Just assume that I understand that a field in quantum field theory is an operator-valued distribution. For simplicity, forget about the distribution and think about a function
$\varphi:M \rightarrow L(H)$
that assigns an operator to each point of spacetime.
Can someone explain to me what (mathematically speaking) physicists mean by "quanta" of this field?
EDIT: (follow up question): if one fixes an inertial system and a point $t_0$ in time (say 4:00pm), then there is "space" and a map
$a^\dagger:\mathbb R^3 \rightarrow L(H) \rightarrow H$
$(x,y,z)\mapsto a^\dagger(t_0,x,y,z) \mapsto a^\dagger(t_0,x,y,z) |0>$
, which creates "quanta". Would it be correct to think about "particles at the point (x,y,z)" then one thinks about this map?