Soft Uncertainty Research Articles

Uncertainty Relations for the Number of Photons and the Phase Operator of an Electromagnetic Field for Phase Quantum Superpositions of Coherent States

We examine various uncertainty relations: (electromagnetic-field phase (the so-called “Pegg–Barnett operator”))–(number of photons), (trigonometric functions of the phase)–(number of photons), and (trigonometric functions of the phase)–(field phase). An electromagnetic field in quantum states of phase superpositions of coherent states, which are general superpositions of two coherent states with identical moduli but arbitrary phases, is considered. The rigorous uncertainty relation (Cauchy inequality) and the soft uncertainty relation (Heisenberg inequality) are examined and compared.

A rigorous uncertainty relation (Cauchy Inequality) for an electromagnetic field in quantum superpositions of coherent states and in a two-photon coherent state

It is shown that the Heisenberg uncertainty relation (or soft uncertainty relation) determined by the commutation properties of operators of electromagnetic field quadratures differs significantly from the Robertson–Schrodinger uncertainty relation (or rigorous uncertainty relation) determined by the quantum correlation properties of field quadratures. In the case of field quantum states, for which mutually noncommuting field operators are quantum-statistically independent or their quantum central correlation moment is zero, the rigorous uncertainty relation makes it possible to measure simultaneously and exactly the observables corresponding to both operators or measure exactly the observable of one of the operators at a finite measurement uncertainty for the other observable. The significant difference between the rigorous and soft uncertainty relations for quantum superpositions of coherent states and the two-photon coherent state of electromagnetic field (which is a state with minimum uncertainty, according to the rigorous uncertainty relation) is analyzed.

The Importance of Information and Participation Principles in Environmental Law in Brazil, the United States and Beyond

This article explores the two different kinds of uncertainty, ‘hard’ uncertainty (unknown unknowns) and ‘soft’ uncertainty (known unknowns), in the context of environmental law decision making. First, it is argued that these different categories should not be treated the same when facing decisions under uncertainty. To deal with these different uncertainties, a tiered risk analysis process is called for, wherein participatory techniques are used both to turn hard uncertainty into (more manageable) soft uncertainty as well as to increase the legitimacy of environmental decision making, even in cases of hard uncertainty. This methodology can and should apply to all instances of domestic, transnational and international environmental law making. This article applies this conceptual platform to analyze how participatory techniques can be factored in to manage uncertainty by reference to two domestic systems – American and Brazilian environmental law – as well as to international (environmental) law. The article concludes that managing uncertainty in the environmental decision‐making process is a procedural justice tool to promote more balanced and equitable outcomes.

Optimal pollution control under imprecise environmental risk and irreversibility

Abstract This paper deals with a model of pollution accumulation in which a catastrophicenvironmental event occurs once the pollution stock exceeds some uncertain criticallevel. This problem is studied in a context of ‘hard uncertainty’ since we considerthat the available knowledge concerning the value taken by the critical pollutionthreshold contains both randomness and imprecision. Such a general form ofknowledge is modelled as a (closed) random interval. This approach ismathematically tractable and amenable to numerical simulations. In thisframework we investigate the effect of hard uncertainty on the optimal pollution/consumption trade-off and we compare the results with those obtained both in thecertainty case and in the case of ‘soft uncertainty’ (where only randomness prevails). 1. IntroductionThis paper presents an optimal pollution control model in which individuals have totrade-off consumption against pollution given that consumption gives rise to pollution.We develop a partial equilibrium model for a polluting economy since we neitherconsider the capital accumulation nor the production process for the final good. Theconsumption decisions are made under uncertainty. Indeed, when the pollution stockexceeds some unknown critical threshold a catastrophic environmental event occurs.Such an environmental catastrophe is irreversible in the sense that once it has occurredthe economy can not recover its initial state. Similar situations have been studiedamong others by Cropper (1976), Conrad (1988), Pethig (1989), and more recently byClarke and Reed (1994) and Tsur and Zemel (1995, 1998).The main difference between these works and ours lies in the form of uncertainty weconsider.FollowingthedistinctionintroducedbyVercelli(1994)weconsiderasituationof ‘hard uncertainty’ which contrasts with the classical situations of ‘soft uncertainty’.A situation of uncertainty is said to be soft whenever the available informationconcerning the value of the critical pollution threshold can be modelled by a unique(additive) probability measure. On the contrary, we talk about hard uncertaintywhen the available information is insufficient or incomplete to be modelled by sucha unique fully reliable probability measure.