The behavior of the coefficients under space-time reflections (which are represented by antiunitaries J) is a bit more involved than the one under translations and boosts.
To study how the coefficients behaves under space-time reflections, we introduce for any contraction C = (m, n,{(`j, rj)}), the “reflected“ contraction CJ given by CJ = (n, m,{(rj −m, `j +n)}). This contraction is the one that is obtained from C by exchanging` withr, andm withn. We also introduce a factor associated withC that will become relevant in later computations:
RC :=
|C|
Y
j=1
1−
m+n
Y
pj=1
Sl(m)
j,pj
. (3.81)
This factor is related in a certain sense to the interaction of the model; indeed we note that in the free case S = 1, one has RC = δ|C|,0, and in the case S = −1 one has RC = 0 when m+n is even.
Lemma 3.10. RC has the property
δC(η,θ)SC(η,θ)RC(η,θ) = (−1)|C|δCJ(θ,η)SCJ(θ,η)RCJ(θ,η), (3.82) Proof. Using Lemma 3.1, we rewrite (3.82) equivalently as
δC(η,θ)Sσ(η)Sρ(θ)RC(η,θ) = (−1)|C|δCJ(θ,η)Sσ0(θ)Sρ0(η)RCJ(θ,η), (3.83) whereσ, ρandσ0, ρ0 correspond to the contractionsC andCJ, respectively, in the sense of Eq. (3.11). We note thatσ is literally the same as in (3.12), instead ρ0 works out to be
ρ0 =
1 . . . m
l|C| . . . l1 1 . . . ˆl . . . m
. (3.84)
By introducing the permutation π =
1 . . .ˆl . . . m l1 . . . l|C|
l|C| . . . l1 1 . . .ˆl . . . m
(3.85)
3.9. BEHAVIOR OF COEFFICIENTS UNDER REFLECTIONS
we notice thatρ0 =σ◦π, and therefore we have thatSρ0(η) =Sπ(ησ)Sσ(η) by Eq. (2.6).
Correspondingly, one finds that Sσ0(θ) = Sτ(θρ)Sρ(θ) where the permutation τ is defined analogously to π. Explicitly, one has
Sπ(ησ) = Indeed, using (3.81), the formula above writes explicitly as:
|CJ| So, to get the equality in the equation above, in particular we have to show that:
m+n To see this, consider the first of the two factors. On the support of δC, we can replace lj withrj, but noting thatlj ≤m and rj > m, so that theS(m) factors are turned into Now we renumber the variables: Therj-th component of (η,θ) isθrj−m, or alternatively speaking, the (rj −m)-th component of (θ,η). We get from there,
Taking the product over all p, inserting into the above, and renaming the product index, we have and that is what we claimed.
We will see now in the following Proposition how the factor RC enters the formula which describes the behaviour of the coefficients fm,n[A] under the action of space-time reflections.
Proposition 3.11. For any A∈ Qω, by Eq. (3.21), we obtain:
fm,n[J A∗J](θ,η) = X
C∈Cm,n
(−1)|C|δCSC(θ,η)h`CJ(η), ArCJ(θ)i. (3.96) Using Prop. 3.5 in the formula above, we find
fm,n[J A∗J](θ,η) = X
C∈Cm,n
C0∈Cn−|C|,m−|C|
(−1)|C|δCSC(θ,η)δC0SC0(ˆη,θ)fˆ n−|C|−|C[A] 0|,m−|C|−|C0|(ˆˆη,ˆˆθ), (3.97)
where ˆθ,ηˆ indicates that the variables contracted in C are left out, instead ˆˆθ,ˆˆη in-dicates that the variables which are contracted C∪C˙ 0J are left out. Now we apply Lemma 3.2 (withC0J in place of C0), and setting D:=C∪C˙ 0J, we reorganize the sum It remains to compute the sum (∗). Using Eq. (3.82), and taking the product with δD into account, we have (here we used the fact that
δC where the last twoS-factors cancel due to the delta distribution.)
and where
To see (3.102), consider the right hand side of the equation above. On the support of δD, we can replace lj0 with rj0, but noting that l0j ≤ n and r0j > n, so that the S(n) factors are turned into their inverse:
m+n
3.9. BEHAVIOR OF COEFFICIENTS UNDER REFLECTIONS
Now we renumber the variables: Ther0j-th component of (η,θ) isθr0
j−n, or alternatively speaking, the (rj0 −n)-th component of (θ,η). We get from there,
Sr(n)0
j,p(η,θ) =Sr(m)0
j−n,p0(θ,η), (3.104)
where p0 =p+m if p≤n, and p0 =p−n if p > n. (Note that if p is “left” then p0 is
“right” and vice versa.)
Taking the product over all p, inserting into the above, and renaming the product index, we have
m+n
Y
p=1
Sl(n)0
j,p(η,θ) =
m+n
Y
p=1
Sr(m)0
j−n,p(θ,η)−1. (3.105) and that is what we claimed in (3.102).
Using the followingdistributional law: Y
a∈A
(αa+βa) = X
B⊂A
Y
b∈B
αb Y
d∈Bc
βd, (3.106)
we find
(∗) = Y
j∈{`j}∪{rj0−n}
1 + 1−aj 1−a−1j
= Y
j∈{`j}∪{r0j−n}
(1−aj) =RD(θ,η). (3.107)
Inserting this result into (3.98) concludes the proof.
Chapter 4
Operators and quadratic forms
Most of the material in the following chapter is due to H. Bostelmann.
4.1 Locality of quadratic forms
In the previous chapter we discussed the existence and uniqueness of the Araki decom-position for any quadratic form A ∈ Qω. So in order to discuss the locality properties of A in terms of the Araki decomposition, we need a notion of locality that is adapted to quadratic forms A∈ Qω. This locality is studied in terms of commutators with the wedge-local field φ, but we have first to clarify in which sense these commutators are well defined.
If f ∈ Dω(R2), then z†(f+) maps Hω,f into Hω,f, so Az†(f+) is well-defined as a quadratic form; indeed, we have the following Lemma:
Lemma 4.1. For f ∈ Dω(R2) one has kAz†(f+)kωk ≤√
k+ 1kf+kω2kAkωk+1. Proof. From (2.41) we have:
kAz†(f+)kωk = 1
2||e−ω(H/µ)QkAz†(f+)Qk||+1
2||QkAz†(f+)Qke−ω(H/µ)|| (4.1) We estimate the first norm on the right hand side of (4.1) as follows:
||e−ω(H/µ)QkAz†(f+)Qk|| = ||e−ω(H/µ)QkAQk+1z†(f+)Qk||
≤ ||e−ω(H/µ)QkAQk+1|| · ||z†(f+)Qk||
≤ ||e−ω(H/µ)Qk+1AQk+1|| · ||f+||2√ k+ 1
≤ ||e−ω(H/µ)Qk+1AQk+1||√
k+ 1kf+kω2. (4.2) And similarly, for the second norm we find:
||QkAz†(f+)Qke−ω(H/µ)|| = ||QkAQk+1e−ω(H/µ)eω(H/µ)z†(f+)Qke−ω(H/µ)||
≤ ||Qk+1AQk+1e−ω(H/µ)|| · ||eω(H/µ)z†(f+)e−ω(H/µ)Qk||
≤ ||Qk+1AQk+1e−ω(H/µ)||√
k+ 1kf+kω2. (4.3) Putting together these two estimates, we find from (4.1) the claimed result.
In analogous way, we have that also the product z†(f+)A, and the products of A with z(f−), φ(f), φ0(f) from the left or the right are well-defined in Qω; this implies that we can define the commutator [A, φ(f)] :=Aφ(f)−φ(f)A ∈ Qω. Given this, we can define our notion of locality as follows.
Definition 4.2. Let A ∈ Qω. We say that A is ω-local in Wx (the right wedge with edge at x) iff
[A, φ(f)] = 0 for all f ∈ Dω(Wx0), as a relation in Qω. (4.4) A is called ω-local in Wx0 iff J A∗J is ω-local in Wx. A is called ω-local in the double cone Ox,y =Wx∩ Wy0 iff it is ω-local in both Wx and Wy0.
In the following lemma we characterize better this notion of locality. Such a Lemma is formulated only for the standard right wedge W, but actually it holds for other regions in analogous way.
Lemma 4.3. Let ω be an indicatrix, and A ∈ Qω. The following conditions are equivalent:
(i) A is ω-local in W.
(ii) [A, φ(f)] = 0 for all f ∈ Dω(W0).
(iii) For every ψ, χ ∈ Hω,f, there exists an indicatrix ω0 such that hψ,[A, φ(f)]χi = 0 for all f ∈ Dω0(W0).
(iv) For every ψ, χ ∈ Hω,f, it holds that hφ(x)ψ, Aχi = hψ, Aφ(x)χi for x ∈ W0, in the sense of tempered distributions.
Proof. First we note that (iv) is well defined: Indeed, since kAkωk < ∞, the matrix elementhψ, Aχiis well defined (by continuous extension) ifψ, χ∈ Hf and at least one of ψ, χis in Hω.
Since φ(f)Hf ⊂ Hf, and since the map S(R2) → H, f 7→ φ(f)χ is continuous with respect to the Schwarz and the Hilbert space norms in the corresponding spaces, the mapf 7→ hψ, Aφ(f)χi is a tempered distribution; the same holds for hφ( ¯f)ψ, Aχi analogously.—Now the equivalence (i)⇔(ii) is true due to the definition; (ii)⇒(iii) is trivial since (iii) is a special case of (ii) where we choose ω0 = ω; (iii)⇒(iv) fol-lows because Dω0(W0) is dense in D(W0) (see [Bj¨o65]); and (iv)⇒(ii) holds because Dω(W0)⊂ S(R2) since D(W0)⊂ S(R2) (and Dω0(W0) is dense in D(W0)).
The notion of ω-locality is clearly weaker than the usual notion of locality. If A is just a quadratic form we cannot write down well-defined commutators ofAfor example with the unitary operators expiφ(f)−, or with a general operator B ∈ A(Wx0), and the notion ofω-locality does not give us any information regarding such commutators.
However,ω-locality is not so weak as it might appear at first glance: In the following section we will try to clarify the relation between ω-locality and the usual notion of locality.