Haag Duality in the Thermal Sector

Haag Dualit y in the Thermal Sector Stefano Galanda 1 ,a , Leonardo Sangaletti 2 ,b 1 Departmen t of Mathematics, Univ ersit y of Y ork - Y ork, United Kingdom. 2 Dipartimen to di Fisica, Universit` a di Geno v a - Via Do decaneso, 33, I-16146 Genov a, Italy . E-mail: a stefano.galanda@y ork.ac.uk, b leonardo.sangaletti@edu.unige.it Abstract. W e pro ve that the net of lo calised von Neumann algebras asso ciated with a real scalar field propagating on Mink o wski spacetime, in the KMS representation, satisfies a generalised version of Haag duality . Our proof com bines ideas from existing arguments for the ground-state representation with purification techniques. 1 In tro duction Haag duality is a cen tral structural prop erty of algebraic quantum field theory (AQFT) as form ulated b y the Haag-Kastler axioms [HK64], expressing a precise correspondence betw een the orthomodular lattices of causally complete spacetime regions and the nets of local algebras. In other terms, it establishes an exact equiv alence betw een spacetime localization and algebraic comm utation relations. In its standard form ulation, Haag duality states that the v on Neumann algebra, in an irreducible represen tation, lo calised in a sufficiently re gular op en region coincides with the commutan t of the algebra asso ciated with its causal complement. This condition ensures that the lo cal algebras are maximal, i.e. no additional observ ables can b e added without v iolating the condition of lo cality . Bey ond its conceptual significance, Haag dualit y plays a crucial technical role in the understanding and pro of of several results in quan tum field theory . Most notably , it is one of the essential prop erties needed to carry the analysis of superse- lection sectors as done b y Doplicher–Haag–Roberts [DHR69a; DHR69b; Dop82; DL83]. Moreov er, it also underpins results on the mo dular structure of lo cal algebras, such as the Bisognano–Wichmann theorem [BW76], thereby linking lo cality with spacetime symmetries. Dualit y for the lo cal net of von Neumann algebras was first rigorously established for free massiv e fields in the v acuum represen tation on Minko wski spacetime. The first proof was giv en by Araki [Ara63; Ara64] who sho wed it for a free real scalar field lo calised in a double cone (causal diamond). Subsequently , alternative pro ofs of the same statement were given by Eck- mann and Osterwalder using T omita–T akesaki mo dular theory for standard subspaces [Ost73; EO73] and by Rieffel [Rie74], see also [GP22] for a review. This result was further extended to bosonic fields with spin by Dell’Antonio in [Del68]. In the analogous case of free fermionic fields, a twisted v ersion of Haag dualit y (b ecause of canonical an ticommutation relations) w as pro v en in the Appendix of [DHR69a]. F urther pro ofs hav e b een given for conformal field theories [BS90; BGL93] and no-go results ha v e b een provided for gauge theories due to the existence of global charges [LR T78]. Finally , these results can b e generalised to globally h yp erbolic spacetimes, in the representation induced by Hadamard states as prov en and pioneered in the construction presented in [V er93; V er97]. Bey ond con tinuum fields, Haag duality in the ground state representation has found important applications in discrete and lattice mo dels of quantum physics, where it serves as a rigorous bridge b etw een AQFT ideas and condensed matter sys- tems. Existing pro ofs treat the case of one-dimensional spin systems [Key+06] and t wo-dimensional lattice models such as the toric co de [Naa12] and more general Kitaev quantum double mo dels for cone-like regions of lo calisation [FN15; OPR25]. These results enabled the classification of any onic sup erselection sectors via a Doplicher–Haag–Roberts–type analysis on the infinite lattice [Oga20; Oga21; Oga22b]. An approximate version of Haag duality [Oga22a] has also b een used to prov e the quan tisation of the Hall conductance for an infinite plane [Bac+25]. Finally , let us mention that Haag dualit y has found an imp ortant application also in the recen t program of formulating quantum information using algebraic metho ds [Lui+25; LSW24]. Ho w ev er, despite its evident relev ance across v arious areas of mathematical physics, a formulation of this dualit y in represen tations other than those induced by pure quasi-free states (such as the ground state) appears, to the b est of our kno wledge, to b e missing. The dualit y relation for reducible representations π β induced b y thermal equilibrium states has recen tly gained some interest [BCS25]. One would exp ect that the usual Haag-Dualit y relation should b e mo dified in a natural wa y to take into accoun t the reducibility of the representation (see [FR19, Def. 37]. The aim of this pap er is to prov e this relation. The paper is organised as follo ws. In Section 2 w e recall the definition of the net of local v on Neumann algebras associated with a free real scalar field in the ground and KMS representations, and we state the main result of our pap er. In Section 3, 1 follo wing the original w orks of Araki and Eckmann–Osterw alder and combining them with the tec hnique of purifying the represen tation induced by a KMS state (see [UMT82] for a review), we pro vide a general pro of of duality in the KMS represen tation. In particular, the reducibility of the original representation gives rise to an additional non-trivial comp onent in the comm utant of a lo cal algebra, b eyond the usual geometric one. A t the one-particle lev el, the real subspaces labeling the lo cal algebras are em b edded into a larger ambien t space via a Bogoljubov transformation. It is therefore necessary to analyze how the relative position of these real subspaces is affected b y the doubling pro cedure used to purify the thermal equilibrium state. Finally , in Section 4 we apply the general results obtained in the previous section to the sp ecific example of a real massive scalar field on Minko wski spacetime. The App endices collect, due to their relev ance for our pro of, the statemen t of the main theorem in [EO73], tw o technical lemmata and a proof of the so called pr e-cyclicity pr op erty of lo cal real subspaces of a scalar field in a thermal representation. 1.1 Notation F or f 1 , f 2 ∈ S ( R 4 , R ) Sch wartz functions, we denote their F ourier and inv erse F ourier transform on Minko wski spacetime M resp ectiv ely b y ˆ f 1 ( p ) : = Z f 1 ( x ) e − ipx d 4 x , ˇ f 2 ( x ) : = 1 (2 π ) 4 Z f 2 ( p ) e ipx d 4 p where in the exp onential px = p µ x µ is the pro duct of the corresp onding four-vectors in the mostly plus conv ention for the metric. F or g 1 , g 2 ∈ S ( R 3 , R ) w e denote the spatial F ourier and spatial inv erse F ourier transform b y F ( g 1 )( p ) = Z g 1 ( x ) e − i px d 3 x , F − 1 ( g 2 )( x ) = 1 (2 π ) 3 Z g 2 ( p ) e i px d 3 p . 2 Setup and results 2.1 Real scalar field The model considered in this pap er is a free scalar field on Mink o wski spacetime. In the C ∗ -algebraic setting, canonical comm utation relations (CCR) are enco ded by the algebraic relations among the generators of the so called Weyl algebr a Definition 1. L et ( D , σ ) b e a r e al symple ctic sp ac e consisting of a r e al line ar ve ctor spac e D and a non-de gener ate symplectic form σ : D × D → R . The Weyl C ∗ -algebr a asso ciate d with the r eal symple ctic sp ac e ( D , σ ) , denote d A ( D , σ ) , is the C ∗ -algebr a gener ate d by symb ols W ( f ) , f ∈ D satisfying the fol lowing r elations for any f , g ∈ D W ( f ) ∗ = W ( − f ) W ( f ) W ( g ) = e − iσ ( f ,g ) 2 W ( f + g ) W (0) = id A ( D ,σ ) and endowe d with a C ∗ -norm (se e [BR79]). It is w ell known that the W eyl C ∗ -algebra exists and is unique up to ∗ -isomorphisms [Sla72]. As w e are interested in studying a free real massive scalar field on Minko wski spacetime M , we consider the bilinear antisymmetric degenerate form on C ∞ c ( M , R ) σ ( f , g ) : = E( f , g ) where E : C ∞ c ( M , R ) × C ∞ c ( M , R ) → R is the unique causal propagator asso ciated with the Klein-Gordon equation with mass m > 0. The pair ( C ∞ c ( M , R ) / ke r(E) , E) defines a symplectic space and w e denote by A ( M ) = A ( C ∞ c ( M , R ) / ke r E , E) the corresp onding W eyl algebra. Sub ordinated to the ab ov e quasi-local definition, w e introduce a lo cal net of C ∗ -algebras [HK64]. This net associates to an y op en, b ounded region of spacetime O ∈ M a C ∗ -algebra A ( O ) via O 7→ A ( O ) = A ( C ∞ c ( O , R ) / k er E , E) , where C ∞ c ( O , R ) denotes the linear space of real v alued test functions whose support is con tained in O . The causal prop erties of E imply that this net is local, i.e [ A ( O 1 ) , A ( O 2 )] = { 0 } if O 1 , O 2 are causally separated. The quasi-lo cal W eyl algebra A ( M ) coincides with C ∗ -inductiv e limit of the net A ( M ) = [ O⊂ M A ( O ) C ∗ , By construction, this net satisfies isotony , i.e. O 1 ⊂ O 2 implies A ( O 1 ) ⊂ A ( O 2 ). 2 A state ov er an abstract C ∗ -algebra, such as the W eyl algebra, is a p ositive, linear and normalised functional ω : A → C . Once a state on the C ∗ -algebra A is chosen, a corresp onding representation of its element as b ounded op erators on a Hilb ert space is obtained via the GNS construction. T o any state ω on A , we asso ciate a triple ( H ω , π ω , Ω ω ) where H ω is a complex Hilb ert space, π ω : A → B ( H ω ) is a ∗ -homomorphism and Ω ω ∈ H ω a cyclic vector that implements the state ω ω ( A ) = (Ω ω , π ω ( A )Ω ω ) H ω , ∀ A ∈ A . This triple is unique up to unitarily equiv alence. If a state ω is quasi-free the corresp onding GNS representation of the W eyl algebra A ( D , σ ) is of F ock type [KW91, Prop. 3.1]. Usually , the free ground state ω ∞ and the free KMS states ω β at inv erse temp erature 0 < β < ∞ defined on the quasi-lo cal W eyl algebra with respect to the free dynamics are assumed to b e quasi-free. W e recall that a dynamics on a C ∗ -algebra A is a one-parameter family of ∗ -automorphism α t , t ∈ R of A . Sp ecifically , the fr e e dynamics on A ( M ) acts on the generators as α t W ( f ) = W ( T t ( f )) ∀ t ∈ R where T t is the one-parameter family of symplectomorphisms of ( C ∞ c ( M / k er E , R ) , E) that implements the time shift isome- tries of M , namely the Killing flow asso ciated with the global timelike killing vector ∂ t T t ( f )( x 0 , x 1 , x 2 , x 3 ) = f ( x 0 − t, x 1 , x 2 , x 3 ) . The GNS representation of A ( M ) induced by ω β can b e obtained b y following [Kay85a; Kay85b] (see also [A W63]). First of all, we recall the definition of gr ound one-p article structur e o ver a symplectic space ( D , σ ) equipp ed with a one-parameter family of linear symplectomorphisms T ( t ) Definition 2 ([Kay85a] Definition 1a ) . A gr ound one-p article structur e ( K ∞ , H ∞ , e − ith ) over ( D , σ, T ( t )) c onsists of a c omplex Hilb ert sp ace H ∞ , a map K ∞ : D → H ∞ and a str ongly c ontinuous unitary gr oup e − ith on H ∞ such that 1. The image of K ∞ is a r e al-line ar subsp ac e of H ∞ and is such that for any f 1 , f 2 ∈ D : 2Im ⟨ K ∞ f 1 , K ∞ f 2 ⟩ H ∞ = σ ( f 1 , f 2 ) . 2. K ∞ D is dense in H ∞ . 3. K ∞ T ( t ) = e − ith K ∞ on D , wher e h is a, possibly unbounde d, strictly p ositive line ar op er ator on H ∞ . 1 The ground one-particle structure is unique up to unitary equiv alence [Kay85a, Thm. 1a.]. It follows that the sym- metric F o ck space constructed on the ground one-particle Hilb ert space H ∞ coincides with the Hilb ert space of the GNS represen tation of A ( D , σ ) induced by the quasi-free ground state ω ∞ H ω ∞ = ∞ M n =0 ( H ∞ ) ⊗ s n . (1) On it, the ground state ω ∞ is implemen ted by the cyclic v acuum vector Ω ∞ ∈ ( H ∞ ) ⊗ s 0 = C . W e now recall the definition of KMS one-p article structur e Definition 3 ([Kay85a] Definition 1b ) . A KMS one-p article structur e ( K β , H β , e − it ˜ h ) over ( D , σ, T ( t )) , c onsists of a c omplex Hilb ert sp ace H β , a map K β : D → H β and a str ongly c ontinuous unitary gr oup e − it ˜ h on H β such that: 1. The image of K β is a r e al-line ar subsp ac e of H β and is such that for any f 1 , f 2 ∈ D 2Im ⟨ K β f 1 , K β f 2 ⟩ H β = σ ( f 1 , f 2 ) . 2. K β D + iK β D is dense in H β . 3. K β T ( t ) = e − it ˜ h K β on D , with ˜ h having trivial kernel and satisfying the one-p article KMS c ondition ⟨ e − it ˜ h K β f 1 , K β f 2 ⟩ H β = ⟨ e − β ˜ h/ 2 K β f 1 , e − it ˜ h e − β ˜ h/ 2 K β f 2 ⟩ H β ∀ t ∈ R , ∀ f 1 , f 2 ∈ D . 1 By strictly p ositive w e mean that its sp ectrum is con tained in (0 , + ∞ ). More generally , an analogous definition can b e giv en requiring h to b e only positive, self-adjoin t and with no zero eigen value; ho wever, for the purp ose of our pro of, the request of strict positivity turns out to be necessary . Therefore, we adopt if from the very beginning. 3 Analogously to the ground state case, the KMS one-particle structure is also unique up to unitary equiv alence [Kay85a, Thm. 1b]. Assume no w that a ground one-particle structure exists and note that the regularity condition K ∞ D ⊂ dom( h − 1 / 2 ) is trivially satisfied since h − 1 is b ounded by assumption. F urther assume that it exists a preferred complex conjugation C on H ∞ suc h that C e − ith = e ith C. Then, the (unique) KMS one-particle structure is obtained by doubling the ground one-particle structure. More in details, we hav e H β = H ∞ ⊕ H ∞ and the action of K β is K β f = C sinh( Z β ) f ⊕ cosh( Z β ) f , f ∈ D , (2) where Z β is defined implicitly on H ∞ b y tanh(Z β ) = e − β h/ 2 . Finally , it holds e − it ˜ h = e ith ⊕ e − ith . Accordingly , the complex Hilb ert space of the GNS representation of A ( D , σ ) induced by the quasi-free state ω β is H ω β = ∞ M n =0 ( H ∞ ⊕ H ∞ ) ⊗ s n , (3) and the v acuum vector Ω β ∈ ( H ∞ ⊕ H ∞ ) ⊗ s 0 = C ⊕ C implements the KMS state ω β on the represented W eyl quasi-lo cal algebra. In addition, the vector state Ω β extends to a pure state on the irreducible C ∗ -algebra B ( H ω β ) and, for this reason, the construction we hav e shortly describ ed is often referred to in the literature as purification. Giv en a representation π ω of the quasi-lo cal W eyl algebra A ( M ) and the corresp onding represented net of C ∗ -algebras, w e construct a net of von Neumann algebras by defining the assignment O 7→ M ω ( O ) : = π ω ( A ( O )) ′′ , where π ω ( A ( O )) ′ denotes the commutan t of π ω ( A ( O )), i.e. the set of op erators in B ( H ω ) that commutes with every element of π ω ( A ( O )). Clearly this net contin ues to satisfy the causal structure, namely M ω ( O 1 ) ⊂ M ω ( O ) ′ if O 1 ⊂ O ′ , where O ′ denotes the interior of the causal complemen t of O . W e also denote by M ω ( M ) = π ω ( A ( M )) ′′ . If the state ω is pure, so that the corresponding GNS representation is irreducible, we hav e π ω ( A ) ′′ = B ( H ω ). This is the case of the v acuum represen tation of the scalar field, for which M ω ∞ ( M ) ≡ M ∞ ( M ) = B ( H ω ∞ ). In this particular mo del, for spacetime regions O ⊂ M corresp onding to causal diamonds, Haag duality holds [Ara64]. Namely M ω ∞ ( O ) ′ ≡ M ∞ ( O ) ′ = M ∞ ( O ′ ) , where M ∞ ( O ′ ) denotes the double comm utant of the quasi-local algebra induced b y the C ∗ -inductiv e limit of the represen ted C ∗ -algebras lo calized in O ′ . 2.2 Main result In the thermal case, instead, the situation is quite differen t. Since the thermal equilibrium state is not pure, the corresp onding GNS representation is reducible. The vector Ω ω β extends the state ω β to a normal state on the v on Neumann algebra M ω β ( M ). If the time ev olution is implemen ted in the represen tation by a strongly con tinuous one-parameter unitary group 2 , the vector state Ω ω β defines a KMS state on M ω β ( M ) ≡ M β ( M ) (see the pro of of [BR81, Cor. 5.3.4.]) and, therefore, it is not only cyclic (b y GNS construction), but also separating for the represented quasi-lo cal von Neumann algebra [BR81, Cor. 5.3.9.]. Consequen tly T omita-T ak esaki mo dular theory for the pair ( M β ( M ) , Ω ω β ) can be constructed and we denote b y J, ∆ the corresp onding mo dular data. The commutan t M β ( M ) ′ is then obtained as M β ( M ) ′ = J M β ( M ) J. Consequen tly , the following relation holds J M β ( M ) J ∨ M β ( O ′ ) =  J M β ( M ) J ∪ M β ( O ′ )  ′′ ⊂ M β ( O ) ′ . Here ∨ denotes the von Neumann algebra generated b y the set-theoretic union, M β ( O ) ≡ M ω β ( O ) and M β ( O ′ ) is defined analogously to the v acuum case. The main goal of this pap er is to identify sufficient conditions that imply the follo wing generalised f orm of Haag duality J M β ( M ) J ∨ M β ( O ′ ) = M β ( O ) ′ , assuming that Haag duality holds in the v acuum representation. Our main theorem is the following 2 Note that this condition is matched if the representation coincides with the one constructed ov er a KMS one-particle structure. 4 Theorem 1. L et O ⊂ M b e a generic op en c ausal diamond 3 on Minkowski sp ac etime M and let O 7→ A ( O ) b e the abstr act net of Weyl C ∗ -algebr as asso ciate d to a fr e e re al massive sc alar field. Consider ω β , for 0 < β < ∞ , the KMS state with r esp e ct to the fr e e dynamics on the quasi-lo c al algebr a A ( M ) and the corr esp ondingly induc e d net of von Neumann algebr as O 7→ M β ( O ) M β ( O ) : = π ω β ( A ( O )) ′′ . Then, gener alise d Haag duality holds in the fol lowing sense: M β ( O ) ′ = M β ( O ′ ) ∨ J M β ( M ) J. 3 General results for the thermal one-particle structure Although the main goal of this pap er is to study the duality of the local net of von Neumann algebras in the KMS rep- resen tation associated with a scalar quan tum field theory on a globally h yp erb olic spacetime manifold, in this section we in v estigate its general prop erties without referring to the sp ecific features of the underlying symplectic space. Our results are mo del-independent in the sense that we analyze only the relative p ositions of the corresp onding one-particle Hilbert space structures and examine how they are affected by a general Bogoliub ov transformation. 3.1 W eyl and Segal formulation of the CCR Since it will b e conv enien t to work within the W eyl formulation of the representation of the CCR algebra, w e shortly recall the standard construction of a real Hilb ert space H starting from a complex Hilb ert space H with complex inner product ⟨· , ·⟩ H (an tilinear in the first en try). The real v ector space H coincides with H as a set, but it is given the structure of a vector space ov er the real field R . The real inner pro duct is ( · , · ) H = Re ⟨· , ·⟩ H and, since obviously ∥ · ∥ H = ∥ · ∥ H , H is complete in the norm topology so that it defines a real Hilbert space. The complex part of the inner product defines a symplectic form σ ( · , · ) H = 2 Im ⟨· , ·⟩ H on H . The real and complex part of the inner product are related by a unique (b y the Riesz representation theorem) canonical complex structure β H : H 7→ H , β H = − β ∗ H (where ∗ denotes here the adjoin t w.r.t. ( · , · ) H ), β 2 H = − 1 H b y 1 2 σ ( v, w ) H = Im ⟨ v , w ⟩ H = − Re ⟨ v , iw ⟩ H = ( v , β H w ) H , ∀ v , w ∈ H . (4) W e finally denote by K a closed real linear subspace of H such that K ⊥ β H K (orthogonal with resp ect to ( · , · ) H ) and H = K ⊕ β H K (this decomp osition alwa ys exists, but is not unique, see [Ara63, F ootnote 13]) 4 . This standard construction applies also to a doubled complex Hilb ert space H ⊕ H with inner pro duct ⟨· , ·⟩ . W e denote b y ( · , · ) the real scalar pro duct on the corresp onding real Hilbert space, whic h clearly coincides with H ⊕ H . The complex structure β H⊕H on H ⊕ H is related to β H b y β H⊕H = β H ⊕ β H . Indeed β H ⊕ β H is ob viously a linear map from H ⊕ H to itself, satisfying ( β H ⊕ β H ) 2 = − 1 , ( β H ⊕ β H ) ∗ = − β H ⊕ β H (with the adjoint defined with resp ect to the inner pro duct ( · , · )) and for every v 1 , v 2 , w 1 , w 2 ∈ H it holds 1 2 σ ( v 1 ⊕ v 2 , w 1 ⊕ w 2 ) : = Im ⟨ v 1 ⊕ v 2 , w 1 ⊕ w 2 ⟩ = Im ⟨ v 1 , w 1 ⟩ H + Im ⟨ v 2 , w 2 ⟩ H = ( v 1 , β H w 1 ) H + ( v 2 , β H w 2 ) H = ( v 1 ⊕ v 2 , β H ⊕ β H ( w 1 ⊕ w 2 )) = ( v 1 ⊕ v 2 , β H⊕H ( w 1 ⊕ w 2 )) . The decomp osition H ⊕ H = ( K ⊕ e K ) ⊕ β H⊕H ( K ⊕ e K ) holds, with ( K ⊕ e K ) ⊥ β H⊕H ( K ⊕ e K ) w.r.t. the real scalar pro duct ( · , · ). F ollowing [Ara63, Sec. 2] we no w make use of the spaces H , H and K to describ e tw o equiv alent form ulations of the F o ck space representation of the canonical commutation relations (CCR). First let us recall that the symmetrised F ock space F s ( H ) is defined as F s ( H ) = ∞ M n =0 H ⊗ s n , 3 Given a generic O 1 ⊂ M the interior of its Cauch y developmen t is a causal diamond. 4 F or A and B linear subspaces of a Hilb ert space C we use the sym b ol A + B to denote the (not necessarily closed) subspace generated by the linear span of A ∪ B . If in addition A ∩ B = { 0 } , we use the notation A ⊕ B . Note that this notation dose not necessarily imply that A ⊥ B ; if A ⊥ B , it holds that A ⊕ B = A ⊕ B . Finally , we also use the notation C ⊕ D to denote the external direct sum of Hilbert spaces and corresp ondingly the notation A ⊕ B for the direct sum of A ⊕ { 0 } ⊂ C ⊕ D and { 0 } ⊕ B ⊂ C ⊕ D . When this notation could be source of confusion, we make use of the symbol ⊕ e to denote the external direct sum. 5 with H ⊗ s 0 = C . On this Hilb ert space we define the usual creation and annihilation op erators a † ( v ) , a ( w ) , v , w ∈ H a † ( v ) v 1 ⊗ s . . . ⊗ s v n = ( n + 1) 1 / 2 v ⊗ s v 1 ⊗ s . . . ⊗ s v n ; a ( w ) v 1 ⊗ s . . . ⊗ s v n = n − 1 / 2 n X i =1 ⟨ w, v i ⟩ H v 1 ⊗ s . . . b v i . . . ⊗ s v n ; a † ( v )Ω = v ; a ( w )Ω = 0 , where Ω ∈ H ⊗ s 0 is the v acuum v ector and only in this case b h i means that v ector is to b e omitted. Note that the annihilation op erator is by definition antilinear in its entry w . The linear extension of these op erators is defined on the dense domain F 0 of v ectors in F s ( H ) having finitely many non v anishing comp onents. On this domain the op erators satisfy the CCR [ a ( w ) , a † ( v )]Ψ = ⟨ w , v ⟩ H Ψ , Ψ ∈ F 0 . (5) The field op erator ϕ ( v ) with dense domain F 0 is defined as ϕ ( v ) = a ( v ) + a † ( v ) . The set F 0 is a dense set of analytic vectors for ϕ ( v ). Therefore, Nelson’s theorem implies that ϕ ( v ) is essentially self-adjoint and w e denote with the same symbol its unique self-adjoint extension. In the Segal formulation of the F ock representation of the CCR we construct the representation starting from the real Hilb ert space H and the complex structure β H asso ciated to H . W e consider the filed op erator ϕ ( v ), but we now identify its argumen t v with an element of H . The commutation relations (5) needs then to b e replaced by [ a ( w ) , a † ( v )]Ψ = ( w , v ) H Ψ + i ( w , β H v ) H Ψ , Ψ ∈ F 0 . The W eyl op erators W ( v ) = e iϕ ( v ) are the unitary operators generated b y the self-adjoin t fields ϕ ( v ) and it follows [BR81, Prop. 5.2.4.] that they satisfy the CCR in the Segal form W ( v ) W ( w ) = W ( v + w ) e − i ( v,β H w ) H = W ( v + w ) e − i 2 σ ( v ,w ) H . As already men tion before, it alwa ys exists a (non unique) rel Hilbert space K ⊂ H suc h that H = K ⊕ β H K . F or an y v 1 , v 2 ∈ K we define the op erators φ ( v 1 ) = a † ( v 1 ) + a ( v 1 ) , π ( v 2 ) = i ( a † ( v 2 ) − a ( v 2 )) and, from now on, w e use the same sym b ol to denote their unique self-adjoint closure. On the subspace F 0 they satisfy the comm utation relations [ φ ( v 1 ) , φ ( v 2 )]Ψ = [ π ( v 1 ) , π ( v 2 )]Ψ = 0 , Ψ ∈ F 0 , [ φ ( v 1 ) , π ( v 2 )]Ψ = 2 i ( v 1 , v 2 ) H Ψ , Ψ ∈ F 0 and consequen tly the unitary op erators U ( v 1 ) : = e iφ ( v 1 ) and V ( v 2 ) = e iπ ( v 2 ) satisfy U ( v 1 ) V ( v 2 ) U ( v 3 ) V ( v 4 ) = U ( v 1 + v 3 ) V ( v 2 + v 4 ) e 2 i ( v 2 ,v 3 ) H , namely the CCR in the W eyl form. Given a generic element v ∈ H , it admits a unique decomp osition v = v 1 + β H v 2 , with v 1 , v 2 ∈ K , and the following equalities relating the field op erators in the Segal and W eyl formulation are easily verified ϕ ( v ) = φ ( v 1 ) + π ( v 2 ) , (6) W ( v ) = U ( v 1 ) V ( v 2 ) e i ( v 1 ,v 2 ) H . (7) Let no w K 1 , K 2 b e tw o linear subspaces of K and H 1 a linear subspace of H . Denoting by R S ( H 1 ) : = { W ( v ); v ∈ H 1 } ′′ ; R F ( K 1 , K 2 ) : = { U ( v 1 ) V ( v 2 ); v 1 ∈ K 1 , v 2 ∈ K 2 } ′′ (8) the Segal and W eyl v on Neumann subalgebra asso ciated to the subspaces, Equation (6) shows that R F ( K 1 , K 2 ) = R S ( K 1 ⊕ β H K 2 ) . (9) Vice versa, it can b e prov en [Ara63, Thm. 3.] that for any H 1 ⊂ H there exists subspaces K , K 1 , K 2 of H such that H = K ⊕ β H K , H 1 = K 1 ⊕ β H K 2 with K 1 , K 2 ⊂ K . Therefore, w e can alw ays decide if to w ork with the W eyl or with Segal form ulation of these subalgebras dep ending on which of the tw o formulations is more conv enient. Finally , Note that the f ollo wing equalities hold R S ( H 1 ) = R S ( H 1 ) , R F ( K 1 , K 2 ) = R F ( K 1 , K 2 ) , (10) where the ov erline denotes the closure in the norm top ology of H . In order to prov e the previous equalities first observe that the op erator W ( w ) is strongly contin uous in w with resp ect to the norm top ology of H , and the op erators U ( v 1 ) , V ( v 2 ) are strongly contin uous in v 1 , v 2 with resp ect to the norm topology of K . This standard result follows from the fact that the self-adjoin t generators ϕ ( v ) , φ ( v 1 ) , π ( v 2 ) are strongly con tin uous in their argumen t on the core F 0 (see for instance [BR81, Prop. 5.2.4., (4)]). Since b y the bicomm utan t theorem a von Neumann algebra is closed in the strong top ology , this observ ation sho ws that the generators of the von Neumann algebras on the left and on the right hand side of Equations (10) coincide. 6 3.2 Results on one-particle Hilb ert spaces and dualit y Let H b e a complex Hilb ert space, h a strictly p ositive (p ossibly unbounded) op erator on H and C a complex conjugation on H ; assume that C and h comm ute, in the sense that C e ith = e − ith C. As mentioned in Section 2.1, if H describ es a ground one-particle Hilb ert space, then under the previous assumptions the KMS one-particle Hilb ert space coincides with H ⊕ H . Since h is strictly p ositive, the op erator e − β h , β ∈ [0 , + ∞ ) is b ounded and self-adjoint. As explained in the previous section, we denote by H the restriction of H to the field R and by β H the induced complex structure on H . Let K ∈ H b e a (non-unique) closed subspace suc h that H = K ⊕ β H K . W e assume that K is invariant under the action of e − β h and C, identified as b ounded self-adjoint op erators on H , i.e. e − β h P K = P K e − β h P K , C P K = P K C P K , (11) where P K denotes the unique orthogonal pro jector on K . This implies that [ e − β h , P K ] = 0 and, b y the sp ectral theorem, [sinh(Z β ) , P K ] = [cosh(Z β ) , P K ] = 0 with cosh(Z β ) = 1 √ 1 − e − β h , sinh(Z β ) = 1 √ e β h − 1 = e − β h 2 √ 1 − e − β h . In particular, the closed subspace K is inv arian t under the action of the op erators sinh(Z β ) , cosh(Z β ). In this setting, w e consider t wo real subspaces K 1 , K 2 ⊂ K and their image in the doubled real Hilbert space H ⊕ H under the action of a b ounded Bogoljub ov transform. First of all, we prov e the following prop osition concerning their orthogonal in K ⊕ K Prop osition 1. L et K i ⊂ K b e a r e al subsp ac e and K ⊥ i ∈ K b e its ortho gonal in K . L et U ( K 1 ) : = n u 1 ⊕ u 2 ∈ K ⊕ K : u 1 = C sinh(Z β ) u, u 2 = cosh(Z β ) u, u ∈ K 1 o . V ( K 2 ) : = n u 1 ⊕ u 2 ∈ K ⊕ K : u 1 = − C sinh(Z β ) u, u 2 = cosh(Z β ) u, u ∈ K 2 o . ˜ V : = n v 1 ⊕ v 2 ∈ K ⊕ K : v 1 = cosh(Z β ) v , v 2 = − C sinh(Z β ) v , v ∈ K o . ˜ U : = n v 1 ⊕ v 2 ∈ K ⊕ K : v 1 = cosh(Z β ) v , v 2 = C sinh(Z β ) v , v ∈ K o . Then U ( K 1 ) ⊥ = V ( K ⊥ 1 ) ⊕ ˜ V , V ( K 2 ) ⊥ = U ( K ⊥ 2 ) ⊕ ˜ U , wher e U ( K 1 ) ⊥ , V ( K 2 ) ⊥ denote the ortho gonal subsp ac es in K ⊕ K . Pr o of. Let us define the real-linear op erator A A : =      1 √ 1 − e − β h − C e − β h 2 √ 1 − e − β h − C e − β h 2 √ 1 − e − β h 1 √ 1 − e − β h      , on the Hilb ert space H ⊕ H . This op erator is b ounded, with ∥ A ∥ op ≤ 2 sup i,j ∈ [1 , 2] ∥ A ij ∥ op = 2 1 √ 1 − e − β m , m = min ρ ( h ) , with ρ ( h ) denoting the sp ectrum of the op erator h . The op erator A is inv ertible with b ounded inv erse A − 1 : =      1 √ 1 − e − β h C e − β h 2 √ 1 − e − β h C e − β h 2 √ 1 − e − β h 1 √ 1 − e − β h ,      , ∥ A − 1 ∥ op ≤ 1 √ 1 − e − β m . F rom now on we identify A, A − 1 with the real-linear op erators that they induce on H ⊕ H . On H ⊕ H the op erators A, A − 1 are self-adjoin t. In addition, since K is an in v ariant subspace of e − β h and C (also iden tified with the real-linear operators induced on H ), it follows by the spectral theorem that K ⊕ K is an inv arian t subspace of A, A − 1 . W e denote by A K , A − 1 K the restriction of A, A − 1 to the Hilb ert space K ⊕ K . Recalling that C e iβ h = e − iβ h C A K U ( K 1 ) = { 0 } ⊕ e K 1 . 7 Its orthogonal in K ⊕ K is ( A K U ( K 1 )) ⊥ = K ⊕ e K ⊥ 1 . W e now apply Lemma 3 to the op erator A K and conclude that U ( K 1 ) ⊥ = A K ( A K U ( K 1 )) ⊥ = V ( K ⊥ 1 ) + ˜ V = V ( K ⊥ 1 ) ⊕ ˜ V , where the last equalit y follows since A is injectiv e. The second equalit y is pro ven in the same w ay , substituting A K with A − 1 K . The pro of of the duality prop erty of a net of W eyl algebras is simplified if the closed real subspaces K 1 , K 2 that lab el the algebra are in generic p osition [EO73]. By this we mean that the intersection of any pair of the real closed subspaces (of K ) K 1 , K 2 , K ⊥ 1 , K ⊥ 2 are trivial, i.e. equal to { 0 } . This condition is generally matched when considering lo cal von Neumann algebras asso ciated to a b ounded op en region O of Mink owski spacetime in the v acuum representation of a free massive real scalar field. How ever, in the present more generic situation the relative p osition b etw een the corresp onding real subspaces is not necessarily preserved. More in details we hav e Prop osition 2. L et K 1 , K 2 ⊂ K b e close d r eal subsp ac es of K . Then U ( K 1 ) , V ( K 2 ) are close d r e al subsp ac es of K ⊕ K and • U ( K 1 ) ∧ V ( K 2 ) = { 0 } ⊕ { 0 } (even if K 1 ∧ K 2 ⊂ { 0 } ). • U ( K 1 ) ∧ V ( K 2 ) ⊥ = { 0 } ⊕ { 0 } iff K 1 ∧ K ⊥ 2 = { 0 } and U ( K 1 ) ⊥ ∧ V ( K 2 ) = { 0 } ⊕ { 0 } iff K ⊥ 1 ∧ K 2 = { 0 } . • K 1 , K 2 in generic p osition  = ⇒ U ( K 1 ) ⊥ ∧ V ( K 2 ) ⊥ = { 0 } ⊕ { 0 } . Pr o of. The subspaces U ( K 1 ) , V ( K 2 ) coincide with A − 1 K ( { 0 } ⊕ K 1 ) , A K ( { 0 } ⊕ K 2 )) and are closed since b oth A K and A − 1 K are b ounded op erators. W e prov e the first statement. The op erators cosh(Z β ) , sinh(Z β ) are obviously injective, so that cosh(Z β ) u = cosh(Z β ) v , u, v ∈ K = ⇒ u = v ; C sinh(Z β ) u = − C sinh(Z β ) v , u, v ∈ K = ⇒ u = − v . By definition of U ( K 1 ) , V ( K 2 ) it follows that they hav e trivial intersection. W e prov e the second statement. Applying the op erator A K on U ( K 1 ) ∧ V ( K 2 ) ⊥ w e obtain A K  U ( K 1 ) ∧ V ( K 2 ) ⊥  = ( { 0 } ⊕ K 1 ) ∧ ( K ⊕ K ⊥ 2 ) = ( { 0 } , K 1 ∧ K ⊥ 2 ) , where the first equalit y follows since A K is injectiv e. Once again, since A K is injectiv e this pro v es the first part of the statemen t. The second part is prov en in the same wa y using the op erator A − 1 K . In order to prov e the final statement we show that ( U ( K 1 ) ⊥ ∧ V ( K 2 ) ⊥ ) ⊥ is not necessarily dense in K ⊕ K K ⊕ K ⊆ U ( K 1 ) + V ( K 2 ) . It is enough to show it in a particular case, so let us assume h to b e further a b ounded op erator. By definition U ( K 1 ) + V ( K 2 ) = n u 1 ⊕ u 2 ∈ K ⊕ K : u 1 = C sinh(Z β )( u − v ) , u 2 = cosh(Z β )( u + v ) , u ∈ K 1 , v ∈ K 2 o Since the subspaces K 1 , K 2 are in generic p osition, there are vectors w ∈ K such that w ∈ K 1 . Consider a vector ψ ∈ K ⊕ K of the form ψ = w 1 ⊕ w 2 ∈ K ⊕ K , w 1 = C sinh(Z β ) w, w 2 = cosh(Z β ) w, w ∈ K 1 . Supp ose that ψ ∈ U ( K 1 ) + V ( K 2 ). It follows that there exist tw o sequences { u i } i ∈ N , { v i } i ∈ N suc h that u i ∈ K 1 , v i ∈ K 2 and C sinh(Z β )( u i − v i ) − − − → i →∞ C sin(Z β ) w, cosh(Z β )( u i + v i ) − − − → i →∞ cosh(Z β ) w. Since h is b ounded, b oth cosh(Z β ) and sinh(Z β ) are b ounded inv ertible op erators. Therefore, both sequences of vectors u i + v i and u i − v i are conv ergen t and so lim i →∞ u i = 1 2 lim i →∞ ( u i + v i + u i − v i ) = 1 2 lim i →∞ ( u i + v i ) + 1 2 lim i →∞ ( u i − v i ) = w. Since w ∈ K 1 , this contradiction prov es that ψ ∈ U ( K 1 ) + V ( K 2 ) and therefore the claim. Remark 1. Note that in the explicit model considered in 4 it is well kno wn that the Reeh-Schlieder prop erty applies [Jak00; SVW02]. Therefore, it follows that the so called pre-cyclicity property is also satisfied [FG89, Thm. 1.2] (see also Theorem C in the App endix), namely U ( K 1 ) + β H⊕H V ( K 2 ) + i ( U ( K 1 ) + β H⊕H V ( K 2 )) H ⊕ H = H ⊕ H , and consequen tly U ( K 1 ) + V ( K 2 ) = K ⊕ K . Here U ( K 1 ) , V ( K 2 ) are the real subspaces that lab el the von Neumann algebra in the thermal representation asso ciated with a op en bounded region of Minko wski. F or consistency with the proof of Prop osition 2 we observe that in the examples the op erator h is not b ounded. 8 F ollowing the discussion in Section 4 and Section 5 of [Ara63], even if the relev ant subspaces are not guaranteed to b e in generic p ositions, for the pro of of duality w e can alwa ys reduce to the case in which this condition is satisfied. In particular, one can prov e that the represented von Neumann algebras admit tensor pro duct decomposition, where the only factor with a non-trivial commutan t consists of (see [Ara63, Lemma 5 . 2] and the discussion after it) R F ( U ( K 1 ) ′ , V ( K 2 ) ′ \U V ) , i.e. the W eyl algebra in W eyl form lab eled by the subspaces U ( K 1 ) ′ , V ( K 2 ) ′ ⊂ U V ⊂ K ⊕ K understoo d as subspaces relative to U V . In the present case, these subspaces are defined as U V : = U ( K 1 ) ∨ V ( K 2 ) : = U ( K 1 ) + V ( K 2 ) U ( K 1 ) ′ : = U ( K 1 ) ∧ U V = U ( K 1 ) V ( K 2 ) ′ : = V ( K 2 ) ∧ U V = V ( K 2 ) . It is easy to chec k that relative to U V the closed subspaces U ( K 1 ) ′ , V ( K 2 ) ′ are in generic p ositions Prop osition 3. The subsp ac es U ( K 1 ) ′ , V ( K 2 ) ′ , ( U ( K 1 ) ′ ) ⊥ ∧ U V , ( V ( K 2 ) ′ ) ⊥ ∧ U V define d ab ove have al l trivial interse ctions. Pr o of. F ollowing the ab ov e definition we hav e ( U ( K 1 ) ′ ) ⊥ ∧ U V : = U ( K 1 ) ⊥ ∧ ( U ( K 1 ) ∨ V ( K 2 )) ( V ( K 2 ) ′ ) ⊥ ∧ U V : = V ( K 2 ) ⊥ ∧ ( U ( K 1 ) ∨ V ( K 2 )) . By Prop osition 2 we know that U ( K 1 ) ′ ∧ V ( K 2 ) ′ = { 0 } ⊕ { 0 } . Using Prop osition 2 and the distributivity of ∧ we also get U ( K 1 ) ′ ∧ (( V ( K 2 ) ′ ) ⊥ ∧ U V ) = ( U ( K 1 ) ∧ V ( K 2 ) ⊥ ) ∧ ( U ( K 1 ) ∨ V ( K 2 )) = { 0 } ⊕ { 0 } and analogously V ( K 2 ) ′ ∧ (( U ( K 1 ) ′ ) ⊥ ∧ U V ) = { 0 } ⊕ { 0 } . Finally , we chec k that  ( U ( K 1 ) ′ ) ⊥ ∧ U V  ∧  ( V ( K 2 ) ′ ) ⊥ ∧ U V  =  U ( K 1 ) ⊥ ∧ V ( K 2 ) ⊥  ∧ ( U ( K 1 ) ∨ V ( K 2 )) = ( U ( K 1 ) ∨ V ( K 2 )) ⊥ ∧ ( U ( K 1 ) ∨ V ( K 2 )) = { 0 } ⊕ { 0 } . Then, duality is prov en by first applying [EO73, Theorem 2] (rep orted here as Theorem 4) R F  U ( K 1 ) ′ , V ( K 2 ) ′ \U V  ′ = R F  ( V ( K 2 ) ′ ) ⊥ ∧ U V , ( U ( K 1 ) ′ ) ⊥ ∧ U V \U V  . (12) and then using the results of [Ara63, Section 5] Theorem 2. The fol lowing e quality holds R F ( U ( K 1 ) , V ( K 2 )) ′ = R F ( U ( K ⊥ 2 ) , V ( K ⊥ 1 )) ∨ R F ( ˜ U , ˜ V ) . (13) Pr o of. By [Ara63, Section 5] Equation (12) implies that R F ( U ( K 1 ) , V ( K 2 )) ′ = R F  V ( K 2 ) ⊥ , U ( K 1 ) ⊥  . Using Prop osition 1, we obtain R F ( U ( K 1 ) , V ( K 2 )) ′ = R F ( U ( K ⊥ 2 ) ⊕ ˜ U , V ( K ⊥ 1 ) ⊕ ˜ V ) . Finally , the real linearity of the fields φ, π (see also [Ara63, Theorem 1’]) implies the statement of the Theorem. Finally , we identify the algebra R ( ˜ U , ˜ V ) with the commutan t R ( U ( K ) , V ( K )) ′ Prop osition 4. The fol lowing e quality holds R F ( U ( K 1 ) , V ( K 2 )) ′ = R F ( U ( K ⊥ 2 ) , V ( K ⊥ 1 )) ∨ R F ( U ( K ) , V ( K )) ′ . (14) 9 Pr o of. By Equation (9) we consider the von Neumann algebra R S ( U ( K ) ⊕ β H⊕H V ( K )) = R F ( U ( K ) , V ( K )). The real linear closed subspace U ( K ) ⊕ β H⊕H V ( K ) ⊂ H ⊕ H is a standard subspace, namely U ( K ) ⊕ β H⊕H V ( K ) + i ( U ( K ) ⊕ β H⊕H V ( K )) H ⊕ H = H ⊕ H , U ( K ) ⊕ β H⊕H V ( K ) ∩ i ( U ( K ) ⊕ β H⊕H V ( K )) = { 0 } ⊕ { 0 } , as prov en in [Kay85a; Kay85b] (see in particular [Kay85a, App. 2] and [Kay85b, Thm. 2], recalling that K ⊕ β H K = H ). W e define the closed, antilinear op erator s ov er H ⊕ H by the following action on its core s : U ( K ) ⊕ β H⊕H V ( K ) + i ( U ( K ) ⊕ β H⊕H V ( K )) → U ( K ) ⊕ β H⊕H V ( K ) + i ( U ( K ) ⊕ β H⊕H V ( K )) , h + ik 7→ h − ik. By the uniqueness of its p olar decomp osition s = j δ 1 / 2 , it follows that 5 δ 1 / 2 = e β h 2 0 0 e − β h 2 ! , j =  0 C C 0  . Since the modular data J, ∆ asso ciated with the W eyl algebra R S ( U ( K ) ⊕ β H⊕H V ( K )) are the second quan tisation of j, δ (see [Ost73]), T omita-T akesaki theorem implies that R S ( U ( K ) ⊕ β H⊕H V ( K )) ′ = J R S ( U ( K ) ⊕ β H⊕H V ( K )) J = R S ( j ( U ( K ) ⊕ β H⊕H V ( K ))) . (15) T o conclude, we now only hav e to note that j ( U ( K ) ⊕ β H⊕H V ( K )) = ˜ U ⊕ β H⊕H ˜ V , since C( K ⊕ β H K ) = K ⊕ β H K . 4 Scalar field on Mink o wski spacetime in the KMS representation In this section we prov e that Haag dualit y holds for the von Neumann algebras of a real massiv e scalar field on Minko wski spacetime in a KMS representation. As op en b ounded regions of spacetime we alwa ys consider causal diamonds. 4.1 F ock-KMS representation of lo cal net W orking on Mink owski spacetime w e make use of F ourier integrals to get explicit expressions for the subspaces introduced in the previous sections. As already mentioned, the GNS representation induced by a quasi-free state coincides with the F ock representation constructed on the corresp onding one-particle Hilb ert space. On Minko wski spacetime, the one-particle ground space is isomorphic to the space of square-integrable functions on the p ositive mass-hyperb oloid H m =  p ∈ R 4 | p ν p ν = − m 2 , p 0 > 0  with resp ect to the Lorentz inv arian t measure µ L H ∞ = L 2 ( H m , d µ L ) , d µ L ( p ) = d 3 p (2 π ) 3 2 ω p , where ω p = p ∥ p ∥ 2 + m 2 and w e fixed the mostly plus conv ention on the signature of the metric. Then, the map K ∞ : C ∞ c ( M , R ) / k er(E) → H ∞ , introduced in Definition 2 and whose image is a dense subspace in H ∞ , acts explicitly as follow K ∞ ( f ) = ˆ f ( p 0 , p )    H m . F ollowing the discussion of Section 3.1, w e turn the complex Hilbert space H ∞ = L 2 ( H m , d µ L ) into a real Hilb ert space H ∞ = L 2 ( H m , d µ L ; R ) ⊕ L 2 ( H m , d µ L ; R ) by iden tifying every complex v alued function f ∈ L 2 ( H m , d µ L ) with its real and imaginary parts wit resp ect to the canonical conjugation operator. On H ∞ , the complex structure β H ∞ that implements the m ultiplication by i and satisfies the relation (4) coincides with the op erator β H ∞ =  0 − 1 1 0  . H ∞ can b e decomp osed as H ∞ = K ∞ ⊕ β H ∞ K ∞ where K ∞ = L 2 s ( H m , d µ L ; R ) ⊕ L 2 a ( H m , d µ L ; R ) and L 2 s / a ( H m , d µ L ; R ) denotes the closed subspace of symmetric/antisymmetric square integrable functions; the tw o subspaces are orthogonal with 5 Note that different conv ention in the definition of s and consequently in the ov erall sign in the operator j , if compared with [Ka y85b], is consistent with the different conv ention we adopted in the definition of the field op erators compared with [Kay85a, Sec. 2.3.]. 10 resp ect the real scalar product because of the symmetry of the co efficient ω − 1 p app earing in the differential of the measure. This decomp osition corresp onds to the decomp osition of the complex Hilb ert space L 2 ( H m , d µ L ) into the direct sum of the subspaces of complex functions that satisfy the relation 6 ¯ ˆ f ( p ) = ˆ f ( − p ) and the relation ¯ ˆ f ( p ) = − ˆ f ( − p ). A unitarily equiv alent wa y to characterize the real Hilb ert space H ∞ [KW91, Prop. 3.1], [Ara64][Sec. 2] is via H ∞ pos , that is the completion of the real vector space C ∞ c ( M , R ) / ke r(E) in the norm induced by the real inner product defined by the symmetric part µ ∞ of the tw o p oint function ω ∞ 2 of the ground state [Ara64, Equation (2 . 6) and (2 . 5)]. On this real Hilb ert space we define a complex structure. It acts on the dense subspace C ∞ c ( M , R ) / k er(E) as the op erator that implements the m ultiplication by i on the F ourier transform ev aluated on the p ositive mass-h yp erb oloid (and therefore by − i on the negative one [Ara64, F o otnote 6]) \ β K ∞ pos f    H m = i ˆ f    H m , f ∈ H ∞ pos . Note that this op erator is a well defined linear map from H ∞ pos to itself and satisfies all the prop erties of a complex structure. Using this complex structure w e turn H ∞ pos in to a complex Hilb ert space H ∞ pos with complex inner pro duct implemen ted b y the ground state tw o point function ω ∞ 2 = µ ∞ + i 2 σ . The isomorphism b etw een the complex Hilb ert spaces H ∞ and H ∞ pos is then implemented on the dense subspace C ∞ c ( M , R ) / k er(E) b y the map (4.1) and we use the same symbol to denote its extension by con tin uit y to H ∞ . The real Hilb ert space H ∞ pos can b e decomp osed in the direct sum of the real subspaces of functions symmetric and an tisymmetric in the time v ariable. Denoting by K ∞ pos the subspace generated b y symmetric test functions in the time v ariable, the space generated b y antisymmetric test functions coincides with β H ∞ pos K ∞ pos so that we can write H ∞ pos = K ∞ pos ⊕ β H ∞ pos K ∞ pos ; the relation K ∞ pos ⊥ β H ∞ pos K ∞ pos directly follo ws from the symmetry of µ ∞ .This decomposition corresponds, via the isomorphism (4.1), to the decomp osition of H ∞ = K ∞ ⊕ β H ∞ K ∞ . Indeed, for an y real v alued function f + ( x ) symmetric in the time v ariable x 0 w e ha v e ¯ ˆ f + ( p 0 , p )    H m = Z f + ( − x 0 , x ) e ix 0 ω p + i x · p d 4 x = ¯ ˆ f + ( p 0 , − p )    H m . Finally , let us define the notation for the tw o following subspaces of K ∞ pos and H ∞ pos K ∞ pos ( O ) : = { f ∈ C ∞ c ( O ) , f ( x 0 , x ) = f ( − x 0 , x ) } H ∞ pos , ( β H ∞ pos K ∞ pos )( O ) : = { f ∈ C ∞ c ( O ) , f ( x 0 , x ) = − f ( − x 0 , x ) } H ∞ pos , H ∞ pos ( O ) : = { f ∈ C ∞ c ( O ) } H ∞ pos . Note that the decomp osition H ∞ pos = K ∞ pos ⊕ β H ∞ pos K ∞ pos is compatible with any choice of localized real subspace H ∞ pos ( O ), since it is alwa ys possible to find subspaces K 1 , K 2 of K ∞ pos suc h that H ∞ pos ( O ) = K 1 ⊕ β H ∞ pos K 2 , namely K 1 = K ∞ pos ( O ) , β ∞ H ∞ pos K 2 = ( β H ∞ pos K ∞ pos )( O ). Relying on these definitions we now iden tify the lo cal von Neumann algebras in the thermal representation of the scalar field, in terms of the corresp onding real subspaces as describ ed in the previous section. F rom now on, the one-particle Hamiltonian h of the ground state is iden tified with the m ultiplication operator ω p acting on H ∞ and C with the usual complex conjugation for complex v alued functions. The same symbol is also used to denote the real linear op erators that they naturally induce on H ∞ . Lemma 1. Let O ⊂ M be a open subset of Minkowski sp acetime and consider the associate d abstr act Weyl C ∗ -algebr a A ( O ) . Then, the c orresp onding lo c al von Neumann algebr a in the GNS r epr esentation induc e d by a quasi-fr e e KMS state ω β at inverse temp er atur e β ∈ (0 , + ∞ ) , is given by M β ( O ) = R F ( U O , V O ) = { U ( u ) , V ( v ); u ∈ U O , v ∈ V O } ′′ , wher e the subsp ac es U O , V O ar e subsp ac es of K ∞ ⊕ K ∞ define d by U O : = n u 1 ⊕ u 2 ∈ K ∞ ⊕ K ∞ : u 1 = C sinh(Z β ) K ∞ f , u 2 = cosh(Z β ) K ∞ f , for f ∈ K ∞ pos ( O ) o , V O : = n v 1 ⊕ v 2 ∈ K ∞ ⊕ K ∞ : v 1 = − β H ∞ C sinh(Z β ) K ∞ g , v 2 = − β H ∞ cosh(Z β ) K ∞ g , for g ∈ ( β H ∞ pos K ∞ pos )( O ) o . The algebr a acts on the Hilb ert sp ac e H ω β given in Equation 3. 6 Note that this unique decomposition is explicitly given by ˆ f = (Re ˆ f s + i Im ˆ f a ) + (Re ˆ f a + i Im ˆ f s ). 11 Pr o of. W e start by sho wing that U O , V O are subspaces of K ∞ ⊕ K ∞ . The map K ∞ maps elemen t of K ∞ pos (resp ectiv ely β H ∞ pos K ∞ pos ) in to elemen ts of K ∞ (resp ectiv ely β H ∞ K ∞ ). The multiplicativ e op erators defined on H ∞ sinh(Z β ) = 1 √ e β ω p − 1 , cosh(Z β ) = 1 √ 1 − e − β ω p . are bounded and, since ω p = ω -p , the real linear op erator that they induce on H ∞ lea v es the subspaces K ∞ , β H ∞ K ∞ in v ariant. Clearly the same holds true for the real linear operator induced by C on H ∞ . Since the complex structure β H ∞ an ti-comm utes with the conjugate linear op erator C and commutes with the linear operator ω p , the inclusion of U O , V O is v erified. In order to construct the algebra M β ( O ) we observe that, since C e iω p = e − iω p C, the F ock representation induced b y the KMS state is consistently obtained making use of the doubling pro cedure describ ed in Section 2.1. The F ock space of the representation coincides with H ω β as given in Equation 3. Then, making use of Equation 2, we obtain the real subspace H β O that lab els the represented algebra in the Segal formulation H β O = n u 1 ⊕ u 2 ∈ H ∞ ⊕ H ∞ : u 1 = C sinh(Z β ) K ∞ f , u 2 = cosh(Z β ) K ∞ f , for f ∈ H ∞ pos ( O ) o . Finally , using Equations (9), (8) w e identify the corresp onding subspaces K ∞ ⊕ K ∞ in the W eyl formulation, concluding the pro of. Remark 2. The real subspaces U O and V O pla y here the role of the real subspaces U ( K 1 ) and V ( K 2 ) in tro duced in Prop osition 1 asso ciated resp ectively with the real subspaces K 1 = K ∞ K ∞ pos ( O ) and K 2 = β H ∞ K ∞ ( β H ∞ pos K ∞ pos )( O ) of the ground state representation. 4.2 Haag dualit y for causal diamonds No w that the explicit form of the lo cal von Neumann algebras M β ( O ) has b een determined, it is the purp ose of this section to pro v e Theorem 1. As it is w ell kno wn [Ara64], for the proof of Haag-Duality certain regularit y prop erties of the bounded open region O ⊂ M pla y an imp ortant role. Therefore, in order to av oid geometrical and top ological complications (see the discussion in [Ara64, Sec. 7]), w e alw ays assume O ⊂ M to b e an op en causal diamond. By definition, O is a globally hyperb olic subset of M , namely it has an em b edded Cauch y hypersurface B . Moreo ver, being also M globally h yp erbolic, this em b edded hypersurface can alwa ys be extended to a Cauch y h yp ersurface of the full spacetime Σ ⊂ M . The Cauch y dev elopmen t of B = Σ ∩ O in M , denoted by C ( B ), coincides with the causal diamond O , namely C ( B ) = O . Note that the equality C ( B ′ ) = O holds for an y other choice B ′ = O ∩ Σ ′ , with B ′ b eing a Cauch y hypersurface of O . F rom now on w e alw ays assume O = C ( B ) ⊂ M to b e a causal diamond. In order to apply the results of Section 3 and pro v e our main theorem, we need to identify the orthogonals of the real subspaces K ∞ pos ( O ) and ( β H ∞ pos K ∞ pos )( O ). This is a known result [Ara64; GP22] of which we rep ort a streamlined pro of for completeness. T o achiev e it, we make use of the globally hyperb olicity of causal diamonds. This allows to establish an isomorphism b etw een K ∞ pos ( O ) and the spaces of initial data on the Cauch y surface B ⊂ O . Let us define the tw o following Hilb ert spaces Definition 4. We define the re al Hilb ert sp ac es of initial c onditions as F φ : = ( S ( R 3 , R ) , ⟨· , ·⟩ φ ) F π : = ( S ( R 3 , R ) , ⟨· , ·⟩ π ) , wher e the c ompletion is taken in the top olo gy induc e d by the c orresp onding inner pr o ducts that, for f , g ∈ S ( R 3 , R ) , ar e define d by ⟨ f , g ⟩ φ : = ⟨ ω − 1 2 f , ω − 1 2 g ⟩ , ⟨ f , g ⟩ π : = ⟨ ω 1 2 f , ω 1 2 g ⟩ . Her e, on the right hand side, the inner pr o duct is the standar d inner pr o duct on L 2 ( R 3 , R ) , amende d by the pr esenc e of multiplic ative op er ators ω α : S ( R 3 , R ) → S ( R 3 , R ) , for α ∈ Q , define d as ω α f : = F − 1  ω α p ˆ f | H m ( p )  . F rom the given definitions of the inner pro ducts the following inclusions F π ⊂ L 2 ( R 3 , R ) ⊂ F φ , 12 follo w. They are implemented by the contin uous maps j 1 : F π → L 2 ( R 3 , R ) , j 2 : L 2 ( R 3 , R ) → F φ . where contin uit y follo ws from the estimates ∥ j 1 f ∥ L 2 ≤ m − 1 2 ∥ f ∥ F π , ∥ j 2 f ∥ F φ ≤ m − 1 2 ∥ f ∥ L 2 . W e recall that the real Hilb ert spaces of initial conditions are isomorphic resp ectively to K ∞ pos and β H ∞ pos K ∞ pos (see Lemma 4), the isomorphisms b eing implemented by the maps δ 0 : K ∞ pos → F φ , δ 1 : β H ∞ pos K ∞ pos → F π . Their explicit action on f ∈ S ( R 4 , R ) ∩ K ∞ pos and g ∈ S ( R 4 , R ) ∩ β H ∞ pos K ∞ pos is giv en b y δ 0 f : = F − 1  ˆ f | H m ( p )  , δ 1 g : = F − 1  ( iω p ) − 1 ˆ g | H m ( p )  (16) and then extended by con tinuit y to K ∞ pos and β H ∞ pos K ∞ pos . If h ∈ C ∞ c ( M , R ) such that supp( h ) ⊂ O , O = C ( B ), then the supp ort prop erties of the causal propagator (see Lemma 4) imply supp  δ 0  K ∞ pos ( O )  ⊂ B , supp  δ 1  ( β H ∞ pos K ∞ pos )( O )  ⊂ B . Therefore we can characterise the real subspaces in terms of initial conditions on a Cauch y surface. W e define the following subspaces of F φ Definition 5 ([Ara64, Eqs. (5 . 1) , (5 . 2)]) . L et O = C ( B ) b e a c ausal diamond. Then, its asso ciate d sp ac es of initial c onditions ar e F R ( B ) : = j 2 L 2 ( B , R ) ∥·∥ φ F I ( B ) : = β π,φ j − 1 1  L 2 ( B , R ) ∩ F π  wher e F R ( B ) , F I ( B ) ⊂ F φ and the c ompletion is with r esp e ct to the top olo gy induc e d by the sc alar pr o duct on F φ . More over, we have intr o duc ed the fol lowing op er ator β π,φ : = δ 0 ◦ β H ∞ pos ◦ δ − 1 1 : L 2 ( R 3 , R ) → F φ . The action of the isomorphism 16 together with de definition of the real subspaces U O and V O in Lemma 1 imply that U O =  f 1 ⊕ f 2 ∈ K ∞ ⊕ K ∞ : f 1 = F ( f )( − p ) √ e β ω p − 1 , f 2 = F ( f )( p ) √ 1 − e − β ω p , f ∈ F R ( B )  , (17) V O =  f 1 ⊕ f 2 ∈ K ∞ ⊕ K ∞ : f 1 = −F ( f )( − p ) √ e β ω p − 1 , f 2 = F ( f )( p ) √ 1 − e − β ω p , f ∈ F I ( B )  . (18) T o determine the orthogonals U ⊥ O , V ⊥ O in K ∞ ⊕ K ∞ w e use the following lemma Lemma 2 ([Ara64, Lem. 2], [GP22, Thm. 4 . 1]) . The fol lowing e qualities hold F R ( B ) ⊥ = F I ( B c ) , F I ( B ) ⊥ = F R ( B c ) , wher e ⊥ denotes the ortho gonal in F φ . F rom this it follows Prop osition 5. Given the subsp ac es U O and V O , their ortho gonal in K ∞ ⊕ K ∞ ar e U ⊥ O = V O ′ ⊕ ˜ V M V ⊥ O = U O ′ ⊕ ˜ U M wher e ˜ U M : = n u 1 ⊕ u 2 ∈ K ∞ ⊕ K ∞ : u 1 = cosh(Z β ) K ∞ f , u 2 = C sinh(Z β ) K ∞ f , for f ∈ K ∞ pos o , ˜ V M : = n v 1 ⊕ v 2 ∈ K ∞ ⊕ K ∞ : v 1 = − β H ∞ cosh(Z β ) K ∞ g , v 2 = − β H ∞ C sinh(Z β ) K ∞ g , for g ∈ ( β H ∞ pos K ∞ pos ) o . 13 Pr o of. The statemen t is pro ved b y applying Lemma 2 and Prop osition 3, where we iden tify K 1 = K ∞ K ∞ pos ( O ) and K 2 = β H ∞ K ∞ ( β H ∞ pos K ∞ pos )( O ). W e finally prov e the main result of this section Theorem 3. L et O ⊂ M be a op en c ausal diamond on Minkowski spac etime M and A ( O ) the c orresp onding abstr act Weyl C ∗ -algebr a of a fr e e r e al massive sc alar field. Consider ω β , for 0 < β < ∞ , the quasi-fr e e KMS state with r esp e ct to the fr e e dynamics on the quasi-lo c al algebr a A ( M ) and let M β ( O ) b e the r epr esente d algebra in the GNS r epr esentation induc e d by the KMS state M β ( O ) : = π ω β ( A ( O )) ′′ . Then, gener alise d Haag duality holds M β ( O ) ′ = M β ( O ′ ) ∨ J M β ( M ) J. Pr o of. As pro v en in Lemma 1 M β ( O ) = R F ( U O , V O ) . Then, b y Theorem 2 and Prop osition 5 we hav e that R F ( U O , V O ) ′ = R F ( U O ′ , V O ′ ) ∨ R F ( ˜ U M , ˜ V M ) . and equiv alently , as prov en in Prop osition 4 R F ( ˜ U M , ˜ V M ) = R F ( U M , V M ) ′ = J R F ( U M , V M ) J. where J is the mo dular conjugation associated with the pair ( M β ( M ) , Ω β ). Ac knowledgmen ts The research of S.G. is funded b y the EPSRC Op en F ello wship EP/Y014510/1 and, for part of this w ork, he also b enefited from a Short-T erm Scientific Mission funded b y COST Action CA21109 – CaLIST A, supp orted by COST (European Co op eration in Science and T ec hnology). S.G. is also grateful to the National Group of Mathematical Ph ysics (GNFM-INdAM). L.S. ac knowledges financial supp ort by Italian Ministry of Universit y and Research through the gran t PRIN 2022ZE8SC4. L.S. would also like to thank Silv ano T osi for the supp ort. Both authors benefited from discussions with Bernard Kay , V alter Moretti, and Nicola Pinamonti, to whom they are grateful. A Ec kmann-Osterw alder theorem W e briefly recall a result in [EO73] that is used in the pro of of our main theorem Theorem 4 ([EO73, Thm. 2]) . L et K b e a r e al Hilb ert spac e and c onsider K 1 , K 2 ⊂ K close d subsp ac es. Denoting by K ⊥ 1 , K ⊥ 2 their ortho gonal complements in K , and assuming that K 1 , K 2 ar e in generic p osition, then it holds R F ( K 1 , K 2 ) ′ = R F ( K ⊥ 2 , K ⊥ 1 ) (19) B T ec hnical Results Lemma 3. Let H b e a Hilb ert sp ac e with sc alar pr o duct ⟨· , ·⟩ and A an oper ator in B ( H ) with b ounde d inverse A − 1 . L et V ⊂ H b e a close d line ar subsp ac e and V ⊥ its ortho gonal. Then the fol lowing r elation holds ( AV ) ⊥ = ( A ∗ ) − 1 V ⊥ . (20) Pr o of. W e hav e v ∈ ( AV ) ⊥ ⇐ ⇒ ⟨ v , Aw ⟩ = 0 , ∀ w ∈ V ⇐ ⇒ ⟨ A ∗ v , w ⟩ = 0 , ∀ w ∈ V ⇐ ⇒ A ∗ v ∈ V ⊥ ⇐ ⇒ v ∈ ( A ∗ ) − 1 V ⊥ , where in the last implication we used the hypothesis to conclude that ( A ∗ ) − 1 = ( A − 1 ) ∗ exists as a b ounded op erator. Note that ( A ∗ ) − 1 V ⊥ is a closed set by the op en mapping theorem. Lemma 4 ([GP22, App. C]) . The r e al subsp ac es K ∞ pos and β H ∞ pos K ∞ pos ar e isomorphic to the r e al Hilb ert sp ac es of initial c onditions. Namely K ∞ pos ∼ = F φ , β H ∞ pos K ∞ pos ∼ = F π . 14 Pr o of. W e prov e this statement by constructing explicitly the isomorphism. Let us define δ 0 : K ∞ pos → F φ , δ 1 : β H ∞ pos K ∞ pos → F π , with explicit action on f ∈ S ( R 4 , R ) ∩ K ∞ pos and g ∈ S ( R 4 , R ) ∩ β H ∞ pos K ∞ pos giv en b y δ 0 f : = F − 1  ˆ f | H m ( p )  , δ 1 g : = F − 1  ( iω p ) − 1 ˆ g | H m ( p )  . W e start proving that they are isometries of Hilb ert spaces and as such their definition can b e extended ov er all of K ∞ pos and β H ∞ pos K ∞ pos . This follows from the following computation for f ∥ δ 0 f ∥ 2 φ = ( ω − 1 2 δ 0 f , ω − 1 2 δ 0 f ) = Z d 3 p ω p    ˆ f | H m ( p )    2 = ∥ f ∥ 2 H ∞ and similarly for g ∥ δ 1 g ∥ 2 φ = ( ω 1 2 δ 1 g , ω 1 2 δ 1 g ) = Z d 3 p ω p | ˆ g | H m ( p ) | 2 = ∥ g ∥ 2 H ∞ . As isometries, δ 0 and δ 1 and their extensions are injectiv e. Let us further sho w that they are surjective maps using the symplectomorphism betw een the symplectic space of initial data and that of sources for the Klein-Gordon equation. Since the op erator P = □ − m 2 is Green hyperb olic, we define the corresp onding causal propagator (Pauli-Jordan function) as the op erator defined on time compact functions E : C ∞ tc ( M , R ) → C ∞ ( M , R ) with the property supp(E( f )) ⊂ J (supp( f )) and suc h that E ◦ P | C ∞ tc = 0 , P ◦ E | C ∞ tc = 0 . F or any h ∈ C ∞ c ( M , R ), w e ha ve (E h )( x ) = Z E( x, y ) h ( y )d 4 y where, using Sc h w artz’s k ernel theorem, we denoted by E( x, y ) the distributional kernel of the causal propagator. Then, denoting by h ± ( x ) = h ( x 0 , x ) ± h ( − x 0 , x ) 2 , w e can explicitly compute (E h )(0 , x ) = 1 (2 π ) 4 Z e i px ˆ E( p ) ˆ h ( p )d 4 p = 1 (2 π ) 3 Z ˆ h − | H m ( p ) d 3 p iω p = ( δ 1 h − )( x ) (21) and − ( ∂ x 0 (E h ))(0 , x ) = ( δ 0 h + )( x ) . (22) On the other hand consider a pair of initial conditions f , g ∈ C ∞ c ( R 3 , R ), whic h is a dense subset of F φ , F π . Resp ectively , iden tify f ∈ F π and g ∈ F φ . W e asso ciate to this pair the corresp onding unique spatially compact solution of the homoge- neous Klein-Gordon equation ϕ ( x 0 , x ) with f , g its initial conditions on the Cauch y hypersurface at x 0 = 0 (see e.g. [Dim80; BGP07]). Then, if w e consider χ ∈ C ∞ ( R , R ) suc h that, for some ϵ > 0, χ ( t ) = 0 if t < − ϵ and χ ( t ) = 1 for t > ϵ , w e can define the test function h ( x 0 , x ) : = P ( χ ( x 0 ) ϕ ( x 0 , x )) whic h, for a different choice of χ ′ with the same ab o v e h yp otheses, determines the same element in the symplectic space C ∞ c ( M , R ) / ker( E). Then, since P ( χ ( x 0 ) ϕ ( x 0 , x )) = − P ((1 − χ ( x 0 )) ϕ ( x 0 , x )), it follows that E h = ϕ and therefore h satisfies b y construction (E h )(0 , x ) = f ( x ) , ( ∂ x 0 (E h ))(0 , x ) = g ( x ) . Namely , f , g are the initial conditions for the solution sourced by h . T ogether with Equations (21),(22) this prov es surjectivit y . C Pre-cyclicit y in the thermal sector Let O b e an non-empty op en subset of Minko wski spacetime M and H β O ⊂ H ∞ ⊕ H ∞ , defined as H β O : = n ψ 1 ⊕ ψ 2 ∈ H ∞ ⊕ H ∞ : ψ 1 = C sinh(Z β ) K ∞ f , ψ 2 = cosh(Z β ) K ∞ f , f ∈ C ∞ c ( O ) o , 15 the real subspace that labels the local von Neumann algebra R S ( H O ) associated with O in the KMS representation of the massiv e scalar field. W e prov e that the closure of H β O is standard H β O + i H β O = H ∞ ⊕ H ∞ . W e follow an argument similar to the one used for the pro of of [BCD21, Prop. 5.6.] (see also the pro of in [San26, Sec. 2.5]). As pro v en in [Ka y85a, App. A.2], the complex span of H β M is dense in H ∞ ⊕ H ∞ H β M + i H β M = H ∞ ⊕ H ∞ . W e sa y that the size of the support of a test function f ∈ C ∞ c ( M ) is smaller than O , and we write supp f ≺ O , if it exists a non empt y I = I 1 × I 2 × I 3 × I 4 subset of R 4 suc h that α u (supp f ) ⊂ O ∀ u ∈ I , α u (supp f ) = { x ∈ M : x − u ∈ supp f } . F or a generic vector Ψ ∈ H β M Ψ = C sinh(Z β ) K ∞ f ⊕ cosh(Z β ) K ∞ f w e denote by Ψ u the corresp onding translated vector Ψ = C sinh(Z β ) K ∞ f u ⊕ cosh(Z β ) K ∞ f u , f u ( x ) = f ( x − u ) . Ev ery test function f ∈ C ∞ c ( M ) can b e decomp osed in a finite sum of test functions whose supp ort has size smaller than O b y considering a sufficien tly fine partition of the iden tity . More in details, it alwa ys exists a sufficiently large N ∈ N and a set of test functions χ j ∈ C ∞ c ( M ) , j ∈ [1 , N ] with supp χ j ⊂ supp f and supp χ i ≺ O suc h that f = (1 − N X j =1 χ j ) f + N X j =1 χ j f , and supp(1 − P N j =1 χ j ) f ≺ O . Therefore we hav e span( H ≺ M + i H ≺ M ) = H ∞ ⊕ H ∞ , where H ≺ M : = n ψ 1 ⊕ ψ 2 ∈ H ∞ ⊕ H ∞ : ψ = C sinh(Z β ) K ∞ f , ξ = cosh(Z β ) K ∞ f , f ∈ C ∞ c ( M ) , supp f ≺ O o . Supp ose now that a v ector Ψ ∈ H ∞ ⊕ H ∞ b elongs to ( H β O + i H β O ) ⊥ (where ⊥ denotes the orthogonal with resp ect to the complex inner product). W e show that Ψ = 0 by showing that it is orthogonal to every v ector in the dense set span( H ≺ M + i H ≺ M ). If Ξ = R (Ξ) + i I (Ξ) is a generic vector in H ≺ M + i H ≺ M , the vector Ξ u defined as follows Ξ u = ( R (Ξ)) u + i ( I (Ξ)) u , b elongs to H β O + i H β O for a non-empty interv al I ⊂ R 4 . Then, if u ∈ I the scalar product ⟨ Ξ u , Ψ ⟩ v anishes by assumption. Thanks to the presence of the operator sinh(Z β ), dominated con vergence theorem can be used to show that the function u 7→ ⟨ Ξ u , Ψ ⟩ is analytic in a complex tub e R 4 + iV β / 2 (see [San26, Sec. 2.5]). Here V β / 2 denotes the conv ex set V β / 2 =  x ∈ R 4 : x ∈ V + ∩ ( β 2 e + V − )  , where V ± = { x ∈ R 4 : ± x 0 > 0 , x µ x µ < 0 } are the future (+) and past ( − ) directed causal cones and e = (1 , 0 , 0 , 0). Analyticit y then implies that ⟨ Ξ u , Ψ ⟩ v anishes for any u ∈ R 4 and, in particular, for u = 0. By conjugate linearity it follows that Ψ is orthogonal to the dense set span( H ≺ M + i H ≺ M ) and so Ψ = 0. In conclusion, since the complex line ar subspace H β O + i H β O has trivial orthogonal, it is dense in H ∞ ⊕ H ∞ . References [Ara63] Huzihiro Araki. “A Lattice of v on Neumann Algebras Assoc iated with the Quantum Theory of a F ree Bose Field”. In: J. Math. Phys. 4.11 (1963), pp. 1343–1362. doi : 10.1063/1.1703912 . [Ara64] Huzihiro Araki. “V on Neumann Algebras of Local Observ ables for F ree Scalar Field”. In: J. Math. Phys. 5.1 (1964), pp. 1–13. doi : 10.1063/1.1704063 . [A W63] H. Araki and E. J. W oo ds. “Representations of the Canonical Commutation Relations Describing a Nonrelativistic Infinite F ree Bose Gas”. In: J. Math. Phys. 4.5 (Ma y 1963), pp. 637–662. issn : 0022- 2488. doi : 10 . 1063 / 1 . 1704002 . eprin t: https : / / pubs . aip . org / aip / jmp / article - pdf / 4 / 5 / 637 / 19220915/637\_1\_online.pdf . url : https://doi.org/10.1063/1.1704002 . 16 [Bac+25] Sven Bachmann et al. “T ensor category describing an yons in the quantum Hall effect and quantization of conductance”. In: R ev. Math. Phys. (2025). doi : 10.1142/S0129055X24610075 . [BCD21] Henning Bostelmann, Daniela Cadamuro, and Simone Del V ecc hio. “Relative Entrop y of Coheren t States on General CCR Algebras”. In: Commun. Math. Phys. 389.1 (Dec. 2021), pp. 661–691. issn : 1432-0916. doi : 10 . 1007 / s00220 - 021 - 04249 - x . url : http: / / dx . doi . org / 10 . 1007 / s00220 - 021 - 04249- x . [BCS25] Henning Bostelmann, Daniela Cadamuro, and Leonardo Sangaletti. “Quan tum L p inequalities in ther- mal states”. In: (No v. 2025). arXiv: 2511.18932 [math-ph] . [BGL93] R. Brunetti, D. Guido, and R. Longo. “Mo dular structure and duality in conformal quan tum field theory”. In: Commun. Math. Phys. 156.1 (1993), pp. 201–219. doi : 10.1007/BF02096738 . [BGP07] Christian B¨ ar, Nicolas Ginoux, and F rank Pf¨ affle. Wave e quations on Lor entzian manifolds and quan- tization . ESI Lectures in Mathematics and Ph ys ics. Europ ean Mathematical So ciety (EMS), Z ¨ uric h, 2007. isbn : 978-3-03719-037-1. url : https://doi.org/10.4171/037 . [BR79] O. Bratteli and D.W. Robinson. Op er ator Algebr as and Quantum Statistic al Me chanics . Op erator Al- gebras and Quantum Statistical Mec hanics Bd. 2. Springer, 1979. isbn : 9783540103813. url : https : //books.google.de/books?id=2h7vAAAAMAAJ . [BR81] Ola Bratteli and Derek W. Robinson. Op er ator A lgebr as and Quantum Statistic al Me chanics . V ol. II. New Y ork: Springer, 1981. [BS90] D. Buchholz and H. Sc h ulz-Mirbac h. “Haag Duality in Conformal Quantum Field Theory”. In: R ev. Math. Phys. 2 (1990), pp. 105–125. doi : 10.1142/S0129055X90000053 . [BW76] J. J Bisognano and E. H. Wic hmann. “On the Duality Condition for Quantum Fields”. In: J. Math. Phys. 17 (1976), pp. 303–321. [Del68] G. F. Dell’Antonio. “Structure of the algebras of some free systems”. In: Commun. Math. Phys. 9 (1968), pp. 81–117. [DHR69a] Sergio Doplic her, Rudolf Haag, and John E. Rob erts. “Fields, observ ables and gauge transformations I”. In: Commun. Math. Phys. 13 (1969), pp. 1–23. doi : 10.1007/BF01645267 . [DHR69b] Sergio Doplic her, Rudolf Haag, and John E. Roberts. “Fields, observ ables and gauge transformations I I”. In: Commun. Math. Phys. 15 (1969), pp. 173–200. doi : 10.1007/BF01645674 . [Dim80] J. Dimo ck. “Algebras of lo cal observ ables on a manifold”. In: Commun. Math. Phys. 77.3 (1980), pp. 219–228. issn : 0010-3616. [DL83] Sergio Doplic her and Rob erto Longo. “Lo cal Aspects of Sup erselection Rules II”. In: Commun. Math. Phys. 88 (1983), pp. 399–409. [Dop82] Sergio Doplicher. “Lo cal Aspects of Superselection Rules”. In: Commun. Math. Phys. 85 (1982), pp. 73– 86. [EO73] Jean-Pierre Eckmann and Konrad Osterwalder. “An application of T omita’s theory of mo dular Hilb ert algebras: Dualit y for free Bose fields”. In: J. F unct. Anal. 13.1 (1973), pp. 1–12. issn : 0022-1236. doi : https : / / doi . org / 10 . 1016 / 0022 - 1236(73 ) 90062 - 1 . url : https : / / www . sciencedirect . com / science/article/pii/0022123673900621 . [F G89] F ranca Figliolini and Daniele Guido. “The T omita op erator for the free scalar field”. In: A nn. Inst. H. Poinc ar´ e Phys. Th ´ eor. 51.4 (1989), pp. 419–435. [FN15] L. Fiedler and P . Naaijkens. “Haag duality for Kitaev’s quantum double mo del for ab elian groups”. In: R ev. Math. Phys. 27 (2015), p. 1550021. issn : 1793-6659. doi : 10.1142/s0129055x1550021x . url : http://dx.doi.org/10.1142/S0129055X1550021X . [FR19] Christopher J. F ewster and Kasia Rejzner. Algebr aic Quantum Field The ory – an intr o duction . 2019. arXiv: 1904.04051 [hep-th] . url : . [GP22] Alan Garbarz and Gabriel Palau. “A note on Haag duality”. In: Nucl. Phys. B 980 (July 2022), p. 115797. issn : 0550-3213. doi : 10 . 1016 / j . nuclphysb . 2022 . 115797 . url : http : / / dx . doi . org/10.1016/j.nuclphysb.2022.115797 . [HK64] Rudolf Haag and D. Kastler. “An algebraic approac h to quantum field theory”. In: J. Math. Phys. 5 (1964), pp. 848–861. 17 [Jak00] Christian D. Jakel. “The Reeh-Schlieder prop erty for thermal field theories”. In: J. Math. Phys. 41 (2000), pp. 1745–1754. doi : 10.1063/1.533208 . arXiv: hep- th/9904049 . [Ka y85a] Bernard S. Kay. “A Uniqueness Result for Quasifree KMS States”. In: Helv. Phys. A cta 58 (1985), p. 1017. [Ka y85b] Bernard S. Kay. “Purification of KMS states”. In: Helv. Phys. A cta 58.6 (1985), pp. 1030–1040. [Key+06] M. Keyl et al. “En tanglement, Haag-dualit y and type properties of infinite quan tum spin c hains”. In: R ev. Math. Phys. 18.9 (2006), pp. 935–970. doi : 10.1142/S0129055X0600284X . [KW91] Bernard S. Kay and Rob ert M. W ald. “Theorems on the uniqueness and thermal properties of stationary , nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon”. In: Physics R ep orts 207.2 (1991), pp. 49–136. issn : 0370-1573. doi : 10.1016/0370- 1573(91)90015- E . [LR T78] P . Leyland, J. E. Rob erts, and D. T estard. “Duality for Quantum F ree Fields”. In: (1978). Unpublished man uscript, Centre de Physique Th´ eorique, CNRS Marseille, July 1978. [LSW24] L. v an Luijk, A. Stottmeister, and H. Wilming. “Uniqueness of Purifications Is Equiv alent to Haag Dualit y”. In: arXiv (2024). [Lui+25] L. v an Luijk et al. “Pure State Entanglemen t and von Neumann Algebras”. In: Commun. Math. Phys. 406.12 (2025), p. 296. doi : 10.1007/s00220- 025- 05465- 5 . [Naa12] P . Naaijk ens. “Haag duality and the distal split prop erty for cones in the toric co de”. In: L ett. Math. Phys. 101 (2012), pp. 341–354. doi : 10.1007/s11005- 012- 0572- 7 . [Oga20] Y oshik o Ogata. “A Z 2 -Index of Symmetry Protected T opological Phases with Time Reversal Symmetry for Quantum Spin Chains”. In: Commun. Math. Phys. 374 (2020), pp. 705–734. doi : 10.1007/s00220- 019- 03521- 5 . [Oga21] Y oshik o Ogata. “An H 3 ( G, T )-v alued index of symmetry-protected topological phases with on-site finite group symmetry for tw o-dimensional quantum spin systems”. In: F orum Math. Pi 9 (2021), e13. doi : 10.1017/fmp.2021.17 . [Oga22a] Y oshiko Ogata. “A deriv ation of braided C*-tensor categories from gapp ed ground states satisfying the appro ximate Haag duality”. In: J. Math. Phys. 63.1 (2022), p. 011902. doi : 10.1063/5.0061785 . [Oga22b] Y oshiko Ogata. “An inv arian t of symmetry protected topological phases with on-site finite group sym- metry for tw o-dimensional fermion systems”. In: Commun. Math. Phys. 395.1 (2022), pp. 405–457. issn : 1432-0916. doi : 10 . 1007 / s00220 - 022 - 04438 - 2 . url : http : / / dx . doi . org / 10 . 1007 / s00220 - 022- 04438- 2 . [OPR25] Y oshiko Ogata, Da vid P´ erez-Garc ´ ıa, and Alb erto Ruiz-de-Alarc´ on. “Haag Dualit y for 2D Quantum Spin Systems”. In: arXiv arXiv:2509.23734 (2025). [Ost73] Konrad Osterw alder. “Dualit y for F ree Bose Fields”. In: Commun. Math. Phys. 29.1 (1973), pp. 1–14. doi : 10.1007/BF01661147 . [Rie74] Marc A. Rieffel. “A commutation theorem and duality for free Bose fields”. In: Commun. Math. Phys. 39.2 (1974), pp. 153–164. doi : 10.1007/BF01608393 . [San26] Leonardo Sangaletti. “On Quantum Liouvillian Densit y Inequalities in the Thermal Sector”. PhD thesis. Univ ersit y of Leipzig, 2026. [Sla72] J. Slawn y. “On factor representations and the C*-algebra of canonical commutation relations”. In: Commun. Math. Phys. 24 (1972), pp. 151–170. doi : 10.1007/BF01878451 . [SVW02] Alexander Strohmaier, Rainer V erc h, and Manfred W ollenberg. “Microlocal analysis of quantum fields on curved space–times: Analytic wa ve fron t sets and Reeh–Schlieder theorems”. In: J. Math. Phys. 43.11 (2002), pp. 5514–5530. issn : 1089-7658. doi : 10 . 1063 / 1 . 1506381 . url : http : / / dx . doi . org / 10.1063/1.1506381 . [UMT82] H. Umezaw a, H. Matsumoto, and M. T achiki. Thermo field dynamics and c ondense d states . 1982. [V er93] Rainer V erc h. “Nuclearity , split prop ert y and duality for the Klein-Gordon field in curved space-time”. In: L ett. Math. Phys. 29 (1993), pp. 297–310. doi : 10.1007/BF00750964 . [V er97] Rainer V erc h. “Con tinuit y of Symplectically Adjoin t Maps and the Algebraic Structure of Hadamard V acuum Represen tations for Quan tum Fields on Curv ed Spacetime”. In: R ev. Math. Phys. 09.05 (1997), pp. 635–674. issn : 1793-6659. doi : 10.1142/s0129055x97000233 . url : http://dx.doi.org/10.1142/ S0129055X97000233 . 18