96
14 Mathematical Description of Infinitely Extended Quantum Systems
where ψ(x) is a field “strictly localized in x (see (13.3)). The algebra generated by
such variables can be taken as the Heisenberg algebra localized in V . Similarly, a
Weyl algebra localized in V is generated by the exponentials of the above localized
canonical variables
U ( f ) = exp[i(a( f ) + a( f )
∗
)], V (g) = exp[a(g) − a(g)
∗
].
Quite generally, the association V → A(V ) realizes the identification of the algebras of observables localized in the volume V as V varies. The consistency of the
physical interpretation requires that such a mapping satisfies the so-called isotony
property, namely A(V 1 ) ⊆ A(V 2 ), whenever V 1 ⊆ V 2 .
The physically motivated concept of localization has an algebraic translation in
terms of commutation relations. For (equal time) space localization, the local structure of the algebras A(V ) is formalized by the property
[A(V ), A(V
)] = 0, if V ∩ V
= ∅.
(14.1)
For relativistic systems, it is more convenient to introduce algebras localized in
bounded (open) space time regions O (usually taken as causally complete as it is the
case of the diamonds or double cones.
72 ) Then, the locality property reads
[A(O 1 ), A(O 2 )] = 0,
(14.2)
whenever O 2 is space-like with respect to O 1 , briefly O 2 ⊂ O
1 ≡ the causal complement of O 1 . For observable algebras, this is the mathematical formulation of Einstein
causality.
73
The union of all A(V )(or A(O)) is called the local algebra
A L ≡ ∪ V A(V), V = V, or = O.
(14.3)
We have already argued before that it is convenient (if not necessary) to have a C
∗ -
algebra and therefore one has to complete A L . As we shall see, this is a delicate
point having deep connections with the dynamics and the physical description of the
system. The most natural and simple choice is to consider the norm closure
A ≡ A L .
(14.4)
72 A set O of points is causally complete if it coincides with its double causal complement, i.e. if
O = (O ) , where O (called the causal complement of O) denotes the set of all points which are
space-like with respect to all points of O.
73 This concept of localization should not be confused with the problems discussed in connection
with the Einstein–Podolski–Rosen paradox, see R. Haag, in The Physicist’s Conception of Nature,
J. Mehra ed., Reidel 1973; Local Quantum Physics, Springer 1992, p. 107; A. S. Wightman, in
Probabilistic Methods in Mathematical Physics, F. Guerra et al. eds., World Scientific 1992.
14 Mathematical Description of Infinitely Extended Quantum Systems
where ψ(x) is a field “strictly localized in x (see (13.3)). The algebra generated by
such variables can be taken as the Heisenberg algebra localized in V . Similarly, a
Weyl algebra localized in V is generated by the exponentials of the above localized
canonical variables
U ( f ) = exp[i(a( f ) + a( f )
∗
)], V (g) = exp[a(g) − a(g)
∗
].
Quite generally, the association V → A(V ) realizes the identification of the algebras of observables localized in the volume V as V varies. The consistency of the
physical interpretation requires that such a mapping satisfies the so-called isotony
property, namely A(V 1 ) ⊆ A(V 2 ), whenever V 1 ⊆ V 2 .
The physically motivated concept of localization has an algebraic translation in
terms of commutation relations. For (equal time) space localization, the local structure of the algebras A(V ) is formalized by the property
[A(V ), A(V
)] = 0, if V ∩ V
= ∅.
(14.1)
For relativistic systems, it is more convenient to introduce algebras localized in
bounded (open) space time regions O (usually taken as causally complete as it is the
case of the diamonds or double cones.
72 ) Then, the locality property reads
[A(O 1 ), A(O 2 )] = 0,
(14.2)
whenever O 2 is space-like with respect to O 1 , briefly O 2 ⊂ O
1 ≡ the causal complement of O 1 . For observable algebras, this is the mathematical formulation of Einstein
causality.
73
The union of all A(V )(or A(O)) is called the local algebra
A L ≡ ∪ V A(V), V = V, or = O.
(14.3)
We have already argued before that it is convenient (if not necessary) to have a C
∗ -
algebra and therefore one has to complete A L . As we shall see, this is a delicate
point having deep connections with the dynamics and the physical description of the
system. The most natural and simple choice is to consider the norm closure
A ≡ A L .
(14.4)
72 A set O of points is causally complete if it coincides with its double causal complement, i.e. if
O = (O ) , where O (called the causal complement of O) denotes the set of all points which are
space-like with respect to all points of O.
73 This concept of localization should not be confused with the problems discussed in connection
with the Einstein–Podolski–Rosen paradox, see R. Haag, in The Physicist’s Conception of Nature,
J. Mehra ed., Reidel 1973; Local Quantum Physics, Springer 1992, p. 107; A. S. Wightman, in
Probabilistic Methods in Mathematical Physics, F. Guerra et al. eds., World Scientific 1992.