We compute the electrical conductivity and shear viscosity at leading order in hot Quantum Electrodynamics. Starting from the Kubo relation for electrical conductivity and shear viscosity, we use diagrammatic methods to write down the appropriate integral equations for bosonic and fermionic effective vertices. We also show how Ward identities can be used to put constraints on these integral equations. One of our main results is an equation relating the kernels of the integral equations with functional derivatives of the full self-energy; it is similar to what is obtained with two-particle-irreducible effective action methods. However, since we use Ward identities as our starting point, gauge invariance is preserved. Using these constraints obtained from Ward identities and also power counting arguments, we select the necessary diagrams that must be resummed at leading order. This includes all non-collinear (corresponding to 2 to 2 scatterings) and collinear (corresponding to 1+N to 2+N collinear scatterings) rungs responsible for the Landau-Pomeranchuk-Migdal effect. We also show the equivalence between our integral equations and the linearized Boltzmann equations of Arnold, Moore and Yaffe obtained using effective kinetic theory.

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The purpose of this thesis is to report a theoretical investigation on the current-current correlation and noise in the tmnsient quantum transport regime. In particular, we calculate current correlations when the bias voltage of a LDL quantum device is suddenly turned off. Namely, we consider the situation that when time t < 0 the device is in a steady-state under bias Vb, when t > 0 the bias is turned off to zero. Under such a bias, the transport current l goes from a finite steady-state value 10 at t < 0 to zero at large times. When electronic structure of the leads as well as well as the device scattering region are to be taken into account, it is a difficult problem to calculate the time dependent current-current correlation. However, for the sharp step-down bias shape, we discover that the time-dependent problem can be solved exactly for non-interacting systems.