Vacuum Energy and the Cosmological Constant

We discuss a numerical relation between the cosmological constant Λ [1] and the vacuum energy arising from the Casimir effect [2]. Many recent theories of the elementary particles invoke the existence of extra dimensions, beyond the familiar 4-dimensional space-time manifold. Often these extra dimensions are compact and have a finite range. In a particular model [3] the range R is of the order of 100 μm 1 mm. It is interesting to consider the vacuum energy, arising from the Casimir effect due to the bounding of the 4-d space by the extra compact dimension(s). In its simplest form, that of parallel plates, the Casimir stress is attractive. However Boyer [4] has shown that a conducting spherical shell of radius a is subject to a repulsive stress due to the electromagnetic zero point energy. This is the desired configuration if the vacuum energy is to be identified with the cosmological constant. Boyer’s calculation leads to an energy density