Een functionele aanpak van het Casimireffect. Van scalairen tot gluonen

Abstract

In this thesis, we aim to set some important steps towards better understanding the role of the Casimir effect in the fundamental mechanism behind color confinement. We investigate the Casimir effect in several settings using a functional methodology. This method is flexible with regard to the specific boundaries and theories of interest, and is gauge invariant by construction. In short, one uses auxiliary fields that act as Lagrange multipliers to include the boundary conditions (BCs) into the action, such that they are naturally lifted into the functional integral. Next, one can extract the Casimir energy from the functional integral by integrating out all the fields. If one first integrates out the original field, one gets an effective boundary theory living in one dimension less. As an alternative way to calculate the Casimir energy, one can compute the boundary-modified field propagator, and then compute the vacuum expectation value (VEV) of the energy-momentum tensor (EMT). In this second approach, one has to be careful to include possible boundary contributions to the EMT. As a warm-up and a first exposition of the functional methods described above, we discuss a massless scalar field with parallel plates with Robin BCs (a linear combination of Dirichlet and Neumann BCs). Depending on the Robin parameters, the Casimir force is attractive, repulsive, or vanishing. Even though the EMT gets some boundary contributions, they vanish after taking the VEV, and thus do not contribute to the Casimir energy. After the scalar warm-up, we turn to an Abelian gauge field with parallel perfect electromagnetic conductors (PEMC) (a linear combination of perfect electric conductors (PEC) and perfect magnetic conductors (PMC)). Again, we find an attractive, repulsive, or vanishing Casimir force depending on the PEMC parameters. Since Maxwell theory is invariant under electromagnetic duality transformations, the Casimir energy is duality invariant and only depends on the difference of the PEMC parameters (in particular, this explains the well-known fact that the Casimir energy for two PEC plates is identical to that of two PMC plates). Here as well, the EMT gets some boundary contributions, but this time, they survive after taking the VEV, therefore being absolutely necessary to arrive at the correct Casimir energy. Instead of mixing PEC and PMC conditions via a linear combination, one can also combine some 4-vector components of PEC with the complementary components of PMC. One particularly interesting choice yields dynamical edge mode (DEM) conditions. These DEM conditions (implicitly) break gauge invariance on the boundary, thus requiring some additional boundary ghost fields in order to restore BRST invariance after quantization. DEM conditions are mostly studied in the context of entanglement entropy, where the edge modes (the would-be gauge degrees of freedom) provide a statistical interpretation for the contact term in the entanglement entropy. Despite their exotic properties, DEM boundaries give rise to the exact same Casimir energy as PMC plates. After having studied Abelian gauge theory, we turn to non-Abelian Yang-Mills (YM) theory. Since we can only feasibly work at the quadratic level, YM theory reduces trivially to (N^2-1) copies of Maxwell theory at that level. To nevertheless introduce some non-Abelian effects, we inspect several non-perturbative models for the infrared (IR) region of non-Abelian YM theory. The simplest such model is the Curci-Ferrari model (CF), which is given by YM theory in Landau gauge, where, after fixing the gauge, an effective mass term that accounts for all missing non-perturbative IR effects is added by hand. In CF theory, the Casimir energy for PEC and PMC plates differs by a constant factor, which gives rise to a van Dam-Veltman-Zakharov-like (vDVZ-like) discontinuity in the massless limit: a discontinuous jump in the PMC Casimir energy when the mass goes from almost zero to exactly zero. We also show that our analytical results are compatible with a variety of recent numerical PEC lattice simulations, in which a non-perturbative mass scale emerges. A slightly more complex non-perturbative extension of YM theory is the Gribov-Zwanziger (GZ) model. The GZ model offers an improvement of the Faddeev-Popov (FP) procedure that takes into account the existence of infinitesimal Gribov copies, and restricts the functional integral to a region without such copies. This restriction is implemented through a gap equation, the solution of which yields a dynamically generated mass scale: the Gribov mass. Similarly to the CF case, we again find a vDVZ-like discontinuity. Since the gap equation comes down to minimizing the vacuum energy, one could expect an interplay between the Gribov mass and the Casimir energy, resulting in a Gribov mass dynamically dependent of the plate separation L. However, such a dependence does not occur because the boundary corrections get dominated by the usual gap equation which lives a dimension higher. While it is often straightforward to show the existence of Gribov copies, explicitly constructing examples of such copies tends to be tricky. It has been done for the unconstrained YM vacuum, and we will discuss such an explicit construction for the YM vacuum with parallel plates inserted. Finally, it should be noted that in YM theory, the boundary conditions are quadratic. To keep the action quadratic, we have restricted ourselves to the linear part of the BCs everywhere above. As a first step towards lifting this restriction, we have studied a scalar toy model with quadratic BCs, and found that the quadratic BCs lead to a dynamical boundary mass for the bulk field.