The infrared behaviour of quantum field theories confined in bounded domains is strongly dependent on the shape and structure of space boundaries. The most significant physical effect arises in the behaviour of the vacuum energy. The Casimir energy can be attractive or repulsive depending on the nature of the boundary. We calculate the vacuum energy for a massless scalar field confined between two homogeneous parallel plates with the most general type of boundary conditions depending on four parameters. The analysis provides a powerful method to identify which boundary conditions generate attractive or repulsive Casimir forces between the plates. In the interface between both regimes we find a very interesting family of boundary conditions which do not induce any type of Casimir force. We also show that the attractive regime holds far beyond identical boundary conditions for the two plates required by the Kenneth–Klich theorem and that the strongest attractive Casimir force appears for periodic boundary conditions whereas the strongest repulsive Casimir force corresponds to anti-periodic boundary conditions. Most of the analysed boundary conditions are new and some of them can be physically implemented with metamaterials

In this paper the quantum vacuum energies induced by massive fluctuations of one real scalar field on a configuration of two partially transparent plates are analyzed. The physical properties of the infinitely thin plates are characterized by two Dirac-$\delta$ potentials. We find that an attractive or repulsive Casimir force arises between the plates when the weights of the $\delta$’s have equal or different sign. If some of the plates absorb fluctuations below some threshold of energy (the corresponding weight is negative), there is the need to set a minimum mass to the scalar field fluctuations to preserve unitarity in the corresponding quantum field theory. Two repulsive $\delta$ interactions are compatible with massless fluctuations. The effect of Dirichlet boundary conditions at the endpoints of the interval $(-a,a)$ on a massless scalar quantum field theory defined on this interval is tantamount to letting the weights of the repulsive $\delta$ interactions go to $+\infty$.

The consistency of quantum field theories defined on domains with external borders imposes very restrictive constraints on the type of boundary conditions that the fields can satisfy. We analyse the global geometrical and topological properties of the space of all possible boundary conditions for scalar quantum field theories. The variation of the Casimir energy under the change of boundary conditions reveals the existence of singularities generically associated to boundary conditions which either involve topology changes of the underlying physical space or edge states with unbounded below classical energy. The effect can be understood in terms of a new type of Maslov index associated to the non-trivial topology of the space of boundary conditions. We also analyze the global aspects of the renormalization group flow, T-duality and the conformal invariance of the corresponding fixed points.

The effects induced by the quantum vacuum fluctuations of one massless real scalar field on a configuration of two partially transparent plates are investigated. The physical properties of the infinitely thin plates are simulated by means of Dirac-$\delta -{\delta }^{\prime }$ point interactions. It is shown that the distortion caused on the fluctuations by this external background gives rise to a generalization of Robin boundary conditions. The $T$ operator for potentials concentrated on points with nondefined parity is evaluated with total generality. The quantum vacuum interaction energy between the two plates is computed in several dimensions using the $TGTG$ formula to find positive, negative, and zero Casimir energies. The parity properties of the $\delta -{\delta }^{\prime }$ potential demands that one distinguish between opposite and identical objects. It is shown that between identical sets of $\delta -{\delta }^{\prime }$ plates, repulsive, attractive, or null quantum vacuum forces arise. However, there is always attraction between a pair of opposite $\delta -{\delta }^{\prime }$ plates.