Abstract
Starting from the construction of the free quantum scalar field of mass mge 0, we give mathematically precise and rigorous versions of three different approaches to computing the Casimir forces between compact obstacles. We then prove that they are equivalent.
Highlights
Casimir interactions are forces between objects such as perfect conductors
This has been carried out for a number of particular geometric situations. Since this method requires knowledge of the spectrum of the Laplace operator in order to perform the analytic continuation, it has long been a very difficult problem to compute the Casimir force in a generic geometric situation even from a non-rigorous point of view. It has been realised by quantum field theorists that the Casimir force can be understood by considering the renormalised stress–energy tensor of the electromagnetic field
Progress was made in the non-rigorous numerical computation of Casimir forces between objects. This approach uses a formalism that relates the Casimir energy to a determinant computed from boundary layer operators
Summary
Casimir interactions are forces between objects such as perfect conductors. They can be either understood as quantum fluctuations of the vacuum or as the total effect of van der Waals forces. Since this method requires knowledge of the spectrum of the Laplace operator in order to perform the analytic continuation, it has long been a very difficult problem to compute the Casimir force in a generic geometric situation even from a non-rigorous point of view Already, it has been realised by quantum field theorists Progress was made in the non-rigorous numerical computation of Casimir forces between objects (see for example [21,22,23]) This approach uses a formalism that relates the Casimir energy to a determinant computed from boundary layer operators.
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