Mark A. Hennings

Criterion-related validity of the last 7-day, short form of the International Physical Activity Questionnaire in Swedish adults

OBJECTIVE To examine the validity of the short, last 7-day, self-administered form of the International Physical Activity Questionnaire (IPAQ). DESIGN All subjects wore an accelerometer for seven consecutive days and completed the IPAQ questionnaire on the eighth day. Criterion validity was assessed by linear regression analysis and by modified Bland-Altman analysis. Specificity and sensitivity were calculated for classifying respondents according to the physical activity guidelines of the American College of Sports Medicine/Centers for Disease Control and Prevention. SETTING Workplaces in Uppsala, Sweden. SUBJECTS One hundred and eighty-five (87 males) participants, aged 20 to 69 years. RESULTS Total self-reported physical activity (PA) (MET-min day(-1)) was significantly correlated with average intensity of activity (counts min(-1)) from accelerometry (r = 0.34, P < 0.001). Gender, age, education and body mass index did not affect this relationship. Further, subcomponents of self-reported PA (time spent sitting, time in PA, time in moderate and vigorous activity (MVPA)) were significantly correlated with objectively measured PA (P < 0.05). Self-reported time in PA was significantly different from time measured by accelerometry (mean difference: -25.9 min day(-1); 95% limits of agreement: -172 to 120 min day(-1); P < 0.001). IPAQ identified 77% (specificity) of those who met the current PA guidelines of accumulating more than 30 min day(-1) in MVPA as determined by accelerometry, whereas only 45% (sensitivity) of those not meeting the guidelines were classified correctly. CONCLUSIONS Our results indicate that the short, last 7-days version of the IPAQ has acceptable criterion validity for use in Swedish adults. However, the IPAQ instrument significantly overestimated self-reported time spent in PA. The specificity to correctly classify people achieving current PA guidelines was acceptable, whereas the sensitivity was low.

The Weyl Quantization of Phase Angle

We suggest a candidate for a non-canonical Hermitian operator corresponding to phase angle, based on the Weyl correspondence rule. The matrix elements of this operator, for harmonic oscillator number states, coincide, in the correspondence limit, with those of a phase operator proposed by Barnett and Pegg, when the dimension of their defining space becomes infinite.

Approximations to the quantum phase operator

In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states Xs( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.

On representing observables in quantum mechanics

There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact poses a serious question on the physical meaning of such a self-adjoint operator and its associated operator measures.

Gauge covariant observables and phase operators

We characterize those Hilbert space operators which are first moments of the gauge covariant positive operator-valued measures introduced by Lahti and Pellonpaa. We consider the suitability of such measures for describing quantum phase, observing that the gauge angle defining the measure is not always closely related to the spectrum of the first moment operator. We take the position that the first moment operator represents the quantal property the measure is (imperfectly) describing. This is in contrast to the viewpoint that positive operator-valued measures are the fundamental quantum observables.

Asymptotics for the Weyl quantized phase

In a previous paper, we considered Weyl quantization of functions of the angle in phase space, in particular a phase operator Delta ( phi ) and the quantized exponentials Delta (e+or-1 phi ). In this paper we consider the first and second moments of these operators with respect to the harmonic oscillator Hermite states hn and the coherent states Phi alpha . Taking asymptotic limits we find, for example, that var[ Delta ( phi ); hn]= pi 2/3+O(log n/n) (n to infinity ) for the variance of Delta ( phi ) in the Hermite states. For the second moment of the phase operator in ||[ Delta ( phi )- theta ] Phi alpha ||2=O(1/| alpha |) as | alpha | tends to infinity, amongst other results.

The pointwise product in Weyl quantization

We study the ⊙-product of Bracken [1], which is the Weyl quantized version of the pointwise product of functions in phase space. We prove that it is not compatible with the algebras of finite rank and Hilbert–Schmidt operators. By solving the linearization problem for the special Hermite functions, we are able to express the ⊙-product in terms of the component operators, mediated by the linearization coefficients. This is applied to finite rank operators and their matrices, and operators whose symbols are radial and angular distributions.

The meaning of phase in the Dicke laser model

Abstract We find that of three phase operator models, only the Weyl quantization of the classical phase angle relates directly to the statistical mechanics treatment of a basic laser model.