102
15 Physically Relevant Representations
III. (Ground state) Inf σ(H ) is a (proper) non-degenerate eigen-value of the Hamiltonian. The corresponding eigenvector Ψ 0 , called the ground state, has the following properties:
i) Ψ 0 is a cyclic vector with respect to the local algebra
ii) Ψ 0 represents the unique translationally invariant state in H π .
Clearly, by a trivial redefinition of H , one can get U (t) Ψ 0 = Ψ 0 .
The ground state condition is obviously satisfied in the free case, described by the
Fock representation, with Ψ 0 being the Fock no-particle state. A physical justification
for the existence of a ground state, in the general case, is that this is the state which
the system should eventually reach, (when subject to small external perturbations),
since the Hamiltonian is bounded from below.
From a mathematical point of view, the cyclicity of the ground state implies that the
physically relevant representations can be obtained, through the GNS construction,
from states (on the quasi-local algebra) invariant under space and time translations,
i.e. from correlation functions invariant under space and time translations.
From a physical point of view, the cyclicity requirement means that all the states
of H π can be approximated, as well as one likes, by local states, in agreement with
the discussion in Sect. 14.1, i.e. the states of H π can be described in terms of local
operations on the ground state. In this picture, the ground state plays the role of
the reference state, all the other states being essentially local modifications of it.
This closely reflects the experimental limitation that, given a reference state, through
physically realizable operations, one has access only to states which differ from it
only locally.
Strictly speaking, the operational identification of the ground state involves some
idealization or extrapolation, since one cannot actually measure or detect the properties of an infinitely extended system at space infinity. The identification of the ground
state is therefore done on the basis of economy of the mathematical description, by
extrapolating at infinity the large distance properties of the system. For example, in
the case of a one-dimensional spin system, if all the relevant states (in a given phase)
have the property that all the spins near the boundary point in the up direction, (as can
be enforced by suitable boundary conditions), then, in the thermodynamical limit,
the most economical description of such states of the system is in terms of (quasi)
local modifications of an infinitely extended homogeneous state, in which all the
spins are in the up direction.
In conclusion, the ground state completely accounts for the large distance
behaviour of the system and this is the only ingredient which involves some extrapolation over the local character of the physically realizable operations.
The uniqueness of the translationally invariant state in any irreducible representation of A follows from asymptotic abelianess. The proof relies on von Neumann’s
bicommutant theorem.
81 Given a
∗ -subalgebra A of B(H) (the set of all bounded operators in H), the commutant, denoted by A
, is the set of all operators in B(H) which
commute with A, and the bicommutant (or double commutant) A
≡ (A
)
is the set
81 For a sketch of the proof see, e.g. [SNS 96].
15 Physically Relevant Representations
III. (Ground state) Inf σ(H ) is a (proper) non-degenerate eigen-value of the Hamiltonian. The corresponding eigenvector Ψ 0 , called the ground state, has the following properties:
i) Ψ 0 is a cyclic vector with respect to the local algebra
ii) Ψ 0 represents the unique translationally invariant state in H π .
Clearly, by a trivial redefinition of H , one can get U (t) Ψ 0 = Ψ 0 .
The ground state condition is obviously satisfied in the free case, described by the
Fock representation, with Ψ 0 being the Fock no-particle state. A physical justification
for the existence of a ground state, in the general case, is that this is the state which
the system should eventually reach, (when subject to small external perturbations),
since the Hamiltonian is bounded from below.
From a mathematical point of view, the cyclicity of the ground state implies that the
physically relevant representations can be obtained, through the GNS construction,
from states (on the quasi-local algebra) invariant under space and time translations,
i.e. from correlation functions invariant under space and time translations.
From a physical point of view, the cyclicity requirement means that all the states
of H π can be approximated, as well as one likes, by local states, in agreement with
the discussion in Sect. 14.1, i.e. the states of H π can be described in terms of local
operations on the ground state. In this picture, the ground state plays the role of
the reference state, all the other states being essentially local modifications of it.
This closely reflects the experimental limitation that, given a reference state, through
physically realizable operations, one has access only to states which differ from it
only locally.
Strictly speaking, the operational identification of the ground state involves some
idealization or extrapolation, since one cannot actually measure or detect the properties of an infinitely extended system at space infinity. The identification of the ground
state is therefore done on the basis of economy of the mathematical description, by
extrapolating at infinity the large distance properties of the system. For example, in
the case of a one-dimensional spin system, if all the relevant states (in a given phase)
have the property that all the spins near the boundary point in the up direction, (as can
be enforced by suitable boundary conditions), then, in the thermodynamical limit,
the most economical description of such states of the system is in terms of (quasi)
local modifications of an infinitely extended homogeneous state, in which all the
spins are in the up direction.
In conclusion, the ground state completely accounts for the large distance
behaviour of the system and this is the only ingredient which involves some extrapolation over the local character of the physically realizable operations.
The uniqueness of the translationally invariant state in any irreducible representation of A follows from asymptotic abelianess. The proof relies on von Neumann’s
bicommutant theorem.
81 Given a
∗ -subalgebra A of B(H) (the set of all bounded operators in H), the commutant, denoted by A
, is the set of all operators in B(H) which
commute with A, and the bicommutant (or double commutant) A
≡ (A
)
is the set
81 For a sketch of the proof see, e.g. [SNS 96].