I know this is a basic question, but why are positive frequency modes so important when expanding a field operator. Furthermore, what do they represent?
A mode with a positive (negative) frequency corresponds to a particle with positive (negative) energy. Obviously, in nature we only see particles with positive energy.
A positive frequency mode of a field operator represents the creation of a positive energy particle. A negative frequency mode of a field operator represents the annihilation of a positive energy antiparticle.