chapter, D will denote the normalized differentiation or momentum operator D = (1/2πi)d/dx. The inequality (5.1) is often written 5.2 σX(f)σD(f)≡∥Xf−⟨Xf,f⟩f∥2∥Df−⟨Df,f⟩f∥2⩾1, \sigma _X (f)\sigma _D (f) \equiv \left\| {Xf - \left\langle {Xf,f} \right\rangle f} \right\|_2 \left\| {Df - \left\langle {Df,f} \right\rangle f} \right\|_2 \geqslant 1,σX​(f)σD​(f)≡∥Xf−⟨Xf,f⟩f∥2​∥Df−⟨Df,f⟩f∥2​⩾1, where X is the multiplication operator f(x) ↦ xf(x). Barnes [22] attributes the first proof of (5.1) to a lecture delivered by Wiener in Hilbert’s mathematical physics seminar in Göttingen in 1925. In fact, Wiener discusses this lecture, though not the explicit proof of (5.1), at length in his autobiography (see [365], p. 105). What distinguished Wiener’s approach, however, was his macroscopic interpretation of (5.1), a point of view that led to further important achievements, including the Landau, Pollack and Slepian “Bell Labs” uncertainty principles in Chapter 3
In connection with the uncertainty principle in quantum mechanics (Heisenberg) or in time-frequency analysis (Heisenberg-Gabor), we study its formulation in terms of entropic inequalities, extending results recently derived by Bialynicki-Birula [1] and Zozor et al. [2]. These results can be considered as generalizations of the Heisenberg inequalities in the sense that they measure the mutual uncertainty of a random variable (or wave function) and its conjugated random variable (or Fourier transformed wave function) through their associated Rényi entropies with conjugated indexes. We consider here the more general case where the entropic indexes are not conjugated, in both cases where the state space is discrete and continuous: we discuss the existence of an uncertainty inequality depending on the location of the entropic indexes α and β in the plane (α, β). Our results explain and extend a recent study by Luis [3], where states with quantum fluctuations below the Gaussian case are discussed at the single point (2, 2).
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.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.
Abstract Uncertainty principles like Heisenberg's assert an inequality obeyed by some measure of joint uncertainty associated with a function and its Fourier transform. The more groups under which that measure is invariant, the more that measure represents an intrinsic property of the underlying object represented by the given function. The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group. A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles. The first member corresponds to Heisenberg's measure but is generally smaller than his although equal to it in special cases.