  We study some kinematical aspects of quantum fields on causal sets. In
particular, we are interested in free scalar fields on a fixed background
causal set. We present various results building up to the study of the
entanglement entropy of de Sitter horizons using causal sets. We begin by
obtaining causal set analogs of Green functions for this field. First we
construct the retarded Green function in a Riemann normal neighborhood (RNN) of
an arbitrary curved spacetime. Then, we show that in de Sitter and patches of
anti-de Sitter spacetimes the construction can be done beyond the RNN. This
allows us to construct the QFT vacuum on the causal set using the
Sorkin-Johnston construction. We calculate the SJ vacuum on a causal set
approximated by de Sitter spacetime, using numerical techniques. We find that
the causal set SJ vacuum does not correspond to any of the known Mottola-Allen
vacua of de Sitter spacetime. This has potential phenomenological consequences
for early universe physics. Finally, we study the spacetime entanglement
entropy for causal set de Sitter horizons. The entanglement entropy of de
Sitter horizons is of particular interest. As in the case of nested causal
diamonds in 2d Minkowski spacetime, we find that the causal set naturally gives
a volume law of entropy, both for nested causal diamonds in 4d Minkowski
spacetime as well as 2d and 4d de Sitter spacetimes. However, an area law
emerges when the high frequency modes in the SJ spectrum are truncated. The
choice of truncation turns out to be non-trivial and we end with several
interesting questions.
