7 research outputs found
Casimir effect due to a single boundary as a manifestation of the Weyl problem
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.Comment: 13 pages, 1 figure, minor changes to the text, extra references
added, version to be published in J. Phys.
Weyl problem and Casimir effects in spherical shell geometry
We compute the generic mode sum that quantifies the effect on the spectrum of
a harmonic field when a spherical shell is inserted into vacuum. This
encompasses a variety of problems including the Weyl spectral problem and the
Casimir effect of quantum electrodynamics. This allows us to resolve several
long-standing controversies regarding the question of universality of the
Casimir self-energy; the resolution comes naturally through the connection to
the Weyl problem. Specifically we demonstrate that in the case of a scalar
field obeying Dirichlet or Neumann boundary conditions on the shell surface the
Casimir self-energy is cutoff-dependent while in the case of the
electromagnetic field perturbed by a conductive shell the Casimir self-energy
is universal. We additionally show that an analog non-relativistic Casimir
effect due to zero-point magnons takes place when a non-magnetic spherical
shell is inserted inside a bulk ferromagnet.Comment: 9 pages, minor changes, additional references added, version to be
published in Phys. Rev.