Now we would like to pass fromω-local quadratic forms to operators which are ω-local and also to investigate the relation of ω-locality to usual notions of locality. We are aiming to characterize closed operators which are affiliated with the local algebras.
We found that this class of operators are still manageable to characterize within the framework of the Araki expansion. On the other hand, smeared pointlike fields are typically closable where they exist, see [FH81, Wol85], and we will also find some closable operators in our models explicitly, see Sec. 9.
4.2. RELATIONS TO USUAL NOTIONS OF LOCALITY
Proposition 4.4. Let R be one of the regions Wx, Wy0, Ox,y for somex, y ∈R2. (i) Let A be a bounded operator; then A is ω-local in R if and only if A∈ A(R).
(ii) Let A be a closed operator with core Hω,f, and Hω,f ⊂domA∗. Suppose that
∀g ∈ DRω(R2) : exp(iφ(g)−)Hω,f ⊂domA. (4.5) Then A is ω-local in R if and only if it is affiliated with A(R).
(iii) In the case S =−1, statement (ii) is true even without the condition (4.5).
Remark on the conditions: We could make the condition (4.5) in the hypothesis (ii) of Prop. 4.4 a bit weaker by requiring that exp(itφ(g)−)Hω,f ⊂domAfor small |t|, and we could also restrict to suppg ⊂ R0. With this, one can see that the proof of this Proposition (see below) proceeds analogously.
Proof. We will prove this proposition only in the case where R is the standard right wedge R = W and its associated algebra A(R) = A(W) = M. The other cases R = Wx or R = Wy0 can be obtained from the case before by applying Poincar´e transformations (and using the property of covariance of the associated algebra); anal-ogously the caseR=Ox,y can be obtained from the caseR=W by applying Poincar´e transformations to the right and left wedges of the intersection.
We notice that (i) is a special case of (ii) since a bounded operator is in particular a closed operator with domain the entire Hilbert space H. Also, obviously a bounded operator is affiliated to the von Neumann algebra A(R) if and only if it is an element of this von Neumann algebra. We will now prove (ii); for this we consider an operator A which is closed and ω-local in W. We need to show that A commutes with the unitaries exp(iφ(g)−) whereg ∈ DωR(W0), in a way that is compatible with the domain of A, in the sense that each unitary exp(iφ(g)−) in A(R0) should carry the domain of A, domA, onto itself and satisfy the commutation relation there (“affiliation” of A with the von Neumann algebra A(R)). So, let g ∈ Dω
R(W0), we consider the “cut-off”
series expansion:
Bn:=
n
X
k=0
ik
k!(φ(g)−)k, (4.6)
with n ∈ N0. This is an operator defined at least on Hω,f, since φ can be applied finitely many times to finite particle vectors; also its adjoint is defined there, since Bn∗ is the same as Bn with i replaced by −i, considering that φ is essentially self-adjoint on the space of vectors of finite particle number.
Since A isω-local in W and since Bn has a series expansion in terms of φ(g), with g ∈ Dω
R(W0), we have, as a consequence of Lemma 4.3(iv), that Bn commutes with A:
Bn∗ψ, Aχ
=
A∗ψ, Bnχ
for all ψ, χ∈ Hω,f, (4.7) where we used that ψ ∈ domA∗, since in the above equation we took A to the left side of the scalar product, applied to ψ. Since ψ and χ are analytic vectors for φ(g), we have for n → ∞ that the exponential series for Bn converges in strong operator topology, namely that Bnχ→Bχ and Bn∗ψ →B∗ψ, whereB := expiφ(g)−.
Equation (4.7) implies in the limit n→ ∞:
B∗ψ, Aχ
=
A∗ψ, Bχ
for all ψ, χ∈ Hω,f. (4.8)
By hypothesis (4.5),Bχ∈domA; this implies that we can take A∗ in (4.8) to the right side of the scalar product, that is hA∗ψ, Bχi=hψ, ABχi. SinceB is bounded, we can do the same withB∗ in (4.8), we have
B∗ψ, Aχ
=
ψ, BAχ
; since ψ can be chosen from a dense set inH, we conclude that
BAχ =ABχ for all χ∈ Hω,f. (4.9)
We would like now to generalize (4.9) to general vectorsχ∈domAand to more general operatorsB in the commutant of M.
First, we consider a general vector χ ∈ domA. Since Hω,f is a core for A, we can find a sequence of vectors (χj) inHω,f such thatχj →χ, and alsoAχj →Aχin Hilbert space norm since Ais a closed operator. Since Aχj →Aχ we compute from Eq. (4.9):
ABχj =BAχj →BAχ. (4.10)
Since B is bounded we have that Bχj → Bχ; moreover since A is closed and Bχj ∈ domA (by (4.5)), we have thatBχ∈domA. Hence, from (4.10) we have that
ABχ=BAχ for all χ∈domA, B = exp(iφ(g)−), g ∈ Dω
R(W0). (4.11) By doing a similar computation as in (4.10), we can see that the same result then holds if B is a finite product of Weyl operators exp(iφ(g)−), or a linear combination of product of Weyl operators, or the strong operator limit of linear combinations of products of Weyl operators. Thus, by the double commutant theorem, (4.11) holds for all B ∈ {exp(iφ(g)−)
g ∈ DωR(W0)}00 = M0. This because by the double commutant theorem the closure of{exp(iφ(g)−)
g ∈ DωR(W0)} in the strong operator topology is equal to the bicommutant of {exp(iφ(g)−)
g ∈ DωR(W0)}, which is in turn equal to M0. Due to (4.11) and the fact that B ∈ M0, this means that A is affiliated with M (we denote this byA η M), as claimed.
For the converse, we consider A η M and g ∈ DR(W0). We need to show that A is ω-local in W. For any t ∈ R, we have that expitφ(g)− ∈ M0, since we can write expitφ(g)− = expiφ(tg)− and we know thatM0 is generated by expiφ(g)− with g ∈ DR(W0). The fact that A is affiliated with M implies that A commutes with expitφ(g)−, g ∈ DR(W0), in the following sense:
∀ψ, χ∈ Hω,f ∀t∈R: he−itφ(g)−ψ, Aχi=hA∗ψ, eitφ(g)−χi. (4.12) To pass from the Weyl operators back to the fieldsφ(which is needed forω-locality), we notice that sinceψ, χare analytic vectors for φ(g), we can think of these exp(itφ(g)−) as series expansion in terms ofφ; in particular, we have that these functions are analytic int and therefore both sides of (4.12) are real analytic int. We can then compute the derivatives at t = 0 of both sides of (4.12) and equal the corresponding derivatives;
hence we find:
∀ψ, χ∈ Hω,f ∀t∈R: hφ(g)ψ, Aχi=hA∗ψ, φ(g)χi. (4.13) This implies that A is ω-local in W by Lemma 4.3(iv). This completes the proof of (ii).
To prove (iii), note that in the case S = −1, the operators φ(g)− are actually bounded operators, and generate the algebra M0 [Lec05]. So, we do not need to consider Weyl operators. We can also restrict tog ∈ DωR(W0) since this space is dense in DR(W0) which is dense in the space of test functions g ∈ S(W0) considered by