Modular Structure of the Weyl Algebra

We study the modular Hamiltonian associated with a Gaussian state on the Weyl algebra. We obtain necessary/sufficient criteria for the local equivalence of Gaussian states, independently of the classical results by Araki and Yamagami, Van Daele, Holevo. We also present a criterion for a Bogoliubov automorphism to be weakly inner in the GNS representation. The main application of our analysis is the description of the vacuum modular Hamiltonian associated with a time-zero interval in the scalar, massive, free QFT in two spacetime dimensions, thus complementing the recent results in higher space dimensions (Longo and Morsella in The massive modular Hamiltonian. arXiv:2012.00565). In particular, we have the formula for the local entropy of a one-dimensional Klein–Gordon wave packet and Araki’s vacuum relative entropy of a coherent state on a double cone von Neumann algebra. Besides, we derive the type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${III}_1$$\end{document}III1 factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.


Introduction
The Heisenberg commutation relations are at the core of Quantum Mechanics. From the mathematical viewpoint, they have a more transparent formulation in Weyl's exponential form. If H is a real linear space equipped with a non-degenerate symplectic form β, one considers the free * -algebra A(H ) linearly generated by the (unitaries) V (h), h ∈ H , that satisfy the commutation relations (CCR) V (h) * = V (−h). The Weyl algebra A(H ) admits a unique C * norm, so its C * completion is a simple C * -algebra, the Weyl C * -algebra C * (H ). The representations, and the states, of A(H ) and of C * (H ) are so in one-to-one correspondence. We refer to [8,14,34] for the basic theory.
For a finite-dimensional H , von Neumann's famous uniqueness theorem shows that all representations of C * (H ), with V (·) weakly continuous, are quasi-equivalent. As is well known, in Quantum Field Theory (QFT) one deals with infinitely many degrees of freedom and many inequivalent representations arise, see [20].
Due to the relations (1), a state on C * (H ) is determined by its value on the Weyl unitaries; a natural class of states is given by the ones with Gaussian kernel. A state ϕ α is called Gaussian, or quasi-free, if with α a real bilinear form α on H , that has to be compatible with β.
Assuming now that H is separating with respect to α, as is the case of a local subspace in QFT, the GNS vector associated with ϕ α is cyclic and separating for the von Neumann algebra A(H ) generated by C * (H ) in the representation. So there is an associated Tomita-Takesaki modular structure, see [41], that we are going to exploit in this paper.
Modular theory is a deep, fundamental operator algebraic structure that is widely known and we refrain from explaining it here, giving for granted the reader to be at least partly familiar with that. We however point out two relevant aspects for our work. The first one is motivational and concerns the growing interest on the modular Hamiltonian in nowadays physical literature, especially in connection with entropy aspects (see e.g. refs in [28]). The other aspect concerns the crucial role taken by the modular theory of standard subspaces, see [27]; this general framework, where Operator Algebras are not immediately visible, reveals a surprisingly rich structure and is suitable for applications of various kind. Most of our paper will deal with standard subspaces.
Our motivation for this paper is the description of the local modular Hamiltonian associated with the free, massive, scalar QFT in 1 + 1 spacetime dimension, in order to complement the higher dimensional results, that were obtained after decades of investigations [30]. We give our formula in Sect. 5.2. Although the present formula could be guessed from the higher dimensional one, its proof is definitely non-trivial because the deformation arguments from the massless case are not directly available now, due to the well known infrared singularities; indeed the free, massless, scalar QFT does not exist in 1 + 1 dimension.
As a consequence, we compute the local entropy of a low dimensional Klein-Gordon wave packet. This gives also Araki's vacuum relative entropy of a coherent state on a local von Neumann algebra the free, massive, scalar QFT, now also in the 1+1 dimension case. We refer to [9,[28][29][30] for background results and explanation of the context. We also show the type I I I 1 factor property for the net of local von Neumann algebras associated with the free, massive, scalar QFT on a low dimensional Minkowski spacetime.
We now briefly describe part of the background of our work. The Canonical Commutation Relations (1) and Anti-Commutation Relations are ubiquitous and intrinsic in Quantum Physics. The study of the corresponding linear symmetries (symplectic transformations, CCR case) is a natural problem; the automorphisms of the associated operator algebras are called Bogoliubov automorphisms, see [14,15]. The classical result of Shale [39] characterises the Bogoliubov automorphisms that are unitarily implementable on the Fock representation. Criteria of unitary implementability in a quasi-free representation were given by Araki and Yamagami [5], van Daele [42] and Holevo [23], these works are independent of the modular theory, although the last two rely on the purification construction, that originated in the classical paper by Powers and Størmer in the CAR case [36]. Woronowicz partly related the purification map to the modular theory and reconsidered the CAR case [43]. However, the modular structure of the Weyl algebra has not been fully exploited so far, although the CCR case is natural to be studied from this point of view.
We work in the context of the standard form of a von Neumann algebra studied by Araki, Connes and Haagerup [3,11,21]. If an automorphism of a von Neumann algebra in standard form is unitarily implementable, then it is canonically implementable; so we know where to look for a possible implementation. This will provide us with a criterion for local normality that is independent of the mentioned previous criteria, we however make use of Shale's criterion. We shall give necessary/sufficient criteria for the quasi-equivalence of Gaussian states in terms of the modular data.
A key point in our analysis concerns the cutting projection on a standard subspace studied in [9]. On one hand, this projection is expressed in terms of the modular data, on the other hand it has a geometric description in the QFT framework. The cutting projection is thus a link between geometry and modular theory, so it gives us a powerful tool.
Among our results, we have indeed necessary/sufficient criteria for the quasi-equivalence of two Gaussian states ϕ α 1 , ϕ α 2 on C * (H ), in terms of the difference of certain functions of the modular Hamiltonians, that are related to the cutting projections. However, our present applications to QFT are based on our general analysis, not directly to the mentioned criteria.
The following diagram illustrates the interplay among the three equivalent structures associated with standard subspaces and the geometric way out to QFT: modular data subspace geometry QFT complex structure cutting projection geometric Our paper is organised as follows. We first study the modular structure of standard subspaces, especially in relations with polarisers and cutting projections. We then study the local normality/weak innerness of Bogoliubov transformations, and the quasiequivalence of Gaussian states, in terms of modular Hamiltonians and other modular data. Finally, we present our mentioned applications in Quantum Field Theory. We also includes appendices, in particular concerning inequalities and functional calculus for real linear operators in the form we shall need. Finally, we point out certain positive selfadjoint extensions of the Laplacian, naturally arising via the inverse Helmholtz operator, that might have their own interest.

Basic Structure
This section contains the analysis of some general, structural aspects related to closed, real linear subspaces of a complex Hilbert space, from the point of view of the modular theory.
2.1. One-particle structure. Let H be a real vector space. A symplectic form β on H is a real, bilinear, anti-symmetric form on H . We shall say that β is non-degenerate on H if We shall say that β is totally degenerate if ker β = H , namely β = 0. A symplectic space is a real linear space H equipped with a symplectic form β.
Given a symplectic space (H, β), a real scalar product α on H is compatible with β (or β is compatible with α) if the inequality holds. Given a compatible α, note that ker β is closed (w.r.t. α), β = 0 on ker β and β is non-degenerate on (ker β) ⊥ . Clearly, β extends to a symplectic form on the completion H of H w.r.t. α, compatible with the extension of α. (However β may be degenerate on H even if β is non-degenerate on H .) A one-particle structure on H associated with the compatible scalar product α (see [24]) is a pair (H, κ), where H is a complex Hilbert space and κ : H → H is a real linear map satisfying WithH the completion ofH w.r.t. α, β extends to a compatible symplectic form onH .
Then κ extends to a real linear mapκ :H → H with (H,κ) a one-particle structure for H . In the following proposition, we shall anticipate a couple of facts explained in later sections. The uniqueness can be found in [24]; the existence is inspired by [34].
so U extends to a unitary operator with the desired property. 2), the orthogonal dilation provides a one-particle structure on H associated with α (Sect. 2.4). If D 2 H = −1, then D H is a complex structure on H , so the identity map is a one-particle structure. Taking the direct sum, we see that a one-particle structure exists if β is non-degenerate.
The existence of a one-particle structure then follows in general because with α(·, ·) = (·, ·), β(·, ·) = (·, ·) We have The operator D H is called the polariser of H . As we have one of our basic relations where E H is the orthogonal projection onto H .

Lemma 2.2. We have
thus H is separating iff ker(D 2 thus H is factorial iff ker(D H ) = {0}.   [25,27,37]. We denote by L H = log H the modular Hamiltonian of H . We often simplify the notation setting L = L H and similarly for other operators.
Assume now H to be standard and factorial. Let E H be the real orthogonal projection from H onto H as above and P H the cutting projection Recall two formulas respectively in [17] and in [9]: more precisely, P H is the closure of the right hand side of (12). These formulas can be written as so give In the following, if T : h ∈ H , thus As Corollary 2.5. We have Proof. By Proposition 2.4 D H = i tanh(L H /2)| H , thus so D 2 H is a bounded selfadjoint operator on H (as real linear operator). Therefore thus (20) holds. By Proposition 2.4 we then have The following corollary follows at once from [31]. The type of a subspace refers to the second quantisation von Neumann algebra.

Orthogonal dilation.
Let H be a real Hilbert space, with real scalar product α, and consider the doubling (direct sum of real Hilbert spaces). We consider a symplectic form β on H , that we assume to be non-degenerate and compatible with α. Let D be the polariser of β on H given by (4). So ker(D) = {0}. We also assume that ker(1 + D 2 ) = {0}, namely (H, α, β) is a factorial abstract subspace (6). Set with V the phase of D in the polar decomposition, D = V |D|; note that V commutes with D, because D is skew-selfadjoint, and V 2 = −1 (see [7,34]). Then ι is a unitary on H and ι 2 = −1, namely ι is a complex structure on H . Let H be the complex Hilbert space given by H and ι. The scalar product of H is given by

Proof. κ(H ) cyclic means that the linear span of H
The proof is then complete by Lemma 2.2.
By the above discussion H ⊂ H is a factorial standard subspace. We call H ⊂ H the orthogonal dilation of (H, β) with respect to α.

Symplectic dilation.
Let (H, α, β) be an abstract factorial standard subspace. Consider the doubled symplectic space With D the polariser of α, let H 0 = ran(D) and set where the matrix entries are defined as real linear operators (H, α) → (H, α) with domain H 0 . Then we have a real scalar productα on H 0 ⊕ H 0 which is compatible withβ. LetĤ be the completion of H 0 ⊕ H 0 with respect toα; thenĤ is a real Hilbert space with scalar product still denoted byα. By (28), (29), ι preservesα, so the closure of ι is a complex structure on H , and ι is the polariser ofα w.r.t.β. Thenβ extends to a symplectic form on H compatible witĥ α. SoĤ is indeed a complex Hilbert space and H ⊂Ĥ is a real linear subspace, where H is identified with H ⊕ 0.
We call H ⊂Ĥ the symplectic dilation of (H, β) with respect to α. Proof. H is complete, thus closed inĤ. Since the polariser of H inĤ is equal to D, the proposition follows by Lemma 2.2.

Bogoliubov Automorphisms
In this section we study symplectic maps that promote to unitarily implementable automorphisms on the Fock space. Given a symplectic space (H, β), we consider the Weyl algebra A(H ) associated with H , namely the free * -algebra complex linearly generated by the Weyl unitaries V (h), h ∈ H , that satisfy the commutation relations The C * envelop of A(H ) is the Weyl C * -algebra C * (H ). If β non-degenerate, there exists a unique C * norm on A(H ) and C * (H ) is a simple C * -algebra.
Let H be a complex Hilbert space and e H be the Bosonic Fock Hilbert space over H. Then we have the Fock representation of C * (H R ) on e H , where H R is H as a real linear space, equipped with the symplectic form β ≡ (·, ·). In the Fock representation, the Weyl unitaries are determined by their action on the vacuum vector e 0 where e h is the coherent vector associated with h. So the Fock vacuum state ϕ = (e 0 , · e 0 ) of C * (H R ) is given by With H any real linear subspace of H, the Fock representation determines a representation of C * (H ) on e H , which is cyclic on e H iff H is a cyclic subspace of H. We denote by A(H ) the von Neumann algebra on e H generated by the image of C * (H ) in this representation. We refer to [8,26,27,32] for details.

Global automorphisms.
Let H be a complex Hilbert space and e H the Fock space as above. A symplectic map T : D(T ) ⊂ H → H is a real linear map with D(T ) and ran(T ) dense, that preserves the imaginary part of the scalar product, thus (T ξ, therefore ker(T ) = {0}, T is closable because T * is densely defined, and T −1 = −i T * i| ran(T ) , so T * | i ran(T ) is a symplectic map too. It also follows that We then have the associated Bogoliubov homomorphism ϑ T of the Weyl algebra A D(T ) onto A ran(T ) : Let T : H → H be a bounded, everywhere defined symplectic map; the criterion of Shale [39] gives a necessary and sufficient condition in order that ϑ T be unitary implementable on e H , under the assumption that T has a bounded inverse: is the commutator and L 2 (H) are the real linear, Hilbert-Schmidt operator on H. Due to the equivalence (33), the assumption T −1 bounded in (34) can be dropped (as we assume that ran(T ) is dense).
We shall deal with symplectic maps that, a priori, are not everywhere defined. However the following holds.

Lemma 3.1. Let T : D(T ) ⊂ H → H be a symplectic map. Then ϑ T is unitarily implementable iff ϑ T is unitarily implementable, where T is the closure of T . In this case, T is bounded.
with ϕ the Fock vacuum state, therefore ||T ξ n || → 0 and T is bounded. If ϑ T is implemented, then ϑ T is obviously implementable by the same unitary. Conversely, assume that ϑ T is implementable by a unitary U on H. So T is bounded. Hence T is a bounded, everywhere defined symplectic map. Let ξ ∈ H and choose a sequence of elements ξ n ∈ D(T ) such that ξ n → ξ . Then

Hilbert-Schmidt perturbations.
Motivated by Shale's criterion, we study here Hilbert-Schmidt conditions related to the symplectic dilation of a symplectic map.
We use the following notations: If H is a complex Hilbert space, L p (H) denotes the space of real linear, densely defined operators T on H that are bounded and the closureT belongs to the Schatten p-ideal with respect to the real part of the scalar product, (the symplectic matrix decomposition). Thus and C 11 is an operator H → H , C 12 is an operator H → H , etc. We want to study the Hilbert-Schmidt condition for C. Note that With D = D H the polariser and J = J H the modular conjugation, the symplectic matrix decomposition of the complex structure is as follows from (27) and the uniqueness of the dilation. Note, in particular, the identity Lemma 3.2. The following symplectic matrix representations hold: Proof. We have the first equality in the lemma follows by matrix multiplication with (36). The second equality is then simply obtained as Last equality follows as Proof. We have thus namely, (39) holds.
Since Ci = −iC, we have and (40) holds. With C j = J C J , we then get Similarly, from (39) we get (42).
With H a standard subspace, a symplectic map of the standard subspace H is a real linear map T : H → H such that if T is invertible, we shall say that T is a symplectic bijection of H . Now, let H be a factorial standard subspace and T : H → H a symplectic bijection. Denote by T the symplectic map T ⊕ J T J : Note that

Corollary 3.4.
We have Proof. We apply Lemma 3.3 with C = T , i . By (39), we get (44). By (40), we get (45). By (41), we get (46). By (42), we get (47). Now, moreover, We conclude that all the four matrix elements in the orthogonal decomposition of Proof. If the assumptions are satisfied, then a) and b) of Proposition 3.5 clearly hold because D and √ 1 + D 2 are bounded.

Finite codimensional subspaces of standard subspaces
where F : H →Ḣ is the orthogonal projection. LetḢ ⊥ ⊂ H be the real orthogonal complement ofḢ in H . We have the matrix where the starred entries have finite rank or co-rank.
and we may apply next lemma because F D H (1 − F)D H |Ḣ is a finite rank operator. Then Similarly as in (50), we have is compact (resp. L p ) by the assumption. Therefore (D 1 − D 2 )F 1 is compact (resp. L p ) because F 1 is bounded, so is compact (resp. L p ) because 1 − F 1 has finite rank.
The converse holds too by reversing the implications.

Local automorphisms.
Let now H k be standard factorial subspaces of the Hilbert spaces H k , k = 1, 2 and T : H 1 → H 2 a symplectic bijection, namely T is real linear, invertible and β 2 (T h, T k) = β 1 (h, k), h, k ∈ H 1 , with β k the symplectic form on H k (the restriction of (·, ·) k to H k , with (·, ·) k the scalar product on H k ). Then T promotes to a * -isomorphism ϑ T between the Weyl C * -algebras C * (H 1 ) and C * (H 2 ) where H k is the symplectic complement of H k in H k and J k = J H k . Then T is a densely defined, real linear, symplectic map with dense range from H 1 to H 2 .

Lemma 3.9. If T i 1 − i 2 T is bounded and densely defined, then T is bounded.
Proof. T is closable by Lemma 3.1 so T i 1 and i 2 T are closable too. By assumptions, there is a bounded, everywhere defined operator C : is a core for P H 1 , as follows by Eq. (12). Indeed, i 1 H 1 = H 1 and J i 1 H 1 = −J H 1 , so the spectral subspaces of H 1 relative to finite closed intervals [a, b] ⊂ (0, 1) ∪ (1, ∞) are in the domain of D(P H 1 ) ∩ D(P i 1 H 1 ) (see [9]). Now, and one easily checks that D is a core for T , similarly as above. It follows that¯ We conclude that    [3,11,21].
(i) ⇔ (iii): Assume (i) and let U T be the vacuum unitary standard implementation ϑ T as above. e J k , the second quantisation of the modular conjugation J k of H k , is the modular conjugation of the von Neumann algebra A k (H ) w.r.t. the vacuum vector e 0 , so we have for all η in the domain of T . Then (iii) holds by Lemma 3.1 and Shale's criterion [39]. Conversely, assuming (iii), by Lemma 3.9 and again by Lemma 3.1 and Shale's criterion, we can find a unitary U such that (51) holds.
Proof. The above conditions are the straightforward generalisations of the conditions a) and b) in Proposition 3.5, so the corollary follows by Proposition 3.10.
Recall that a real linear map T : H 1 → H 2 is symplectic iff T * D 2 = D 1 T −1 , so the conditions in the above corollary take a different form by inserting this relation.

Gaussian States, Modular Hamiltonian, Quasi-equivalence
Let (H, β) be a symplectic space. With α a real scalar product on H compatible with β, let κ α : H → H α be the one-particle structure associated with α (Proposition 2.1).
Let e H α be the Bose Fock Hilbert space over H α and denote by V α (·) the Weyl unitaries acting on e H α and by e 0 the vacuum vector of e H α , thus V (h) → V α (h) gives a representation of C * (H ) on e H α (see for example [26]). By (31), we have Proposition 4.1. There exists a unique state ϕ α on C * (H ) such that With {H ϕ α , π ϕ α , ξ ϕ α } the GNS triple associated with ϕ α , the vector ξ ϕ α is separating for the von Neumann algebra A(H ) = π ϕ α C * (H ) iff the completionH of H is a separating subspace, namely ker(D 2H + 1) = {0}.
Proof. Equation (52) shows that there exists a state ϕ a such that (53) holds. Moreover (53) determines ϕ α because the linear span of the Weyl unitaries is a dense subalgebra of C * (H ).

As κ α (H ) is cyclic in H α , κ α (H ) is a standard subspace of H α iff κ α (H ) is separating.
On the other hand, e 0 is cyclic and separating for the von Neumann algebra generated by the V α (h)'s, h ∈ H , iff κ α (H ) is a standard subspace of H, see [26]. The proposition then follows by the uniqueness of the GNS representation.
The state ϕ α determined by (53) is well known and is called the Gaussian, or quasifree, state associated with α, see [14,34]. It is usually defined by showing directly, by positivity, that the Gaussian kernel (53) defines a state.
We summarise in the following diagram the two above considered, unitarily equivalent constructions with the GNS representation of a Gaussian state: As a consequence, if H is a standard subspace, the modular group σ ϕ α of ϕ α on C * (H ) is given by , therefore the study of the modular structure of A(H ) can be reduced to the study of the modular structure of H .
The following quasi-equivalence criterion is related to the analysis in [5,23,42], although we do not rely on their work.

Theorem 4.2. Let
and hold, where D k is the polariser of (H, α k , β).
Proof. Let H k be the symplectic dilation of (H, β k ) with respect to α k ; so H ⊂ H k is a factorial standard subspace. We have spelled out the conditions for the symplectic map I :Ĥ →Ĥ to promote a unitary between the Fock spaces over H 1 and H 2 (I is the identity on H ⊕ H as vector spaces). Shale's criterion gives that entails the statement of the theorem by Proposition 3.5.
We now consider the property that is that is We write α 1 ≈ α 2 if Property (56) holds.
To end our proof, we now show that (67) implies , therefore F is uniformly Lipschitz. Since 0 is not in the point spectrum of L k , it follows by Cor. 6.5 that (67) implies (58), namely α 1 ≈ α 2 . Now, if A 1 , A 2 are bounded, real linear operators on H with trivial kernel, we have on the domain of the right hand side operator, thus We then have: then the Gaussian states ϕ α 1 and ϕ α 2 on C * (H ) are quasi-equivalent.
The above corollary suggests that ϕ α 1 and ϕ α 2 are quasi-equivalent if P 1 i 1 | H − P 2 i 2 | H is compact with proper values decaying sufficiently fast.

Weakly inner Bogoliubov automorphisms.
In this section, we study the condition for a real linear, symplectic bijection of a standard space to give rise to a weakly inner automorphism in the representation associated with a given Gaussian state.
Let H ⊂ H be a factorial standard subspace of the complex Hilbert space H, T : H → H a symplectic bijection and ϑ T the associated Bogoliubov automorphism of the Weyl algebra A(H ). Denote by A(H ) the weak closure of A(H ) on e H as in previous sections.
We consider the real linear map on H given bŷ

Proof. Since A(H ) is the commutant of A(H ), ϑ T extends to an inner automorphism of A(H ) if and only if the Bogoliubov automorphism associated withT is unitarily
implementable on e H . Therefore the equivalence (i) ⇔ (ii) follows by Shale's criterion and Lemma 3.1.
(ii) ⇔ (iii) follows again by Shale's criterion, Lemma 3.1 and the obvious adaptation of Lemma 3.9.
Set now T = 1 + X andX = X ⊕ 0 on H + H . In the symplectic matrix decomposition, we haveX With C = [X , i], we apply Lemma 3.3. Then Note that  Proof. The theorem follows now by Lemma (4.7).

QFT and the Modular Hamiltonian
We now work out the studied abstract structure, within the context of Quantum Field Theory. We then provide a couple of applications of our results.
5.1. One-particle space of the free scalar QFT. This section concerns the one-particle space of the free scalar QFT, especially in the low dimensional case. Although we are primarily interested in the low dimensional case in this paper, we start by describing the higher dimensional case in order to clarify the general picture. In the following, d is the space dimension, so R d is the time-zero space of the Minkowski spacetime R d+1 , cf. [30].
is a unitary operator. Then with ı 2 m = −1, namely a complex structure on H m that so becomes a complex Hilbert space H m with the imaginary part of the scalar product given by which is independent of m ≥ 0 (where (·, ·) is the L 2 scalar product).
With B the unit ball of R d , we shall denote by H Here C ∞ 0 (B) denotes the space of real C ∞ function on R d with compact support in B. The H m (B)'s, m ≥ 0, are the same linear space with the same Hilbert space topologies (see e.g. [30]). We shall often identify these spaces as topological vector spaces.
In the following, we consider the abstract standard spaces (H, α m , β) where H = H m (B), β is the symplectic form on H given by (79) and α m is the real scalar product on H as a real subspace of We have the following commutative diagram where χ B is the multiplication operator by the characteristic function of B in L 2 (R d ), i.e. the orthogonal projection L 2 (R d ) → L 2 (B), and ι 1 , ι 2 are natural embeddings.
We need a couple of lemmas in order to conclude our proof.
Proof. By (80) we have with a(s) = m 2 √ s 2 + m 2 /( √ s 2 + m 2 +s), so and 1/a are bounded continuous functions on R d . Therefore with A the multiplication operator by a, a bounded linear operator with bounded inverse. So the Laplacian on B with external boundary condition (6.3). We conclude that by Corollary 6.7.

Case d = 1
• Case m > 0. In this case the one-particle Hilbert space is defined exactly as in the higher dimensional case. In particular H  (−1, 1). • Case m = 0. H 1/2 0 is defined as in the higher dimensional case (76): We now seṫ H 0 (B) is a standard subspace ofḢ 0 . Note that, in the massless case, our notation is unconventional:Ḣ 0 is the usual one-particle space and H 0 has not been defined yet. See also [6,12] for related structures.

The modular
Hamiltonian, d = 1. We now describe the modular Hamiltonian associated with the unit double cone in the free, scalar QFT on the 1 + 1 dimensional Minkowski spacetime. Recall that the modular Hamiltonian on the Fock space is the second quantisation of the modular Hamiltonian on the one-particle space, that will therefore be the subject of our analysis. In this subsection B = (−1, 1). Proof. The proof that the natural, real linear identifications of theḢ m (B)'s preserve the Hilbert space topology is a simple adaptation of the one given in the higher dimensional case, see [30]. We have seen in Proposition 5.4 thatḢ m (B) is a standard subspace ofḢ m . The factoriality ofḢ 0 (B) follows, for example, by [22]. Now, the identification ofḢ m (B) withḢ 0 (B) preserves the symplectic form. Since the factoriality is equivalent to the non-degeneracy of the symplectic form, alsoḢ m (B) is factorial.  Proof toḢ m (B) , the symplectic complement ofḢ m (B) inḢ m ; so, by Lemma 5.6, and this shows thatṖ m is a diagonal matrix of the form (90).
We then have The equation P + f = χ B f , with f in the domain of P + , follows by similar arguments. Proof. By Lemma 5.7, we havė We have to show thatṖ − μ m −Ṗ − μ 0 :Ḣ is L 1 . Similarly as above, we have the following commutative diagraṁ Here ι 1 is the restriction toḢ

m = 0
In the massless case, the modular group associated with the unit, timezero interval B acts geometrically on the spacetime double cone spanned by B [22]. We have: Theorem 5.9. In the free scalar, massless, quantum field theory in 1 + 1 spacetime dimension, the modular Hamiltonian log˙ B,0 associated with the unit interval B, that is with the standard subspaceḢ 0 (B) ⊂Ḣ 0 , is given by Setting log˙ B,0 = −2πȦ 0 andȦ 0 ≡ −ı 0K0 , we have thatK 0 is essentially skewselfadjoint on S×Ṡ.
The formula is obtained as in [30], with obvious modifications.

m
(m > 0); the domain D(K B m ) is defined in [30], K B m is Hermitian on C ∞ 0 (B) 2 (proved to be essentially skew-selfadjoint in the case d ≥ 2 in [30]).
Here, G B m : H where ∇ 2 m is the Laplacian on B with external boundary conditions in Appendix 6.3.

m ≥ 0 We now set
is in L p , p > 1. By (93) and (92), the operator (96) is equal to the sum of two operators that are both in L p , p > 1, see [30]. The is in L p , p > 1, by Corollary 6.5, so it is compact. By Lemma 5.10, also is compact. Set by (98)

Local entropy of a
Klein-Gordon wave packet, d = 1. Although this section contains a main application of our paper, we shall be very short on its background as this is explained in details in [9,30].
Here, H = H m (B), H is the modular operator and P H is the cutting projection associated with H . is the vector f ⊕ g ∈ H m = H 1/2 Recall that the time-zero energy density of is given by T (m) Theorem 5.13. The entropy S of the Klein-Gordon wave in the unit interval (−1, 1) at time t = 0 is given by where G m is the Green function for the Helmholtz operator, G m (x) = 1 2m e −m|x| .
Proof. The proof follows the one in the higher dimensional case; this is possible as we now have the formula for the local modular Hamiltonian.
Note that the above results have a straightforward version with B replaced by any other interval, same as [30].

Further consequences in QFT.
In this section, we provide a few direct consequences in second quantisation of our results.  [4]) between the vacuum state ϕ and the coherent state ϕ associated with the one-particle wave ∈ H m is given by (100).

Local entropy of coherent states
Proof. The case d ≥ 2 is proved in [30]. By applying Theorem 5.13, the corollary follows now in the d = 1 case too as in [9,29].
The formula for S is the same in the massless case, provided one deals with restricted Cauchy data as above, in order that ∈ H 0 , see [28,Sect. 4]. See also [10] for a discussion on relative entropy in a curved spacetime setting.

Type I I I 1 property
We show here the type I I I 1 factor property (see [41]) for the local von Neumann algebras associated with free, scalar QFT. In the massless case, this follows from [22]; in the massive case from [16], if d > 1. A m (B) is a factor because the symplectic form on H m (B) is non-degenerate. Concerning the type I I I 1 property, by [17] it suffices to show that the additive subgroup of R generated by sp e (log B,m ) is equal to R, with sp e denoting the essential spectrum. Due to the relation (9), sp e (log B,m ) is symmetric, so it is enough to show that sp e (tanh 2

Proof.
Let T be a real linear, bounded operator on H . We denote byŤ its promotion to H C : namelyŤ is the unique complex linear operator on H C that restricts to T on H . Then ||Ť || = ||T || because Note that for every f ∈ B.
Proof. The statement holds if f (x) = e i x because T is the infinitesimal skew-selfadjoint generator of e is A | H = e is X | H . So it holds if f is the Fourier transform of a real L 1function g as Then (102) holds for every continuous function with compact support f ∈ B, as it can be uniformly approximated by functions as above by the Stone-Weierstrass theorem.
Let now f be any function in B and fix two vectors ξ, η ∈ H . There exists a uniformly bounded sequence of continuous functions f n ∈ B with compact support such that f n → f almost everywhere with respect to the spectral measures of A and X associated with ξ, η. Then by the Lebesgue dominated convergence theorem, that concludes our proof because ξ, η are arbitrary.

Operator Lipschitz perturbations.
The next theorem is due to Potatov and Sukochev [35]. Note that, in Thm. 6.3, it suffices to assume that (A 1 − A 2 )| D ∈ L p (H) with D a core for A 1 or A 2 , since then D is a core for both A 1 or A 2 and D(A 1 ) = D(A 2 ) because A 1 − A 2 is bounded.
The following corollary was communicated to us by F. Sukochev. Corollary 6.4. Let A k be a selfadjoint operator on the Hilbert space H k , k = 1, 2, and suppose that H 1 and H 2 are the same topological vector space, that we call H. Then p > 1, for every uniformly Lipschitz function f on R.
Proof. Let C : H 1 → H 2 be the complex linear identification of H 1 and H 2 as topological vector spaces. So C is a bounded operator with bounded inverse C −1 . Then we have to show that or, equivalently, that (H 1 , H 2 ).
With K = H 1 ⊕ H 2 , the operator A = A 1 ⊕ A 2 is selfadjoint on K. Set V = 0 0 C 0 ; then so we have to show that that follows by [35,Eq. (14)].
We need a certain real version of Corollary 6.4.
p > 1, for every uniformly Lipschitz function f on R such that f (−x) = − f (x).
Proof. Let H k C be the usual complexification of the real Hilbert space H k . Then H 1C and H 2C are equivalent complex Hilbert spaces. Let A k be the selfadjoint extension of X k to H k C as above; by Proposition 6.2, we have

Extensions of the Laplacian via Helmholtz operator.
Let H be a Hilbert space, K a closed subspace and A : D(A) ⊂ H → H a positive selfadjoint linear operator.
is dense in K and denote by A 0 the restriction of A to D 0 , as operator K → K. Clearly A 0 is a positive Hermitian operator on K. We want to study the selfadjoint extensions of A 0 . Choose m > 0, then (A + m 2 ) −1 is a bounded selfadjoint operator on H whose norm is ||(A + m 2 ) −1 || ≤ 1/m 2 . With E the orthogonal projection of H onto K, set Then T is a bounded, selfadjoint operator on K and ||T || ≤ 1/m 2 . We have We note the following. • By theorems of von Neumann, Krein, Friedrichs et al. (see [1,2,38]), every positive selfadjoint extension of A 0 lies between A min and A max , where where A min and A max are respectively the Krein and the Friedrichs extension of A 0 on K. In particular, this problem is studied e.g. [33]. Denote by C m the space of all f ∈ C ∞ (∂ B) such that f c exists with f c and partial derivatives of all order tending to zero as r = |x| → +∞ faster than any inverse power of r . In this case the solution f c is unique by the maximum principle. For completeness, we sketch the following proposition, although it is not needed in this form in the paper (we need Corollary 6.7). Proof. Let A 0 = −∇ 2 +m 2 on C ∞ 0 (B); then A min = −∇ 2 D +m 2 and A max = −∇ 2 K +m 2 , where ∇ 2 D and ∇ 2 K are the Dirichlet and the Krein Laplacian. Now ∇ 2 D satisfies the Weyl asymptotic, so (∇ 2 D −m 2 ) −1 ∈ L p iff p > d/2, see [13]. Moreover, the same asymptotic hold for (∇ 2 K −m 2 ) −1 , see [19]. By the min-max principle (see [38,Sect. 12.1]), the same asymptotic holds for every positive, selfadjoint extension of the Laplacian on C ∞ 0 (B), in particular for ∇ m = E(∇ 2 − m 2 ) −1 | L 2 (B) , so our statement holds.