104
15 Physically Relevant Representations
The above equation (15.2) displays the fact that the ground state accounts for the
large distance behaviour of the observables.
Since irreducibility of the GNS representation defined by a state Ω is equivalent
to Ω being a pure state, the physically relevant irreducible representations of A
are also called pure phases; in fact, they describe the pure phases of the system at
zero temperature, in the standard sense of statistical mechanics. By the discussion
of Sect. 14.1, they can also be interpreted as describing disjoint worlds.
Thus, as a consequence of asymptotic abelianess, which is crucially related to the
local structure of the observables, different translationally invariant pure states on the
observable algebra identify different phases or disjoint worlds, each characterized
by different large distance (weak) limits of the observables.
15 Physically Relevant Representations
The above equation (15.2) displays the fact that the ground state accounts for the
large distance behaviour of the observables.
Since irreducibility of the GNS representation defined by a state Ω is equivalent
to Ω being a pure state, the physically relevant irreducible representations of A
are also called pure phases; in fact, they describe the pure phases of the system at
zero temperature, in the standard sense of statistical mechanics. By the discussion
of Sect. 14.1, they can also be interpreted as describing disjoint worlds.
Thus, as a consequence of asymptotic abelianess, which is crucially related to the
local structure of the observables, different translationally invariant pure states on the
observable algebra identify different phases or disjoint worlds, each characterized
by different large distance (weak) limits of the observables.