Abstract:
Nonperturbative approximation schemes based on two-particle irreducible (2PI) effective actions provide an important means for our current understanding of (non-)equilibrium quantum field theory. A remarkable property is their renormalizability, since these approximations involve selective summations to infinite perturbative orders. In this paper we show how to renormalize all $n$-point functions of the theory, which are given by derivatives of the 2PI-resummed effective action $\Gamma[\phi]$ for scalar fields $\phi$. This provides a complete description in terms of the generating functional for renormalized proper vertices, which extends previous prescriptions in the literature on the renormalization for 2PI effective actions. The importance of the 2PI-resummed generating functional for proper vertices stems from the fact that the latter respect all symmetry properties of the theory and, in particular, Goldstone's theorem in the phase with spontaneous symmetry breaking. This is important in view of the application of these techniques to gauge theories, where Ward identities play a crucial role.

Abstract:
We show that the lowest nontrivial truncation of the two-particle irreducible (2PI) effective action correctly determines transport coefficients in a weak coupling or 1/N expansion at leading (logarithmic) order in several relativistic field theories. In particular, we consider a single real scalar field with cubic and quartic interactions in the loop expansion, the O(N) model in the 2PI-1/N expansion, and QED with a single and many fermion fields. Therefore, these truncations will provide a correct description, to leading (logarithmic) order, of the long time behavior of these systems, i.e. the approach to equilibrium. This supports the promising results obtained for the dynamics of quantum fields out of equilibrium using 2PI effective action techniques.

Abstract:
We discuss the application of two-particle-irreducible (2PI) functional techniques to gauge theories, focusing on the issue of non-perturbative renormalization. In particular, we show how to renormalize the photon and fermion propagators of QED obtained from a systematic loop expansion of the 2PI effective action. At any finite order, this implies introducing new counterterms as compared to the usual ones in perturbation theory. We show that these new counterterms are consistent with the 2PI Ward identities and are systematically of higher order than the approximation order, which guarantees the convergence of the approximation scheme. Our analysis can be applied to any theory with linearly realized gauge symmetry. This is for instance the case of QCD quantized in the background field gauge.

Abstract:
In this article we calculate the electrical conductivity in QED using the 2PI effective action. We use a modified version of the usual 2PI effective action which is defined with respect to self-consistent solutions of the 2-point functions. We show that the green functions obtained from this modified effective action satisfy ward identities and that the conductivity obtained from the kubo relation is gauge invariant. We work to 3-loop order in the modified 2PI effective action and show explicitly that the resulting expression for the conductivity contains the square of the amplitude that corresponds to all binary collision and production processes.

Abstract:
Four-dimensional Einstein's General Relativity is shown to arise from a gauge theory for the conformal group, SO(4,2). The theory is constructed from a topological dimensional reduction of the six-dimensional Euler density integrated over a manifold with a four-dimensional topological defect. The resulting action is a four-dimensional theory defined by a gauged Wess-Zumino-Witten term. An ansatz is found which reduces the full set of field equations to those of Einstein's General Relativity. When the same ansatz is replaced in the action, the gauged WZW term reduces to the Einstein-Hilbert action. Furthermore, the unique coupling constant in the action can be shown to take integer values if the fields are allowed to be analytically continued to complex values.

Abstract:
We summarize our recent work [1-3] concerning the formulation of two-particle-irreducible (2PI) functional techniques for abelian gauge field theories.

Abstract:
We discuss the computation of transport coefficients in large N_f QCD and the O(N) model for massive particles. The calculation is organized using the 1/N expansion of the 2PI effective action to next-to-leading order. For the gauge theory, we verify gauge fixing independence and consistency with the Ward identity. In the gauge theory, we find a nontrivial dependence on the fermion mass.

Abstract:
We discuss the formulation of the prototype gauge field theory, QED, in the context of two-particle-irreducible (2PI) functional techniques with particular emphasis on the issues of renormalization and gauge symmetry. We show how to renormalize all $n$-point vertex functions of the (gauge-fixed) theory at any approximation order in the 2PI loop-expansion by properly adjusting a finite set of local counterterms consistent with the underlying gauge symmetry. The paper is divided in three parts: a self-contained presentation of the main results and their possible implementation for practical applications; a detailed analysis of ultraviolet divergences and their removal; a number of appendices collecting technical details.

Abstract:
High-temperature resummed perturbation theory is plagued by poor convergence properties. The problem appears for theories with bosonic field content such as QCD, QED or scalar theories. We calculate the pressure as well as other thermodynamic quantities at high temperature for a scalar one-component field theory, solving a three-loop 2PI effective action numerically without further approximations. We present a detailed comparison with the two-loop approximation. One observes a strongly improved convergence behavior as compared to perturbative approaches. The renormalization employed in this work extends previous prescriptions, and is sufficient to determine all counterterms required for the theory in the symmetric as well as the spontaneously broken phase.

Abstract:
Topological objects can influence each other if the underlying homotopy groups are non-Abelian. Under such circumstances, the topological charge of each individual object is no longer a conserved quantity and can be transformed to each other. Yet, we can identify the conservation law by considering the back-action of topological influence. We develop a general theory of topological influence and back-action based on the commutators of the underlying homotopy groups. We illustrate the case of the topological influence of a half-quantum vortex on the sign change of a point defect and point out that the topological back-action from the point defect is such twisting of the vortex that the total twist of the vortex line carries the change in the point-defect charge to conserve the total charge. We use this theory to classify charge transfers in condensed matter systems and show that a non-Abelian charge transfer can be realized in a spin-2 Bose-Einstein condensate.