The Bisognano-Wichmann property for asymptotically complete massless QFT

We prove the Bisognano-Wichmann property for asymptotically complete Haag-Kastler theories of massless particles. These particles should either be scalar or appear as a direct sum of two opposite integer helicities, thus, e.g., photons are covered. The argument relies on a modularity condition formulated recently by one of us (VM) and on the Buchholz' scattering theory of massless particles.


Introduction
For any von Neumann algebra with a faithful state Tomita-Takesaki theory gives a natural dynamics constructed using the complex structure of the algebra. It was shown by Bisognano and Wichmann that for the algebra of the Wightman fields localised in a spacelike wedge this latter dynamics coincides with the Lorentz boosts in the direction of this wedge [BW76]. Furthermore, in the presence of the Bisognano-Wichmann (B-W) property the full global symmetry of the model is contained in the modular structure of the net reducing the dichotomy between symmetries and algebras to an inclusion [BGL95]. The B-W property is also important for many other reasons, ranging from the intrinsic meaning of the CPT symmetry [GL95] to a construction of interacting models [Le08] and to entanglement theory [W18]. While its formulation is most natural in the algebraic (Haag-Kastler) setting, and it is known to hold in all the 'physical' examples, its general proof in this framework is missing to date. The reason is the broadness of the Haag-Kastler setting which admits also non-physical counterexamples to the B-W property. For example, when an infinite family of massive spinorial or infinite spin particles occurs [LMR16,Mo18]. Thus it is important to find natural assumptions which exclude such pathological cases.
For massless theories the assumption of global conformal invariance implies the B-W property as shown in [BGL93]. A search for an algebraic sufficient condition for 1 the B-W property, not relying on conformal covariance, was started in [Mo18,Mo17] at the level of one particle nets. Here a criterion on the covariant representation called the modularity condition was shown to give the B-W property of the one particle net. For massive theories which are asymptotically complete, the paper by Mund [Mu01] gives the B-W property. This paper exploits a result of Buchholz and Epstein [BE85] in order to study geometrically the analytic extension of one parameter boosts and identify it with the associated modular operator. This allows to verify the Bisognano-Wichmann property on the one particle subspace and conclude it for the full interacting net by asymptotic completeness and wedge localization of the modular operator. Unfortunately this method does not apply to massless theories as the argument of Buchholz and Epstein requires a mass gap.
In the present paper we prove the B-W property for massless bosonic theories which are asymptotically complete, by combining some ideas contained in the works mentioned above. First, we identify the modular operator and the boost generator associated to the same wedge at the single-particle level. To this end, we verify the modularity condition introduced in [Mo17, Mo18] and thus avoid the use of the Buchholz-Epstein result. Our argument requires that the representation of the Poincaré group is either scalar or a direct sum of two representations with opposite integer helicities. Thereby we show that the modularity condition applies to a large family of massless representations, including higher helicity. Next, we show that the B-W property holds on the entire Hilbert space by using scattering theory and the assumption of asymptotic completeness. We recall that scattering theory for massless bosons was developed in [Bu77] and various simplifications have been found meanwhile. In the present paper we use the variant from [AD17] which is based on novel ergodic theorem arguments and on uniform energy bounds on asymptotic fields from [Bu90, He14.1]. Our results extend the range of validity of the B-W property and reconfirm its status as a generic property of physically reasonable models.
Our paper is organized as follows: In Sect. 2 we state our main result after the necessary preparations. In Sect. 3 we recall some relevant facts from scattering theory of massless particles, the theory of standard subspaces and one-particle nets, and representations of the Poincaré group. In Sect. 4 the modularity condition is stated and verified for one-particle massless nets with arbitrary integer spin. In Sect. 5 the result is generalised to an arbitrary number of particles using scattering theory and the assumption of asymptotic completeness.

Local nets and the Bisognano-Wichmann property
Let R 1+3 be the Minkowski spacetime. We denote by K the family of double cones O ⊂ R 1+3 ordered by inclusion and write O ′ for the causal complement of O in R 1+3 . Furthermore, let P ↑ + = R 4 ⋊ SL(2, C) denote the covering group of the proper ortochronous Poincaré group P ↑ + . We denote with Λ : P ↑ + → P ↑ + the covering map.
4. Cyclicity of the vacuum: there is a unique (up to a phase) unit vector Ω ∈ H, the physical vacuum state, which is U-invariant and cyclic for the global algebra A := O⊂R 1+3 A(O) · of the net.

Locality:
A local net of von Neumann algebras will be denoted by (A, U, Ω).
For future reference, we set for any region U ⊂ R 1+3 and we refer to A loc := A loc (R 1+3 ) as the algebra of strictly local operators. In order to introduce the B-W property, we need some geometric preliminaries: a wedge shaped region W ⊂ R 1+3 is an open region of the form gW 1 where g ∈ P ↑ + and W 1 = {x ∈ R 1+3 : |x 0 | < x 1 }. The set of wedges is denoted by W. Note that if W ∈ W, then W ′ ∈ W, where prime denotes here the spacelike complement. It is possible to associate to any wedge a one-parameter group of boosts Λ W fixing the wedge W by the following formula for W 1 and the covariant action of the Poincaré group on the set of wedges. 3 We call W α = {x ∈ R 1+3 : |x 0 | < x α }, α = 1, 2, 3, the wedge in the x α direction and with R α , Λ α are the one-parameter groups of rotations and boosts, respectively, fixing W α . Their unique one parameter group lifts to SL(2, C) are denoted r α and λ α . In general λ W will denote the one parameter group lift of Λ W . Note that λ α (t) = e t 2 σα and r α (θ) = e i θ 2 σα where t, θ ∈ R and σ α are the Pauli matrices. In particular one has that r α (2π) = −I =: r(2π).
For any W ∈ W we define A(W ) according to (2.2). It is well known that the vacuum is cyclic and separating for A(W ) thus the Tomita-Takesaki theory gives the corresponding modular evolution R ∋ t → ∆ it W .
Definition 2.2. We say that a local net (A, U, Ω) satisfies the Bisognano-Wichmann property if for all W ∈ W, t ∈ R,

Massless Wigner particles and asymptotic nets
Scattering theory of massless Wigner particles was developed by Buchholz [Bu77,Bu75], both in the bosonic and fermionic case. Recently the bosonic case was simplified in [AD17]. We collect below the main results in this subject following [AD17]. We first introduce the single-particle subspace. where 1 {0} (M) denotes the spectral projection of the mass operator M := (P 0 ) 2 − P P P 2 corresponding to the eigenvalue zero. We say that these particles have helicities h 1 , h 2 , h 3 . . . ∈ Z if U| H (1) is a finite or infinite multiple of the direct sum of the corresponding zero mass representations.
It is well known that to any local theory containing massless particles one can associate an asymptotic (free) theory [Bu77]. We outline now this construction following [AD17]. For the unitary representation of translations U| R 4 we shall write U(x) = e i(P 0 x 0 −P ·x) and for translates of observables A ∈ A the notations α x (A) := A(x) := U(x)AU(x) * are used. If g ∈ L 1 (R 4 ), then A(g) := A(x)g(x)d 4 x denotes the operator A smeared with the function g. Moreover, we set A loc,0 := {A ∈ A loc : x → A(x) smooth in norm}.
(2.5) This is a weakly dense * -subalgebra of A loc , as can be seen by smearing local operators with delta-approximating functions. Next, we specify the following Poincaré invariant subset of C ∞ 0 (R 4 ) and define A C * := Span A C * , (2.8) (2.9) Now we move on to the construction of asymptotic fields of massless particles. For any A ∈ A C * and f ∈ C ∞ (S 2 ), we set as in [Bu77,Bu82] A t {f } := −2 t dω(n) f (n) ∂ 0 A(t, tn). (2.10) Here dω(n) = sin ν dνdϕ 4π is the normalized, invariant measure on S 2 and ∂ 0 A := ∂ s (e isH Ae −isH )| s=0 . In order to improve the convergence in the limit of large t, we proceed to time averages of A t {f }, namelȳ Here for non-negative h ∈ C ∞ 0 (R), supported in the interval [−1, 1] and normalized so that dt h(t) = 1, we set h t (t ′ ) = t −ε h(t −ε (t ′ − t)) with t ≥ 1 and 0 <ε < 1. It turns out that these limits exist on all vectors from the domain where D((P 0 ) n ) is the domain of self-adjointness of (P 0 ) n .
Lemma 2.4. Let A ∈ A C * (O) and f ∈ C ∞ (S 2 ). Then, the limit exists for Ψ ∈ D P 0 and is again an element of D P 0 .
The operators A out {f } are constructed in such a way that they create singleparticle states from the vacuum, namely where P (1) is the projection on the single-particle subspace H (1) . Vectors of the form (2.14) span a dense subspace of H (1) , even in the case f ≡ 1. Furthermore, if A out {f }, A ′out {f ′ } are two asymptotic fields as specified above, then as operators on D P 0 . For f ≡ 1 the operators A out {f } appearing in Lemma 2.4 are denoted A out and are called the asymptotic fields. For A = A * these operators are essentially self-adjoint on D(P 0 ) and their self-adjoint extensions are denoted by the same symbol. For any O ∈ K we introduce the von Neumann algebra: The triple (A out , U, Ω) satisfies all the properties from Definition 2.1, except, perhaps, for the cyclicity of the vacuum. If the latter property also holds, then we say that the theory (A, U, Ω) is asymptotically complete. Clearly, for the definition of asymptotic completeness the case f ≡ 1 suffices. However, the operators A out {f } for other choices of f will be needed in Sect. 5 at the technical level. For this reason we collected their properties above. Now we are ready to state the main result of this paper: Theorem 2.5. Let (A, U, Ω) be a local net containing massless particles with helicity zero or with helicities (h, −h) for some h ∈ N. If this net is asymptotically complete, then it satisfies the Bisognano-Wichmann property.
Even if the original net (A, U, Ω) is not asymptotically complete, we can set H out := A out Ω and define the asymptotic net (A out | H out , U| H out , Ω) which is asymptotically complete by construction. In view of the commutation relations (2.15), this net can be considered free, but it is not automatically the net of the corresponding textbook free field theory 1 . By Theorem 2.5, this net satisfies the Bisognano-Wichmann property if it contains massless Wigner particles with helicity zero or (h, −h), h ∈ N. We note that the local nets satisfying the assumptions of Theorem 2.5 are a posteriori in the setting of [GL95]. Indeed, the modular covariance is an obvious consequence of the Bisognano-Wichmann property and the Reeh-Schlieder property for spacelike cones follows from the spectrum condition and cyclicity of the vacuum under A (cf. [Bu75,Appendix]). From the spin-statistics theorem of [GL95] it follows that such nets are actually covariant under P ↑ + . Furthermore, by the CPT theorems of this reference, the unitary representation U of P ↑ + extends to an (anti-)unitary covariant representation of P + (the group generated by P ↑ + and the PT operator Θ) as follows: Corollary 2.6. Let (A, U, Ω) be a local net as in Theorem 2.5. Then U extends to an (anti-)unitary representation of the Poincaré group P + by where Θx = −x with x ∈ R 1+3 , J W 1 is the modular conjugation associated to (A(W 1 ), Ω) and R 1 (π) is the rotation by π around the first axis.

Scattering states of massless particles
In this subsection we provide some preparatory information about the Hilbert space of scattering states H out := A out Ω introduced above. Namely, we extract the creation and annihilation parts of the asymptotic fields (2.13) in order to facilitate the construction of scattering states. We still follow [AD17] which in turn relied here on [DH15]. Let θ ∈ C ∞ (R), 0 ≤ θ ≤ 1, be supported in (0, ∞) and equal to one on (1, ∞). Moreover, let β ∈ C ∞ 0 (R 4 ), 0 ≤ β ≤ 1, be equal to one in some neighbourhood of zero and satisfy β(−p) = β(p). Furthermore, for a parameter 1 ≤ r < ∞ and a future oriented timelike unit vector n we define η ±,r (p) := θ(±r(n µ p µ ))β(r −1 p), (3.1) where tilde denotes the Fourier transform. As r → ∞ these functions approximate the characteristic functions of the positive/negative energy half planes { p ∈ R 4 : ±n µ p µ ≥ 0 }. We also haveη ±,r = η ∓,r . Note that the family of functions η ±,r , as specified above, is invariant under Lorentz transformations.
. Suppose that the timelike unit vectors n entering the definition of A and of η ±,r coincide. Then: (a) The limits A out {f } ± Ψ := lim r→∞ A out {f }(η ±,r )Ψ, Ψ ∈ D P 0 , exist and define the creation and annihilation parts of A out {f } as operators on D P 0 . A out {f } ± do not depend on the choice of the functions θ and β in (3.1) within the specified restrictions.
Making use of Proposition 3.1 and of (2.15) we also obtain on D P 0 and the commutators of pairs of creation (resp. annihilation) operators vanish. The following definition of scattering states is slightly more general than in [Bu77, AD17], as we do not assume f ≡ 1. The proof is an obvious application of the canonical commutation relations (3.2).
Proposition 3.2. [Bu77, AD17] The states Ψ out := A out 1 {f 1 } + . . . A out n {f n } + Ω have the following properties: (a) Ψ out depends only on the single-particle states where S n is the set of all permutations of (1, . . . , n).
The subspace of H spanned by vectors of the form Ψ out = Φ 1 out × · · · out × Φ n for fixed n will be denoted H (n) . We note that where H (0) = CΩ and H (1) was introduced in Definition 2.3. The last equality in (3.3) follows from density of vectors of the form (2.14) in H (1) and from the canonical commutation relations (3.2) by standard Fock space arguments. Clearly, H out is naturally isomorphic to the symmetric Fock space over H (1) , denoted Γ(H (1) ).

Standard subspaces
We recall here some elements of the theory of standard subspaces following [Lo]. In the later part of this subsection we also provide several results which we were not able to find in literature and that will be needed in our investigation.
Given a standard subspace H the associated Tomita operator S H is defined to be the closed anti-linear involution with domain H + iH, given by: The polar decomposition where ⊥ R denotes the orthogonal complement in H with respect to the real part of the scalar product on H. H ′ is a closed, real linear subspace of H. It is a fact that H is cyclic (resp. separating) iff H ′ is separating (resp. cyclic), thus H is standard iff H ′ is standard and in this case We recall that the one-parameter, strongly continuous group t → ∆ it H is called the modular group of H and There is a 1-1 correspondence between Tomita operators and standard subspaces.
Proposition 3.3. [Lo]. The map is a bijection between the set of standard subspaces of H and the set of closed, densely defined, anti-linear involutions on H.
The following are three basic results on standard subspaces. Lemma 3.5. [Lo]. Let H ⊂ H be a standard subspace, and K ⊂ H be a closed, real linear subspace of H. If ∆ it H K = K, ∀t ∈ R, then K is a standard subspace of K := K + iK and ∆ H | K is the modular operator of K on K. Moreover, if K is a cyclic subspace of H, then H = K.
Theorem 3.6. [Lo]. Let H ⊂ H be a standard subspace, and U(t) = e itP be a one-parameter unitary group on H with a generator ±P > 0, such that ∀t, s ∈ R. (3.5) We note that the above result is a variant of the Borchers theorem [Bo92,Fl98] for standard subspaces.
The following three lemmas, which we could not find in the literature, will be needed to analyze the subspaces H (1) (W ) defined in (5.2) below. They concern decompositions of standard subspaces w.r.t. projections E commuting with S H . Since S H is unbounded and not self-adjoint, we mean here that E commutes with J H and bounded Borel functions of where χ n is the characteristic function of [−n, n] and we made use of the fact that Proof. H is defined to be the kernel of 1 − S H . Now since E commutes with S H , for every ξ ∈ H, Eξ ∈ Ker(1 − S H ), thus Eξ ∈ H (cf. computation (3.6) above). As the same argument applies to (1 − E) and ξ = Eξ + (1 − E)ξ, we have the claim. The last statement is obvious.
Lemma 3.8. Let H ⊂ H be a standard subspace and E = E 2 = E * be a projection commuting with S H . Then

One particle nets
Let U be a unitary representation of the Poincaré group P ↑ + on a Hilbert space H. We shall call a U-covariant (or Poincaré covariant) net of standard subspaces on wedges a map H : associating to every wedge in R 1+3 a closed real linear subspace of H, satisfying the following properties 2 : 3. Positivity of the energy: the joint spectrum of translations in U is contained in the forward lightcone We shall indicate a U-covariant net H of standard subspaces on wedges satisfying 1.-5. with the couple (U, H). This is the setting in which we are going to study the following property: The next property is a completeness property for a model in the sense of the causal structure and, by Lemma 3.5, is a consequence of the locality and the B-W properties (see e.g. [Mo18]).
Denote by P 0 , P be the generators of translations in the representation U and M = (P 0 ) 2 − P 2 the resulting mass operator. Then Theorem 3.6 has the following corollary, which is well known in the context of nets of von Neumann algebras. Proof. Consider the wedge W 1 , defined as in Sect. 2.1, and the associated standard subspace H(W 1 ). Translations in direction of the axes x 2 and x 3 fix W 1 . In particular the generators of the associated translation group P 2 and P 3 , respectively, commute strongly with ∆ H(W 1 ) and J H(W 1 ) by Lemma 3.4. Lightlike translations of the form a ± (t) = (±t, t, 0, 0) with t ≥ 0 have generators P ± := (±P 0 − P 1 ) s.t. ±P ± ≥ 0 and U(a ± (t))H(W 1 ) ⊂ H(W 1 ) for t ≥ 0. By the Borchers theorem for standard subspaces (Theorem 3.6) we have that U(a ± ) have the commutation relations as in equation (3.5): where f is any bounded Borel function. The implications above follow by approximating f pointwise with Schwartz-class functions (which gives strong convergence of the corresponding operators) and using the Fourier transform. Now P 2 = M 2 = −(P + P − + P 2 2 + P 2 3 ) and for any real Borel function g it is easy to check, using the above relations, that g(M 2 ) commutes with ∆ H(W ) and J H(W ) . Indeed, by approximating g pointwise by Schwartz-class functions, applying the Fourier transform and using that P 2 , P 3 commute strongly with ∆ H(W ) and J H(W ) it suffices to verify that This is achieved by approximating e −iP + P − t pointwise by linear combinations of expressions of the form f + (P + )f − (P − ), where f + , f − are bounded Borel functions, and applying the relations above. For a general wedge W , let g ∈ P ↑ + s.t. W = gW 1 . Then, by Lemma 3.4, Clearly P 2 = U(g)P 2 U(g) * , thus P 2 commutes with J H(W ) , ∆ H(W ) , S H(W ) for every W ∈ W in the same sense as discussed above.

Induced representations: the Poincaré group and its subgroups
Our group theoretic considerations in the remaining part of Sect. 3 and in Sect. 4 are based on the standing assumption that all the representations of topological groups on Hilbert spaces are strongly continuous. Let G be a locally compact group, N a nontrivial closed normal abelian subgroup and H another closed subgroup such that G = N ⋊ H 4 . Assume that the action of G onN, the dual group of N, obtained by conjugation, is regular (cf. [Fol16] Sect. 6.6 and Definition C.1). Let p ∈N, Ω p be the orbit under the G-dual action 5 , with x ∈ N, p ∈N and g ∈ G, Stab p and Stab p be the stabilizers of the point p under the action of H and G. Stab p is called the little group. Let χ p be the character associated to p ∈N.
Every unitary irreducible representation of G is obtained by induction in the following way (see e.g. [Fol16] Sect. 6) where V and χ p · V are unitary representations of the little group Stab p and of Stab p , respectively, and the following proposition holds: are equivalent if and only if p and q belongs to the same orbit, say p = g q, and V and V ′ • ad g −1 are equivalent representations of Stab p for some g ∈ G.
is an irreducible representation of G then the spectral measure of W | N is concentrated on the orbit o = Gp (cf. Proposition 6.36 [Fol16]). References for general induced representations are for instance [Fol16,Ki76,BR86].
The Poincaré group. The Minkowski space R 1+3 is the 4-dimensional real vector space endowed with the metric tensor η = diag(1, −1, −1, −1). The Lorentz group L is the group of linear transformations L s.t. L T ηL = η. Let L ↑ + be the connected component of the identity of the Lorentz group and L ↑ + = SL(2, C) its universal covering group. Let P ↑ + = R 1+3 ⋊ SL(2, C) be the universal covering of the Poincaré group P ↑ + = R 1+3 ⋊ L ↑ + (the inhomogeneous symmetry group of R 1+3 ) and Λ be the covering map. First of all, we recall that to any 4-vector is 1-1 associated a 2 × 2 matrix where σ i are the Pauli matrices. Real vectors define Hermitian matrices. If A ∈ SL(2, C), and Λ : P ↑ + → P ↑ + is the covering homomorphism, then the Poincaré action is ruled by the following relation (3.8) Let U be a unitary strongly continuous representation of the Poincaré group, then the representation of an x-translation has the form U(x) = e iP x where P is a vector of four self-adjoint operators and P x is obtained through the Minkowski product. Every g ∈ L ↑ + acts on an x-translation by the adjoint action, namely gxg −1 = Λ(g)x. Let Sp (P ) be the joint spectrum of generators of translations and p be a point in the spectrum, then we have the character χ p (x) = e ipx . As in the general case, the dual action on the momentum space is defined s.t. χ p (Λ(g) · x) = χ p ′ (x) and it is easy to see that p ′ = Λ(g) −1 p, where the latter is the matrix-vector multiplication. Clearly the adjoint action of the translations act trivially on themselves, hence on their dual (see e.g. [BR86]).
Positive energy massless representations of the Poincaré group. Let χ q , q = 0, be a character of the translation group. We shall call Stab q and Stab q the stabilizers of the point q through the L ↑ + and P ↑ + actions, respectively. The latter is Stab q = R 1+3 ⋊ Stab q , where Stab q shall be called as above the little group. Any massless, unitary, positive energy representation of P ↑ + is obtained starting with the character associated to q := (1, 1, 0, 0) ∈ ∂V + (∂V + {0} is an L ↑ + -orbit) and inducing by a unitary representation of the Stab q group. Note that a Stab q representation is of the form where V is the unitary representation of Stab q . The little group Stab q is isomorphic to E(2) = R 2 ⋊ T, where T is the unit circle. Note that r 1 (θ) generate T. E(2) is the double cover of E(2) = R 2 ⋊ SO(2), which is the group of Euclidean motions in two dimensions 6 . Irreducible representations V of E(2) fit in one of the following two classes: (See e.g. [Va85] and [BR86, page 520]) (a) The restriction of V to R 2 is trivial; (b) The restriction of V to R 2 is non-trivial.
Irreducible representations of E(2) in class (a) are labelled by half-integers h, called the helicity parameters. Irreducible representations in class (b) are labelled by κ > 0, the radius of a circle in R 2 , namely the joint spectrum of the E(2)-translations, and a Bose/Fermi alternative parameter ǫ ∈ {0, 1 2 }.
be a unitary representation of P ↑ + induced from the representation χ q · V of Stab q . We say that U has finite helicity or infinite spin if V has the form (a) or (b), respectively.

G W and related subgroups of the Poincaré group and their representations
In this subsection we introduce certain subgroups of P ↑ + which will be needed in Sect. 4 below to formulate a criterion for the B-W property. We refer to Sect. 2.1 for the definitions of the wedges W i , i = 1, 2, 3, and the set of wedges W. We also recall from this subsection that R i , Λ i ∈ L ↑ + and r i , λ i ∈ SL(2, C) denote the one-parameter families of rotations and boosts preserving the wedges W i , i = 1, 2, 3.
Definition 3.12. We denote with where R 1+3 is the translation group and G 0 3 , R 1+3 denotes the group generated by G 0 3 and R 1+3 .
W are defined by the transitive action of P ↑ + on wedges. We will denote the massless orbits of the R 1+3 translation characters under the G 3 action with In the present paper, we will be interested in orbits σ r with r > 0 since σ ± 0 have null measure w.r.t. the Lorentz invariant measure on ∂V + . Finally, we warn the reader thatG 3 ,G W do not denote the covering groups of G 3 , G W .
Note that G 0 3 = λ 3 , r 3 , r(2π) . Indeed, any SL(2, C) element implementing a Poincaré transformation can be decomposed by the polar decomposition A = U A · T A (see e.g. [Mor06]), where U A is a rotation and T A a boost. Let Λ : SL(2, C) → L ↑ + be the covering map, then Λ(A) Next, we note that G 0 3 andG 0 3 share the same orbits in ∂V + as the following remark explains.
can be obtained as a composition of a Λ 3 -boost of parameter t p and a R 3 -rotation of parameter θ p as Λ 3 (t p )R 3 (θ p )(p 0 , p 1 , p 2 , p 3 ) = Λ 3 (t p )(p 0 , p 1 , −p 2 , p 3 ) = (p 0 , p 1 , −p 2 , −p 3 ) (3.10) for all the orbits except for σ ± 0 , where σ ± 0 appeared in Definition 3.12. Clearly t p and θ p depend on p and the orbits excluded by this geometrical fact have null measure w.r.t. the Lorentz invariant measure on ∂V + . The discussion does not change if p is considered as an element of R 1+3 or of its dual.
By (3.10), we deduce that almost all G 0 3 orbits on ∂V + are preserved by the R 1 (π)action. Furthermore, we note that R 1 (π) sends W 3 onto W ′ 3 . Thus any transformation R ∈ P ↑ + such that RW 3 = W ′ 3 also preserves the G 0 3 orbits on ∂V + as well as R 1 (π), since R 1 (π)R ∈ G 0 3 . We have just seen that it is possible to pointwise reconstruct a transformation sending W to W ′ just starting with elements in G 0 W . With the help of the modularity condition (cf. Definition 4.1 and Theorem 4.2 below), this gives the proof of the B-W property in the scalar case in [Mo18].
As the regularity of the action of G 0 3 andG 0 3 on R 1+3 is verified in Appendix C, we can apply the theory of induced representations to G 3 = R 1+3 ⋊G 0 3 andG 3 = R 1+3 ⋊G 0 3 . Choose a point q r on each massless, positive energy orbit σ r of G 0 3 on the dual of R 1+3 . Up to a null measure set in ∂V + , the stabilizer of q r in G 3 is R 1+3 × r(2π) (cf. the definition of the orbits σ r and of G 0 3 ). Thus there exist only two irreducible representations of G 3 induced by χ qr , namely where V n (r(2π)) = (−1) n and n = 0, 1, cf. Proposition 3.11. They correspond to bosonic and fermionic representations of G 3 . Again by Remark 3.13 the subgroupsG 3 andG 0 3 share the same orbits σ r with r > 0 on ∂V + (up to a null measure set in ∂V + ). On the other hand the stabilizer of the point q r = (r, r, 0, 0) ∈ σ r inG 3 is R 1+3 ⋊ r 1 (π) .
Thus the little group of q r , the subgroup ofG 0 3 fixing q r , is Z 4 . We have four irreducible representations of Z 4 indexed by the representation of the generator, namely V n (r 1 (π)) = i n , with n = 0, 1, 2, 3. Correspondingly, we have four induced representations ofG 3 associated to each orbit, namely W r,n = IndG 3 R 1+3 ⋊Z 4 (χ qr · V n ) (3.12) acting on the Hilbert spaces H r,n . Note that we called V 2h the representation of character 2h, trivial on translations of E(2), and V n the representation of character n of Z 4 . That this is not an abuse of notation is justified by the fact that r 1 (π) ∈ E(2) and when we restrict V 2h to the group r 1 (π) , we get V 2h (r 1 (π)) = e i π 2 2h = i 2h = V n (r 1 (π)), V 2h (r(2π)) = (−1) n = V n (r(2π)) and it is enough to consider 2h in Z 4 in the first case or 2h in Z 2 in the second. Furthermore W r,n as a representation ofG 3 restricts to W r,m with m ≡ n (mod 2) as a representation of G 3 , cf. Lemma 4.9 in the next section. We will refer to (3.11) and (3.12) as massless representation of G 3 andG 3 , respectively, since the translation spectrum lies on the boundary of the forward lightcone.

Modularity condition and U h restriction
In this section we show that any local net of standard subspaces, covariant under a finite or infinite multiple of U h ⊕ U −h , h ∈ Z, satisfies the B-W property. The analysis is based on the following modularity condition: Note that (MC) depends neither on the choice of r W nor on W. Indeed ifr W ∈ P ↑ + is another transformation such that Λ(r W )W = W ′ then r W ·r W ∈ G W and if (MC) holds for U(r W ), then it holds for U(r W ). By transitivity of the P ↑ + action on wedges it is not restrictive to fix a wedge region W . Condition (MC) can be straightforwardly stated when just a representation ofG 3 is taken into account. We remark that the class of nets in Theorem 4.2, transforming under P ↑ + , is more general than the class of bosonic nets we defined in Sect. 3.3. The theorem applies to the families of Poincaré representations covered by the following two results.   (ii) Let K be a Hilbert space, then (MC) holds for U ⊗ 1 K ∈ B(H ⊗ K).
(iii) Let U 2 be a unitary representation ofG 3 s.t. U 2 is unitarily equivalent to U 1 . Then U 2 satisfies (MC).
The representations W r,n ofG 3 , restricted to G 3 , give W r,m , where m ≡ n(mod 2), see Lemma 4.9 below. Thus they are disjoint for different r, by the disjointness of the respective orbits σ r , cf. Proposition 3.11. Furthermore, since W r,n is irreducible, W r,n (G 3 ) ′ = C · 1 and W r,n (r 1 (π)) ∈ W r,n (G 3 ) ′′ = B(H r,n ). We deduce that W r,n satisfies (MC).
Corollary 4.5. Let W r,n be the irreducibleG 3 -representation of radius r > 0. Then W r,n satisfies (MC).
The following proposition ensures that also a direct integral of masslessG 3 representations satisfies (MC).
One can reduce the argument to the cases U r | G W = W r,0 or U r | G W = W r,1 by Proposition 4.4 (i). We give details on the proof in Appendix B.
Now we want to verify the modularity condition (MC) for a large family of massless bosonic representations of the physically relevant form U h ⊕ U −h for integer helicities h. In order to prove the result, we want to disintegrate the restriction of U h to thẽ G 3 subgroup, to check the condition (MC) on the disintegration and to apply Theorem 4.2. The Poincaré representations are obtained by induction and the Mackey subgroup theorem teaches how to make the disintegration for such kind of representation.
Let H 1 and H 2 be subgroups of a locally compact group G. Then H 1 \G/H 2 is the double coset, i.e., the set of the equivalence classes [g] = H 1 gH 2 , with g ∈ G.
Definition 4.7. [Ma52] Let G be a separable locally compact group.
Closed subgroups H 1 and H 2 of G are said to be regularly related if there exists a sequence E 0 , E 1 , E 2 , . . . of measurable subsets of G each of which is a union of double cosets in H 1 \G/H 2 such that E 0 has Haar measure zero and each double coset not in E 0 is the intersection of the E j which contain it.
Because of the correspondence between orbits of G/H 2 under H 1 and double cosets H 1 \G/H 2 , H 1 and H 2 are regularly related if and only if the orbits, (i.e., the double cosets) outside of a certain set of measure zero form the equivalence classes of a measurable equivalence relation. Given a topological standard measure space X, an equivalence relation ∼ and the quotient map s : X → Y = X/ ∼, the equivalence relation is said to be measurable if there exists a countable family {F n } n∈N of subsets of the quotient space Y , s.t. s −1 (F n ) is measurable and each point in Y is the intersection of all the F n ′ , n ′ ∈ N, containing this point.
Consider the map s : G → H 1 \G/H 2 carrying each element of G into its double coset. Then equip H 1 \G/H 2 with the quotient topology given by s and consider a finite measure µ on G which is in the same measure class 7 as the Haar measure. It is possible to define a measureμ on the Borel sets of H 1 \G/H 2 byμ(F ) = µ(s −1 (F )).
We shall callμ an admissible measure in H 1 \G/H 2 . The definition is well posed since any two of such measures have the same null measure sets.
General theory of induced representations can be found for instance in [Ma52,Fol16,BR86]. We recall the Mackey's subgroup theorem.
Theorem 4.8 (Mackey's subgroup theorem). [Ma52]. Let H 1 , H 2 be closed subgroups regularly related in G. Let π be a strongly continuous representation of H 1 . For each g ∈ G consider H g = H 2 ∩ (g −1 H 1 g) and set Hg (π • ad g). Then V g is determined to within equivalence by the double coset [g] to which g belongs. If ν is an admissible measure on H 1 \G/H 2 , then (4.1) An immediate application of Theorem 4.8 to the restriction of W r,n to G 3 is the following lemma, which entered into the proof of Corollary 4.5 above.
Lemma 4.9. The restriction of theG 3 representation W r,n to G 3 is W r,m , where m ≡ n (mod 2).
Of central importance for our analysis is the following proposition. Proof. The h-helicity representation U h is induced by the stabilizer Stab q 1 of the point q 1 = (1, 1, 0, 0). Again Stab q 1 is isomorphic to R 1+3 ⋊ E(2). In Theorem 4.8 we can consider G = P ↑ + , H 1 = R 1+3 ⋊ E(2) and we want to study the restriction of U h to H 2 =G 3 . We postpone the proof of the fact that H 1 and H 2 are regularly related.
Let us now describe the equivalence class of an admissible measure (cf. Definition 4.7). Starting with a finite measure µ on G in the equivalence class of the Haar measure, we induce a measure on R + ≃H 1 \G/H 2 , which we prove to be in the measure class of the Lebesgue measure. Indeed, let W r be the representation W r −1 ,2h and W 0 be V [r 2 (−π/2)] , we get the formula: (4.3) Finite helicity representations U h extend to the conformal group (cf. [Ma77]). In particular, by dilation covariance, we have that U(δ(t))U h U(δ(t)) * ≃ U h hence U(δ(t))U h |G 3 U(δ(t)) * ≃ U h |G 3 . Dilations change the unitary class of W r,2h dilating the radius of the representation 9 , since (4.4) Similarly as in Lemma 4.1 in [LMR16], this is a consequence of the following computation: where q e t r −1 := (e t r −1 , e t r −1 , 0, 0) and in the second step we used Lemma 4.1 of [LMR16]. An analogous computation can be done for W 0 in order to show that W 0 • ad(δ(t)) ≃ W 0 : since ad(λ 3 ) does not change the unitary equivalence class of thẽ G 3 -representations where we used (χ q 0 · χ 2h ) • ad(λ 3 (t)) • ad(δ(t)) = (χ λ 3 (−t)δ(t)q 0 · χ 2h ) = χ q 0 · χ 2h referring again to Lemma 4.1 of [LMR16]. Therefore, where µ t (r):=µ(e −t r). We show that µ is equivalent to the Lebesgue measure: assume by contradiction that there exists a set E ⊂ R + \{0} such that µ(E) > 0 but µ t (E) = 0 and consider the multiplication operator by the projection P E := ⊕ E dµ(r) ∈ U h (G 3 ) ′ . Then we have that the subrepresentation P E U h |G 3 P E is not contained in U(δ(t))U h |G 3 U(δ(t)) * (representations of radius r ∈ E have measure zero in the latter representation). In particular for every t ∈ R, µ t is equivalent to µ, hence to the Lebesgue measure on R + \{0} up to a possible singular measure in 0 (see Proposition 11 of [Bo04]). Since σ 0 has null measure in the joint spectrum of translation in U h , by comparing the translation spectrum in left and right side of (4.3), {r = 0} has null µ−measure. Now the statement of the theorem is obtained by a change of variables r → 1 r . By Propositions 4.6, 4.10 and 4.4 (ii) we conclude the modularity condition for finite helicity representations: Corollary 4.11. For every h ∈ Z 2 , U h and its multiples satisfy (MC). Proposition 4.12. If h is an integer, namely U h is bosonic, then any finite or infinite multiple of U h ⊕ U −h satisfies (MC).
The main result of this section is a corollary of Theorem 4.2: Theorem 4.13. Every net of real subspaces H undergoing the action of a finite or infinite multiple of U = U h ⊕ U −h , where h ∈ Z, satisfies the B-W and the duality properties.
A final remark on finite helicity one particle nets is the following. Massless nonzero finite helicity representations of the Poincaré group have to be properly coupled in order to act consistently on a net of standard subspaces on spacelike cones 10 . Indeed, by Theorem 4.2, U h satisfies the modularity condition (MC) and any net of standard subspaces it acts covariantly on satisfies the B-W property. Following [GL95], when the B-W property holds then by spacelike cone localization property one deduces that U h is covariant under the action of the wedge modular conjugations, namely U h extends to an (anti-)unitary representation of the group P + = P ↑ + , Θ , where Θ is the space and time reflection. The extension is unique up to unitary equivalence (Proposition 2.3 of [LMPR] or [NO17] for an abstract discussion). This is not possible for the irreducible finite helicity representations as they are not induced by a selfconjugate representation of the little group (cf. for instance [Va85]). In particular any anti-unitary operator implementing the PT symmetry (no charge C considered in this one particle setting) takes U h into U −h .

Bisognano-Wichmann property and asymptotic completeness
In this section we apply Theorem 4.13 to a concrete one-particle net of standard subspaces in the subspace H (1) from Definition 2.3 and then verify the Bisognano-Wichmann property on H out using scattering theory.
Remark 5.2. The locality property for the net W → H (1) (W ) can be extracted from [Bu77], where was obtained for A 1 , A 2 localized in spacelike separated double cones by the JLD technique. In our context the same property follows from the Borchers theorem and Lemma 3.9.
Proof. (ii) For helicities as in the statement of the proposition we obtain the B-W property for the one-particle net from Theorem 4.13.
(iii) Let ξ i ∈ H (1) (W ), i = 1, 2, and P (1) A i,n Ω, n ∈ N, be the corresponding approximating sequences with A * i,n = A i,n . Then After this preparation we give a massless version of Lemma 6 and Proposition 7 from [Mu01] and thereby conclude the proof of the Bisognano-Wichmann property for asymptotically complete massless theories, as stated in Theorem 2.5.
Lemma 5.3. In each n-particle subspace H (n) , n ≥ 2, there is a total set of scattering states Ψ out := A out n {f n } + . . . A out Proof. We consider an arbitrary scattering state Ψ out = (Φ n out × · · · out × Φ 1 ) constructed using A 1 , . . . , A n localized in some arbitrary double conesÕ 1 , . . . ,Õ n and arbitrary smooth functions f 1 , . . . f n on S 2 . By cutting the sphere of velocities into small slices with planes orthogonal to the 1-st axis, we can approximate Ψ out with linear combinations of scattering states Ψ out 1 such that the projections of suppf i on the 1-st axis are disjoint. (Cf. Proposition 3.2, formula (2.14) and the absolute continuity of the momentum spectral measure [BF82]). Let f i 0 be such function that Up to numbering, these scattering states satisfy the condition from the lemma concerning velocities, but possibly not the condition concerning localisation regions. Therefore, using again Proposition 3.2, formula (2.14) and the Reeh-Schlieder property for wedges, we approximate each Ψ out . By changing the numbering, we obtain the claim.
Proposition 5.4. If the unitary groups R ∋ s →∆ is A(W 1 ) and R ∋ s → U(λ W 1 (−2πs)) coincide on H (1) , they also coincide on the subspace H out of scattering states.
Proof.Let V s := ∆ is A(W 1 ) U(Λ W 1 (2πs)). By induction over the particle number n, we show that V s is the unity on each H (n) . Let Ψ out = (Φ n out × · · · out × Φ 1 ) be a scattering state as in Lemma 5.3 with Φ i = A out i {f i }Ω. For n = 0 we have Φ out = Ω and the statement follows from the invariance of the vacuum under U( · ) and s → ∆ is A(W 1 ) . For n = 1 the statement holds by Proposition 5.1 (ii). Now let n ≥ 2 be arbitrary and suppose that the statement holds for n ′ < n. Making use of Proposition 3.1 (d), we write where, by the canonical commutation relations for the asymptotic creation/annihilation operators (cf. formula (3.2)), the compensating vectorΨ out has components only in H (ℓ) for ℓ < n. Thus we have, by the induction hypothesis, V sΨ out =Ψ out , and it suffices to considerΨ out := A out n {f n } . . . A out 1 {f 1 }Ω. We can write where in the last step we used the induction hypothesis. (We also used that the state A out n−1 {f n−1 } . . . A out 1 {f 1 }Ω is in D P 0 , cf. Proposition 3.1, and the fact that due to the Borchers theorem, V s leaves D P 0 invariant. Consequently, the limit A out n {f n } still exists in the last line above). Using again that V s commutes with translations, we (5.10) Concerning (5.10), we use that (A ′ n ) out {f n }Ω = V s A out n {f n }V −1 s Ω = A out n {f n }Ω, since V s preserves both the vacuum and single-particle subspace. This vector coincides witĥ Ψ out provided that we can show for all i = 1, 2, . . . , n−1. This follows from formulas (2.14), (2.15) and the disjointness of supports of f n , f i (cf. Lemma 5.3). Thus to conclude the proof of the proposition we have to show that the terms in (5.9) are zero. Since A ′ n is only wedge-local, we cannot use property (2.15) and we need to proceed via a direct computation: For unit vectors Ψ 1 , Ψ 2 ∈ D P 0 we write verifying this property at the single particle level. For this purpose, the single-particle subspace H (1) is equipped with the structure of a local net of standard subspaces. This net is covariant w.r.t. a representation U (1) := U| H (1) of the Poincaré group, which is a direct sum of two representations of opposite integer helicities, i.e., U (1) = U h ⊕ U −h (or a multiple thereof). Then we verified the modularity condition for the B-W property U (1) (r W ) ∈ U (1) (G W ) ′′ , where G W is the subgroup of Poincaré transformations preserving a wedge W and r W maps W to the opposite wedge. This technically demanding step was accomplished by showing that U h and U −h have the same restriction to the groupG W generated by G W and r W . Hence U (1) |G W = U h |G W ⊗ 1 and the modularity condition could be concluded from earlier results [Mo18]. Given the B-W property at the single-particle level, the B-W property of the full theory was verified using scattering theory and asymptotic completeness. In this part we adapted the arguments of Mund [Mu01] to the massless case.
A natural question for future research is a generalization of our arguments to particles with half-integer helicities. The obstruction comes from the fact that in this case U h and U −h are unitarily equivalent when restricted to G 3 but have disjoint restrictions toG 3 and one cannot apply Proposition 4.4 (iii). Another future research direction is to relax the assumption of asymptotic completeness. We remark that in the vacuum sector of QED asymptotic completeness of photons can be assumed only below a certain energy threshold, excluding the electron-positron pair production. It is an interesting question how to prove the B-W property in this physically relevant situation. In this context we remark that our results give the B-W property of the net of asymptotic photon fields of QED, defined at the end of Sect. 2.2. This net plays an important role in the study of infrared problems (see e.g. [Bu77, BD84, AD17]) and we hope that our results will also find applications there.

A Direct integral representations
We suggest [Tak02,Dix81,MT18] as further references for basic definitions.

B Proof of Proposition 4.6
For fundamental concepts on direct integral of representation see Appendix A.
The function of the translation generators P 2 1 + P 2 2 is a Casimir operator for G 3 and decomposes according to µ, i.e., P 2 1 + P 2 2 = ⊕ R + r 2 · 1dµ(r) and P 2 1 + P 2 2 is affiliated to U(G 3 ) ′′ . By definition, bounded functions of r = P 2 1 + P 2 2 generate the diagonal algebra D. Thus D is contained in the center of U(G 3 ) ′′ , hence any operator in U(G 3 ) ′ is decomposable since it commutes with D.
We now pick T ∈ U(G 3 ) ′ then T is a decomposable operator, namely, Assume that there exists a positive measure set I s.t. T (r) is not in U r (G 3 ) ′ for r ∈ I. Let χ be a characteristic function of I. Then ⊕ R + T (r)χ(r)dµ(r) is not in the commutant of U(G 3 ), which is a contradiction. We conclude that and thus by [Tak02,Theorem 8.18]. Now we recall that U r satisfies (MC) by Corollary 4.5. Since, by assumption, U(r 1 (π)) = ⊕ R + U r (r 1 (π))dµ(r), we obtain from (B.3) that U satisfies (MC).

C Regularity of the actions on R 1+3
Definition C.1. Let G be a locally compact, σ-compact, group and N be a normal abelian subgroup, then the (dual-)action of G onN is regular if R1. the orbit space is countably separated, namely there exists a countable family {E n } n∈N of G-invariant Borel sets inN s.t. each orbit inN is the intersection of all E n that contain it, R2. each orbit is relatively open in its closure.
Here we check that the action of G 0 3 andG 0 3 on R 1+3 is regular according to the previous definition.
In the following we shall say that a family of sets selects an orbit o if the latter is the intersection of all the set of the family containing o. The set selecting an orbit of a group will be invariant under the group action. All the families we will consider will be countable as well as their union.
Firstly, any orbit of G 0 3 orG 0 3 is contained in a Lorentz orbit in R 1+3 . The family in A selects the Lorentz orbits. The orbit in the origin is selected by O. Now G 0 3 andG 0 3 share the same massive orbits, contained in p 2 = m 2 , m > 0, that can be selected by considering A and E families. Now consider a massless orbit in the forward lightcone. If for every p ∈ o, p 2 1 + p 2 2 > 0 then it is both a G 0 3 andG 0 3 orbit (cf. Remark 3.13) and can be selected by the families A and E. If p 2 1 + p 2 2 = 0 then the two G 0 3 orbits {p : 0 < p 0 = p 3 } and {p : 0 < p 0 = −p 3 } are selected by K ±,+ . If p 2 1 + p 2 2 = 0 then thẽ G 0 3 orbit is selected byK + . We argue analogously for the backward lightcone, referring to sets K ±,− ,K − . Now consider imaginary mass orbits, contained in p 2 = −m 2 and assume that p 2 1 + p 2 2 = r 2 . We have three cases: • r 2 < −m 2 . In this case we have two branches of the hyperboloid p 2 0 − p 2 3 = m 2 + r 2 < 0 that become two G 0 3 orbits and a uniqueG 0 3 orbit. The G 0 3 andG 0 • r 2 = −m 2 . G 0 3 orbits are selected by Z a 1 ,a 2 ,±,± or X a 1 ,a 2 .G 0 3 orbits are selected byZ a 1 ,a 2 ,± .