  In a quantum many-body system where the Hamiltonian and the order operator do
not commute, it often happens that the unique ground state of a finite system
exhibits long-range order (LRO) but does not show spontaneous symmetry breaking
(SSB). Typical examples include antiferromagnetic quantum spin systems with
Neel order, and lattice boson systems which exhibit Bose-Einstein condensation.
By extending and improving previous results by Horsch and von der Linden and by
Koma and Tasaki, we here develop a fully rigorous and almost complete theory
about the relation between LRO and SSB in the ground state of a finite system
with continuous symmetry. We show that a ground state with LRO but without SSB
is inevitably accompanied by a series of energy eigenstates, known as the
"tower" of states, which have extremely low excitation energies. More
importantly, we also prove that one gets a physically realistic "ground state"
by taking a superposition of these low energy excited states.
  The present paper is written in a self-contained manner, and does not require
any knowledge about the previous works on the subject.
