# 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 recent results in higher space dimensions. 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 III_1 factor property. Incidentally, we run across certain positive selfadjoint extensions of the Laplacian, with outer boundary conditions, seemingly not considered so far.

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## 1 Introduction

The Heisenberg commutation relations are at the core of Quantum Mechanics. From the mathematical viewpoint, they have a more transparent formulations in Weyl’s the exponential. If is a real linear space equipped with a non-degenerate symplectic form , one considers the free -algebra linearly generated by the (unitaries) , , that satisfy the commutation relations (CCR)

 V(h+k)=eiβ(h,k)V(h)V(k),h,k∈H, (1)

. The Weyl algebra admits a unique norm, so its completion is a simple -algebra, the Weyl -algebra . The representations, and the states, of and of are so in one-to-one correspondence. We refer to [7, 33, 13] for the basic theory.

For a finite-dimensional , von Neumann’s famous uniqueness theorem shows that all representations of , with

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

[19].

Due to the relations (1), a state on is determined by its value on the Weyl unitaries; a natural class of states is given by the ones with Gaussian kernel. A state is called Gaussian, or quasi-free, if

 φα(V(h))=e−12α(h,h),

with a real bilinear form on , that has to be compatible with .

Assuming now that is separating with respect to

, as is the case of a local subspace in QFT, the GNS vector associated with

is cyclic and separating for the von Neumann algebra generated by in the representation. So there is an associated Tomita-Takesaki modular structure, see [40], 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 [27]). The other aspect concerns the crucial role taken by the modular theory of standard subspaces, see [26]; 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 spacetime dimension, in order to complement the higher dimensional results, that were obtained after decades of investigations [29]. We give our formula in Section 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 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 dimension case. We refer to [27, 28, 8, 29] for background results and explanation of the context. We also show the type 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 out 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 transformation, CCR case) is a natural problem; the automorphisms of the associated operator algebras are called Bogoliubov automorphisms, see [14, 13]. The classical result of Shale [38] 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 [4], van Daele [41] and Holevo [22], 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 CAR case [35]. Woronowicz partly related the purification map to the modular theory and reconsidered the CAR case [42]. 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 [2, 10, 20]. 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 [8]. 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 , on , 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:

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 quasi-equivalence 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.

## 2 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 be a real vector space. A symplectic form on is a real, bilinear, anti-symmetric form on . We shall say that is non degenerate on if

 kerβ≡{h∈H:β(h,k)=0, ∀k∈H}={0}.

We shall say that is totally degenerate if , namely . A symplectic space is a real linear space equipped with a symplectic form .

Given a symplectic space , a real scalar product on is compatible with (or is compatible with ) if the inequality

 β(h,k)2≤α(h,h)α(k,k) ,h,k∈H, (2)

holds. Given a compatible , note that is closed (w.r.t. ), on and is non-degenerate on . Clearly, extends to a symplectic form on the completion of w.r.t. , compatible with the extension of . (However may be degenerate on even if is non-degenerate on .)

A one-particle structure on associated with the compatible scalar product (see [23]) is a pair , where is a complex Hilbert space and is a real linear map satisfying

• and ,  ,

• is dense in .

Note that is injective because

 h∈H, κ(h)=0⇒R(κ(h),κ(h))=0⇒α(h,h)=0⇒h=0. (3)

With the completion of w.r.t. , extends to a compatible symplectic form on . Then extends to a real linear map with a one-particle structure for .

In the following proposition, we shall anticipate a couple of facts explained in later sections. The uniqueness can be found in [23]; the existence is inspired by [33].

###### Proposition 2.1.

Let be a symplectic space with a compatible scalar product . There exists a one-particle structure on associated with . It is unique, modulo unitary equivalence; namely, if is another one-particle structure on , there exists a unitary such that the following diagram commutes:

###### Proof.

Uniqueness. The linear map is well defined by on by (3). Moreover, it extends to a complex linear map and is isometric because

 ||κ(h)+iκ(k)||2=||κ(h)||2+||κ(k)||2+2R(κ(h),iκ(k))=||κ(h)||2+||κ(k)||2−2I(κ(h),κ(k))=α(h,h)+α(k,k)−2β(h,k)=||κ′(h)+iκ′(k)||2,

so extends to a unitary operator with the desired property.

Existence. By replacing with its completion w.r.t. , we may assume that is complete. Suppose first that is totally degenerate, i.e. , and let the usual complexification of , namely as real Hilbert space with complex structure given by the matrix . Then is a one-particle structure on associated with .

Suppose now that is non-degenerate and consider the polariser (Sect. 2.2). If , i.e. is separating (see Lemma 2.2), the orthogonal dilation provides a one-particle structure on associated with (Sect. 2.4). If , then is a complex structure on , so the identity map is a one particle structure. Taking the direct sum, we see that a one particle structure exists if is non degenerate.

The existence of a one particle structure then follows in general because , where the restriction of to is totally degenerate and to is non-degenerate.

### 2.2 Polariser

Let be a closed, real linear subspace of the complex Hilbert space . By the Riesz lemma, there exists a unique bounded, real linear operator on such that

 β(h,k)=α(h,DHk),h,k∈H, (4)

with ,

We have

 ||DH||≤1,D∗H=−DH.

The operator is called the polariser of . As

 I(h,k)=−R(h,ik)=−R(h,EHik),h,k∈H,

we have one of our basic relations

 DH=−EHi|H, (5)

where is the orthogonal projection onto .

Let be the symplectic complement of . We shall say that is factorial if .

###### Lemma 2.2.

We have

 ker(D2H+1)=H∩iH, (6)

thus is separating iff . Furthermore,

 ker(DH)=kerβ=H∩H′. (7)

thus is factorial iff ker.

###### Proof.

As , with the orthogonal projection of onto (5), we have

 D2H=EHiEHi|H=−EHEiH|H (8)

so, if ,

 (D2H+1)h=0⇔EHEiHh=h⇔h∈H∩iH,

showing the first part of the lemma.

Last assertion follows as

 kerβ=ran(DH)⊥=ker(D∗H)=ker(DH)

and clearly .

.

###### Proof.

Let , thus , so and this implies because . Thus , so ; hence . So .

Conversely, assume that ; then , so . Finally, assume the equality to hold. Then , so , hence , so , namely .

### 2.3 Standard subspaces

Let be a complex Hilbert space and a closed, real linear subspace. We say that is cyclic if is dense in , separating if , standard is if it is both cyclic and separating.

Let be a closed, real linear subspace of and on , where is the complex scalar product on ; then is a symplectic form on that makes it a symplectic space. Moreover, is a compatible real scalar product on .

An abstract standard subspace (or simply ) is a real Hilbert space , where is the real scalar product and a symplectic compatible with , with separating, that is , with the polariser of , see Lemma 2.2.

By Proposition 2.1, an abstract standard subspace can be uniquely identified, up to unitary equivalence, with a standard subspace of a complex Hilbert space as above.

We shall say that the abstract standard subspace is factorial if , namely is non degenerate.

In view of the above explanations, we shall often directly deal with standard subspaces of a complex Hilbert space .

Given a standard subspace of , we shall denote by and the modular conjugation and the modular operator of ; they are defined by the polar decomposition of the closed, densely defined, anti-linear involution on

 SH:h+ik↦h−ik,h,k∈H.

is a non-singular, positive selfadjoint operator, is an anti-unitary involution and we have

 JHΔHJH=Δ−1H. (9)

The fundamental relations are

 ΔisHH=H,JHH=H′,s∈R,

see [36, 24, 26]. We denote by

 LH=logΔH

the modular Hamiltonian of . We often simplify the notation setting and similarly for other operators.

Assume now to be standard and factorial. Let be the real orthogonal projection from onto as above and the cutting projection

 PH:h+h′↦h,h∈H,h′∈H′. (10)

is a closed, densely defined, real linear operator with domain .

Recall two formulas respectively in [16] and in [8]:

 EH=(1+ΔH)−1+JHΔ1/2H(1+ΔH)−1, (11) PH=(1−ΔH)−1+JHΔ1/2H(1−ΔH)−1; (12)

more precisely, is the closure of the right hand side of (12).

These formulas can be written as

 EH=(1+SH)(1+ΔH)−1, (13) PH=(1+SH)(1−ΔH)−1, (14)

so give

 PH=EH(1+ΔH)(1−ΔH)−1=EHcoth(LH/2). (15)

In the following, if is a real linear operator, is the restriction of to , that we may consider also as operator if , as it will be clear from the context.

###### Proposition 2.4.

Let be a factorial standard subspace. The polariser of and its inverse are given by

 DH=−EHi|H=i(ΔH−1)(ΔH+1)−1|H=itanh(LH/2)|H, (16) D−1H=PHi|H=−i(ΔH+1)(ΔH−1)−1|H=icoth(LH/2)|H. (17)

As a consequence,

is a skew-selfadjoint real linear operator on

.

###### Proof.

As , eq. (11) gives

 EH=(1+ΔH)−1+ΔH(1+ΔH)−1JΔ1/2H,

therefore

 EHih=((1+ΔH)−1+ΔH(1+ΔH)−1SH)ih=(1+ΔH)−1ih−ΔH(1+ΔH)−1ih=(1−ΔH)(1+ΔH)−1ih, (18)

, thus

 EHi|H=(1−ΔH)(1+ΔH)−1i|H. (19)

As (5), eq. (16) is proved.

Concerning formula (17), since is left invariant by , from (15) we get

 PHi|H=EHcoth(LH/2)i|H=icoth(LH/2)|H=i(ΔH+1)(ΔH−1)−1|H.

So is skew-selfadjoint because is globally -invariant, [29, Prop. 2.2].

###### Corollary 2.5.

We have

 √1+D2H=2(Δ1/2H+Δ−1/2H)−1|H=1cosh(LH/2)∣∣H. (20)
 D−1H√1+D2H=2i(Δ1/2H−Δ−1/2H)−1|H=i1sinh(LH/2)∣∣H; (21)
###### Proof.

By Prop. 2.4 , thus

 D2H=−tanh2(LH/2)|H, (22)

so is a bounded selfadjoint operator on (as real linear operator). Therefore

 1+D2H=(1−tanh2(LH/2)|H)∣∣H=1cosh2(LH/2)∣∣H, (23)

thus (20) holds.

By Prop. 2.4 we then have

 D−1H√1+D2H=icoth(LH/2)cosh(LH/2)∣∣H=i1sinh(LH/2)∣∣H.

The following corollary follows at once from [30]. The type of a subspace refers to the second quantisation von Neumann algebra.

###### Corollary 2.6.

We have

 EHEH′|H=1+D2H. (24)

Therefore, is a type subspace iff is a trace class operator.

###### Proof.

By [30, Lemma 2.4], we have ; by (23), we have

 4ΔH(1+ΔH)−2|H=1cosh2(LH/2)∣∣H=1+D2H.

The corollary thus follows by [30, Cor. 2.6].

By (24) and (8), we have the nice identity

 EHEH′|H+EHEiH|H=1. (25)

Let be abstract standard subspaces, , and suppose that is equivalent to , thus there exists a bounded, positive linear map with bounded inverse such that . Then

 α1(h,D1k)=β(h,k)=α2(h,D2k)=α1(h,TD2k),

thus .

### 2.4 Orthogonal dilation

Let be a real Hilbert space, with real scalar product , and consider the doubling

 ˜H=H⊕H

(direct sum of real Hilbert spaces). We consider a symplectic form on , that we assume to be non degenerate and compatible with . Let be the polariser of on given by (4). So . We also assume that , namely is a factorial abstract subspace (6). Set

 ι=[DV√1+D2V√1+D2−D], (26)

with the phase of in the polar decomposition, ; note that commutes with , because is skew-selfadjoint, and (see [33, 6]). Then is a unitary on and , namely is a complex structure on .

Let be the complex Hilbert space given by and . The scalar product of is given by

 (h1⊕h2,k1⊕k2)=˜α(h1⊕h2,k1⊕k2)+i˜β(h1⊕h2,k1⊕k2)

with and .

The embedding

 κ:h↦κ(h)≡h⊕0

satisfies in Sect. 2.1, that is and

 ˜β(κ(h),κ(k))=˜α(h⊕0,ι(k⊕0))=˜α(h⊕0,Dk⊕V√1+D2k))=α(h,Dk)=β(h,k),

.

###### Lemma 2.7.

cyclic and separating in , so is a one particle structure for with respect to and is a factorial subspace.

###### Proof.

cyclic means that the linear span of and is dense in . As

 ι(h⊕0)=Dh⊕−V√1+D2h,

is cyclic iff is dense, thus iff . The proof is then complete by Lemma 2.2.

By the above discussion is a factorial standard subspace. We call the orthogonal dilation of with respect to .

### 2.5 Symplectic dilation

Let be an abstract factorial standard subspace. Consider the doubled symplectic space , where .

With the polariser of , let and set

 ι=[D−1D−1√1+D2D−1√1+D2−D−1], (27)

where the matrix entries are defined as real linear operators with domain . Then

 ι2=−1

on . A direct calculation shows that

 ^β(ιξ,ιη)≡^β(ξ,η),ξ,η∈H0⊕H0; (28)

setting

 ^α(ξ,η)≡^β(ξ,ιη),ξ,η∈H0⊕H0, (29)

we have real scalar product on which is compatible with . Let be the completion of with respect to ; then is a real Hilbert space with scalar product still denoted by .

By (28), (29), preserves , so the closure of is a complex structure on , and is the polariser of w.r.t. . Then extends to a symplectic form on compatible with . So is indeed a complex Hilbert space and is a real linear subspace of , where is identified with .

We call the symplectic dilation of with respect to .

###### Proposition 2.8.

is a factorial standard subspace of the symplectic dilation . Therefore the symplectic and the orthogonal dilations are unitarily equivalent.

###### Proof.

is complete, thus closed in . Since the polariser of in is equal to , the proposition follows by Lemma 2.2.

## 3 Bogoliubov automorphisms

In this section we study symplectic maps that promote to unitarily implementable automorphisms on the Fock space.

Given a symplectic space , we consider the Weyl algebra associated with , namely the free -algebra complex linearly generated by the Weyl unitaries , , that satisfy the commutation relations

 V(h+k)=eiβ(h,k)V(h)V(k),V(h)∗=V(−h), h,k∈H.

The envelop of is the Weyl -algebra. If non degenerate, there exists a unique norm on and is a simple -algebra.

Let be a complex Hilbert space and be the Bosonic Fock Hilbert space over . Then we have the Fock representation of on , where is as a real linear space, equipped with the symplectic for . In the Fock representation, the Weyl unitaries are determined by their action on the vacuum vector

 V(h)e0=e−12(h,h)eh, h∈H, (30)

where is the coherent vector associated with . So the Fock vacuum state of is given by

 φ(V(h))=e−12||h||2,h∈H. (31)

With any real linear subspace of , the Fock representation determines a representation of on , which is cyclic on iff is a cyclic subspace of . We denote by the von Neumann algebra on generated by the image of in this representation. We refer to [7, 31, 25, 26] for details.

### 3.1 Global automorphisms

Let be a complex Hilbert space and the Fock space as above. A symplectic map is a real linear map with and ran dense, that preserves the imaginary part of the scalar product, thus , .

Let be a symplectic map. Then

 R(iTξ,Tη)=R(iξ,η),ξ,η∈D(T),

thus and for all , namely

 T∗iT=i|D(T), (32)

therefore ker, is closable because is densely defined, and , so is a symplectic map too. It also follows that

 T bounded⟺T∗ bounded⟺T−1 bounded. (33)

We then have the associated Bogoliubov homomorphism of the Weyl algebra onto :

 ϑT:V(ξ)↦V(Tξ), ξ∈D(T).

Let be a bounded, everywhere defined symplectic map; the criterion of Shale [38] gives a necessary an sufficient condition in order that be unitary implementable on , under the assumption that has a bounded inverse:

 ϑT unitary implementable ⟺ T∗T−1∈L2(H) ⟺ [T,i]∈L2(H), (34)

where is the commutator and are the real linear, Hilbert-Schmidt operator on .

Due to the equivalence (33), the assumption bounded in (34) can be dropped (as we assume that ran is dense).

We shall deal with symplectic maps that, a priori, are not everywhere defined. However the following holds.

###### Lemma 3.1.

Let be a symplectic map. Then is unitarily implementable iff is unitarily implementable, where is the closure of . In this case, is bounded.

###### Proof.

First we show that, if is implemented by a unitary on , then is bounded. Indeed, if is a sequence of vectors with , then strongly, thus , so

 φ((V(Tξn))=e−12||Tξn||2→1,

with the Fock vacuum state, therefore and is bounded.

If is implemented, then is obviously implementable by the same unitary. Conversely, assume that is implementable by a unitary on . So is bounded. Hence is a bounded, everywhere defined symplectic map. Let and choose a sequence of elements such that . Then

 ϑ¯¯¯T(V(ξ))=V(¯¯¯¯Tξ)=limnV(Tξn)=limnUV(ξn)U∗=UV(ξ)U∗,

so is implemented by .

### 3.2 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 is a complex Hilbert space, denotes the space of real linear, densely defined operators on that are bounded and the closure belongs to the Schatten -ideal with respect to the real part of the scalar product, . If are complex Hilbert spaces, means . If is a standard subspace, means that is a real linear, everywhere defined operator on in the Schatten -ideal with respect to the real part of the scalar product. Similarly, means .

Let now be a factorial standard subspace of the Hilbert space and a real linear operator. As is the linear direct sum of and , we may write as a matrix of operators

 C=[C11C12C21C22] (35)

(the symplectic matrix decomposition). Thus

 C11=PHC|H,  C12=PHC|H′,…

and is an operator , is an operator , etc.

We want to study the Hilbert-Schmidt condition for . Note that

 C∈L2(H)⟺EHCEH∈L2(H),  EHCEH⊥∈L2(H)…

With the polariser and the modular conjugation, the symplectic matrix decomposition of the complex structure is

 i=[D−1D−1√1+D2J−JD−1√1+D2−JD−1J], (36)

as follows from (27) and the uniqueness of the dilation. Note, in particular, the identity

 PH′i|H=−JD−1√1+D2. (37)
###### Lemma 3.2.

The following symplectic matrix representations hold:

 EH=[1√1+D2J00],EH⊥=[0−√1+D2J01],EH′=[00J√1+D21].
###### Proof.

We have

 EHi=[−D000] (38)

because is equal to on and zero on . As , the first equality in the lemma follows by matrix multiplication with (36). The second equality is then simply obtained as

 EH⊥=1−EH=[0−√1+D2J01].

Last equality follows as

 EH′=JEHJ

and the symplectic matrix decomposition of is .

###### Lemma 3.3.

Let be a symplectic map such that , with symplectic matrix decomposition (35). We have

 EHC|H=C11+√1+D2JC21, (39) EHCi|H′=DC12, (40) EH′iC|H=JD