  The Casimir self-energy of a boundary is ultraviolet-divergent. In many cases
the divergences can be eliminated by methods such as zeta-function
regularization or through physical arguments (ultraviolet transparency of the
boundary would provide a cutoff). Using the example of a massless scalar field
theory with a single Dirichlet boundary we explore the relationship between
such approaches, with the goal of better understanding the origin of the
divergences. We are guided by the insight due to Dowker and Kennedy (1978) and
Deutsch and Candelas (1979), that the divergences represent measurable effects
that can be interpreted with the aid of the theory of the asymptotic
distribution of eigenvalues of the Laplacian discussed by Weyl. In many cases
the Casimir self-energy is the sum of cutoff-dependent (Weyl) terms having
geometrical origin, and an "intrinsic" term that is independent of the cutoff.
The Weyl terms make a measurable contribution to the physical situation even
when regularization methods succeed in isolating the intrinsic part.
Regularization methods fail when the Weyl terms and intrinsic parts of the
Casimir effect cannot be clearly separated. Specifically, we demonstrate that
the Casimir self-energy of a smooth boundary in two dimensions is a sum of two
Weyl terms (exhibiting quadratic and logarithmic cutoff dependence), a
geometrical term that is independent of cutoff, and a non-geometrical intrinsic
term. As by-products we resolve the puzzle of the divergent Casimir force on a
ring and correct the sign of the coefficient of linear tension of the Dirichlet
line predicted in earlier treatments.
