SL(n, R) IN PARTICLE PHYSICS AND GRAVITY — DECONTRACTION FORMULA AND UNITARY IRREDUCIBLE REPRESENTATIONS

Abstract

SL(n, R) and Diff(n, R) groups play a prominent role in various particle physics and gravity theories, notably in chromogravity (that models the IR region of QCD), gauge affine generalizations of general relativity, and pD-branes. Applications of these groups require a knowledge of their features and especially rely on the unitary irreducible representation details. Lie algebra, topology and unitary representation issues of the covering groups of the SL(n, R) and Diff(n, R) groups with respect to their maximal compact SO(n) subgroups are considered. Topological properties determining spinorial representations of these groups are reviewed. An especial attention is paid to the fact that, contrary to other classical Lie algebras, the SL(n, R), n ≥ 3 covering groups are groups of infinite matrices, as are all their spinorial representations. A notion of Lie algebra decontraction, also known as the Gell-Mann formula, that plays a role of an inverse to the Inonu–Wigner contraction, is recalled. Contrary to orthogonal type of algebras, the decontraction formula has a limited validity. The validity domain of this formula for sl(n, R) algebras contracted with respect to their so(n) subalgebras is outlined. A recent generalization of the decontraction formula, that applies to all SL(n, R) covering group representations, as well as an explicit closed expression of all non-compact sl(n, R) operators matrix elements for all representations is presented. A construction of the unitary sl(n, R) representations is discussed within a framework than combines the Harish-Chandra results and a method of fulfilling the unitarity requirements in Hilbert spaces with non-trivial scalar product kernel.