(Non)existence of static scalar field configurations in finite systems

Abstract

Derrick's theorem on the nonexistence of stable time-independent scalar field configurations [G. H. Derrick, J. Math. Phys. 5, 1252 (1964)] is generalized to finite systems of arbitrary dimension. It is shown that the "dilation" argument underlying the theorem hinges upon the fulfillment of specific Neumann boundary conditions, providing thus new means of evading it without resorting to time-dependence or additional fields of higher spin. The theorem in its original form is only recovered when the boundary conditions are such that both the gradient and potential energies vanish at the boundaries, in which case it establishes the nonexistence of stable time-independent solutions in finite systems of more than two spatial dimensions.