The Bisognano–Wichmann Property for Asymptotically Complete Massless QFT

Abstract

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.

1 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 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]. 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. We stress the fact that even if the main result is obtained for bosonic theories, the analysis of the modularity condition is done including fermionic representations.

2 Framework and Results

2.1 Local nets and the Bisognano–Wichmann property

Let ${\mathbb {R}}^{1+3}$ be the Minkowski spacetime. We denote by ${\mathcal {K}}$ the family of double cones $\mathcal {O}\subset {\mathbb {R}}^{1+3}$ ordered by inclusion and write $\mathcal {O}'$ for the causal complement of $\mathcal {O}$ in ${\mathbb {R}}^{1+3}$. Furthermore, let $\widetilde{\mathcal {P}}_+^\uparrow ={\mathbb {R}}^4\rtimes {\mathrm {SL}(2,{{\mathbb {C}}})}$ denote the covering group of the proper ortochronous Poincaré group ${\mathcal {P}}_+^\uparrow$. We denote with $\Lambda :{\widetilde{{\mathcal {P}}}}_+^\uparrow \rightarrow {\mathcal {P}}_+^\uparrow$ the covering map.

Definition 2.1

Let $\mathfrak {H}$ be a fixed Hilbert space. We say that ${\mathcal {K}}\ni \mathcal {O}\mapsto {\mathcal {A}}(\mathcal {O})\subset B(\mathfrak {H})$ is a local net of von Neumann algebras in a vacuum representation if the following properties hold:

• 1. Isotony: ${\mathcal {A}}(\mathcal {O}_1)\subset {\mathcal {A}}(\mathcal {O}_2)$ for $\mathcal {O}_{1}\subset \mathcal {O}_2$.

• 2. Poincaré covariance: there is a continuous unitary representation $U$ of $\widetilde{\mathcal {P}}_+^\uparrow$ such that

  $$\begin{aligned} U(g){\mathcal {A}}(\mathcal {O})U(g)^*={\mathcal {A}}({\Lambda (g)} \mathcal {O})\quad \mathrm {for}\quad g \in \widetilde{\mathcal {P}}_+^\uparrow . \end{aligned}$$

  (2.1)

  We assume that U factors on ${{\mathcal {P}}}_+^\uparrow$.

• 3. Positivity of the energy: the joint spectrum of translations in U is contained in the forward lightcone $V_+=\{ p\in {{\mathbb {R}}}^{1+3}: p^0\ge 0, \, p^2=(p,p)\ge 0\}$.

• 4. Cyclicity of the vacuum: there is a unique (up to a phase) unit vector $\Omega \in \mathfrak {H}$, the physical vacuum state, which is U-invariant and cyclic for the global algebra ${\mathcal {A}}:=\overline{\bigcup _{{{\mathcal {O}}}\subset {{\mathbb {R}}}^{1+3}}{{\mathcal {A}}}({{\mathcal {O}}})}^{\Vert \,\cdot \,\Vert }$ of the net.

• 5. Locality: $\, {\mathcal {A}}(\mathcal {O}_{1})\subseteq {\mathcal {A}}(\mathcal {O}_{2})'$ for $\mathcal {O}_{1}\subset \mathcal {O}_{2}'$.

A local net of von Neumann algebras will be denoted by $({\mathcal {A}},U, \Omega )$.

For future reference, we set for any region ${\mathcal {U}}\subset {\mathbb {R}}^{1+3}$

$$\begin{aligned} {\mathcal {A}}_{\mathrm {loc}}(\mathcal {U}):=\bigcup _{\mathcal {O}\subset {\mathcal {U}}} {\mathcal {A}}(\mathcal {O}) \quad \text { and } \quad {\mathcal {A}}(\mathcal {U}):={\mathcal {A}}_{\mathrm {loc}}({\mathcal {U}})'' \end{aligned}$$

(2.2)

and we refer to ${\mathcal {A}}_{\mathrm {loc}}:={\mathcal {A}}_{\mathrm {loc}}({\mathbb {R}}^{1+3})$ as the algebra of strictly local operators.

In order to introduce the B–W property, we need some geometric preliminaries: a wedge region $W\subset {{\mathbb {R}}}^{1+3}$ is an open region of the form $gW_1$ where $g\in {{\mathcal {P}}_+^\uparrow }$ and $W_1=\{x\in {{\mathbb {R}}}^{1+3}: |x_0|<x_1\}$. The set of wedges is denoted by ${{\mathcal {W}}}$. Note that if $W\in {{\mathcal {W}}}$, then $W'\in {{\mathcal {W}}}$. It is possible to associate to any wedge a one-parameter group of boosts $\Lambda _W$ fixing the wedge W by the following formula for $W_1$

$$\begin{aligned} {\mathbb {R}}\ni t\rightarrow \Lambda _{W_{1}}(t):=\begin{pmatrix} \cosh (t)&{}\quad \sinh ( t)&{}\quad 0&{} \quad 0\\ \sinh ( t)&{}\quad \cosh (t)&{}\quad 0&{}\quad 0\\ 0&{} \quad 0&{}\quad 1&{}\quad 0\\ 0&{} 0&{} 0&{} 1 \end{pmatrix} \end{aligned}$$

(2.3)

and the covariant action of the Poincaré group on the set of wedges.

We call $W_\alpha =\{x\in {{\mathbb {R}}}^{1+3}: |x_0|<x_\alpha \}$, $\alpha =1,2,3$, the wedge in the $x_\alpha$ direction and $R_\alpha$, $\Lambda _\alpha$ are the one-parameter groups of rotations and boosts, respectively, fixing $W_\alpha$. Their unique one parameter group lifts to ${\mathrm {SL}(2,{{\mathbb {C}}})}$ are denoted $r_\alpha$ and $\lambda _\alpha$. In general $\lambda _W$ will denote the one parameter group lift of $\Lambda _W$. Note that $\lambda _\alpha (t)=e^{\frac{t}{2}\,\sigma _\alpha }$ and $r_\alpha (\theta )=e^{i\frac{\theta }{2} \sigma _\alpha }$ where $t,\theta \in {{\mathbb {R}}}$ and $\sigma _\alpha$ are the Pauli matrices. In particular one has that $r_\alpha (2\pi )=-I=: r(2\pi )$.

For any $W\in {{\mathcal {W}}}$ we define ${\mathcal {A}}(W)$ according to (2.2). It is well known that the vacuum is cyclic and separating for ${\mathcal {A}}(W)$ thus the Tomita-Takesaki theory gives the corresponding modular evolution ${\mathbb {R}}\ni t\mapsto \Delta _{W}^{it}$.

Definition 2.2

We say that a local net $({\mathcal {A}},U, \Omega )$ satisfies the Bisognano–Wichmann property if for all $W\in {{\mathcal {W}}}$, $t\in {\mathbb {R}}$,

$$\begin{aligned} U(\lambda _W(2\pi t))=\Delta ^{-it}_{W}. \end{aligned}$$

In this discussion we will deal with bosonic net, namely U factorizing on ${{\mathcal {P}}_+^\uparrow }$. References for nets of von Neumann algebras including fermionic representations are [LMR16, Mo18].

2.2 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]. For the unitary representation of translations $U|_{{\mathbb {R}}^4}$ we shall write $U(x)=e^{i(P^0x^0-{\varvec{P}}\cdot {\varvec{x}})}$, where $P^0, {\varvec{P}}$ are the energy-momentum operators. Now we introduce the single-particle subspace.

Definition 2.3

A local net $({\mathcal {A}},U, \Omega )$ describes massless Wigner particles if $\mathfrak {H}$ contains a subspace $\mathfrak {H}^{(1)}\ne \{0\}$ s.t.

$$\begin{aligned} \mathrm {Ran}{\varvec{1}}_{\{0\}}( M)={\mathbb {C}}\Omega \oplus \mathfrak {H}^{(1)}, \end{aligned}$$

(2.4)

where ${\varvec{1}}_{\{0\}}(M)$ denotes the spectral projection of the mass operator $M:=\sqrt{(P^0)^2-\pmb {P}^2}$ corresponding to the eigenvalue zero. We say that these particles have helicities $h_1,h_2,h_3\ldots \in {\mathbb {Z}}$ if $U |_{\mathfrak {H}^{(1)}}$ 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 translates of observables $A\in {\mathcal {A}}$ the notations $\alpha _x(A):=A(x):=U(x)AU(x)^*$ are used. If $g\in L^1({\mathbb {R}}^4)$, then $A(g):=\int A(x) g(x) d^4x$ denotes the operator A smeared with the function g. Moreover, we set

$$\begin{aligned} {\mathcal {A}}_{\mathrm {loc},0}:=\{A\in {\mathcal {A}}_{\mathrm {loc}} \,:\, x\mapsto A(x)\, \text { smooth in norm}\}. \end{aligned}$$

(2.5)

This is a weakly dense $*$-subalgebra of ${\mathcal {A}}_{\mathrm {loc}}$, as can be seen by smearing local operators with delta-approximating functions. Next, we specify the following Poincaré invariant subset of $C_0^{\infty }({\mathbb {R}}^4)$

$$\begin{aligned} C_*({\mathbb {R}}^4):=\{ (n_{\mu }\partial ^{\mu })^{5}g \,:\, g\in C_0^{\infty }({\mathbb {R}}^4), \,\, n_0=\sqrt{1+{\varvec{n}}^2}\,\}, \end{aligned}$$

(2.6)

where ${\varvec{n}}$ is not fixed, and define

$$\begin{aligned} {\mathcal {A}}_{C_*}&:=\{\, B(g) \,:\, B\in {\mathcal {A}}_{\mathrm {loc},0}, \ g\in C_*({\mathbb {R}}^4) \,\}, \end{aligned}$$

(2.7)

$$\begin{aligned} {\mathcal {A}}^{C_*}&:=\mathrm {Span}\, {\mathcal {A}}_{C_*}, \end{aligned}$$

(2.8)

$$\begin{aligned} {\mathcal {A}}_{C_*}(\mathcal {O})&:={\mathcal {A}}_{C_*}\cap {\mathcal {A}}(\mathcal {O}), \quad {\mathcal {A}}^{C_*}(\mathcal {O}):={\mathcal {A}}^{C_*}\cap {\mathcal {A}}(\mathcal {O}), \quad \mathcal {O}\in {\mathcal {K}}. \end{aligned}$$

(2.9)

Now we move on to the construction of asymptotic fields of massless particles. For any $A\in {\mathcal {A}}^{C_*}$ and $f\in C^{\infty }(S^2)$, we set as in [Bu77, Bu82]

$$\begin{aligned} A_t\{f\}:=-2\,t\int d\omega ({\varvec{n}})\,f({\varvec{n}})\,\partial _0A(t,t{\varvec{n}}). \end{aligned}$$

(2.10)

Here $d\omega ({\varvec{n}})=\frac{\sin \nu \,d\nu d\varphi }{4\pi }$ is the normalized, invariant measure on $S^2$ and $\partial _0A:=\partial _s(\mathrm {e}^{\mathrm {i}sP^0}A\mathrm {e}^{-\mathrm {i}sP^0})|_{s=0}$. In order to improve the convergence in the limit of large t, we proceed to time averages of $A_t\{f\}$, namely

$$\begin{aligned} {\bar{A}}_t\{f\}:=\int \,dt'\,h_t(t')\,A_{t'}\{f\}. \end{aligned}$$

(2.11)

Here for non-negative $h\in C_0^{\infty }({\mathbb {R}})$, supported in the interval $[-1,1]$ and normalized so that $\int dt\, h(t)=1$, we set $h_t(t')=t^{-{{\bar{\varepsilon }}}} h(t^{-{{\bar{\varepsilon }}}}(t'-t))$ with $t\ge 1$ and $0<{\bar{\varepsilon }}<1$. It turns out that these limits exist on all vectors from the domain

$$\begin{aligned} D_{P^0}:=\bigcap _{n\ge 1} D((P^{0})^n), \end{aligned}$$

(2.12)

where $D((P^{0})^n)$ is the domain of self-adjointness of $(P^{0})^n$.

Lemma 2.4

Let $A\in {\mathcal {A}}^{C_*}(\mathcal {O})$ and $f\in C^{\infty }(S^2)$. Then, the limit

$$\begin{aligned} A^{\mathrm {out}}\{f\}\Psi =\lim _{t\rightarrow \infty }{\bar{A}}_t\{f\}\Psi \end{aligned}$$

(2.13)

exists for $\Psi \in D_{P^0}$ and is again an element of $D_{P^0}$.

The operators $A^{\mathrm {out}}\{f\}$ are constructed in such a way that they create single-particle states from the vacuum, namely

$$\begin{aligned} A^{\mathrm {out}}\{f\}\Omega =P^{(1)} f\left( \tfrac{{\varvec{P}}}{|{\varvec{P}}|}\right) A\Omega , \end{aligned}$$

(2.14)

where $P^{(1)}$ is the projection on the single-particle subspace $\mathfrak {H}^{(1)}$. Vectors of the form (2.14) span a dense subspace of $\mathfrak {H}^{(1)}$, even in the case $f\equiv 1$. Furthermore, if $A^{\mathrm {out}}\{f\}$, $A'^{\mathrm {out}}\{f'\}$ are two asymptotic fields as specified above, then

$$\begin{aligned} \,[A^{\mathrm {out}}\{f\}, A'^{\mathrm {out}}\{f'\}]=\langle \Omega ,[A^{\mathrm {out}}\{f\}, A'^{\mathrm {out}}\{f'\}]\Omega \rangle 1_{\mathfrak {H}} \end{aligned}$$

(2.15)

as operators on $D_{P^0}$. For $f\equiv 1$ the operators $A^{\mathrm {out}}\{f\}$ appearing in Lemma 2.4 are denoted $A^{\mathrm {out}}$ 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 $\mathcal {O}\in {\mathcal {K}}$ we introduce the von Neumann algebra:

$$\begin{aligned} {\mathcal {A}}^{\mathrm {out}}(\mathcal {O}):=\{\, e^{i A^{\mathrm {out}}} \,:\, A\in {\mathcal {A}}^{C_*}(\mathcal {O}), \ A^*=A \,\}''. \end{aligned}$$

(2.16)

The triple $({\mathcal {A}}^{\mathrm {out}},U, \Omega )$ 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 $({\mathcal {A}},U, \Omega )$ is asymptotically complete. Clearly, for the definition of asymptotic completeness the case $f\equiv 1$ suffices. However, the operators $A^{\mathrm {out}}\{ f\}$ 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 $({\mathcal {A}}, U,\Omega )$ be a local net containing massless particles with helicity zero or with helicities $(h, -h)$ for some $h\in {\mathbb {N}}$. If this net is asymptotically complete, then it satisfies the Bisognano–Wichmann property.

Proof

By Theorem 4.13 and Proposition 5.1 we obtain the equality $U(\lambda _W(2\pi t))=\Delta ^{-it}_{W}$ from Definition 2.2 on the single-particle subspace (Definition 2.3). Then, using the assumption of asymptotic completeness and Proposition 5.4 we extend this equality to the entire Hilbert space. $\quad \square$

Even if the original net $({\mathcal {A}}, U,\Omega )$ is not asymptotically complete, we can set $\mathfrak {H}^{\mathrm {out}}:={ \overline{{\mathcal {A}}^{\mathrm {out}}\Omega }}$ and define the asymptotic net $({\mathcal {A}}^{\mathrm {out}} |_{ \mathfrak {H}^{\mathrm {out}} }, U |_{ \mathfrak {H}^{\mathrm {out}} }, \Omega )$ 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 $(h, -h)$, $h\in {\mathbb {N}}$.

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 ${\mathcal {A}}$ (cf. [Bu75, Appendix]). From the spin-statistics theorem of [GL95] it follows that such nets are actually covariant under ${{\mathcal {P}}_+^\uparrow }$. Furthermore, by the CPT theorems of this reference, the unitary representation U of ${{\mathcal {P}}_+^\uparrow }$ extends to an (anti-)unitary covariant representation of ${{\mathcal {P}}}_+$ (the group generated by ${{\mathcal {P}}_+^\uparrow }$ and the PT operator $\Theta$) as follows:

Corollary 2.6

Let $({\mathcal {A}}, U,\Omega )$ be a local net as in Theorem 2.5. Then U extends to an (anti-)unitary representation of the Poincaré group ${{\mathcal {P}}}_+$ by

$$\begin{aligned} J_{W_1}U(R_1(\pi ))=U(\Theta ) \end{aligned}$$

where $\Theta { x}=-{ x}$ with ${ x}\in {{\mathbb {R}}}^{1+3}$, $J_{W_1}$ is the modular conjugation associated to $({\mathcal {A}}(W_1),\Omega )$ and $R_1(\pi )$ is the rotation by $\pi$ around the first axis.

As a consequence the assumption to have a theory containing opposite helicities is forced by the occurrence of the B–W property. Indeed, h and $-h$ helicity representations are conjugated through the PCT symmetry and both have to appear under PCT covariance.

3 Preliminaries

3.1 Scattering states of massless particles

In this subsection we provide some preparatory information about the Hilbert space of scattering states $\mathfrak {H}^{\mathrm {out}}:={ \overline{{\mathcal {A}}^{\mathrm {out}}\Omega }}$ 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 $\theta \in C^{\infty }({\mathbb {R}})$, $0\le \theta \le 1$, be supported in $(0,\infty )$ and equal to one on $(1,\infty )$. Moreover, let $\beta \in C_0^{\infty }({\mathbb {R}}^4)$, $0\le \beta \le 1$, be equal to one in some neighbourhood of zero and satisfy $\beta (-p)=\beta (p)$. Furthermore, for a parameter $1\le r<\infty$ and a future oriented timelike unit vector n we define

$$\begin{aligned} \widetilde{\eta }_{\pm ,r}(p):=\theta (\pm r (n_{\mu }p^{\mu }))\beta (r^{-1} p), \end{aligned}$$

(3.1)

where tilde denotes the Fourier transform. As $r\rightarrow \infty$ these functions approximate the characteristic functions of the positive/negative energy half planes $\{\, p\in {\mathbb {R}}^4 \,:\, \pm n_{\mu }p^{\mu }\ge 0\,\}$. We also have ${\bar{\eta }}_{\pm ,r}=\eta _{\mp ,r}$. Note that the family of functions $\eta _{\pm ,r}$, as specified above, is invariant under Lorentz transformations.

Proposition 3.1

[Bu77, AD17]. Let $A\in {\mathcal {A}}_{C_*}$, $f\in C^{\infty }(S^2)$. Suppose that the timelike unit vectors n entering the definition of A and of $\eta _{\pm ,r}$ coincide. Then:

• (a) The limits $A^{\mathrm {out}}\{f\}^{\pm }\Psi :=\lim _{r\rightarrow \infty }A^{\mathrm {out}}\{f\}(\eta _{\pm ,r})\Psi$, $\Psi \in D_{P^0}$, exist and define the creation and annihilation parts of $A^{\mathrm {out}}\{f\}$ as operators on $D_{P^0}$. $A^{\mathrm {out}}\{f\}^{\pm }$ do not depend on the choice of the functions $\theta$ and $\beta$ in (3.1) within the specified restrictions.

• (b) $(A^{\mathrm {out}}\{f\}^{\pm })^*|_{D_{P^{0}} }=A^{*\mathrm {out}}\{{{\bar{f}}}\}^{\mp }$. In particular, $A^{\mathrm {out}}\{f\}^{\pm }$ are closable operators.

• (c) $A^{\mathrm {out}}\{f\}^{\pm }D_{P^0}\subset D_{P^0}$.

• (d) $A^{\mathrm {out}}\{f\}=A^{\mathrm {out}}\{f\}^{+}+A^{\mathrm {out}}\{f\}^{-}$ on $D_{P^0}$.

Making use of Proposition 3.1 and of (2.15) we also obtain on $D_{P^0}$

$$\begin{aligned} \,[A^{\mathrm {out}}\{f\}^-, A'^{\mathrm {out}}\{f'\}^+ ]=\langle A^{*\mathrm {out}}\{{\bar{f}}\}^+\Omega , A'^{\mathrm {out}}\{f'\}^+ \Omega \rangle 1_{\mathfrak {H}} \end{aligned}$$

(3.2)

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 $f \equiv 1$. The proof is an obvious application of the canonical commutation relations (3.2).

Proposition 3.2

[Bu77, AD17]. The states $\Psi ^{\mathrm {out}}:= A_1^{\mathrm {out}}\{f_1\}^+ \ldots A_n^{\mathrm {out}}\{ f_n\}^+\Omega$ have the following properties:

• (a) $\Psi ^{\mathrm {out}}$ depends only on the single-particle states $\Phi _i=A_i^{\mathrm {out}}\{f\}\Omega \in \mathfrak {H}^{(1)}$. Therefore, we write $\Psi ^{\mathrm {out}}=\Phi _1\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi _n$.

• (b) $\langle \Phi _1\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi _n,\Phi '_1\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi '_{n'}\rangle = \delta _{n,n'}\sum _{\sigma \in \mathfrak {S}_n}\langle \Phi _1, \Phi '_{\sigma _1}\rangle \ldots \langle \Phi _n, \Phi '_{\sigma _n}\rangle$, where $\mathfrak {S}_n$ is the set of all permutations of $(1,\ldots , n)$.

The subspace of $\mathfrak {H}$ spanned by vectors of the form $\Psi ^{\mathrm {out}}=\Phi _1\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi _n$ for fixed n will be denoted $\mathfrak {H}^{(n)}$. We note that

$$\begin{aligned} \mathfrak {H}^{\mathrm {out}}:={ {\overline{{\mathcal {A}}^{\mathrm {out}} \Omega }}}=\bigoplus _{n\ge 0}\mathfrak {H}^{(n)}, \end{aligned}$$

(3.3)

where $\mathfrak {H}^{(0)}={\mathbb {C}}\Omega$ and $\mathfrak {H}^{(1)}$ was introduced in Definition 2.3. The last equality in (3.3) follows from density of vectors of the form (2.14) in $\mathfrak {H}^{(1)}$ and from the canonical commutation relations (3.2) by standard Fock space arguments. Clearly, $\mathfrak {H}^{\mathrm {out}}$ is naturally isomorphic to the symmetric Fock space over $\mathfrak {H}^{(1)}$, denoted $\Gamma (\mathfrak {H}^{(1)})$.

3.2 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 the literature and that will be needed in our investigation.

A real linear, closed subspace H of a complex Hilbert space ${{\mathcal {H}}}$ is called cyclic if $H+iH$ is dense in ${{\mathcal {H}}}$, separating if $H\cap iH=\{0\}$ and standard if it is cyclic and separating.

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:

$$\begin{aligned} S_H:H+iH\ni \xi + i\eta \mapsto \xi - i\eta \in H+iH, \qquad \xi ,\eta \in H. \end{aligned}$$

The polar decomposition

$$\begin{aligned} S_H = J_H\Delta _H^{1/2} \end{aligned}$$

defines the positive self-adjoint modular operator $\Delta _H$ and the anti-unitary modular conjugation $J_H$. $\Delta _H$ is invertible and $J_H\Delta _H J_H=\Delta _H^{-1}.$

Let H be a real linear subspace of ${{\mathcal {H}}}$, the symplectic complement of H is defined by

$$\begin{aligned} H' := \{\xi \in {{\mathcal {H}}}\ :\ { \mathrm {Im}\langle \xi ,\eta \rangle }=0, \forall \eta \in H\} = (iH)^{\bot _{{\mathbb {R}}}}\ , \end{aligned}$$

where $\bot _{{\mathbb {R}}}$ denotes the orthogonal complement in ${{\mathcal {H}}}$ with respect to the real part of the scalar product on ${{\mathcal {H}}}$. $H'$ is a closed, real linear subspace of ${{\mathcal {H}}}$. 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

$$\begin{aligned} S_{H'} = S^*_H \ , \end{aligned}$$

with $J_{H'}=J_{H}$ and $\Delta _{H'}=\Delta ^{-1}_{H}=J_{H}\Delta _{H} J_{H}$ [Lo]. Furthermore, if H is standard, then $H = H''$. We recall that the one-parameter, strongly continuous group $t\mapsto \Delta _H^{it}$ is called the modular group of H and

$$\begin{aligned} \Delta _H^{it}H = H, \quad J_H H = H' \ ,\qquad t\in {{\mathbb {R}}}\ . \end{aligned}$$

There is a 1-1 correspondence between Tomita operators and standard subspaces.

Proposition 3.3

[Lo]. The map

$$\begin{aligned} H\longmapsto S_H \end{aligned}$$

(3.4)

is a bijection between the set of standard subspaces of ${{\mathcal {H}}}$ and the set of closed, densely defined, anti-linear involutions on ${{\mathcal {H}}}$.

The following are three basic results on standard subspaces.

Lemma 3.4

[Mo18]. Let $H,K\subset {{\mathcal {H}}}$ be standard subspaces and $U\in {{\mathcal {U}}}({{\mathcal {H}}})$ be a unitary operator on ${{\mathcal {H}}}$ such that $UH=K$. Then $U\Delta _H U^*=\Delta _K$ and $UJ_HU^*=J_K$.

Lemma 3.5

[Lo]. Let $H\subset {{\mathcal {H}}}$ be a standard subspace, and $K\subset H$ be a closed, real linear subspace of H. If $\Delta _H^{it}K=K$, $\forall t\in {{\mathbb {R}}}$, then K is a standard subspace of ${{\mathcal {K}}}:= \overline{K+iK}$ and $\Delta _H |_K$ is the modular operator of K on ${{\mathcal {K}}}$. Moreover, if K is a cyclic subspace of ${{\mathcal {H}}}$, then $H=K$.

Theorem 3.6

[Lo]. Let $H\subset {{\mathcal {H}}}$ be a standard subspace, and $U(t)=e^{itP}$ be a one-parameter unitary group on ${{\mathcal {H}}}$ with a generator $\pm P>0$, such that $U(t)H\subset H$, $\forall t\ge 0$. Then,

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta _H^{is}U(t)\Delta _H^{-is}= U(e^{\mp 2\pi s}t)&{}\\ J_{H}U(t)J_{H}=U(-t)\end{array}\right. \qquad \forall t,s\in {{\mathbb {R}}}. \end{aligned}$$

(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 $\Delta _{H}$. If $\xi '\in D(S_{H}^*)=H'+iH'$ and $\xi \in D(S_{H})=H+iH$, then, for such E

$$\begin{aligned} \langle \xi ', S_{H}E\xi \rangle&=\overline{\langle S_{H}^*\xi ', E\xi \rangle }= \overline{\langle J_{H}\Delta ^{-1/2}_{H}\xi ', E\xi \rangle }= \lim _{n\rightarrow \infty } \langle \chi _n(\Delta ^{-1/2}_{H}) { \Delta ^{-1/2}_{H}} \xi ', E J_{H} \xi \rangle \nonumber \\&= \lim _{n\rightarrow \infty } \langle \xi ', E \chi _n(\Delta ^{-1/2}_{H}){ \Delta ^{-1/2}_{H}} J_{H} \xi \rangle = \langle \xi ', E \Delta ^{-1/2}_{H} J_{H} \xi \rangle = \langle \xi ', E S_{H} \xi \rangle , \end{aligned}$$

(3.6)

where $\chi _n$ is the characteristic function of $[-n,n]$ and we made use of the fact that $\xi ', J_{H}\xi \in D( \Delta ^{-1/2}_{H})$ to control the limit $n\rightarrow \infty$.

Lemma 3.7

Let $H\subset {{\mathcal {H}}}$ be a standard subspace and $E=E^2=E^*$ be a projection commuting with $S_H$. Then $H=EH \oplus (1-E)H$. Furthermore, EH and $(1-E)H$ are standard in $E{{\mathcal {H}}}$ and $(1-E){{\mathcal {H}}}$, respectively.

Proof

H is defined to be the kernel of $1-S_H$. Now since E commutes with $S_H$, for every $\xi \in H$, $E\xi \in \mathrm {Ker} (1-S_H)$, thus $E\xi \in H$ (cf. computation (3.6) above). As the same argument applies to $(1-E)$ and $\xi =E\xi +(1-E)\xi$, we have the claim. The last statement is obvious. $\quad \square$

Lemma 3.8

Let $H\subset {{\mathcal {H}}}$ be a standard subspace and $E=E^2=E^*$ be a projection commuting with $S_H$. Then $H'= EH'\oplus (1-E)H'$. Furthermore, $EH'$ and $(1-E)H'$ are standard in $E{{\mathcal {H}}}$ and $(1-E){{\mathcal {H}}}$, respectively.

Proof

If $E\in {{\mathcal {B}}}({{\mathcal {H}}})$ is a projection commuting with $S_H$, then E also commutes with $S_{H'}=J_H\Delta _H^{-1/2}=\Delta ^{1/2}_H {J_H}$ and the decomposition $H'= EH'\oplus (1-E)H'$ follows as in Lemma 3.7. We note that $S_H$ and $S_{H'}$ and their polar decompositions decompose through E. (Clearly, $S_{EH}=S_{H} |_{E\mathcal {H}}$ and $S_{(1-E)H}=S_{H} |_{(1-E)\mathcal {H}}$). Consequently, $(EH)'=EH'$ and $((1-E)H)'=(1-E)H'$ on $E{{\mathcal {H}}}$ and $(1-E){{\mathcal {H}}}$, respectively. Since EH and $(1-E)H$ are cyclic and separating in $E{{\mathcal {H}}}$ and $(1-E){{\mathcal {H}}}$, the claim follows. $\quad \square$

An immediate consequence is:

Lemma 3.9

Let $H,K\subset {{\mathcal {H}}}$ standard subspaces and E a projection satisfying the assumptions of Lemma 3.8 w.r.t. H and K. Assume that $K\subset H'$. Then $EK\subset EH'$ and $(1-E)K\subset (1-E)H'$.

Proof

Since $H'=EH'\oplus (1-E)H'$ and $K=EK\oplus (1-E)K$, for every $\xi \in K$ we have $E\xi \in K$, thus $E\xi \in EH'$. We conclude that $EK\subset EH'$ and analogously $(1-E)K\subset (1-E)H'$. $\square$

3.3 One particle nets

Let U be a unitary representation of the Poincaré group ${\widetilde{{\mathcal {P}}}_+^\uparrow }$ on a Hilbert space ${{\mathcal {H}}}$. We shall call a U -covariant (or Poincaré covariant) net of standard subspaces on wedges a map

$$\begin{aligned} H:{{\mathcal {W}}}\ni W\longmapsto H(W)\subset {{\mathcal {H}}}, \end{aligned}$$

associating to every wedge in ${{\mathbb {R}}}^{1+3}$ a closed real linear subspace of ${{\mathcal {H}}}$, satisfying the following properties:²

• 1. Isotony: If $W_1,W_2\in {{\mathcal {W}}}$ and $W_1\subset W_2$ then $H(W_1)\subset H(W_2)$;

• 2. Poincaré covariance: $U(g)H(W)=H({\Lambda (g)}W),$ $\forall g\in {{\widetilde{{\mathcal {P}}}_+^\uparrow }},\,\forall W\in {{\mathcal {W}}}$. We assume that U factors on ${{\mathcal {P}}_+^\uparrow }$;

• 3. Positivity of the energy: the joint spectrum of translations in U is contained in the forward lightcone $V_+=\{\, p\in {{\mathbb {R}}}^{1+3}: p^0\ge 0, p^2=(p,p)\ge 0\}$;

• 4. Cyclicity: if $W\in {{\mathcal {W}}}$, then H(W) is a cyclic subspace of ${{\mathcal {H}}}$;

• 5. Locality: if $W_1\subset W_2'$ then $H(W_1)\subset H(W_2)'.$

Since $H(W')\subset H(W)'$ then by cyclicity H(W) are also separating, thus standard. 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:

• 6. Bisognano–Wichmann property: if $W\in {{\mathcal {W}}}$, then $U(\lambda _W(2\pi t))=\Delta ^{-it}_{H(W)},$ $\forall t\in {{\mathbb {R}}}.$

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]).

• 7. Duality property: if $W\in {{\mathcal {W}}}$, then $H(W)'=H(W')$.

References for the axioms including fermionic representations are [LMR16, Mo18].

Recall that $P^0, {\varvec{P}}$ are the generators of translations in the representation U and $M=\sqrt{(P^0)^2-{\varvec{P}}^2}$ the mass operator. Then Theorem 3.6 has the following corollary, which is well known in the context of nets of von Neumann algebras.

Corollary 3.10

For any wedge W, the mass operator M commutes strongly³ with $\Delta _{H(W)}$ and $J_{H(W)}$. Its real bounded Borel functions commute weakly with $S_{H(W)}$ on domains specified as in (3.6).

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 $\Delta _{H(W_1)}$ and $J_{H(W_1)}$ by Lemma 3.4. Lightlike translations of the form $a_\pm (t)=(\pm t, t, 0,0)$ with $t\ge 0$ have generators $P_\pm :=( \pm P_0-P_1)$ s.t. $\pm P_{\pm }\ge 0$ and $U(a_\pm (t))H(W_1)\subset H(W_1)$ for $t\ge 0$. By the Borchers theorem for standard subspaces (Theorem 3.6) we have that $U(a_\pm )$ have the commutation relations as in equation (3.5):

$$\begin{aligned} \Delta _{H(W_1)}^{is}U(a_\pm (t))\Delta _{H(W_1)}^{-is}= & {} U(a_\pm (e^{\mp 2\pi s}t))\Rightarrow {\Delta _{H(W_1)}^{is}f(P_\pm )\Delta _{H(W_1)}^{-is}=f(e^{\mp 2\pi s}P_\pm )} \\ J_{H(W_1)}U(a_\pm (t))J_{H(W_1)}= & {} U({ a}_\pm (-t))\Rightarrow J_{H(W_1)}{f(P_\pm )^{{ *}}} J_{H(W_1)}=f(P_\pm ), \end{aligned}$$

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 $P^2=M^2=-(P_+P_-+P^2_2+P_3^2)$ and for any real Borel function g it is easy to check, using the above relations, that $g(M^2)$ commutes with $\Delta _{H(W)}$ 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 $\Delta _{H(W)}$ and $J_{H(W)}$ it suffices to verify that

$$\begin{aligned} \Delta _{H(W_1)}^{is}e^{-i P_+P_- t} \Delta _{H(W_1)}^{-is}=e^{-i P_+P_- t}, \quad J_{H(W_1)}e^{-i P_+P_- t} J_{H(W_1)}=e^{i P_+P_- t}. \end{aligned}$$

This is achieved by approximating $e^{-i P_+P_- t}$ 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 $g\in {\mathcal {P}}_+^\uparrow$ s.t. $W=gW_1$. Then, by Lemma 3.4, $J_{H(W)}= U(g)J_{H(W_1)}U(g)^*$, $\Delta _{H(W)}= U(g)\Delta _{H(W_1)}U(g)^*$ and thus $S_W=U(g)S_{H(W_1)}U(g)^*$. Clearly $P^2=U(g)P^2U(g)^*$, thus $P^2$ commutes with $J_{H(W)}, \Delta _{H(W)}, S_{H(W)}$ for every $W\in {{\mathcal {W}}}$ in the same sense as discussed above. $\quad \square$

3.4 Induced representations: the Poincaré group and its sub-groups

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 $G=N\rtimes H$⁴. Assume that the action of G on ${\hat{N}}$, the dual group of N, obtained by conjugation, is regular (cf. [Fol16] Sect. 6.6 and Definition C.1). Let $p\in {\hat{N}}$, $\Omega _p$ be the orbit under the G-dual action,⁵ with $x\in N$, $p\in {\hat{N}}$ and $g\in G$, ${{\,\mathrm{Stab}\,}}_p$ and $\overline{{{\,\mathrm{Stab}\,}}}_p$ be the stabilizers of the point p under the action of H and G. ${{\,\mathrm{Stab}\,}}_p$ is called the little group. Let $\chi _p$ be the character associated to $p\in {\hat{N}}$.

Every unitary irreducible representation of G is obtained by induction in the following way (see e.g. [Fol16] Sect. 6)

$$\begin{aligned} {{\,\mathrm{Ind}\,}}_{\overline{{{\,\mathrm{Stab}\,}}}_p}^G ({\chi _p}\cdot V), \end{aligned}$$

(3.7)

where V and ${\chi _p}\,\cdot \,V$ are unitary representation s of the little group ${{\,\mathrm{Stab}\,}}_{p}$ and of $\overline{{{\,\mathrm{Stab}\,}}}_p$, respectively, and the following proposition holds:

Proposition 3.11

[Fol16]. Let $G=N\rtimes H$ as above. Every unitary irreducible representation of G is equivalent to one of the form (3.7). Furthermore ${{\,\mathrm{Ind}\,}}_{\overline{{{\,\mathrm{Stab}\,}}}_p}^G ({\chi _p}\cdot V)$ and ${{\,\mathrm{Ind}\,}}_{\overline{{{\,\mathrm{Stab}\,}}}_q}^G ({\chi _q}\cdot V')$ are equivalent if and only if p and q belongs to the same orbit, say ${p=g\, q}$, and V and $V'\circ ad_{g^{-1}}$ are equivalent representations of ${{\,\mathrm{Stab}\,}}_p$ for some $g\in G$.

If $W={{\,\mathrm{Ind}\,}}_{\overline{{{\,\mathrm{Stab}\,}}}_p}^G (\chi _p\cdot V)$ 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 ${{\mathbb {R}}}^{1+3}$ is the 4-dimensional real vector space endowed with the metric tensor $\eta =\mathrm {diag}(1,-1,-1,-1)$. The Lorentz group ${{\mathcal {L}}}$ is the group of linear transformations L s.t. $L^T\eta L=\eta$. Let ${{\mathcal {L}}}_+^\uparrow$ be the connected component of the identity of the Lorentz group and ${\widetilde{{{\mathcal {L}}}}}_+^\uparrow ={\mathrm {SL}(2,{{\mathbb {C}}})}$ its universal covering group. Let $\widetilde{{\mathcal {P}}}_+^\uparrow ={{\mathbb {R}}}^{1+3} \rtimes {\mathrm {SL}(2,{{\mathbb {C}}})}$ be the universal covering of the Poincaré group ${{\mathcal {P}}_+^\uparrow }={{\mathbb {R}}}^{1+3} \rtimes {{\mathcal {L}}}_+^\uparrow$ (the inhomogeneous symmetry group of ${{\mathbb {R}}}^{1+3}$) and $\Lambda$ be the covering map. First of all, we recall that to any 4-vector is 1-1 associated a $2\times 2$ matrix

$$\begin{aligned} x_\sim = x_0\,{{\varvec{1}}}+\sum _{i=1,2,3}x_i\,\sigma _i=\left( \begin{array}{cc} x_0+x_3&{}x_1-ix_2\\ x_1+ix_2&{}x_0-x_3 \end{array} \right) , \end{aligned}$$

where $\sigma _i$ are the Pauli matrices. Real vectors define Hermitian matrices. If $A\in {\mathrm {SL}(2,{{\mathbb {C}}})}$ then the Poincaré action is ruled by the following relation

$$\begin{aligned} ({\Lambda (A)x})_\sim =A x_\sim A^*. \end{aligned}$$

(3.8)

Let U be a unitary strongly continuous representation of the Poincaré group, then the representation of an x-translation has the form $U(x)=e^{iPx}$ where P is a vector of four self-adjoint operators and Px is obtained through the Minkowski product. Every $g\in {\widetilde{{{\mathcal {L}}}}}_+^\uparrow$ acts on an x-translation by the adjoint action, namely $gxg^{-1}=\Lambda (g)x$. Let $\mathrm {Sp}\,(P)$ be the joint spectrum of generators of translations and p be a point in the spectrum, then we have the character $\chi _p(x)=e^{ipx}$. As in the general case, the dual action on the momentum space is defined s.t. $\chi _p(\Lambda (g)\cdot x)=\chi _{p'}(x)$ and it is easy to see that $p'=\Lambda (g)^{-1}p$, 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 $\chi _q$, $q\ne 0$, be a character of the translation group. We shall call ${{\,\mathrm{Stab}\,}}_q$ and ${\overline{{{\,\mathrm{Stab}\,}}}}_q$ the stabilizers of the point q through the ${\widetilde{{{\mathcal {L}}}}}_{+}^\uparrow$ and $\widetilde{{\mathcal {P}}}_+^\uparrow$ actions, respectively. The latter is $\overline{{{\,\mathrm{Stab}\,}}}_q={{\mathbb {R}}}^{1+3}\rtimes {{\,\mathrm{Stab}\,}}_q$, where ${{\,\mathrm{Stab}\,}}_q$ shall be called as above the little group. Any massless, unitary, positive energy representation of $\widetilde{{\mathcal {P}}}_+^\uparrow$ is obtained starting with the character associated to $q:= (1,1,0,0)\in \partial V_+$ ($\partial V_+\smallsetminus \{0\}$ is an ${{\mathcal {L}}}_+^\uparrow$-orbit) and inducing by a unitary representation of the ${\overline{{{\,\mathrm{Stab}\,}}}}_q$ group. Note that a $\overline{{{\,\mathrm{Stab}\,}}_q}$ representation is of the form

$$\begin{aligned} {{\mathbb {R}}}^{1+3}{\rtimes }{{\,\mathrm{Stab}\,}}_q\ni (x,\sigma )\mapsto {\chi _q}(x)V(\sigma ), \end{aligned}$$

where V is the unitary representation of ${{\,\mathrm{Stab}\,}}_q$. The little group ${{\,\mathrm{Stab}\,}}_q$ is isomorphic to ${\widetilde{\mathrm {E}}}(2)={{\mathbb {R}}}^2 \rtimes {\mathbb {T}}$, where ${\mathbb {T}}$ is the unit circle. Note that $r_1(\theta )$ generate ${\mathbb {T}}.$ ${\widetilde{\mathrm {E}}}(2)$ is the double cover of $\mathrm {E}(2)={{\mathbb {R}}}^2 \rtimes \mathrm {{ S}O}(2)$, which is the group of Euclidean motions in two dimensions.⁶ Irreducible representations V of ${\widetilde{\mathrm {E}}}(2)$ fit in one of the following two classes: (See e.g. [Va85] and [BR86, page 520])

(a):

The restriction of V to ${{\mathbb {R}}}^2$ is trivial;

(b):

The restriction of V to ${{\mathbb {R}}}^2$ is non-trivial.

Irreducible representations of ${\widetilde{\mathrm {E}}}(2)$ in class (a) are labelled by half-integers h, called the helicity parameters.

Let

$$\begin{aligned} U={{\,\mathrm{Ind}\,}}_{{\overline{{{\,\mathrm{Stab}\,}}}}_q }^{{ \widetilde{{\mathcal {P}}}_+^\uparrow }}{({\chi _q}\cdot V)} \end{aligned}$$

be a unitary representation of $\widetilde{{\mathcal {P}}}_+^\uparrow$ induced from the representation ${\chi _q}\cdot V$ of ${\overline{{{\,\mathrm{Stab}\,}}}}_q$. We say that U has finite helicity if V has the form (a). If V belongs to the class (b) then it is called infinite spin but such a family of representations will not be studied in this paper.

An irreducible finite helicity representation is of the form

$$\begin{aligned} U_{h}={{\,\mathrm{Ind}\,}}_{{\overline{{{\,\mathrm{Stab}\,}}}}_q} ^{{ \widetilde{{\mathcal {P}}}_+^\uparrow }} ({\chi _q}\, \cdot \, V_{2h}) ,\qquad h\in \frac{{{\mathbb {Z}}}}{2}, \end{aligned}$$

(3.9)

where $V_{2h}(y,g)=(2h)(g)$, $(y,g)\in {{\mathbb {R}}}^2 \rtimes {\mathbb {T}}$ and 2h is the one dimensional representation of ${\mathbb {T}}$ of character $h\in {{\mathbb {Z}}}/2$. In particular, $r_1\in {{\,\mathrm{Stab}\,}}_q$ and $(2h)(r_1(\varphi ))=e^{i\frac{\varphi }{2} 2h}$ for a $\varphi \in [0,4\pi ]$. Integer and half-integer values of h discern, respectively, bosonic and fermionic representation of ${\widetilde{\mathrm {E}}}(2)$, hence, by induction, of $\widetilde{{\mathcal {P}}}_+^\uparrow$.

3.5 $G_W$ and related subgroups of the Poincaré group and their representations

In this subsection we introduce certain subgroups of $\widetilde{{\mathcal {P}}}_+^\uparrow$ 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 ${{\mathcal {W}}}$. We also recall from this subsection that $R_i, \Lambda _i\in {\mathcal {L}}_+^{\uparrow }$ and $r_i, \lambda _i\in \mathrm {SL}(2,{{\mathbb {C}}})$ denote the one-parameter families of rotations and boosts preserving the wedges $W_i$, $i=1,2,3$.

Definition 3.12

We denote with

• $G_3^0$ the subgroup of $A\in {\mathrm {SL}(2,{{\mathbb {C}}})}$ s.t. $\Lambda (A)W_3=W_3$.

• $G_3=\langle G_3^0,{{\mathbb {R}}}^{1+3}\rangle$, where ${{\mathbb {R}}}^{1+3}$ is the translation group and $\langle G_3^0,{{\mathbb {R}}}^{1+3}\rangle$ denotes the group generated by $G_3^0$ and ${{\mathbb {R}}}^{1+3}$.

• ${\tilde{G}}_3^0=\langle G_3^0, r_1(\pi )\rangle$, ${\tilde{G}}_3=\langle G_3, r_1(\pi )\rangle ={{\mathbb {R}}}^{1+3} \rtimes {\tilde{G}}_3^0$.

For a general wedge $W\in {{\mathcal {W}}}$, $G_W^0$, $G_W$, $\tilde{G}_{W}$ and $\tilde{G}_{W}^0$ are defined by the transitive action of ${{\mathcal {P}}_+^\uparrow }$ on wedges. We will denote the massless orbits of the ${{\mathbb {R}}}^{1+3}$ translation characters under the $G_3$ action with $\sigma _r=\{p=(p_0,{\varvec{p}}= (p_1,p_2,p_3))\in {{\mathbb {R}}}^{1+3}: p_1^2+p_2^2=r^2, p_0^2={\varvec{p}}^2, { p_0>0}\}$, $\sigma _0^\pm =\{p\in {{\mathbb {R}}}^{1+3}: p_1=0=p_{2}, p_0=\pm p_3, { p_0>0}\}$ and $\sigma _0=\sigma _0^+\cup \sigma _0^-$. In the present paper, we will be interested in orbits $\sigma _r$ with $r>0$ since $\sigma _0^\pm$ have null measure w.r.t. the Lorentz invariant measure on $\partial V_+$. Finally, we warn the reader that ${\tilde{G}}_3, \tilde{G}_{W}$ do not denote the covering groups of $G_3, G_{W}$.

Note that $G_3^0=\langle \lambda _3,\,r_3, r(2\pi )\rangle$. Indeed, any $\mathrm {SL}(2,{{\mathbb {C}}})$ element implementing a Poincaré transformation can be decomposed by the polar decomposition $A=U_A\cdot T_A$ (see e.g. [Mor06]), where $U_A$ is a rotation and $T_A$ a boost. Then $\Lambda (A)W_3=W_3$ iff $U_A ^{-1}W_3=T_AW_3$. Assume that there exists a transformation A such that $U_A ^{-1}W_3 \ne W_3\ne T_AW_3$ but $U_A ^{-1}W_3=T_AW_3$. Consider the edge of the wedge $E{ :=}\{x\in {{\mathbb {R}}}^{1+3}: x_0=0=x_3\}$. Then, $U_{A}^{-1}E=\{x\in {{\mathbb {R}}}^{1+3}: x_0=0=(U_{ A} x)_3\}$ cannot be equal to $T_{A} E=\{x\in {{\mathbb {R}}}^{1+3}: ( T^{-1}_{A} x)_0=0=( T_{A}^{-1}x)_3\}$. In particular ${ U_A ^{-1}}W_3= W_3=T_AW_3$.

Next, we note that $G_3^0$ and ${\tilde{G}}_3^0$ share the same orbits in ${ \partial } V_+$ as the following remark explains.

Remark 3.13

[Mo18]. Fix $p=(p_0,p_1,p_2,p_3)\in (\partial V_+ \smallsetminus \{0\} ):=\{p\in {{\mathbb {R}}}^{1+3}:p^2=0, p_0>0\}$. The $R_1(\pi )$-rotation

$$\begin{aligned} R_1(\pi )p=(p_0,p_1,-p_2,-p_3) \end{aligned}$$

can be obtained as a composition of a $\Lambda _3$-boost of parameter $t_p$ and a $R_3$-rotation of parameter $\theta _p$ as

$$\begin{aligned} \Lambda _3(t_p)R_3(\theta _p)(p_0,p_1,p_2,p_3)= \Lambda _3(t_p)(p_0,p_1,-p_2,p_3)=(p_0,p_1,-p_2,-p_3) \end{aligned}$$

(3.10)

for all the orbits except for $\sigma _0^{\pm }$, where $\sigma _0^{\pm }$ appeared in Definition 3.12. Clearly $t_p$ and $\theta _p$ depend on p and the orbits excluded by this geometrical fact have null measure w.r.t. the Lorentz invariant measure on $\partial V_+$. The discussion does not change if p is considered as an element of ${{\mathbb {R}}}^{1+3}$ or of its dual.

By (3.10), we deduce that almost all $G_3^0$ orbits on $\partial V_+$ are preserved by the $R_1(\pi )$-action. Furthermore, we note that $R_1(\pi )$ sends $W_3$ onto $W_3'$. Thus any transformation $R\in {{\mathcal {P}}_+^\uparrow }$ such that $RW_3=W_3'$ also preserves the $G_3^0$ orbits on $\partial V_+$ as well as $R_1(\pi )$, since $R_1(\pi )R\in G_3^0$. We have just seen that it is possible to pointwise reconstruct a transformation sending W to $W'$ just starting with elements in $G_W^0$. 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 $G_3^0$ and $\tilde{G}^0_3$ on ${\mathbb {R}}^{1+3}$ is verified in Appendix C, we can apply the theory of induced representations to $G_3={{\mathbb {R}}}^{1+3}\rtimes G^0_3$ and $\tilde{G}_3 = {{\mathbb {R}}}^{1+3}\rtimes \tilde{G}^0_3$. Choose a point $q_r$ on each massless, positive energy orbit $\sigma _r$ of $G_3^0$ on the dual of ${{\mathbb {R}}}^{1+3}$. Up to a null measure set in $\partial V_+$, the stabilizer of $q_r$ in $G_3$ is ${{\mathbb {R}}}^{1+3}\times \langle r(2\pi ) \rangle$ (cf.  the definition of the orbits $\sigma _r$ and of $G_3^0$). Thus there exist only two irreducible representations of $G_3$ induced by $\chi _{q_r}$, namely

$$\begin{aligned} W_{r,n}=\displaystyle {{{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^{3+1}\times \langle r(2\pi ) \rangle }^{G_3}(\chi _{q_r}\cdot V_n )}, \end{aligned}$$

(3.11)

where $V_n(r(2\pi ))=(-1)^n$ and $n=0,1$, cf. Proposition 3.11. They correspond to bosonic and fermionic representations of $G_3$.

Again by Remark 3.13 the subgroups ${\tilde{G}}_3$ and ${\tilde{G}}_3^0$ share the same orbits $\sigma _r$ with $r>0$ on $\partial V_+$ (up to a null measure set in $\partial V_+$). On the other hand the stabilizer of the point $q_r=(r,r,0,0)\in \sigma _r$ in ${\tilde{G}}_3$ is ${{\mathbb {R}}}^{1+3}\rtimes \langle r_1(\pi ) \rangle$.

Thus the little group of $q_r$, the subgroup of ${\tilde{G}}_3^0$ fixing $q_r$, is ${{\mathbb {Z}}}_4$. We have four irreducible representations of ${{\mathbb {Z}}}_4$ indexed by the representation of the generator, namely $V_n(r_1(\pi ))=i^n$, with $n=0,1,2,3.$ Correspondingly, we have four induced representations of ${\tilde{G}}_3$ associated to each orbit, namely

$$\begin{aligned} W_{r,n}={{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^{1+3}\rtimes {{\mathbb {Z}}}_4}^{{\tilde{G}}_3} (\chi _{q_r}\cdot V_n ) \end{aligned}$$

(3.12)

acting on the Hilbert spaces $\mathcal {H}_{r,n}$. Note that we called $V_{2h}$ the representation of character 2h, trivial on translations of ${\widetilde{\mathrm {E}}}(2)$, and $V_n$ the representation of character n of ${{\mathbb {Z}}}_4$. That this is not an abuse of notation is justified by the fact that $r_1({\pi })\in {\widetilde{\mathrm {E}}}(2)$ and when we restrict $V_{2h}$ to the group $\langle r_1(\pi )\rangle$, we get

$$\begin{aligned} V_{2h}(r_1(\pi ))=e^{i\frac{\pi }{2} 2h}=i^{2h}=V_n(r_1(\pi )),\quad V_{2h}(r(2\pi ))=(-1)^n=V_n(r(2\pi )) \end{aligned}$$

and it is enough to consider $n \equiv 2h$ in ${{\mathbb {Z}}}_4$ in the first case or $n \equiv 2h$ in ${{\mathbb {Z}}}_2$ in the second. Furthermore $W_{r,n}$ as a representation of ${\tilde{G}}_3$ restricts to $W_{r,m}$ with $m\,{\equiv }\, n\, (\mathrm {mod} \,\, 2)$ 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 $G_3$ and ${\tilde{G}}_3$, respectively, since the translation spectrum lies on the boundary of the forward lightcone.

4 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 $U_{h}\oplus U_{-h}$, $h\in {\mathbb {Z}}$, satisfies the B–W property. The analysis is based on the following modularity condition:

Definition 4.1

[Mo18]. A (unitary, positive energy) $\widetilde{{\mathcal {P}}}_+^\uparrow$ representation is said to be modular if, for any U-covariant net of standard subspaces H, we have that (U, H) satisfies the B–W and duality properties.

Let $W\in {{\mathcal {W}}}$. A unitary, positive energy $\widetilde{{\mathcal {P}}}_+^\uparrow$-representation U satisfies the modularity condition if for an element $r_W\in \widetilde{{\mathcal {P}}}_+^\uparrow$ such that $\Lambda (r_W)W=W'$ we have that

$$\begin{aligned} {U(r_W)\in U(G_W)''}. \end{aligned}$$

(MC)

Note that (MC) depends neither on the choice of $r_W$ nor on W. Indeed if ${\tilde{r}}_W\in \widetilde{{\mathcal {P}}}_+^\uparrow$ is another transformation such that $\Lambda ({\tilde{r}}_W) W=W'$ then $r_W\cdot {\tilde{r}}_W\in G_W$ and if (MC) holds for $U(r_W)$, then it holds for $U({\tilde{r}}_W)$. By transitivity of the $\widetilde{{\mathcal {P}}}_+^\uparrow$ action on wedges it is not restrictive to fix a wedge region W. Condition (MC) can be straightforwardly stated when just a representation of ${\tilde{G}}_3$ is taken into account.

Theorem 4.2

[Mo18]. Let U be a positive energy unitary representation of the Poincaré group $\widetilde{{\mathcal {P}}}_+^\uparrow$. If condition (MC) holds for U, then any local U-covariant net of standard subspaces H, namely any pair (U, H) as in Sect. 3.3, satisfies B–W and duality properties. In particular U is modular.

We remark that the class of nets in Theorem 4.2, transforming under $\widetilde{{\mathcal {P}}}_+^\uparrow$, is more general than the class of bosonic nets we defined in Sect. 3.3. Since Theorem 4.2 applies also if fermionic representations are included, the forthcoming analysis of the modularity condition is done for general $\widetilde{{\mathcal {P}}}_+^\uparrow$-unitary positive energy representations. The theorem applies to the families of Poincaré representations covered by the following two results.

Proposition 4.3

[Mo18]. Let U be an irreducible scalar massive or massless representation of the Poincaré group, then U satisfies (MC).

The proof of the above proposition adapts in irreducible finite helicity case. Furthermore, this section contains a general and independent proof of the modularity condition for finite helicity representations $U_h$, $h\in {\mathbb {Z}}/2$, and their multiplicities, see Corollary 4.11. On the other hand in order to have a consistent net of real subspaces one needs to couple h and $-h$ helicities, for $h\ne 0$, cf. comments after Theorem 4.13. It is not known how to apply the argument in [Mo18] to this and more general cases. The main technical result of the present paper consists in resolving this difficulty in the bosonic case, see Proposition 4.12 and Theorem 4.13.

Proposition 4.4

Let U, $U_1$ be unitary positive energy representations of ${\tilde{G}}_3$ on ${{\mathcal {H}}}$, satisfying (MC).

• (i) Let E be the projection on the subspace, where $U(r(2\pi ))=1$. Then U satisfies (MC) iff both $EU(\cdot )E$ and $(1-E) U(\cdot )(1-E)$ satisfy (MC).

• (ii) Let ${{\mathcal {K}}}$ be a Hilbert space, then (MC) holds for $U\otimes 1_{{\mathcal {K}}}\in {{\mathcal {B}}}({{\mathcal {H}}}\otimes {{\mathcal {K}}})$.

• (iii) Let $U_2$ be a unitary representation of ${\tilde{G}}_3$ s.t. $U_2$ is unitarily equivalent to $U_1$. Then $U_2$ satisfies (MC).

Proof

• (i) Since $r(2\pi )$ commutes with every element in ${\tilde{G}}_{3}$, the spectral projection E of $U(r(2\pi ))$ decomposes U into disjoint representations $U(\cdot )=EU(\cdot )\oplus (1-E) U(\cdot )$. The thesis follows since $U(G_3)''=(EU(G_3))''\oplus ((1-E)U(G_3))''$.

• (ii) Is proved in [Mo18].

• (iii) Let W be a unitary s.t. $WU_2(g)W^*=U_1(g)$, $g\in {\tilde{G}}_3$. It is easy to see that $WU_2(G_3)'W^*=U_1(G_3)'$ and $WU_2(G_3)''W^*= U_1(G_3)''$. If $WU_2(r_1(\pi ))W^*=U_1(r_1(\pi ))$, then

  $$\begin{aligned} U_1(r_1(\pi ))\in U_1(G_3)''\Rightarrow WU_2(r_1(\pi ))W^*\in WU_2(G_3)''W^*\Rightarrow U_2(r_1(\pi ))\in U_2(G_3)'', \end{aligned}$$

  which concludes the proof. $\quad \square$

The representations $W_{r,n}$ of $\tilde{G}_3$, restricted to $G_3$, give $W_{r,m}$, where $m\,{\equiv }\,n (\mathrm {mod}\,\, 2$), see Lemma 4.9 below. Thus they are disjoint for different r, by the disjointness of the respective orbits $\sigma _r$, cf. Proposition 3.11. Furthermore, since $W_{r,n}$ is irreducible, $W_{r,n}(G_3)'={{\mathbb {C}}}\cdot \mathbf{1 }$ and $W_{r,n}(r_1(\pi ))\in W_{r,n}(G_3)''= B(\mathcal {H}_{r,n})$. We deduce that $W_{r,n}$ satisfies (MC).

Corollary 4.5

Let $W_{r,n}$ be the irreducible ${\tilde{G}_3}$-representation of radius $r>0$. Then $W_{r,n}$ satisfies (MC).

The following proposition ensures that also a direct integral of massless ${\tilde{G}}_3$ representations satisfies (MC).

Proposition 4.6

Let $\mu$ be a positive Borel measure on ${{\mathbb {R}}}^+$. Assume that $U_{r}$ are multiples of the massless $W_{r,n}$, $n=0,1,2,3$, representations of $\tilde{G}_{W}$. Then $U=\int ^{{ \oplus }}_{{{\mathbb {R}}}^+} U_{r} d\mu (r)$ 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 $U_h\oplus U_{-h}$ for integer helicities h. In order to prove the result, we want to disintegrate the restriction of $U_h$ to the ${\tilde{G}}_3$ 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 $H_1\backslash G/H_2$ is the double coset, i.e., the set of the equivalence classes $[g]=H_1gH_2$, with $g\in G$.

Definition 4.7

[Ma57] 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 $E_0, E_1, E_2, \ldots$ of measurable subsets of G each of which is a union of double cosets in $H_1\backslash G/H_2$ 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 $H_1\backslash G/H_2$, $H_1$ 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 $\sim$ and the quotient map $s: X\rightarrow Y=X/\sim$, the equivalence relation is said to be measurable if there exists a countable family $\{F_n\}_{n\in {{\mathbb {N}}}}$ 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 $F_{n'}$, $n'\in {\mathbb {N}}$, containing this point.

Consider the map $s: G\rightarrow H_1\backslash G/H_2$ carrying each element of G into its double coset. Then equip $H_1\backslash G/H_2$ with the quotient topology given by s and consider a finite measure $\mu$ on G which is in the same measure class⁷ as the Haar measure. It is possible to define a measure ${\bar{\mu }}$ on the Borel sets of $H_1\backslash G/H_2$ by ${\bar{\mu }}(F)=\mu (s^{-1}(F))$. We shall call ${\bar{\mu }}$ an admissible measure in $H_1\backslash G/H_2$. 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 [Ma57, Fol16, BR86]. We recall the Mackey’s subgroup theorem.

Theorem 4.8

(Mackey’s subgroup theorem). [Ma57]. Let $H_1,\, H_2$ be closed subgroups regularly related in G. Let $\pi$ be a strongly continuous representation of $H_1$. For each $g\in G$ consider $H_g=H_2\cap (g^{-1}H_1g)$ and set

$$\begin{aligned} \mathcal {V}_g=\mathrm {Ind}_{H_g}^{H_2}(\pi \circ {{\text {ad}}}\, g). \end{aligned}$$

Then $\mathcal {V}_g$ is determined to within equivalence by the double coset [g] to which g belongs. If $\nu$ is an admissible measure on $H_1\backslash G/H_2,$ then

$$\begin{aligned} (\mathrm {Ind}_{H_1}^G\pi )|_{H_2}\simeq \int ^{\oplus }_{H_1\backslash G/H_2}\mathcal {V}_{[g]}\,d\nu ([g]). \end{aligned}$$

(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 the ${\tilde{G}}_3$ representation $W_{r,n}$ to $G_3$ is $W_{r,m}$, where $\displaystyle {m\,{\equiv }\, n\, (\mathrm {mod}\,\, 2)}.$

Of central importance for our analysis is the following proposition.

Proposition 4.10

$\displaystyle {U_h|_{{\tilde{G}}_3}\simeq \int _{{{\mathbb {R}}}^+}^{\oplus }W_{r,2h}dr}$, where 2h in the right-hand side has to be considered modulo 4.

Proof

The h-helicity representation $U_h$ is induced by the stabilizer ${\overline{{{\,\mathrm{Stab}\,}}}}_{q_1}$ of the point ${q_1}=(1,1,0,0)$. Again ${\overline{{{\,\mathrm{Stab}\,}}}}_{q_1}$ is isomorphic to ${{\mathbb {R}}}^{1+3} \rtimes {\widetilde{\mathrm {E}}}(2)$. In Theorem 4.8 we can consider $G=\widetilde{{\mathcal {P}}}_+^\uparrow$, $H_1= {{\mathbb {R}}}^{1+3}\rtimes {\widetilde{\mathrm {E}}}(2)$ and we want to study the restriction of $U_h$ to $H_2=\tilde{G}_3$. We postpone the proof of the fact that $H_1$ and $H_2$ are regularly related.

Let us now compute $H_g$ and $\mathcal {V}_g$ for several choices of g. First, for $g={{\varvec{1}}}$, $H_{g={{\varvec{1}}}}={{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle \subset {{\mathbb {R}}}^{4}\rtimes {\widetilde{\mathrm {E}}}(2)$. Here we made use of the fact that $\tilde{G}_3^0=\langle \lambda _3,\,r_3, r_1(\pi )\rangle$ and $r_1(\pi )\in {{\,\mathrm{Stab}\,}}_{q_1}={\widetilde{\mathrm {E}}}(2)$. Hence

$$\begin{aligned} \mathcal {V}_{{{\varvec{1}}}}={\mathrm {Ind}}_{{{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle }^{{\tilde{G}}_3}(\chi _{q_1}{ \cdot }V_{2h})= W_{1,2h}, \end{aligned}$$

by (3.12) and the comment below this formula.

Next, we consider $g={\lambda _1(\ln r)}\in \widetilde{{\mathcal {P}}}_+^\uparrow$, where $\lambda _1$ is the lift of $\Lambda _1$, the boost in the $x_1$-direction, to $\mathrm {SL}(2,{\mathbb {C}})$, see Sect. 2.1. Setting $t\mapsto {\Lambda }_1(t)=\Lambda (\lambda _1(t))$ and noting that $q_r= \Lambda _1(\ln \,r){q_1}=r\cdot {q_1}$, the intersection $H_g:=H_2\cap (g^{-1}H_{1}g)$ satisfies

$$\begin{aligned} H_g=H_{{{\varvec{1}}}}\;\; \text { since }\;\; \tilde{G}_3\cap \left( {\lambda }_1\left( {\ln r}\right) {^{-1}}\,\overline{{{\,\mathrm{Stab}\,}}}_{q_1} \,{\lambda _1}\left( {\ln r}\right) \right) =\tilde{G}_3\cap \overline{{{\,\mathrm{Stab}\,}}}_{q_{r^{-1}}}={{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle . \end{aligned}$$

Hence,

$$\begin{aligned} \begin{aligned} \mathcal {V}_{\Lambda _1(\ln r)}&={{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle }^{\tilde{G}_3}\left( (\chi _{q_1}{ \cdot }V_{2h})\circ {{\text {ad}}}\, { \lambda _1}\left( {\ln r}\right) \right) \\ {}&={{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle }^{\tilde{G}_3}(\chi _{q_{r^{-1}}}{ \cdot } V_{2h})=W_{r^{-1},2h} \end{aligned} \end{aligned}$$

(4.2)

because ${\lambda _1}\left( {\ln r}\right)$ commutes with $r_1(\pi )$ (see comment on dual action in Sect. 3.4).

Finally, we consider $g=r_2(-\pi /2)$. We note for future reference that $\Lambda (g)^{-1} q_1\in \sigma _0$ since $\Lambda (r_2(\pi /2))q_1=R_2(\pi /2)q_1 =(1,0,0,1)=:q_0\in \sigma _0$. We have

$$\begin{aligned} H_{r_2(-\pi /2)}= & {} {\tilde{G}}_3\cap r_2(\pi /2)\,{\overline{{{\,\mathrm{Stab}\,}}}}_{q_1}\,r_2(\pi /2)^{-1}={\tilde{G}}_3\cap \overline{{{\,\mathrm{Stab}\,}}}_{q_0}={{\mathbb {R}}}^{1+3}\rtimes \langle r_3\rangle , \\ \mathcal {V}_{r_2(-\pi /2)}= & {} {{\,\mathrm{Ind}\,}}_{H_{r_2({ -}\pi /2)}}^{{\tilde{G}}_3}\left( (\chi _{q_1}{ \cdot }V_{2h})\circ {{\text {ad}}}\,{r_2(\pi /2)^{-1}}\right) ={{\,\mathrm{Ind}\,}}_{H_{r_2({ -}\pi /2)}}^{{\tilde{G}}_3}(\chi _{q_0} \cdot \chi _{2h}) \end{aligned}$$

and the joint spectrum of translations is supported in $\sigma _0$. Here $\chi _{2h}$ is the 2h–character representation of $r_{ 3}(\,\cdot \,)$, $\chi _{2h}(r_{ 3}(\theta ))=e^{i\frac{\theta }{2} 2h}$, $h\in \frac{{{\mathbb {Z}}}}{2}$.

Let us now show that $[{\lambda }_1(t)]$ and $[r_2(-\pi /{ 2})]$ cover all the equivalence classes in $H_1\backslash G/H_2$. Let $g_1,g_2\in \widetilde{{\mathcal {P}}}_+^\uparrow$ and assume that ${p}_a$ and ${p}_b$ are points on the same massless ${ \tilde{G}_3}$-orbit⁸ s.t. ${\Lambda }(g_1^{-1}){q_1}= {p}_a$ and ${\Lambda }(g_2^{-1}){q_1}= {p}_b$. That is, there exists $x\in { \tilde{G}_3}$ such that ${p}_b={\Lambda }(x) {p}_a$. Then $g_1x^{-1}g_2^{-1}=\mathrm {s}\in { \overline{{{\,\mathrm{Stab}\,}}}_{q_1} }$, thus $g_2=\mathrm {s}^{-1}g_1x^{-1}$ and $g_2$ belongs to $[g_1]$, the double coset of $g_1$ in $H_1\backslash \widetilde{{\mathcal {P}}}_+^\uparrow /H_2$. In particular, for every $g\in G$ for which there exists $t_g\in {{\mathbb {R}}}$ such that ${ \Lambda }( g^{-1}){q_1}$ and $\Lambda _{1}(-\ln t_g){q_1}$ belong to the same orbit $\sigma _{t_g^{-1}}$, we have $g\in [ \lambda _1\left( {\ln t_g}\right) ]$. All the other $g\in G$ such that $\Lambda (g)^{-1}q_1\in \sigma _0$ belong to the double coset $[r_2(-\pi /2)]$. This follows from the fact that $\Lambda (r_2(-\pi /2))^{-1}q_1= q_0\in \sigma _0$ which was mentioned above.

Now we will verify that the sets $H_1$ and $H_2$ are regularly related. For this purpose, we identify $H_1\backslash G/H_2$ with ${{\mathbb {R}}}^+{=[0,+\infty )}$ when $[{\lambda _1}\left( {\ln r}\right) ]\in H_1\backslash G/H_2$ is identified with $r^{-1}\in {{\mathbb {R}}}^+$ and $[r_2(-\pi /2)]$ with 0. We consider ${{\mathbb {R}}}^+$ with the relative topology inherited from ${{\mathbb {R}}}$. Open intervals with respect to the relative topology of ${\mathbb {R}}^+$ are open sets with respect to the quotient topology given by the map $s:G\rightarrow H_1\backslash G/H_2$ defined through the previous identification. Indeed, let $\epsilon >0$ and take an element g in the preimage $s^{-1}\left( I\right)$, $I:={{\mathbb {R}}}^+\cap (a,b), a, b, \in {\mathbb {R}}$. Then the set $\mathcal {N}_g^{\epsilon }=\{g'\in G: { \Lambda ({g'}^{-1})}{q_1}\in B_\epsilon ( { \Lambda ({g}^{-1}) }{q_1})\}$ is an open neighbourhood of g: as the inverse image of the open set $B_\epsilon ( { \Lambda ({g}^{-1}) }{q_1})$ under the continuous mapping $g'\mapsto \Lambda (g'{}^{-1})q_1$, the set $\mathcal {N}^{\epsilon }_g$ is open and contains g. For sufficiently small $\epsilon$ it is contained in $s^{-1}{(I)}$, since if ${ \Lambda ({g}^{-1}) }{q_1}\in \sigma _r$, then, for every $g'\in {{\mathcal {N}}}_g^{\epsilon }$, ${ \Lambda ({g'}^{-1})}{q_1}\in \sigma _{r'}$ with $r,r'\in {I}$ by continuity of the Poincaré action on Minkowski space. One can further see that the relative and the quotient topologies on ${\mathbb {R}}^{+}$ are equivalent since the map s is open and continuous with respect to the relative topology. With the identification $H_1\backslash G/H_2\simeq {{\mathbb {R}}}^{+}$, it is easy to see that $H_1$ and $H_2$ are regularly related by using intervals $(q-\frac{1}{n},q+\frac{1}{n})\cap {{\mathbb {R}}}^+$, $n\in {\mathbb {N}}$, with rational center contained in ${{\mathbb {R}}}^{+}$. (The second part of Definition 4.7 is used here).

Let us now describe the equivalence class of an admissible measure (cf. Definition 4.7). Starting with a finite measure $\mu$ on G in the equivalence class of the Haar measure, we induce a measure on ${{\mathbb {R}}}^+ { \simeq } H_1\backslash G/H_2$, 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 ${{\mathcal {V}}}_{[r_2(-\pi /2)]}$, we get the formula:

$$\begin{aligned} U_{h}|_{{\tilde{G}}_3}\simeq \int _{{{\mathbb {R}}}^+}^\oplus {d\mu (r)} W_{r}. \end{aligned}$$

(4.3)

Finite helicity representations $U_h$ extend to the conformal group (cf. [Ma77]). In particular, by dilation covariance, we have that $U(\delta (t))U_h U(\delta (t))^*\simeq U_h$ hence $U(\delta (t))U_h|_{{\tilde{G}}_3} U(\delta (t))^*\simeq U_h|_{{\tilde{G}}_3}$. Dilations change the unitary class of $W_{r,2h}$ dilating the radius of the representation⁹, since

$$\begin{aligned} U(\delta (t))U_h|_{{\tilde{G}}_3} U(\delta (t))^*= U_h \circ \mathrm {ad}(\delta (t))|_{\tilde{G}_3}=\int _{{\mathbb {R}}^+}^{\oplus } d\mu (r) W_{e^{-t} r}. \end{aligned}$$

(4.4)

Similarly as in Lemma 4.1 in [LMR16], this is a consequence of the following computation:

$$\begin{aligned} W_r\circ \mathrm {ad}(\delta (t))&= W_{r^{-1},2h} \circ \mathrm {ad}(\delta (t)) = {{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle }^{\tilde{G}_3}\left( (\chi _{q_1}{\cdot } V_{2h})\circ {{\text {ad}}}\, { \lambda }_1\left( {\ln r}\right) \circ \mathrm {ad}(\delta (t)) \right) \nonumber \\&={{\,\mathrm{Ind}\,}}_{{{\mathbb {R}}}^4\rtimes \langle r_1(\pi )\rangle }^{\tilde{G}_3}(\chi _{q_{ e^t r^{-1}}}{ \cdot } V_{2h})=W_{e^tr^{-1},2h}=W_{e^{-t}r}, \end{aligned}$$

(4.5)

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 $W_0\circ { \mathrm {ad}}(\delta (t))\simeq W_0$: since ${ \mathrm {ad}}(\lambda _3)$ does not change the unitary equivalence class of the ${\tilde{G}}_3$-representations

$$\begin{aligned} \mathcal {V}_{[r_2(-\pi /2)]}\circ { \mathrm {ad}}(\lambda _3(t))\circ { \mathrm {ad}}(\delta (t))&={{\,\mathrm{Ind}\,}}_{H_{r_2(\pi /2)}}^{{\tilde{G}}_3}(\chi _{q_0} \cdot \chi _{2h})\circ { \mathrm {ad}}(\lambda _3(t))\circ { \mathrm {ad}}(\delta (t))\\&={{\,\mathrm{Ind}\,}}_{H_{r_2(\pi /2)}}^{{\tilde{G}}_3}(\chi _{q_0} \cdot \chi _{2h})= \mathcal {V}_{[r_2(-\pi /2)]}, \end{aligned}$$

where we used $(\chi _{q_0} \cdot \chi _{2h})\circ { \mathrm {ad}}(\lambda _3(t))\circ { \mathrm {ad}}(\delta (t))=(\chi _{\lambda _3(-t) \delta (t)q_0} \cdot \chi _{2h})=\chi _{q_0} \cdot \chi _{2h}$ referring again to Lemma 4.1 of [LMR16]. Therefore,

$$\begin{aligned} \int _{{{\mathbb {R}}}^+}^\oplus d\mu (r)W_{r}\simeq \int _{{\mathbb {R}}^+}^{\oplus } {d\mu (r)}W_{e^tr}=\int _{{\mathbb {R}}^+}^{\oplus } {d\mu _t(r)}W_{r}, \end{aligned}$$

where $\mu _t(r){ :=}\mu (e^{-t}r)$. We show that $\mu$ is equivalent to the Lebesgue measure: assume by contradiction that there exists a set $E\subset {{\mathbb {R}}}^+\backslash \{0\}$ such that $\mu (E)>0$ but $\mu _t(E)=0$ and consider the multiplication operator by the projection $P_E:={\int ^{\oplus }_Ed\mu (r)} \in U_{h}({\tilde{G}}_3)'$. Then we have that the subrepresentation $P_EU_h|_{{\tilde{G}}_3} P_E$ is not contained in $U(\delta (t))U_h|_{{\tilde{G}}_3} U(\delta (t))^*$ (representations of radius $r\in E$ have measure zero in the latter representation). In particular for every $t\in {{\mathbb {R}}}$, $\mu _t$ is equivalent to $\mu$, hence to the Lebesgue measure on ${{\mathbb {R}}}^+\backslash \{0\}$ up to a possible singular measure in 0 (see Proposition 11 of [Bo04]). Since $\sigma _0$ 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 $\mu -$measure. Now the statement of the theorem is obtained by a change of variables $r\mapsto \frac{1}{r}$. $\quad \square$

By Propositions 4.6,  4.10 and  4.4 (ii) we conclude the modularity condition for finite helicity representations:

Corollary 4.11

For every $h\in \frac{{{\mathbb {Z}}}}{2}$, $U_h$ 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 $U_h\oplus U_{-h}$ satisfies (MC).

Proof

By Proposition 4.10$U_h$ and $U_{-h}$ have unitarily equivalent restrictions to ${{\tilde{G}}_3}$. Indeed, since h is supposed to be integer, 2h is equal to 0 or 2 modulo 4. Clearly $0\equiv -0$ (mod 4) and $2\equiv -2$ (mod 4). By Proposition 4.6 and Proposition 4.4 (ii),(iii), the direct sum $U_h\oplus U_{-h}$ satisfies (MC). Any multiple of U satisfies (MC) again by Proposition 4.4(ii). $\quad \square$

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 $U=U_h\oplus U_{-h}$, where $h\in {\mathbb {Z}}$, satisfies the B–W and the duality properties.

A final remark on finite helicity one particle nets is the following. Massless non-zero 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 $\widetilde{{\mathcal {P}}}_+=\langle \widetilde{{\mathcal {P}}}_+^\uparrow ,\Theta \rangle$, where $\Theta$ 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 (non-zero) 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 $U_{-h}$.

5 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 $\mathfrak {H}^{(1)}$ from Definition 2.3 and then verify the Bisognano–Wichmann property on $\mathfrak {H}^{\mathrm {out}}$ using scattering theory.

Let ${\mathcal {A}}_{\mathrm {sa}}(W)\subset {\mathcal {A}}(W)$ be the subspace of self-adjoint operators. It is well known and easy to check that if $({\mathcal {A}},U, \Omega )$ is a local net of von Neumann algebras in the sense of Definition 2.1, then

$$\begin{aligned} H(W):={ \overline{ \{\, A\Omega \,:\, A\in {\mathcal {A}}_{\mathrm {sa}}(W)\,\}}}, \end{aligned}$$

(5.1)

is a net of standard subspaces on wedges w.r.t. U, i.e., it satisfies properties 1.-5. of Sect. 3.3. Motivated by formula (2.14), we define for any $W\in \mathcal {W}$ the following real subspace of $\mathfrak {H}^{(1)}$

$$\begin{aligned} H^{(1)}(W):=P^{(1)}H(W). \end{aligned}$$

(5.2)

Furthermore, we recall that by the Borchers theorem $\Delta _{{\mathcal {A}}(W)}, J_{{\mathcal {A}}(W)}, S_{{\mathcal {A}}(W)}$ commute with the mass operator and therefore can be restricted to the domain

$$\begin{aligned} D:=\{\, P^{(1)}A\Omega \,:\, A\in {\mathcal {A}}(W)\,\}, \end{aligned}$$

(5.3)

which is dense in $\mathfrak {H}^{(1)}$. (This is proven analogously as Corollary 3.10). The following result holds:

Proposition 5.1

Let $({{\mathcal {A}}}, U, \Omega )$ be a local net of von Neumann algebras describing massless particles.

• (i) The map $H^{(1)}(W)=P^{(1)}H(W)$, defined in (5.1)–(5.2) above, gives a one particle net of standard subspaces on wedges w.r.t. $U^{(1)}:=P^{(1)}UP^{(1)}$, in the sense of properties 1.-5. of Sect. 3.3.

• (ii) If the theory $({\mathcal {A}}, U,\Omega )$ contains massless Wigner particles with helicity zero or with helicities $(h,-h)$, $h\in {\mathbb {N}}$, then the one-particle net $W\mapsto H^{(1)}(W)$ satisfies the B–W and duality properties.

• (iii) $J_{H^{(1)}(W)}=J_{{\mathcal {A}}(W)}P^{(1)}$ and $\Delta _{H^{(1)}(W)}^{it}=\Delta ^{it}_{{\mathcal {A}}(W)} P^{(1)}$ for $t\in {\mathbb {R}}$.

Remark 5.2

The locality property for the net $W\mapsto H^{(1)}(W)$ can be extracted from [Bu77], where

$$\begin{aligned} \langle \Omega , A_1P^{(1)}A_2\Omega \rangle = \langle \Omega , A_2P^{(1)}A_1\Omega \rangle \end{aligned}$$

(5.4)

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

• (i) By Lemmas 3.7, 3.8 and 3.9 we have that $H(W)= P^{(1)}H(W)\oplus (1- P^{(1)})H(W)$ and $W\mapsto P^{(1)}H(W)$ defines a local net of standard subspaces on $\mathfrak {H}^{(1)}:=P^{(1)} \mathfrak {H}$. It transforms under the massless Poincaré representation $P^{(1)}UP^{(1)}$ and satisfies the assumptions 1.-5 in Sect. 3.3.

• (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 $\xi _i\in H^{(1)}(W)$, $i=1,2$, and $P^{(1)}A_{i,n}\Omega$, $n\in {\mathbb {N}}$, be the corresponding approximating sequences with $A_{i,n}^*=A_{i,n}$. Then

  $$\begin{aligned} S_{H^{(1)}(W)}(\xi _1+i\xi _2)=\xi _1-i\xi _2&=\lim _{n\rightarrow \infty } P^{(1)} (A_{1,n}\Omega -iA_{2,n})\Omega \nonumber \\&=\lim _{n\rightarrow \infty } P^{(1)} S_{{\mathcal {A}}(W)}(A_{1,n}\Omega +iA_{2,n})\Omega \nonumber \\&=\lim _{n\rightarrow \infty } S_{{\mathcal {A}}(W)}P^{(1)} (A_{1,n}\Omega +iA_{2,n})\Omega . \end{aligned}$$

  (5.5)

  Hence $H^{ (1)}(W)+iH^{ (1)}(W)$ belongs to the domain of the closure of $S_{{\mathcal {A}}(W)}P^{(1)}$ and the latter operator coincides with $S_{H(W)}$ on $H^{ (1)}(W)+iH^{ (1)}(W)$. By the uniqueness of the polar decomposition, we have $J_{H^{(1)}(W)}=J_{{\mathcal {A}}(W)} P^{(1)}$ and $\Delta ^{it}_{ H^{(1)}(W) } = \Delta _{{\mathcal {A}}(W)} ^{it} P^{(1)}$. $\quad \square$

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 $\mathfrak {H}^{(n)}$, $n\ge 2$, there is a total set of scattering states $\Psi ^{\mathrm {out}}:= A_n^{\mathrm {out}}\{f_n\}^+ \ldots A_1^{\mathrm {out}}\{ f_1\}^+\Omega$ with the localisation regions $\mathcal {O}_i$ of $A_i$ chosen s.t. $\mathcal {O}_n\subset W_{1}$. Furthermore, $({\varvec{v}}_i-{\varvec{v}}_n)_1<0$ for ${\varvec{v}}_i\in \mathrm {supp}\, f_i$ and ${\varvec{v}}_n\in \mathrm {supp}\, f_n$ for $i\ne n$.

Proof

We consider an arbitrary scattering state $\Psi ^{\mathrm {out}}=(\Phi _n\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi _1)$ constructed using $A_1, \ldots , A_n$ localized in some arbitrary double cones $\tilde{\mathcal {O}}_1, \ldots , \tilde{\mathcal {O}}_n$ and arbitrary smooth functions $f_1, \ldots f_n$ on $S^2$. By cutting the sphere of velocities into small slices with planes orthogonal to the 1-st axis, we can approximate $\Psi ^{\mathrm {out}}$ with linear combinations of scattering states $\Psi ^{\mathrm {out}}_1$ such that the projections of $\mathrm {supp}f_i$ 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

$$\begin{aligned} ({\varvec{v}}_i-{\varvec{v}}_{i_0})_1<0 \end{aligned}$$

(5.6)

for all ${\varvec{v}}_{i_0}\in \mathrm {supp}\, f_{i_0}$ and ${\varvec{v}}_i\in \mathrm {supp}\, f_i$, $i\ne i_0$. 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 $\Psi ^{\mathrm {out}}_1$ by linear combinations of vectors $\Psi ^{\mathrm {out}}_2$ constructed using $A'_i$ localised in double cones $\mathcal {O}_i$ s.t. $\mathcal {O}_{i_0}\in W_{1}$. Due to the canonical commutation relations of asymptotic fields (cf. formula (3.2) above) each such vector $\Psi ^{\mathrm {out}}_2=(\Phi '_n\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi '_1)$ coincides with $(\Phi '_{i_0}\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi '_n\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi '_1)$. By changing the numbering, we obtain the claim. $\quad \square$

Proposition 5.4

If the unitary groups ${ {\mathbb {R}}\ni s\mapsto } \Delta ^{is}_{{\mathcal {A}}(W_{1})}$ and ${\mathbb {R}}\ni s\mapsto U(\lambda _{W_1}(-2\pi s))$ coincide on $\mathfrak {H}^{(1)}$, they also coincide on the subspace $\mathfrak {H}^{\mathrm {out}}$ of scattering states.

Proof

Let $V_s:= \Delta ^{is}_{{\mathcal {A}}(W_{1})} U(\Lambda _{W_1}(2\pi s))$. By induction over the particle number n, we show that $V_s$ is the unity on each $\mathfrak {H}^{(n)}$. Let $\Psi ^{\mathrm {out}}=(\Phi _n\overset{\mathrm {out}}{\times }\cdots \overset{\mathrm {out}}{\times }\Phi _1)$ be a scattering state as in Lemma 5.3 with $\Phi _i=A_i^{\mathrm {out}}\{f_i\}\Omega$. For $n=0$ we have $\Phi ^{\mathrm {out}}=\Omega$ and the statement follows from the invariance of the vacuum under $U(\, \cdot \,)$ and $s\mapsto \Delta ^{is}_{{\mathcal {A}}(W_{1})}$. For $n=1$ the statement holds by Proposition 5.1 (ii). Now let $n\ge 2$ be arbitrary and suppose that the statement holds for $n'<n$. Making use of Proposition 3.1 (d), we write

$$\begin{aligned} \Psi ^{\mathrm {out}}~= & {} ~A_n^{\mathrm {out}}\{f_n\}^+\ldots A_1^{\mathrm {out}}\{f_1\}^+\Omega \nonumber \\= & {} ~ (A_n^{\mathrm {out}}\{f_n\}^++ A_n^{\mathrm {out}}\{f_n\}^-) \ldots (A_1^{\mathrm {out}}\{f_1\}^+ + A_1^{\mathrm {out}}\{f_1\}^-)\Omega +{\check{\Psi }}^{\mathrm {out}}\nonumber \\= & {} A_n^{\mathrm {out}}\{f_n\}\ldots A_1^{\mathrm {out}}\{f_1\}\Omega +{\check{\Psi }}^{\mathrm {out}}, \end{aligned}$$

(5.7)

where, by the canonical commutation relations for the asymptotic creation/annihilation operators (cf. formula (3.2)), the compensating vector ${\check{\Psi }}^{\mathrm {out}}$ has components only in $\mathfrak {H}^{(\ell )}$ for $\ell <n$. Thus we have, by the induction hypothesis, $V_s {\check{\Psi }}^{\mathrm {out}}={\check{\Psi }}^{\mathrm {out}}$, and it suffices to consider ${\hat{\Psi }}^{\mathrm {out}}:=A_n^{\mathrm {out}}\{f_n\}\ldots A_1^{\mathrm {out}}\{f_1\}\Omega$. We can write

$$\begin{aligned} V(s){\hat{\Psi }}^{\mathrm {out}}= & {} V_sA_n^{\mathrm {out}}\{f_n\}\ldots A_1^{\mathrm {out}}\{f_1\}\Omega \nonumber \\= & {} V_sA_n^{\mathrm {out}}\{f_n\} V_s^{-1} V_{s} A_{n-1}^{\mathrm {out}}\{f_{n-1}\} \ldots A_1^{\mathrm {out}}\{f_1\}\Omega \nonumber \\= & {} V_sA_n^{\mathrm {out}}\{f_n\} V_s^{-1} A_{n-1}^{\mathrm {out}}\{f_{n-1}\} \ldots A_1^{\mathrm {out}}\{f_1\}\Omega , \end{aligned}$$

(5.8)

where in the last step we used the induction hypothesis. (We also used that the state $A_{n-1}^{\mathrm {out}}\{f_{n-1}\} \ldots A_1^{\mathrm {out}}\{f_1\}\Omega$ 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 $A_n^{\mathrm {out}}\{f_n\}$ still exists in the last line above). Using again that $V_s$ commutes with translations, we have $V_{s}A_n^{\mathrm {out}}\{f_n\} V_{s}^{-1}= (A_n')^{\mathrm {out}}\{f_n\}$, where $A_n':=V_sA_nV_s^{-1}$ is localized in $W_{1}$. Thus we can write

$$\begin{aligned} V(s){\hat{\Psi }}^{\mathrm {out}}= & {} (A_n')^{\mathrm {out}}\{f_n\} A_{n-1}^{\mathrm {out}}\{f_{n-1}\} \ldots A_1^{\mathrm {out}}\{f_1\}\Omega \nonumber \\= & {} \sum _{i=n-1}^1 A_{n-1}^{\mathrm {out}}\{f_{n-1}\} \ldots [(A_n')^{\mathrm {out}}\{f_n\}, A_i^{\mathrm {out}}\{f_i\}] \ldots A_1^{\mathrm {out}}\{f_1\}\Omega \quad \end{aligned}$$

(5.9)

$$\begin{aligned}&+ A_{n-1}^{\mathrm {out}}\{f_n\}\ldots A_1^{\mathrm {out}}\{f_1\} (A_n')^{\mathrm {out}}\{f_n\}\Omega . \end{aligned}$$

(5.10)

Concerning (5.10), we use that $(A_n')^{\mathrm {out}}\{f_n\}\Omega =V_{s}A_n^{\mathrm {out}}\{f_n\} V_{s}^{-1}\Omega =A_n^{\mathrm {out}}\{f_n\}\Omega$, since $V_s$ preserves both the vacuum and single-particle subspace. This vector coincides with ${\hat{\Psi }}^{\mathrm {out}}$ provided that we can show

$$\begin{aligned}{}[A_n^{\mathrm {out}}\{f_n\}, A_i^{\mathrm {out}}\{f_i\}]=0 \end{aligned}$$

(5.11)

for all $i=1,2,\ldots , n-1$. 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 $\Psi _1,\Psi _2\in D_{P^0}$ we write

$$\begin{aligned}&|\langle \Psi _1, [(A_n')^{\mathrm {out}}\{f_n\}, A_i^{\mathrm {out}}\{f_i\}]\Psi _2\rangle |\nonumber \\&\le \overline{\lim }_{t\rightarrow \infty } \int dt'_n dt'_i d\omega ({\varvec{n}}_n)d\omega ({\varvec{n}}_i) \, h_t(t'_n)h_t(t'_i) |f_n({\varvec{n}}_n)| |f_i({\varvec{n}}_i) 4t'_nt'_i\times \nonumber \\&\quad \times \Vert [\partial _0A'_n(t'_n, t'_n{\varvec{n}}_n), \partial _0A_i(t'_i, t'_i{\varvec{n}}_i)]\Vert \nonumber \\&\le \overline{\lim }_{t\rightarrow \infty } \int dt'_n dt'_i d\omega ({\varvec{n}}_n)d\omega ({\varvec{n}}_i) \, h_t(t'_n)h_t(t'_i) |f_n({\varvec{n}}_n)| |f_i({\varvec{n}}_i) 4t'_nt'_i\times \nonumber \\&\quad \times \Vert [\partial _0A'_n, \partial _0A_i(t'_i-t'_n, t'_i{\varvec{n}}_i -t'_n{\varvec{n}}_n )]\Vert \nonumber \\&\le \overline{\lim }_{t\rightarrow \infty } \int dt''_n dt''_i d\omega ({\varvec{n}}_n)d\omega ({\varvec{n}}_i) \, \nonumber \\&\qquad \times { t^{2\varepsilon }}h(t''_n)h(t''_i) |f_n({\varvec{n}}_n)| |f_i({\varvec{n}}_i) |4(t+t^{\varepsilon }t''_i) (t+t^{\varepsilon }t''_n) \times \nonumber \\&\quad \times \Vert [\partial _0A'_n, \partial _0A_i(t^{\varepsilon } (t''_i-t''_n), (t+t^{\varepsilon }t''_i){\varvec{n}}_i -(t+t^{\varepsilon }t''_n){\varvec{n}}_n )]\Vert . \end{aligned}$$

(5.12)

Making use of Lemma 5.3, we conclude that the last expression is zero for sufficiently large t. Indeed, $t( {\varvec{n}}_i-{\varvec{n}}_n)_1<0$, $0<\varepsilon <1$ and $t_i'', t_n''$ are restricted to unit balls around zero. Hence $\partial _0A_i(t^{\varepsilon } (t''_i-t''_n), (t+t^{\varepsilon }t''_i){\varvec{n}}_i -(t+t^{\varepsilon }t''_n){\varvec{n}}_n )$ is eventually localized in the left wedge. $\quad \square$

6 Conclusion and Outlook

In this paper we proved the Bisognano–Wichmann property for asymptotically complete theories of massless particles with integer helicities. The argument starts from verifying this property at the single particle level. For this purpose, the single-particle subspace $\mathfrak {H}^{(1)}$ is equipped with the structure of a local net of standard subspaces. This net is covariant w.r.t. a representation $U^{(1)}:=U |_{ \mathfrak {H}^{(1)} }$ of the Poincaré group, which is a direct sum of two representations of opposite integer helicities, i.e., $U^{(1)}=U_{h}\oplus U_{-h}$ (or a multiple thereof). Then we verified the modularity condition for the B–W property $U^{(1)}(r_W)\in U^{(1)}(G_W)''$, 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 group $\tilde{G}_W$ generated by $G_W$ and $r_W$. Hence $U^{(1)}{ |_{\tilde{G}_W}}= { U_h|_{\tilde{G}_W}}\otimes 1$ 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 to ${\tilde{G}}_3$ 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.

Appendices

Direct Integral Representations

We suggest [Tak02, Dix81, MT19] as further references for basic definitions.

Given a field of Hilbert spaces $\gamma \mapsto {{\mathcal {H}}}(\gamma )$ on a standard measure space $(\Gamma ,\mu )$, the direct integral Hilbert space $\int _\Gamma ^\oplus {{\mathcal {H}}}(\gamma ) d\mu (\gamma )$ is defined if the field is $\mu$-measurable. This definition requires and depends on the choice of a linear $\gamma$-pointwise dense subspace $\mathcal {S}$ of the topological product $\Pi _{\gamma \in \Gamma }{{\mathcal {H}}}(\gamma )$ which selects a family of $\mu$-measurable vector fields. (cf. [Dix81, Part II, Sect. II.1.3, Definition 1]). Note that given a sequence of measurable vector field $\xi _n$ $\mu$-a.e. pointwise converging to $\xi$, namely $\Vert (\xi _n)(\gamma )-(\xi )(\gamma )\Vert _{ \gamma } \rightarrow 0$ for $\mu$-a.e. $\gamma \in \Gamma$, we obtain that $\xi$ is a $\mu$-measurable vector field. We also recall that a vector field of bounded operators $\gamma \mapsto T(\gamma )\in {{\mathcal {B}}}({{\mathcal {H}}}(\gamma ))$ is $\mu$-measurable if for any $\mu$-measurable field $\gamma \mapsto \xi (\gamma )\in {{\mathcal {H}}}(\gamma )$ we have that $\gamma \mapsto T(\gamma )\xi (\gamma )\in {{\mathcal {H}}}(\gamma )$ is $\mu$-measurable. In this case we define

$$\begin{aligned} T\xi :=\int ^\oplus _\Gamma T(\gamma )\xi (\gamma )d\mu (\gamma )\in \int _\Gamma ^\oplus {{\mathcal {H}}}(\gamma )d\mu (\gamma ). \end{aligned}$$

We write this operator as $T=\int _\Gamma ^\oplus T(\gamma ) d\mu (\gamma )$ and T is called the direct integral of ${ \gamma \mapsto } T(\gamma ).$ Furthermore, we have that $\gamma \mapsto \Vert T(\gamma )\Vert _{ \gamma }$ is measurable and $\Vert T\Vert =\sup _{\gamma \in \Gamma }\Vert T(\gamma )\Vert _\gamma$. The operators of this form are said to be decomposable. If $T(\gamma )$ is a scalar for any $\gamma { \in \Gamma }$, then T is said to be a diagonal operator. The algebra generated by the diagonal operators is called the diagonal algebra. Note that any operator $T\in {{\mathcal {B}}}({{\mathcal {H}}})$ is decomposable iff T commutes with the diagonal algebra (cf. [Tak02] IV.8, Corollary 8.16). A field of von Neumann algebras $\gamma \mapsto {{\mathcal {M}}}(\gamma )\subset {{\mathcal {B}}}({{\mathcal {H}}}(\gamma ))$ is said to be measurable if there exists a countable family $\{x_n(\gamma )\}_{n\in {{\mathbb {N}}}}$ of measurable fields of operators s.t. ${{\mathcal {M}}}(\gamma )$ is generated by $x_n(\gamma )$ for (a.e.) $\gamma \in \Gamma$. Then it is possible to define ${{\mathcal {M}}}=\int _\Gamma ^\oplus {{\mathcal {M}}}(\gamma ) d\mu (\gamma )$. Given a $C^*$-algebra ${{\mathcal {A}}}$, a field of continuous representations $\gamma \mapsto \pi (\gamma )$ is said to be measurable if for any $A\in {{\mathcal {A}}}$, the operator field $\gamma \mapsto \pi (\gamma )(A)$ is measurable. Then one can define the representation $\int _\Gamma ^\oplus \pi (\gamma ){d\mu (\gamma )}$ acting on $\int _\Gamma ^\oplus {{\mathcal {H}}}(\gamma )d\mu (\gamma )$.

Proof of Proposition 4.6

For fundamental concepts on direct integrals of representations see Appendix A.

The function of the translation generators $P_1^2+P_2^2$ is a Casimir operator for $G_3$ and decomposes according to $\mu$, i.e., $P_1^2+P_2^2=\int ^{\oplus }_{{\mathbb {R}}{^+}}r^2\cdot 1 d\mu (r)$ and $P_1^2+P_2^2$ is affiliated to $U(G_3)''$. By definition, bounded functions of $r={\sqrt{P_1^2+P_2^2}}$ generate the diagonal algebra $\mathcal {D}$. Thus $\mathcal {D}$ is contained in the center of $U(G_3)''$, hence any operator in $U(G_3)'$ is decomposable since it commutes with $\mathcal {D}$.

We now pick $T\in U(G_3)'$ then T is a decomposable operator, namely,

$$\begin{aligned} T =\int _{{\mathbb {R}}^+}^\oplus \, T(r) d\mu (r). \end{aligned}$$

(B.1)

Assume that there exists a positive measure set I s.t. T(r) is not in $U_r(G_3)'$ for $r\in I$. Let $\chi$ be a characteristic function of I. Then $\int _{{\mathbb {R}}^+}^\oplus \, T(r) \chi (r)d\mu (r)$ is not in the commutant of $U(G_3)$, which is a contradiction. We conclude that

$$\begin{aligned} U(G_3)'=\int _{{\mathbb {R}}{^+}}^\oplus U_r(G_3)' d\mu (r) \end{aligned}$$

(B.2)

and thus

$$\begin{aligned} U(G_3)''=\int _{{{\mathbb {R}}^+}}^\oplus U_r(G_3)'' d\mu (r) \end{aligned}$$

(B.3)

by [Tak02, Theorem 8.18]. Now we recall that $U_r$ satisfies (MC) by Corollary 4.5. Since, by assumption, $U(r_1(\pi ))=\int _{{{\mathbb {R}}^+}}^\oplus U_r(r_1(\pi )) d\mu (r)$, we obtain from (B.3) that U satisfies (MC).

Regularity of the Actions on ${{\mathbb {R}}}^{{ 1+3}}$

Definition C.1

Let G be a locally compact, $\sigma$-compact, group and N be a normal abelian subgroup, then the (dual) action of G on ${\hat{N}}$ is regular if

• R1. the orbit space is countably separated, namely there exists a countable family $\{E_n\}_{n\in {{\mathbb {N}}}}$ of G-invariant Borel sets in ${\hat{N}}$ s.t. each orbit in ${\hat{N}}$ 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 $G_3^0$ and ${\tilde{G}}_3^0$ on ${{\mathbb {R}}}^{{ 1+3}}$ is regular according to the previous definition.

R1. Let $\mathbf{o }$ be a $G_3^0$ or a ${\tilde{G}}_3^0$ orbit on ${{\mathbb {R}}}^{1+3}$. Then $\mathbf{o }$ can be obtained by intersection of the subsets of the following countable family of Borel subsets containing $\mathbf{o }$. For $a_1,a_2,b_1,b_2\in {\mathbb {Q}}$, $c,d\in {\mathbb {Q}}^{\ge 0}$, consider the following sets:

• $A_{a_1,b_1}=\{p: a_1\le p^2\le b_1\}$ and $A_{a_1,b_1}^\pm =A_{a_1,b_1}\cap \{p:\pm p_0>0\}$ if $a_1,b_1\ge 0$,

• $E_{c,d}=\{p:c\le p_1^2+p_2^2\le d\}$,

• $F_{a_2,b_2}=\{p:a_2\le p_0^2-p_3^2\le b_2\}$,

• $F_{a_2,b_2}^\pm =\{p:a_2\le p_0^2-p_3^2\le b_2, \pm p_3>0\}$ if $a_2, b_2<0$,

• $K^{\pm ,\pm }=\{p=(p_0,0,0,p_3): p_0=\pm p_3, \pm p_0>0\}$,

• ${\tilde{K}}^{\pm }=\{p=(p_0,0,0,p_3): p_0=\pm { |p_3|}, \pm p_0>0\}$,

where in the case of $K^{\pm ,\pm }$ the two signs are uncorrelated. We shall denote with ${{\mathcal {A}}}, {{\mathcal {E}}}, {{\mathcal {F}}}, {{\mathcal {F}}}^{\pm },{{\mathcal {K}}}^{\pm \pm }, {\tilde{{{\mathcal {K}}}}}^\pm$ the countable families of the above sets with the corresponding letters. We also define the sets:

• $O=\{p:p=0\}$,

• $Z_{a_1,b_1,\pm ,\pm }=\{p:a_1\le p^2\le b_1<0, p_0^2-p_3^2=0, \pm p_3>0, \pm p_0> 0\}$,

• ${\tilde{Z}}_{a_1,b_1,\pm }=\{p:a_1\le p^2\le b_1<0, p_0^2-p_3^2=0, \pm p_0> 0\}$,

• $X_{a_1,b_1}=\{p:a_1\le p^2\le b_1<0, p_0=p_3=0\}$.

In the following we shall say that a family of sets selects an orbit $\mathbf{o }$ if the latter is the intersection of all the set of the family containing $\mathbf{o }$. 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 $G_3^0$ or ${\tilde{G}}_3^0$ is contained in a Lorentz orbit in ${{\mathbb {R}}}^{1+3}$. The family in ${{\mathcal {A}}}$ selects the Lorentz orbits. The orbit in the origin is selected by O. Now $G_3^0$ and ${\tilde{G}}_3^0$ share the same massive orbits, contained in $p^2=m^2$, $m>0$, that can be selected by considering ${{\mathcal {A}}}$ and ${{\mathcal {E}}}$ families. Now consider a massless orbit in the forward lightcone. If for every $p\in \mathbf{o }$, $p_1^2+p_2^2>0$ then it is both a $G_3^0$ and ${\tilde{G}}_3^0$ orbit (cf. Remark 3.13) and can be selected by the families ${{\mathcal {A}}}$ and ${{\mathcal {E}}}$. If $p_1^2+p_2^2=0$ then the two $G_3^0$ orbits $\{p:0<p_0=p_3\}$ and $\{p:0<p_0=-p_3\}$ are selected by ${ K}^{\pm ,{ +}}$. If $p_1^2+p_2^2=0$ then the ${\tilde{G}}_3^0$ orbit is selected by ${ {\tilde{K}}^+}$. We argue analogously for the backward lightcone, referring to sets $K^{\pm ,-}, \tilde{K}^-$. Now consider imaginary mass orbits, contained in $p^2=-m^2$ and assume that ${ p_1^2+p_2^2=r^2}$. We have three cases:

• $r^2<-m^2$. In this case we have two branches of the hyperboloid $p_0^2-p_3^2=m^2+r^2<0$ that become two $G_3^0$ orbits and a unique ${\tilde{G}}_3^0$ orbit. The $G_3^0$ and ${\tilde{G}}_3^0$ orbits are selected by ${{\mathcal {A}}},{{\mathcal {E}}},{{\mathcal {F}}}^\pm$ and ${{\mathcal {A}}},{{\mathcal {E}}},{{\mathcal {F}}}$, respectively.

• $r^2=-m^2$. $G_3^0$ orbits are selected by $Z_{a_1,a_2,\pm ,\pm }$ or $X_{a_1,a_2}$. ${\tilde{G}}_3^0$ orbits are selected by ${\tilde{Z}}_{a_1,a_2,\pm }$.

• $r^2>-m^2$. $G_3$ and ${\tilde{G}}_3$ share the same orbits selected by ${{\mathcal {A}}}, {{\mathcal {E}}}, {{\mathcal {F}}}\cap U^\pm$, where $U^\pm = \{p:\pm p_0\ge 0\}$.

R2. trivially holds.