From photons to fractons: Casimir effect in higher-rank electrodynamics and its fractonic branch

The Casimir effect probes the modes that can propagate between boundaries. This makes it useful for fracton gauge theories, where restricted mobility follows from the simultaneous conservation of charge and dipole moment and changes the physical sector of the theory. We compare ordinary Maxwell electrodynamics with a Lorentz-covariant higher-rank theory of a symmetric tensor gauge field and with the scalar-charge fractonic branch selected from it by a preferred timelike background. The comparison shows why the distinction matters. In the full covariant theory, the Euclidean slab problem contains three independently propagating Maxwell-like modes and two coupled vector sectors that do not reduce to independent ordinary modes. Their boundary determinants are well defined, but a Lorentzian analysis shows that the vector sector has an unbounded Hamiltonian, so these determinants do not describe the vacuum energy of a stable fractonic phase. On the fractonic branch, the Gauss constraint removes that vector sector and leaves three stable massless modes: two transverse-traceless tensor modes and one scalar trace mode. For ideal reflectors, the physical Casimir energy is therefore 3/2 of the corresponding Maxwell result, for both equal and complementary boundary conditions. The calculation isolates the effect of the fractonic restriction on the radiative spectrum and clarifies which features of the full higher-rank boundary problem survive in the physical fracton sector.