Boundary effects on bose-einstein condensation in ultra-static space-times

In this theses relativistic boson anti-boson system, inside d dimensional box B with volume V and surface area A, which is a sub-manifold of d dimensional manifold M is studied. Mellin-Barnes Transformations and short time asymptotic of heat kernel are used as mathematical tools. Both Neumann and Dirichlet boundary conditions are considered. Analysis are done for the free energy and charge. Charge is expressed as a function of temperature. Effects of boundaries on temperature is calculated per- turbatively. Same methods are employed for studying the neutral gas; free energy and number of particles are studied and again the effect of boundary on temperature is calculated. Apart from these, upper and lower bounds for depletion coefficient of ideal relativistic bose gas in thermodynamic limit is given with the help of Li-Yau and Colbois-Maerten bounds.