112
17 Examples
The first non-kinematical question is the existence of the time evolution and
the stability of A under it. To this purpose, as discussed before, we replace the
formal (ill defined) Hamiltonian (17.1) with the (infrared regularized) finite volume
Hamiltonian
H V ≡ −J
(i, j)∈V
σ i σ j ,
(17.2)
which is well defined since it involves only a finite number of terms. Then, we
consider the finite volume dynamics
α
V
t (A) ≡ e
i H V t A e
−i H V t
, A ∈ A L
(17.3)
and try to define the infinite volume dynamics as a limit of α
V
t when V → ∞.
The idea is that for A ∈ A(V 0 ), V 0 fixed, the above transformation α
V
t (A) becomes
independent of V for V large enough, thanks to (14.1) and the nearest neighbor
coupling. Thus, the limit of α
V
t (A) exists in norm and it defines a time evolution α t
as an automorphism of the local algebra, which can be extended to an automorphism
of the quasi-local algebra A, since it is norm preserving. The existence of the time
evolution as a norm limit of finite volume dynamics has been proved quite generally
for any lattice spin Hamiltonian with short-range interactions, i.e. with absolutely
summable spin interaction potentials.
90 For example, in d = 3 space dimensions,
such short-range interactions include Hamiltonians of the form
H =
i, j
J i j σ i · σ j .
provided the potential J i j decreases at least as |i − j|
−3−ε , ε > 0.
Essentially the same logic applies to Hamiltonians which are local functions of
canonical variables or fields, which satisfy the locality condition (14.1) or (14.2),
respectively. This is the case of UV regularized quantum field theories, for which the
infinite volume limit of the time evolution of local operators can be proved to exist
by locality.
91
90 D.W. Robinson, Comm. Math. Phys. 7, 337 (1968). For the convenience of the reader, also due
to the conceptual relevance of the result, a sketch of the proof is given in Appendix 7.3.
91 M. Guenin, Comm. Math. Phys. 1, 127 (1966); I.E. Segal, Proc. Natl. Acad. Sci. USA, 57, 1178
(1967).
17 Examples
The first non-kinematical question is the existence of the time evolution and
the stability of A under it. To this purpose, as discussed before, we replace the
formal (ill defined) Hamiltonian (17.1) with the (infrared regularized) finite volume
Hamiltonian
H V ≡ −J
(i, j)∈V
σ i σ j ,
(17.2)
which is well defined since it involves only a finite number of terms. Then, we
consider the finite volume dynamics
α
V
t (A) ≡ e
i H V t A e
−i H V t
, A ∈ A L
(17.3)
and try to define the infinite volume dynamics as a limit of α
V
t when V → ∞.
The idea is that for A ∈ A(V 0 ), V 0 fixed, the above transformation α
V
t (A) becomes
independent of V for V large enough, thanks to (14.1) and the nearest neighbor
coupling. Thus, the limit of α
V
t (A) exists in norm and it defines a time evolution α t
as an automorphism of the local algebra, which can be extended to an automorphism
of the quasi-local algebra A, since it is norm preserving. The existence of the time
evolution as a norm limit of finite volume dynamics has been proved quite generally
for any lattice spin Hamiltonian with short-range interactions, i.e. with absolutely
summable spin interaction potentials.
90 For example, in d = 3 space dimensions,
such short-range interactions include Hamiltonians of the form
H =
i, j
J i j σ i · σ j .
provided the potential J i j decreases at least as |i − j|
−3−ε , ε > 0.
Essentially the same logic applies to Hamiltonians which are local functions of
canonical variables or fields, which satisfy the locality condition (14.1) or (14.2),
respectively. This is the case of UV regularized quantum field theories, for which the
infinite volume limit of the time evolution of local operators can be proved to exist
by locality.
91
90 D.W. Robinson, Comm. Math. Phys. 7, 337 (1968). For the convenience of the reader, also due
to the conceptual relevance of the result, a sketch of the proof is given in Appendix 7.3.
91 M. Guenin, Comm. Math. Phys. 1, 127 (1966); I.E. Segal, Proc. Natl. Acad. Sci. USA, 57, 1178
(1967).