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A Heisenberg Bound for Stationary Time Series

Eric C. Blankmeyer

Published Jan 1, 1999• 0 citations
OpenAlex

Abstract

Heisenberg's principle of indeterminacy is applied to stationary time series models. The position and velocity of a forecast are defined and are shown to be imperfectly correlated. Then a first-order autoregression is used to illustrate the trade-off between precision of position and precision of velocity. A counterpart of Planck's constant is identified, and the Heisenberg bound is derived for several autoregressive moving- average models. The time-energy version of the Heisenberg principle is discussed in the context of a stationary model in continuous time.

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Heisenberg's principle of indeterminacy is applied to stationary time series models. The position and velocity of a forecast are defined and are shown to be imperfectly correlated. Then a first-order autoregression is used to illustrate the trade-off between precision of position and precision of velocity. A counterpart of Planck's constant is identified, and the Heisenberg bound is derived for several autoregressive moving- average models. The time-energy version of the Heisenberg principle is discussed in the context of a stationary model in continuous time.
The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck's constant $\ensuremath{\Elzxh}/2$ as demonstrated by Heisenberg's thought experiment using a \ensuremath{\gamma}-ray microscope. Here it is shown that this common assumption is not universally true: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck's constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg's lower bound for the noise-disturbance product is violated even by a nearly nondisturbing precise position measurement. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices.
Abstract The uncertainty principle lies at the heart of quantum mechanics, as it describes the fundamental trade-off between the precision of position and momentum measurements. In this work, we study the quantum particle in the Boltzmann states and derive a refined lower bound on the product of and . Our new bound is expressed in terms of the ratio between and the thermal de Broglie wavelength, and provides a valuable tool for characterizing thermodynamic precision. We apply our results to the Brownian oscillator system, where we compare our new bound with the well-known Heisenberg uncertainty principle. Our analysis shows that our new bound offers a more precise measure of the thermodynamic limits of precision.
Game theory and quantum mechanics are two of the twentieth century's most productive mathematical frameworks. Both describe systems navigating spaces of possibility under irreducible constraints, and both harbour quantities that cannot be made simultaneously sharp. Their formal meetings since 1999 --- quantum games and quantum models of cognition --- have borrowed quantum hardware or formalism without asking why the borrowing works. This paper proposes that the answer lies in Gradient Indeterminacy: regions and gradients, rather than points, constrain any pair of variables whose joint variation is limited by the nature of a system. Strategic spaces contain three kinds of such pairs. Contending pairs, such as pressure and flexibility, trade off with a strict floor on their product. Cooperating pairs, such as being informed and being unpredictable, can grow together but cannot vanish together. Consequent pairs, such as present action and long-term outcome, are linked by a delayed tendency whose bandwidth is bounded. Nash equilibria become states that saturate a contending relation, with a non-zero residual pressure. In a repeated game, past a threshold in the weight of the future, a contending pair becomes cooperating; the change shows hysteresis --- cooperation, like trust, remembers. The finite game is written as a film in the qualitative notation of the corpus. Critical social transitions become drain vortices whose first symptom is the widening of a consequent bandwidth. Against two studies and public index series, the evidence is firm on an absence --- autocorrelation does not rise before most financial crashes --- and suggests that variance rises at the scale over which a crisis is prepared; the corpus's D function is applied to these series for the first time. The argument is carried by an imagined correspondence between Heisenberg and Nash, and by a game of Hex played between them by mail and left unfinished.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.
The uncertainty relation formulated by Heisenberg in 1927 describes a trade-off between the error of a measurement of one observable and the disturbance caused on another complementary observable so that their product should be no less than a limit set by Planck's constant. In 1980, Braginsky, Vorontsov, and Thorne claimed that this relation leads to a sensitivity limit for gravitational wave detectors. However, in 1988 a model of position measurement was constructed that breaks both this limit and Heisenberg's relation. Here, we discuss the problems as to how we reformulate Heisenberg's relation to be universally valid and how we experimentally quantify the error and the disturbance to refute the old relation and to confirm the new relation.