14.1 Local Structure
97
The norm closure leads to the smallest C
∗ -algebra generated by strictly local elements, all other topologies, like the (ultra-)strong and the (ultra-)weak being weaker,
and therefore it gives the C
∗ -algebra with best localization properties. For this reason,
the norm closure A is called the quasi-local algebra.
Since the time evolution is one of the possible physically realizable operations,
in order to have a consistent physical picture, the algebra of observables, and consequently its localization properties, must be stable under time evolution. We shall
therefore take for granted that the time evolution defines a one-parameter group
α t , t ∈ R of
∗ -automorphisms of the algebra of observables. Furthermore, we shall
restrict our attention to systems for which also the space translations α x , define
∗ -automorphisms of the observable algebra.
For systems with a dynamics characterized by a finite propagation speed, the norm
closure of A L is stable under time evolution and therefore the quasi-local algebra is a
good candidate for the algebra of observables. This is the case of lattice spin systems
with short-range interactions
74 as well as the case of relativistic systems, since for
them the causality requirement for the observables implies that under time evolution
strictly local algebras are mapped into strictly local ones.
On the other hand, for non-relativistic systems the speed of propagation is in
general infinite (even for the free Schrödinger propagator) and therefore some delocalization is unavoidable. Operators which are localized in a bounded region V at
the initial time will not be so at any subsequent time. Therefore, the non-relativistic
approximation necessarily requires a weaker form of locality, and, consequently, one
should take as relevant algebra A a larger completion of A L .
75 We shall return to
this point later in Appendix A below.
14.2 Asymptotic Abelianess
Independently from the possible delocalization induced by the dynamics, strong
physical reasons require that the algebra A of observables (or of the canonical
variables) has at least the following (asymptotic) localization property, namely ∀A,
B ∈ A, putting A x ≡ α x (A),
lim
|x|→∞
[A x , B] = 0.
(14.5)
Such a property is called asymptotic abelianess (in space). The physical meaning
of such a property is rather transparent, since it states that the measurement of the
observable A becomes compatible with the measurement of the observable B, in the
74 See O. Bratteli and D. W. Robinson, loc. cit. Vol. II, Sect. 6.2. For the convenience of the reader,
a brief account is presented in the Appendix, Sect. 17.3.
75 D. A. Dubin and G. L. Sewell, Jour, Math. Phys. 11, 2290 (1970); G. L. Sewell, Comm. Math.
Phys. 33, 43 (1973); G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); J. Math.
Phys. 28, 622 (1987).
97
The norm closure leads to the smallest C
∗ -algebra generated by strictly local elements, all other topologies, like the (ultra-)strong and the (ultra-)weak being weaker,
and therefore it gives the C
∗ -algebra with best localization properties. For this reason,
the norm closure A is called the quasi-local algebra.
Since the time evolution is one of the possible physically realizable operations,
in order to have a consistent physical picture, the algebra of observables, and consequently its localization properties, must be stable under time evolution. We shall
therefore take for granted that the time evolution defines a one-parameter group
α t , t ∈ R of
∗ -automorphisms of the algebra of observables. Furthermore, we shall
restrict our attention to systems for which also the space translations α x , define
∗ -automorphisms of the observable algebra.
For systems with a dynamics characterized by a finite propagation speed, the norm
closure of A L is stable under time evolution and therefore the quasi-local algebra is a
good candidate for the algebra of observables. This is the case of lattice spin systems
with short-range interactions
74 as well as the case of relativistic systems, since for
them the causality requirement for the observables implies that under time evolution
strictly local algebras are mapped into strictly local ones.
On the other hand, for non-relativistic systems the speed of propagation is in
general infinite (even for the free Schrödinger propagator) and therefore some delocalization is unavoidable. Operators which are localized in a bounded region V at
the initial time will not be so at any subsequent time. Therefore, the non-relativistic
approximation necessarily requires a weaker form of locality, and, consequently, one
should take as relevant algebra A a larger completion of A L .
75 We shall return to
this point later in Appendix A below.
14.2 Asymptotic Abelianess
Independently from the possible delocalization induced by the dynamics, strong
physical reasons require that the algebra A of observables (or of the canonical
variables) has at least the following (asymptotic) localization property, namely ∀A,
B ∈ A, putting A x ≡ α x (A),
lim
|x|→∞
[A x , B] = 0.
(14.5)
Such a property is called asymptotic abelianess (in space). The physical meaning
of such a property is rather transparent, since it states that the measurement of the
observable A becomes compatible with the measurement of the observable B, in the
74 See O. Bratteli and D. W. Robinson, loc. cit. Vol. II, Sect. 6.2. For the convenience of the reader,
a brief account is presented in the Appendix, Sect. 17.3.
75 D. A. Dubin and G. L. Sewell, Jour, Math. Phys. 11, 2290 (1970); G. L. Sewell, Comm. Math.
Phys. 33, 43 (1973); G. Morchio and F. Strocchi, Comm. Math. Phys. 99, 153 (1985); J. Math.
Phys. 28, 622 (1987).