Carrollian quantum states and flat space holography

We study free Carrollian quantum field theories from an algebraic perspective and explore their implications for flat space holography. As explicit examples, we construct the electric and magnetic Carrollian Weyl algebras obtained from Carroll limits of the relativistic scalar field and analyze their states, including vacuum and thermal configurations. For the massive electric theory, we find a regular Carroll-invariant vacuum state and a regular KMS state, yielding a consistent Carrollian thermodynamic system. By contrast, the massless electric and magnetic theories are more subtle: depending on the quantization, they admit either no regular distinguished vacuum or only nonregular Carroll-invariant ground states, while still supporting nonregular thermal states. We further analyze alternative classes of states in the massless electric theory, including spatially homogeneous quasifree pure states and Sorkin--Johnston states. Motivated by these results, we discuss consequences for flat space holography. We construct a well-defined state relevant for Carrollian holography whose Hilbert-space representation factorizes into a standard Fock sector and a nonseparable zero-mode sector, thereby highlighting the role of infrared degrees of freedom in the boundary theory.