In this section we recall from Section X.7 of [RS2] some basic properties of scalar free field theory of mass m ≥ 0 in s space dimensions. The single-particle space of this theory is L2(Rs, dsp). On this space there act the multiplication operatorsω(~p) =p
|~p|2+m2 and
B.2. Scalar Free Field Theory and Related Models 77 p1, . . . , ps which are self-adjoint on a suitable dense domain. The unitary representation of the Poincar´e groupP+↑ ∋(x,Λ)→U1(x,Λ), acting on the single-particle space, is given by
(U1(x,Λ)f)(~p) =ei(ω(~p)x0−~p~x)f(Λ−1~p), f ∈L2(Rs, dsp), (B.2.1) where Λ−1~pis the spatial part of the four-vector Λ−1(ω(~p), ~p). The full Hilbert spaceHof the theory is the symmetric Fock space overL2(Rs, dsp). By the method of second quanti-zation, we obtain the unitary representation of the Poincar´e group U(x,Λ) = Γ(U1(x,Λ)) which implements the corresponding family of automorphisms acting on B(H)
α(x,Λ)(·) =U(x,Λ) · U(x,Λ)∗. (B.2.2) The Hamiltonian H =dΓ(ω), and the momentum operators Pi =dΓ(pi), i = 1,2, . . . , s are defined on a suitable domain in H. The joint spectrum of this family of commuting, self-adjoint operators is contained in the closed forward light cone.
The elements of H have the form Ψ = {Ψn}∞n=0, where Ψn is an n-particle vector.
We denote by DF the dense subspace of finite particle vectors, consisting of such Ψ that Ψn 6= 0 only for finitely many n. On this subspace we define the annihilation operator a(f),f ∈L2(Rs, dsp), given by the formula
(a(f)Ψ)n(~p1, . . . , ~pn) =√ n+ 1
Z
dspf¯(~p)Ψn+1(~p, ~p1, . . . , ~pn) (B.2.3) and its adjoint a∗(f). With the help of these operators we construct the (time zero) canonical fields and momenta for any g∈S(Rs)
φ+(g) = 1
√2 a∗(ω−12g) +˜ a(ω−12g)˜¯
, (B.2.4)
φ−(g) = 1
√2 a∗(iω12˜g) +a(iω12˜¯g)
. (B.2.5)
The algebra A(O(r)) of observables localized in the double coneO(r), whose base is the s-dimensional ballOr of radiusr, is given by
A(O(r)) ={ei√2φ+(F+)+i√2φ−(F−)|F± ∈D(Or)R}′′, (B.2.6) where D(Or)R is the space of real-valued test functions supported in this ball. Let us mention an equivalent definition of the local algebra which is more convenient for some purposes: We define the following (non-closed) subspaces in L2(Rs, dsp)
L˚±r =ω∓12D(e Or), (B.2.7) where tilde denotes the Fourier transform. We denote by J the complex conjugation in configuration space and set
L˚r= (1 +J) ˚L+r + (1−J) ˚L−r. (B.2.8) Then every f ∈L˚r has the formf =f++if−, where
f˜±=ω∓21F˜±, (B.2.9)
for someF±∈D(Or)R. For anyf ∈L˚r we define the Weyl operator
W(f) :=ei(a∗(f)+a(f))=ei√2φ+(F+)+i√2φ−(F−). (B.2.10) In view of the second equality and definition (B.2.6) there holds
A(O(r)) ={W(f)|f ∈L˚r}′′. (B.2.11) With the help of the translation automorphisms αx, introduced above, we define local algebras attached to double cones centered at any point x of spacetime
A(O(r) +x) =αx(A(O(r))). (B.2.12) Now for any open, bounded regionO we set
A(O) =
[
r,x O(r)+x⊂O
A(O(r) +x) ′′
, (B.2.13)
obtaining the local net A. The global algebra, denoted by the same symbol, is the C∗ -inductive limit of all such local algebras. It is well known that the triple (A, α,H) satisfies the postulates 1-5 from Section 1.6 [RS2].
We can immediately construct two related theories: First, we define the local algebra generated by the derivatives of the free field
A(d)(O(r)) ={ei
√2φ+(Ps
j=1∂xjFj+)+i√
2φ−(F−)
|Fj+, F−∈D(Or)R, j ∈ {1, . . . , s} }′′, (B.2.14) which is clearly a subalgebra of A(O(r)). In the massive case one can show that A(d)(O(r)) = A(O(r)), making use of the equation of motion. In the massless case, however, the inclusion is proper. The global C∗-algebra, constructed as in the case of the full theory, is denoted by A(d) and the theory (A(d), α,H) satisfies the postulates from Section 1.6.
The second example is the even part of scalar free field theory. Here the local algebra, attached to the double coneO(r), is given by
A(e)(O(r)) ={cos(√
2φ+(F+) +√
2φ−(F−))|F+, F−∈D(Or)R}′′. (B.2.15) The corresponding local netA(e), constructed as above, gives rise to the theory (A(e), α,H) which satisfies all the postulates from Section 1.6 except for irreducibility. In order to ensure this latter property, we represent the algebraA(e) on the subspaceH(e)inHspanned by vectors with even particle numbers. We defineA(e) =A(e)|H(e),U(x,Λ) =U(x,Λ)|H(e) and α(x,Λ)(·) =U(x,Λ) · U(x,Λ)−1. The resulting theory (A(e), α,H(e)) satisfies all the general postulates.
For future convenience we discuss the relation between the theory (A(e), α,H), acting on the full Fock space, and the theory (A(e), α,H(e)) acting onH(e). First, we define the C∗-representation π(e) :A(e) →A(e) given by
π(e)(A) =A:=A|H(e), A∈A(e). (B.2.16)
B.2. Scalar Free Field Theory and Related Models 79 Due to the Reeh-Schlieder property of the vacuum and the fact that local operators are norm dense in A we obtain that π(e) is a faithful representation. Thus there holds, by Proposition 2.3.3 of [BR],
kAk=kAk, A∈A(e). (B.2.17)
Next, we note that the natural embedding H(e) ֒→ H induces the embedding of the preduals ι(e):T(e)֒→ T, whereT(e)=B(H(e))∗. There clearly holds
ι(e)(ϕ)(A) =ϕ(A), A∈A(e), ϕ∈ T(e), (B.2.18) and it is easy to see that ι(e) is an isometry: The absolute value of any ϕ ∈ T(e) can be expressed as |ϕ| = P
ipi|Ψi)(Ψi|, where pi > 0 and Ψi ∈ H(e) form an orthonormal system. Completing it to an orthonormal basis in H, we obtain
kϕk= Tr|ϕ|= Tr|ι(e)(ϕ)|=kι(e)(ϕ)k. (B.2.19) Thus, making use of the fact that the embedding H(e) ֒→ H preserves the energy of a vector, we obtain that
ι(e):TE,1(e) → TE,1, (B.2.20) what implies, together with relation (B.2.18), that kAkE,2≤ kAkE,2 for square-integrable operators A∈Aˆ(e).
Our next goal is to describe the field content of these theories. (Cf. Section 2.3 for a general discussion of this concept). For this purpose we introduce the domainDS ={Ψ∈ DF|Ψn∈S(Rs×n)} on which there acts the annihilation operator of the mode~p∈Rs
(a(~p)Ψ)n(~p1, . . . , ~pn) =√
n+ 1Ψn+1(~p, ~p1, . . . , ~pn). (B.2.21) Clearly, on DS there holds the equality
a(f) = Z
dspf(~¯p)a(~p), f ∈S(Rs). (B.2.22) The adjointa∗(~p) is only a quadratic form onDS×DS. The Hamiltonian can be expressed as a quadratic form on this domain by
H= Z
dsp ω(~p)a∗(~p)a(~p). (B.2.23) Similarly, we define the pointlike localized canonical fields and momenta as quadratic forms on DS×DS
φ+(~x) = 1 (2π)s/2
Z dsp
p2ω(~p) e−i~p~xa∗(~p) +ei~p~xa(~p)
, (B.2.24)
φ−(~x) = i (2π)s/2
Z dsp
rω(~p)
2 e−i~p~xa∗(~p)−ei~p~xa(~p)
. (B.2.25)
This terminology is justified by the fact that on this domain there holds for anyg∈S(Rs) φ±(g) =
Z
dsx φ±(~x)g(~x). (B.2.26)
Moreover, for any s-indexκ, (see Section B.1 for our multiindex notation), we define the derivatives ∂κφ±:=∂κφ±(~x)|~x=0 and the corresponding Wick monomials
:∂κ+1φ+. . . ∂κ+k+φ+∂κ−1φ−. . . ∂κ−k−φ−: (B.2.27) which are also quadratic forms onDS×DS. The Wick powers are defined by the standard prescription consisting in shifting all the creation operators to the left disregarding the commutators1.
It is a well known fact (see e.g. [Bos00]) that the Wick monomials can be extended by continuity to bounded functionals on T∞ and the field content of scalar free field theory is given by
ΦFH= Span{:∂κ+1φ+. . . ∂κ+k+φ+∂κ−1φ−. . . ∂κ−k−φ−: |κ±j± ∈Ns0, j±∈ {1, . . . , k±}, k±∈N0}(B.2.28) i.e. it consists of finite linear combinations of the Wick monomials. Furthermore, it can be extracted from Section 7.4.2. of [Bos00] that the field content of the sub-theory (A(d), α,H) generated by the derivatives of the field has the form
Φ(d)FH= Span{:∂κ+1φ+. . . ∂κ+k+φ+∂κ−1φ−. . . ∂κ−k−φ−: |κ±j± ∈Ns0, j± ∈ {1, . . . , k±}, k±∈N0,|κ+j+|>0}.(B.2.29) (As mentioned above, in the massive case one can show that Φ(d)FH = ΦFH, making use of the equation of motion. In the massless case, however, Φ(d)FH is a proper subspace of ΦFH). Finally, the even part (A(e), α,H(e)) of scalar free field theory has the following field content
Φ(e)FH= Span{:∂κ+1φ+. . . ∂κ+k+φ+∂κ−1φ−. . . ∂κ−k−φ−:|κ±j± ∈Ns0, j± ∈ {1, . . . , k±}, k±∈N0, k++k− is even },(B.2.30) where the underlining indicates that the Wick monomials act on the Hilbert space H(e).
To close the present section, we show that ConditionT, stated in Section 2.3, holds in massive scalar free field theory and its even part.
Theorem B.2.1. Massive scalar free field theory and its even part satisfy Condition T for any dimension of space s≥1.
Proof. First, we consider the full massive scalar free field theory. The (0,0)-component of the stress-energy tensor T00∈ΦFH is given by
T00= 1
2 :φ2−: +1 2
Xs
j=1
: (∂jφ+)2: +1
2m2 :φ2+:. (B.2.31) The standard computation, which makes use of representation (B.2.23) of the Hamiltonian, gives for Ψ,Φ∈DS Z
dsx(Ψ|T00(~x)Φ) = (Ψ|HΦ). (B.2.32)
1We warn the reader that the Wick ordering is not linear on the algebra of (smeared) creation and annihilation operators.
B.3. Special Functionals on Local Algebra 81