Integrability for Relativistic Spin Networks

The evaluation of relativistic spin networks plays a fundamental role in the Barrett-Crane state sum model of Lorentzian quantum gravity in 4 dimensions. A relativistic spin network is a graph labelled by unitary irreducible representations of the Lorentz group appearing in the direct integral decomposition of the space of L^2 functions on three-dimensional hyperbolic space. To `evaluate' such a spin network we must do an integral; if this integral converges we say the spin network is `integrable'. Here we show that a large class of relativistic spin networks are integrable, including any whose underlying graph is the 4-simplex (the complete graph on 5 vertices). This proves a conjecture of Barrett and Crane, whose validity is required for the convergence of their state sum model.

Gender and East Asian Welfare States: from Confucianism to Gender Equality

How can we understand the gender logic underpinning the welfare states/systems of East Asia? Does the comparative literature, which has largely been concerned with western Welfare states, whether in The Three Worlds of Welfare Capitalism (Esping-Andersen 1990), or in gender-based analysis of the male breadwinner model (Lewis 1992, 2001, 2006), have anything to offer in understanding the gender assumptions underpinning East Asian welfare states? Are the welfare systems of East Asian countries distinctive, with Confucian assumptions hidden beneath the surface commitment to gender equality? We will use the (mainly western) comparative literature, but argue that Confucian influences remain important, with strong assumptions of family, market and voluntary sector responsibility rather than state responsibility, strong expectations of women’s obligations, without compensating rights, a hierarchy of gender and age, and a highly distinctive, vertical family structure, in which women are subject to parents-in-law. In rapidly changing economies, these social characteristics are changing too. But they still put powerful pressures on women to conform to expectations about care, while weakening their rights to security and support. Nowhere do welfare states’ promises bring gender equality in practice. Even in Scandinavian countries women earn less, care more, and have less power than men. We shall compare East Asian countries (Japan, Korea, Taiwan where possible) with some Western ones, to argue that some major comparative data (e.g. OECD) show the extreme situation of women in these countries. Some fine new qualitative studies give us a close insight into the experience of mothers, including lone and married mothers, which help us to understand how far the gender assumptions of welfare states are from Scandinavia’s dual earner model. There are signs of change in society as well as in economy, and room for optimism that women’s involvement in social movements and academic enquiry may be challenging Confucian gender hierarchies.