We study logarithmic conformal field theories (LCFTs) through the introduction of nilpotent conformal weights. Using this device, we derive the properties of LCFT's such as the transformation laws, singular vectors and the structure of correlation functions. We discuss the emergence of an extra energy momentum tensor, which is the logarithmic partner of the energy momentum tensor.

BackgroundPerformance indices provide quantitative measures for the quality of motion, and therefore, assist in analyzing and monitoring patients’ progress. Measurement of performance indices requires costly devices, such as motion capture systems. Recent developments of sensors for game controllers, such as Microsoft Kinect, have motivated many researchers to develop affordable systems for performance measurement applicable to home and clinical care. In this work, the capability of Kinect in finding motion performance indices was assessed by analyzing intra-session and inter-session test–retest reliability.MethodEighteen stroke patients and twelve healthy subjects participated in this investigation. The intra-session and inter-session reliability of eight performance indices, namely mean velocity (MV), normalized mean speed (NMS), normalized speed peaks (NSP), logarithm of dimensionless jerk (LJ), curvature (C), spectral arc length (SAL), shoulder angle (SA), and elbow angle (EA), were assessed using intra-class correlation coefficient (ICC), standard error of measurement (SEM) and coefficient of variation (CV).ResultsThe results showed that, among the performance indices, MV, LJ, C, SA and EA have more than 0.9 ICC together with an acceptable SEM and CV in both stroke patients and healthy subjects. Comparing the results of different therapy sessions showed that MV, LJ and C are more sensitive than other indices, and hence, more capable of reflecting the progress of a patient during the rehabilitation process.ConclusionThe results of this study shows acceptable reliability and sensitivity across the sessions for MV, LJ and C measured by Kinect for both healthy subjects and stroke patients. The results are promising for the development of home-based rehabilitation systems, which can analyze patient’s movements using Kinect as an affordable motion capture sensor.

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We show that logarithmic conformal field theories may be derived using nilpotent scale transformation. Using such nilpotent weights we derive properties of LCFT's, such as two and three point correlation functions solely from symmetry arguments. Singular vectors and the Kac determinant may also be obtained using these nilpotent variables, hence the structure of the four point functions can also derived. This leads to non homogeneous hypergeometric functions. We can construct "superfields" using a nilpotent variable. Using this construct we show that the superfield of conformal weight zero, composed of the identity and the pseudo identity is related to a superfield of conformal dimension two, which comprises of energy momentum tensor and its logarithmic partner. This device also allows us to derive the operator product expansion for logarithmic operators. Finally we consider LCFT's near a boundary.