Statistical Equilibrium in Quantum Gravity: Gibbs states in Group Field Theory

Gibbs states are known to play a crucial role in the statistical description of a system with a large number of degrees of freedom. They are expected to be vital also in a quantum gravitational system with many underlying fundamental discrete degrees of freedom. However, due to the absence of well-defined concepts of time and energy in background independent settings, formulating statistical equilibrium in such cases is an open issue. This is even more so in a quantum gravity context that is not based on any of the usual spacetime structures, but on non-spatiotemporal degrees of freedom. In this paper, after having clarified general notions of statistical equilibrium, on which two different construction procedures for Gibbs states can be based, we focus on the group field theory formalism for quantum gravity, whose technical features prove advantageous to the task. We use the operator formulation of group field theory to define its statistical mechanical framework, based on which we construct three concrete examples of Gibbs states. The first is a Gibbs state with respect to a geometric volume operator, which is shown to support condensation to a low-spin phase. This state is not based on a pre-defined symmetry of the system and its construction is via Jaynes' entropy maximisation principle. The second are Gibbs states encoding structural equilibrium with respect to internal translations on the GFT base manifold, and defined via the KMS condition. The third are Gibbs states encoding relational equilibrium with respect to a clock Hamiltonian, obtained by deparametrization with respect to coupled scalar matter fields.Comment: v2 31 pages; typos corrected; section 2 modified substantially for clarity; minor modifications in the abstract and introduction; arguments and results unchange

Identifying targeting with nonparametric methods: An application to an Indian microfinance program

We discuss nonparametric methods and statistical tests that are appropriate to assess poverty targeting in public programs. These methods explicitly account for the possibility that the population distributions of participants and non-participants cross. Crossing points provide us with upper bounds on the income of those who have been excluded from the program. Applying these methods to data from a microfinance program in the state of Jharkhand in India, we find evidence that very poorest households are largely excluded from the program.

Poverty targeting in public programs: A comparison of alternative nonparametric methods

Very poor households may be excluded from public programs intended for their benefit for a variety of reasons as lack information, a permanent residence or membershiip in social networks. We are interested in methods of testing for such exclusion based on independently drawn samples of program participants and non-participants. We discuss three alternative nonparametric procedures; sign tests, tests for stochastic dominance and a test for distribution crossing. In the cases where there is a poverty threshold below which program participation is difficult, our simulation results suggests that the last of these test procedures is the most powrful. we apply this test to data from a microfinance program in India and find evidence that the poorest households in the area were largely outside the program.

Statistical equilibrium of tetrahedra from maximum entropy principle

Discrete formulations of (quantum) gravity in four spacetime dimensions build space out of tetrahedra. We investigate a statistical mechanical system of tetrahedra from a many-body point of view based on non-local, combinatorial gluing constraints that are modelled as multi-particle interactions. We focus on Gibbs equilibrium states, constructed using Jaynes' principle of constrained maximisation of entropy, which has been shown recently to play an important role in characterising equilibrium in background independent systems. We apply this principle first to classical systems of many tetrahedra using different examples of geometrically motivated constraints. Then for a system of quantum tetrahedra, we show that the quantum statistical partition function of a Gibbs state with respect to some constraint operator can be reinterpreted as a partition function for a quantum field theory of tetrahedra, taking the form of a group field theory.Comment: v3 published version; v2 18 pages, 4 figures, improved text in sections IIIC & IVB, minor changes elsewher

Poverty targeting in public programs: A comparison of some nonparametric tests and their application to Indian microfinance.

Many popular social programs have limited coverage among households at the very bottom of the income and wealth distribution. If a program reaches the poor, but neglects the destitute, the (pre-program) income distribution of participants and non-participants will cross. We are interested in the statis-tical methods that can be used to test for this particular pattern of program participation. Our numerical simulations suggest that recently developed tests for distribution crossing are powerful even when the two distributions under study are fairly similar and they can be usefully combined with more stan-dard quantile tests to characterize program participation among the very poor. We apply this approach to data on household expenditures and membership of micro-credit groups in India and find that participation among the poorest households in the study area was lower than that of slightly richer households.