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Register a new userScalar Field with Robin Boundary Conditions in the Worldline Formalism
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The worldline formalism has been widely used to compute physical quantities in quantum field theory. However, applications of this formalism to quantum fields in the presence of boundaries have been studied only recently. In this article we show how to compute in the worldline approach the heat kernel expansion for a scalar field with boundary conditions of Robin type. In order to describe how this mechanism works, we compute the contributions due to the boundary conditions to the coefficients A_1, A_{3/2} and A_2 of the heat kernel expansion of a scalar field on the positive real line.
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The interaction of a quantum field with a background containing a Dirac delta function with support on a surface of codimension 1 represents a particular kind of matching conditions on that surface for the field. In this article we show that the worldline formalism can be applied to this model. We obtain the asymptotic expansion of the heat-kernel corresponding to a scalar field on $mathbb{R}^{d+1}$ in the presence of an arbitrary regular potential and subject to this kind of matching conditions on a flat surface. We also consider two such surfaces and compute their Casimir attraction due to the vacuum fluctuations of a massive scalar field weakly coupled to the corresponding Dirac deltas.
We study the Casimir effect for scalar fields with general curvature coupling subject to mixed boundary conditions $(1+beta_{m}n^{mu}partial_{mu})phi =0$ at $x=a_{m}$ on one ($m=1$) and two ($m=1,2$) parallel plates at a distance $aequiv a_{2}-a_{1}$ from each other. Making use of the generalized Abel-Plana formula previously established by one of the authors cite{Sahrev}, the Casimir energy densities are obtained as functions of $beta_{1}$ and of $beta_{1}$,$beta_{2}$,$a$, respectively. In the case of two parallel plates, a decomposition of the total Casimir energy into volumic and superficial contributions is provided. The possibility of finding a vanishing energy for particular parameter choices is shown, and the existence of a minimum to the surface part is also observed. We show that there is a region in the space of parameters defining the boundary conditions in which the Casimir forces are repulsive for small distances and attractive for large distances. This yields to an interesting possibility for stabilizing the distance between the plates by using the vacuum forces.
We explore boundary scattering in the sine-Gordon model with a non-integrable family of Robin boundary conditions. The soliton content of the field after collision is analysed using a numerical implementation of the direct scattering problem associated with the inverse scattering method. We find that an antikink may be reflected into various combinations of an antikink, a kink, and one or more breathers, depending on the values of the initial antikink velocity and a parameter associated with the boundary condition. In addition we observe regions with an intricate resonance structure arising from the creation of an intermediate breather whose recollision with the boundary is highly dependent on the breather phase.
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 non defined parity is computed with total generality. The quantum vacuum interaction energy between the two plates is computed using the $TGTG$ formula to find positive, negative, and zero Casimir energies. The parity properties of the $delta-delta^prime$ potential allow repulsive quantum vacuum force between identical plates.
We use the image charge method to compute the trace of the heat kernel for a scalar field on a flat manifold with boundary, representing the trace by means of a worldline path integral and obtain useful non-iterative master formulae for n insertions of the scalar potential. We discuss possible extensions of the method.
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