Functionals and the Quantum Master Equation

The quantum master equation is usually formulated in terms of functionals of the components of mappings from a space-time manifold M into a finite-dimensional vector space. The master equation is the sum of two terms one of which is the anti-bracket (odd Poisson bracket) of functionals and the other is the Laplacian of a functional. Both of these terms seem to depend on the fact that the mappings on which the functionals act are vector-valued. It turns out that neither this Laplacian nor the anti-bracket is well-defined for sections of an arbitrary vector bundle. We show that if the functionals are permitted to have their values in an appropriate graded tensor algebra whose factors are the dual of the space of smooth functions on M, then both the anti-bracket and the Laplace operator can be invariantly defined. Additionally, one obtains a new anti-bracket for ordinary functionals.Comment: 21 pages, Late

Broadening the phenotypic and molecular spectrum of FINCA syndrome: Biallelic NHLRC2 variants in 15 novel individuals

FINCA syndrome [MIM: 618278] is an autosomal recessive multisystem disorder characterized by fibrosis, neurodegeneration and cerebral angiomatosis. To date, 13 patients from nine families with biallelic NHLRC2 variants have been published. In all of them, the recurrent missense variant p.(Asp148Tyr) was detected on at least one allele. Common manifestations included lung or muscle fibrosis, respiratory distress, developmental delay, neuromuscular symptoms and seizures often followed by early death due to rapid disease progression.Here, we present 15 individuals from 12 families with an overlapping phenotype associated with nine novel NHLRC2 variants identified by exome analysis. All patients described here presented with moderate to severe global developmental delay and variable disease progression. Seizures, truncal hypotonia and movement disorders were frequently observed. Notably, we also present the first eight cases in which the recurrent p.(Asp148Tyr) variant was not detected in either homozygous or compound heterozygous state.We cloned and expressed all novel and most previously published non-truncating variants in HEK293-cells. From the results of these functional studies, we propose a potential genotype-phenotype correlation, with a greater reduction in protein expression being associated with a more severe phenotype.Taken together, our findings broaden the known phenotypic and molecular spectrum and emphasize that NHLRC2-related disease should be considered in patients presenting with intellectual disability, movement disorders, neuroregression and epilepsy with or without pulmonary involvement