Ladder operators with no vacuum, their coherent states, and an application to graphene

In literature ladder operators of different nature exist. The most famous are those obeying canonical (anti-) commutation relations, but they are not the only ones. In our knowledge, all ladder operators have a common feature: the lowering operators annihilate a non zero vector, the {\em vacuum}. This is connected to the fact that operators of these kind are often used in factorizing some positive operators, or some operators which are { bounded from below}. This is the case, of course, of the harmonic oscillator, but not only. In this paper we discuss what happens when considering lowering operators with no vacua. In particular, after a general analysis of this situation, we propose a possible construction of coherent states, and we apply our construction to graphene.