Recovering the QNEC from the ANEC

Abstract

We study the relative entropy in QFT comparing the vacuum state to a special family of purifications determined by an input state and constructed using relative modular flow. We use this to prove a conjecture by Wall that relates the shape derivative of relative entropy to a variational expression over the averaged null energy (ANE) of possible purifications. This variational expression can be used to easily prove the quantum null energy condition (QNEC). We formulate Wall’s conjecture as a theorem pertaining to operator algebras satisfying the properties of a half-sided modular inclusion, with the additional assumption that the input state has finite averaged null energy. We also give a new derivation of the strong superadditivity property of relative entropy in this context. We speculate about possible connections to the recent methods used to strengthen monotonicity of relative entropy with recovery maps.

Relative Modular Operator

In this appendix we collect various formula related to the relative modular operator. We are particularly interested in defining these objects when the vector states are not necessarily cyclic and separating. These considerations are standard and can be found in the Appendix of [44]. We warn the reader that we have a different convention for labeling our Sand$\varDelta$relative modular operators - the state labels are switched. This convention was used in [56] and we stick with this.

Take $\psi$ to not be cyclic and separating. This means that there could be some $\alpha \in {\mathcal {A}}$ such that $\alpha | \psi \rangle =0$ (not separating) and also that ${\mathcal {A}} | \psi \rangle$ may generate a proper subspace of ${\mathcal {H}}$ instead of the full Hilbert space (not cyclic). To describe this situation we define support projections as the minimal projectors that satisfy:

$$\begin{aligned} s^{\mathcal {A}}(\psi ) | \psi \rangle =&| \psi \rangle \qquad s^{\mathcal {A}}(\psi ) \in {\mathcal {A}} \end{aligned}$$

(276)

$$\begin{aligned} s^\mathcal {A'}(\psi ) | \psi \rangle =&| \psi \rangle \qquad s^\mathcal {A'}(\psi ) \in {\mathcal {A}}'. \end{aligned}$$

(277)

An equivalent definition follows from finding the projector onto the following subspaces:

$$\begin{aligned} \left[ {\mathcal {A}}' | \psi \rangle \right]&= \pi (\psi ){\mathcal {H}} \subset {\mathcal {H}} \end{aligned}$$

(278)

$$\begin{aligned} \left[ {\mathcal {A}} | \psi \rangle \right]&= \pi '(\psi ){\mathcal {H}} \subset {\mathcal {H}}. \end{aligned}$$

(279)

These are seen to be equivalent as follows. Firstly the $\pi (\psi )$ commutes with ${\mathcal {A}}'$ since for arbitrary state $|\phi _i\rangle \in {\mathcal {H}}$ and for all $\alpha ' \in {\mathcal {A}}'$:

$$\begin{aligned} \langle \phi _1 | \left[ \pi _{\mathcal {A}}(\psi ), \alpha ' \right] | \phi _2 \rangle&= \langle \chi _1 | \alpha ' | \phi _2 \rangle - \langle \phi _1 | \alpha ' | \chi _2 \rangle \qquad&\left( | \chi _i \rangle = \pi (\psi ) | \phi _i \rangle \right) \end{aligned}$$

(280)

$$\begin{aligned}&= \langle \chi _1 | \alpha ' | \chi _2 \rangle - \langle \chi _1 | \alpha ' | \chi _2 \rangle = 0 \qquad&\left( \alpha ' | \chi _i \rangle = \pi (\psi ) \alpha ' | \chi _i \rangle \right) \end{aligned}$$

(281)

so $\pi$ is in ${\mathcal {A}}$. Secondly it is the minimal such projector leaving $\psi$ invariant since if it were not there would be another projector $\pi _2 \in {\mathcal {A}}$ with $\pi _2 {\mathcal {H}} \subset \pi (\psi ) {\mathcal {H}}$ which also leaves invariant the subspace:

$$\begin{aligned} \left[ A' | \psi \rangle \right] = \left[ A' \pi _2 | \psi \rangle \right] = \pi _2 \left[ A' | \psi \rangle \right] \end{aligned}$$

(282)

such that $\pi _2 {\mathcal {H}} \subset \pi (\psi ) {\mathcal {H}} = \pi _2 \pi (\psi ) {\mathcal {H}} \subset \pi _2 {\mathcal {H}}$ implying that $\pi _2 = \pi (\psi )$. Thus $\pi = s^{\mathcal {A}}$ and similarly for the commutant.

Note that if $\pi (\psi )$ is not the unit operator, then $\psi$ is not cyclic for ${\mathcal {A}}'$ and $(1-\pi (\psi ))$ annihilates $\psi$ which means that $\psi$ is not separating for ${\mathcal {A}}$. That is the lack of either of these two properties exchange under ${\mathcal {A}} \leftrightarrow {\mathcal {A}}'$.

We now move to the modular operators. We will consider two state $\psi ,\phi$ neither of which needs to be cyclic and separating. We start with the definition of the Tomita operators:

$$\begin{aligned} S_{\psi |\phi }\left( \alpha | \psi \rangle + | \chi ' \rangle \right)&= \pi (\psi ) \alpha ^\dagger | \phi \rangle \qquad \forall \,\, \chi ' \in (1-\pi '(\psi )) {\mathcal {H}} \end{aligned}$$

(283)

$$\begin{aligned} S_{\phi |\psi } \left( \alpha | \phi \rangle + |\xi '\rangle \right)&= \pi (\phi ) \alpha ^\dagger | \psi \rangle \qquad \forall \, \, \xi ' \in (1-\pi '(\phi )) {\mathcal {H}} \end{aligned}$$

(284)

for $\alpha \in {\mathcal {A}}$. Note that, for the first equation above, if both $\alpha | \psi \rangle =0$ and $| \chi ' \rangle =0$ vanish then $\alpha [ {\mathcal {A}}' | \psi \rangle ] = 0 \implies \pi (\psi ) \alpha ^\dagger = 0$ so 0 is mapped to 0 as is necessary for a linear operator. The Tomita operators are closable as defined and we will use the same symbol for the closure as the original operator. The support of these operators is:

$$\begin{aligned} \mathrm{supp} ( S_{\psi |\phi }, \, S_{\phi |\psi }^\dagger ) = \pi '(\psi ) \pi (\phi ) {\mathcal {H}} \qquad \mathrm{supp} ( S_{\phi |\psi }, \, S_{\psi |\phi }^\dagger ) = \pi (\psi ) \pi '(\phi ) {\mathcal {H}}. \end{aligned}$$

(285)

Applying the definitions twice we have:

$$\begin{aligned} S_{\psi |\phi } S_{\phi |\psi }&= \pi (\psi ) \pi '(\phi ) \end{aligned}$$

(286)

$$\begin{aligned} S_{\phi |\psi } S_{\psi |\phi }&= \pi '(\psi ) \pi (\phi ). \end{aligned}$$

(287)

For the commutant algebra we have:

$$\begin{aligned} S_{\psi |\phi }'\left( \alpha ' | \psi \rangle + | \chi \rangle \right)&= \pi '(\psi ) (\alpha ')^\dagger | \phi \rangle \qquad \forall \, \, \chi \in (1-\pi (\psi )) {\mathcal {H}} \end{aligned}$$

(288)

$$\begin{aligned} S_{\phi |\psi }' ( \alpha ' | \phi \rangle + | \xi \rangle )&= \pi '(\phi ) (\alpha ')^\dagger | \psi \rangle \qquad \forall \, \, \chi \in (1-\pi (\phi )) {\mathcal {H}} \end{aligned}$$

(289)

for $\alpha ' \in {\mathcal {A}}'$ with support that is complementary to (285). And similar equations hold for the commutant as in (286). Now consider:

$$\begin{aligned}&\left( \beta ' | \psi \rangle + | \chi \rangle , S_{\psi |\phi }\left( \alpha | \psi \rangle + | \chi ' \rangle \right) \right) = \left( \beta ' | \psi \rangle + | \chi \rangle , \pi (\psi ) \alpha ^\dagger | \phi \rangle \right) \end{aligned}$$

(290)

$$\begin{aligned}&\quad = \left( \beta ' | \psi \rangle , \alpha ^\dagger | \phi \rangle \right) = \left( \alpha | \psi \rangle , (\beta ')^\dagger | \phi \rangle \right) = \left( \alpha | \psi \rangle + | \chi '\rangle , \pi '(\psi ) (\beta ')^\dagger | \phi \rangle \right) \end{aligned}$$

(291)

$$\begin{aligned}&\quad = \left( \alpha | \psi \rangle + | \chi '\rangle , S_{\psi |\phi }'\left( \beta ' | \psi \rangle + | \chi \rangle \right) \right) \end{aligned}$$

(292)

which means that (because the above states are dense on the appropriate support)

$$\begin{aligned} S_{\psi |\phi }' = S_{\psi |\phi }^\dagger \,, \qquad S_{\phi |\psi }' = S_{\phi |\psi }^\dagger \end{aligned}$$

(293)

where the later equation follows a similar analysis.

We move now to the relative modular operators. Consider the positive self adjoint operators:

$$\begin{aligned} \varDelta _{\psi |\phi } = S_{\psi |\phi }^\dagger S_{\psi |\phi } \qquad \varDelta _{\phi |\psi } = S_{\phi |\psi }^\dagger S_{\phi |\psi } \end{aligned}$$

(294)

with support

$$\begin{aligned} \mathrm{supp} ( \varDelta _{\psi |\phi } ) = \pi '(\psi ) \pi (\phi ) {\mathcal {H}} \qquad \mathrm{supp} ( \varDelta _{\phi |\psi } ) = \pi (\psi ) \pi '(\phi ) {\mathcal {H}}. \end{aligned}$$

(295)

For the commutant algebra we learn that:

$$\begin{aligned} \varDelta _{\psi |\phi }' =&(S_{\psi |\phi }')^\dagger S_{\psi |\phi }' = S_{\psi |\phi } S_{\psi |\phi }^\dagger \quad \implies \quad \varDelta _{\psi |\phi }' \varDelta _{\phi |\psi } = \pi (\psi ) \pi '(\phi ) \end{aligned}$$

(296)

$$\begin{aligned} \varDelta _{\phi |\psi }' =&S_{\phi |\psi } S_{\phi |\psi }^\dagger \quad \implies \quad \varDelta _{\phi |\psi }' \varDelta _{\psi |\phi } = \pi '(\psi ) \pi (\phi ). \end{aligned}$$

(297)

We can define powers of the modular operators $\varDelta _{\psi |\phi }^z$ etc. to be zero when acting on $(1-\pi '(\psi ) \pi (\phi )) {\mathcal {H}}$ and to be the usual power when acting on the support of $\varDelta _{\psi |\phi }$ which for example means that $\varDelta _{\psi |\phi }^0 =\pi '(\psi ) \pi (\phi )$. So for example we have:

$$\begin{aligned} (\varDelta _{\psi |\phi }')^{z} \varDelta _{\phi |\psi }^{-z} = \pi (\psi ) \pi '(\phi ). \end{aligned}$$

(298)

Furthermore we can apply polar decompositions to the Tomita operators, where the anti-linear part is not unitary, but rather a partial anti-linear isometry with the support and range of the Tomita operators:

$$\begin{aligned} S_{\psi |\phi } = J_{\psi |\phi } \varDelta _{\psi |\phi }^{1/2} \qquad \qquad \mathrm{etc} \end{aligned}$$

(299)

where:

$$\begin{aligned} J_{\psi |\phi }^\dagger J_{\psi |\phi }&= \pi '(\psi ) \pi (\phi ) \qquad J_{\psi |\phi } J_{\psi |\phi }^\dagger = \pi (\psi ) \pi '(\phi ) \end{aligned}$$

(300)

$$\begin{aligned} J_{\phi |\psi }^\dagger J_{\phi |\psi }&= \pi (\psi )\pi '(\phi ) \qquad J_{\phi |\psi } J_{\phi |\psi }^\dagger = \pi '(\psi ) \pi (\phi ) \qquad \qquad \mathrm{etc} \end{aligned}$$

(301)

(with appropriate support and range.) Plugging back into the Tomita operators and (294) we have:

$$\begin{aligned} J_{\psi |\phi } \varDelta ^{1/2}_{\psi |\phi } J_{\phi |\psi } \varDelta ^{1/2}_{\phi |\psi } = \pi (\psi ) \pi '(\phi ) \quad \implies \quad J_{\psi |\phi } J_{\phi |\psi } ( J_{\phi |\psi }^\dagger \varDelta _{\psi |\phi }^{1/2} J_{\phi |\psi } ) = \varDelta _{\phi |\psi }^{-1/2} \end{aligned}$$

(302)

which by the uniqueness of the polar decomposition implies that:

$$\begin{aligned} J_{\psi |\phi } J_{\phi |\psi } = \pi (\psi ) \pi '(\phi ) \qquad J_{\psi |\phi } = J_{\phi |\psi }^\dagger \qquad J_{\psi |\phi } \varDelta _{\psi |\phi }^{1/2} J_{\phi |\psi } = \varDelta _{\phi |\psi }^{-1/2} \end{aligned}$$

(303)

where the last equation implies that:

$$\begin{aligned} J_{\psi |\phi } \varDelta _{\psi |\phi }^{is} = \varDelta _{\phi |\psi }^{is} J_{\psi |\phi } \end{aligned}$$

(304)

by the anti-linearity of J.

For the complement we use (293) to derive:

$$\begin{aligned} J_{\psi |\phi }' = J_{\phi |\psi } \qquad (\varDelta _{\psi |\phi }')^z = \varDelta _{\phi |\psi }^{-z}. \end{aligned}$$

(305)

In order to understand relative modular flow and cocycles we have to apply the Connes $2\times 2$ or $3\times 3$ trick which we present here in a vector language. Consider the Hilbert space:

$$\begin{aligned} {\mathcal {H}}_{\mathrm{tot}} = {\mathcal {H}}_L \otimes {\mathcal {H}}_R \otimes {\mathcal {H}}_{QFT} \end{aligned}$$

(306)

where ${\mathcal {H}}_{L,R}$ are both simple n-dimensional qunit Hilbert spaces with basis $|i \rangle ; i = 0,1, \ldots , n-1$. In this Hilbert space we consider the state:

$$\begin{aligned} | \varPsi \rangle = \sum _{i=1}^{n} \frac{1}{\sqrt{n}} | i_L i_R \rangle \otimes | \phi _i \rangle \end{aligned}$$

(307)

where $| \phi _i \rangle$ are vector states in the QFT Hilbert space that need not be cyclic and separating. We will actually consider the state as living in the subspace projected by the support projectors of the states $\phi _i$:

$$\begin{aligned} \widetilde{{\mathcal {H}}} = \left[ | i_L j_R \rangle \otimes \pi '(\phi _i) \pi (\phi _j) {\mathcal {H}}_{QFT}: i,j = 0,1, \ldots n-1 \right] \subset {\mathcal {H}}_{\mathrm{tot}} \end{aligned}$$

(308)

where the square brackets means the linear span. We now consider the algebra of operators acting on this new Hilbert space:

$$\begin{aligned} \gamma \in \mathbb {A}\,: \qquad \gamma = \sum _{ij} 1_L \otimes \left( | i \rangle \langle j | \right) _R \otimes c_{ij} \qquad c_{ij} \in \pi (\phi _i) {\mathcal {A}}\, \pi (\phi _j). \end{aligned}$$

(309)

The commutant is:

$$\begin{aligned} \gamma ' \in \mathbb {A}'\,: \qquad \gamma ' = \sum _{ij} \left( | i \rangle \langle j | \right) _L \otimes 1_R \otimes c'_{ij} \qquad c_{ij}' \in \pi '(\phi _i) {\mathcal {A}}'\, \pi '(\phi _j) \end{aligned}$$

(310)

and $\varPhi$ is cyclic and separating for these algebras after we project to the subspace $\tilde{{\mathcal {H}}}$. The generalized Tomita operator is:

$$\begin{aligned} \mathbb {S} \gamma | \varPsi \rangle = \gamma ^\dagger | \varPsi \rangle \end{aligned}$$

(311)

from which one finds:

$$\begin{aligned} \mathbb {S} = \sum _{ij} | j_L i_R \rangle \langle i_L j_R | \otimes {\tilde{S}}_{i|j} \,, \qquad {\tilde{S}}_{i|j} c_{ji} | \phi _i \rangle = c_{ji}^\dagger | \phi _j \rangle . \end{aligned}$$

(312)

Note that this later Tomita operator acts between Hilbert spaces:

$$\begin{aligned} {\tilde{S}}_{i|j}\, : \pi '(\phi _i) \pi (\phi _j) {\mathcal {H}}_{QFT} \rightarrow \pi '(\phi _j) \pi (\phi _i) {\mathcal {H}}_{QFT}. \end{aligned}$$

(313)

We can relate this to our original definition of the Tomita operators by passing back to the original unprojected Hilbert spaces setting $c_{ji} = \pi (\phi _j) \alpha \pi (\phi _i)$:

$$\begin{aligned} {\tilde{S}}_{i|j} \pi (\phi _j) \alpha | \phi _i \rangle = \pi (\phi _i) \alpha ^\dagger | \phi _j \rangle . \end{aligned}$$

(314)

We can lose the $\pi (\phi _j)$ on the left hand side since the orthogonal part is killed on the right hand side anyway. Or in other words we can extend the definition ${\tilde{S}}_{i|j}$ to the larger Hilbert space consistent with the right hand side by dropping this projector and also demanding:

$$\begin{aligned}&S_{i|j} \alpha | \phi _i \rangle = \pi (\phi _i) \alpha ^\dagger | \phi _j \rangle \end{aligned}$$

(315)

$$\begin{aligned}&S_{i|j} (1- \pi '(\phi _i) ) {\mathcal {H}}_{QFT} = 0 \end{aligned}$$

(316)

which is the same definition we gave for $i=\psi$ and $j=\phi$ in (283).

The modular operator for the $n \times n$ Hilbert space is:

$$\begin{aligned} \varvec{\Delta } = \mathbb {S}^\dagger \mathbb {S} = \sum _{ij} | i_L j_R \rangle \langle i_L j_R | \otimes {\tilde{\varDelta }}_{i|j}\,, \qquad {\tilde{\varDelta }}_{i|j} = {\tilde{S}}_{i|j}^\dagger {\tilde{S}}_{i|j} \end{aligned}$$

(317)

and the modular conjugation operator is:

$$\begin{aligned} \mathbb {J} = \sum _{ij} | j_L i_R \rangle \langle i_L j_R | \otimes {\tilde{J}}_{i|j} \qquad {\tilde{S}}_{i|j} = {\tilde{J}}_{i|j} {\tilde{\varDelta }}_{i|j}^{1/2} \end{aligned}$$

(318)

where $\mathbb {J}^2 = 1,\, \mathbb { J} \varvec{\Delta } \mathbb {J} = \varvec{\Delta }^{-1}$. The projected modular operators act between the following Hilbert spaces:

$$\begin{aligned} {\tilde{\varDelta }}_{i|j}\, : \pi '(\phi _i) \pi (\phi _j) {\mathcal {H}}_{QFT}&\rightarrow \pi '(\phi _i) \pi (\phi _j) {\mathcal {H}}_{QFT} \end{aligned}$$

(319)

$$\begin{aligned} {\tilde{J}}_{i|j}\, : \pi '(\phi _i) \pi (\phi _j) {\mathcal {H}}_{QFT}&\rightarrow \pi '(\phi _j) \pi (\phi _i) {\mathcal {H}}_{QFT} \end{aligned}$$

(320)

and can be extended to ${\mathcal {H}}_{QFT}$ as we did with the Tomita operators.

We now apply the results of Tomita-Takesaki theory to these new modular operators. That is we know that $\mathbb {A} = \varvec{\Delta }^{is} \mathbb {A} \varvec{\Delta }^{-is}$ and $\mathbb {A}' = \mathbb {J} \mathbb {A} \mathbb {J}$. Computing:

$$\begin{aligned} \varvec{\Delta }^{is} \gamma \varvec{\Delta }^{-is}&= \sum _k \sum _{ij} ( | k \rangle \langle k | )_L \otimes ( | i \rangle \langle j | )_R \otimes {\tilde{\varDelta }}_{k|i}^{is} c_{ij} {\tilde{\varDelta }}_{k|j}^{-is} \end{aligned}$$

(321)

$$\begin{aligned} \mathbb {J} \gamma \mathbb {J}&= \sum _k \sum _{ij} ( | i \rangle \langle j |)_L \otimes (| k \rangle \langle k |)_R \otimes {\tilde{J}}_{k | i } c_{ij} {\tilde{J}}_{j | k}. \end{aligned}$$

(322)

This is only consistent with the form of the algebra given in (309) if we have:

$$\begin{aligned} {\tilde{\varDelta }}_{k|i}^{is} c_{ij} {\tilde{\varDelta }}_{k|j}^{-is} \, \in \pi (\phi _i) {\mathcal {A}} \pi (\phi _j)\,, \qquad c_{ij} \in \pi (\phi _i) {\mathcal {A}} \pi (\phi _j) \end{aligned}$$

(323)

and the flowed operator is the same operator for all k. Extending these statements to the full Hilbert space by setting the flowed and conjugated operators to zero away from the support we find that:

$$\begin{aligned} \varDelta _{k|i}^{is} \alpha \varDelta _{k|j}^{-is} \, \in {\mathcal {A}} \pi '(\phi _k)\,, \qquad \alpha \in {\mathcal {A}} \end{aligned}$$

(324)

where we are forced to add $\pi '(\phi _k)$ so that it vanishes away from this support. For example this is consistent with $s \rightarrow 0$ where we find $\pi (\phi _i) \alpha \pi (\phi _j) \pi '(\phi _k)$. Note that we can pick $\alpha$ in ${\mathcal {A}}$ rather than the projected algebra since the modular operators above anyway apply this projection. Note the flowed operator is now marginally not independent of k due to $\pi '(\phi _k)$.

If we set $i=j$ this defines the standard modular automorphism group but now for a non-cyclic and separating vector:

$$\begin{aligned} \varDelta _{i}^{is} \alpha \varDelta _i^{-is} = \sigma _s^{\phi _i}(\alpha ) \pi '(\phi _i) \,, \qquad \sigma _s^{\phi _i}(\alpha ) = \varDelta _{\varOmega |i}^{is} \alpha \varDelta _{\varOmega |i}^{-is} \end{aligned}$$

(325)

where in the later equation we have used a cyclic and separating vector $\varOmega$ to define this flow.

For $\alpha = 1$ we can define the operator in ${\mathcal {A}}$ that is independent of k as the cocycle:

$$\begin{aligned} \varDelta _{k|i}^{is} \varDelta _{k|j}^{-is} \equiv (D \phi _i : D\phi _j )_s \pi '(\phi _k) \,, \qquad (D \phi _i : D\phi _j )_s \in {\mathcal {A}} \end{aligned}$$

(326)

which can be extracted by picking $\phi _k$ to be a cyclic and separating vector. The cocycle satisfies:

$$\begin{aligned} (D \phi _i : D\phi _j)^\dagger _s&= (D \phi _j : D\phi _i)_s \end{aligned}$$

(327)

$$\begin{aligned} (D \phi _i : D\phi _j)^\dagger _s (D \phi _i : D\phi _j)_s&= \sigma _s^{\phi _i}( \pi (\phi _j)) \end{aligned}$$

(328)

$$\begin{aligned} (D \phi _i : D\phi _j)_s (D \phi _i : D\phi _j)^\dagger _s&= \sigma _s^{\phi _j}(\pi (\phi _i)) \end{aligned}$$

(329)

where the right hand side of the later two equations are projection operators if $[ \pi (\phi _i), \pi (\phi _j)] =0$ which means that for states with commuting support projectors the cocycle is a partial isometry.

There is the following relation on triples of the cocycle:

$$\begin{aligned} (D \phi _1 : D\phi _2)_s (D\phi _2:D\phi _3)_s = (D\phi _1:D\phi _3)_s \qquad \mathrm{if} \,\,\,\, \begin{matrix} \pi (\phi _1)\pi (\phi _2) = \pi (\phi _1) \,\, \mathrm{or} \\ \pi (\phi _2)\pi (\phi _3) = \pi (\phi _3) \end{matrix} \end{aligned}$$

(330)

where the later conditions on the projectors can be guaranteed by demanding:

$$\begin{aligned} \pi (\phi _1){\mathcal {H}} \subset \pi (\phi _2){\mathcal {H}} \,\, \mathrm{or} \,\, \pi (\phi _3){\mathcal {H}} \subset \pi (\phi _2){\mathcal {H}}. \end{aligned}$$

(331)

Similary for the modular conjugation operators we have:

$$\begin{aligned} {\tilde{J}}_{k|i} c_{ij} {\tilde{J}}_{j|k} \, \in \pi '(\phi _i) {\mathcal {A}}' \pi '(\phi _j)\,, \qquad c_{ij} \in \pi (\phi _i) {\mathcal {A}} \pi (\phi _j) \end{aligned}$$

(332)

which extends to the ${\mathcal {H}}_{QFT}$ in the usual way with:

$$\begin{aligned} J_{k|i} \alpha J_{j|k} \in {\mathcal {A}}' \pi (\phi _k)\,, \qquad \alpha \in {\mathcal {A}} \end{aligned}$$

(333)

and where apart from the projector $\pi (\phi _k)$ this is the same operator independent of k. We define the non relative modular conjugation action as:

$$\begin{aligned} J_i \alpha J_i = j^{\phi _i}(\alpha ) \pi (\phi _i) \,,\qquad j^{\phi _i}(\alpha ) = J_{\varOmega |i} \alpha J_{i|\varOmega }. \end{aligned}$$

(334)

The equivalent of the cocycles are the following linear operators:

$$\begin{aligned} J_{k|i} J_{j|k} = \varTheta _{i|j}' \pi (\phi _k)\,, \qquad \varTheta _{i|j}' \in {\mathcal {A}}' \end{aligned}$$

(335)

which satisfies:

$$\begin{aligned} (\varTheta '_{i|j})^\dagger = (\varTheta '_{j|i}) \,, \quad (\varTheta '_{i|j})^\dagger (\varTheta '_{i|j}) = j^{\phi _j}(\pi (\phi _i)) \,, \quad (\varTheta '_{i|j}) (\varTheta '_{i|j})^\dagger = j^{\phi _i}(\pi (\phi _j)) \end{aligned}$$

(336)

where this is then a partial isometry if $[\pi (\phi _i),\pi (\phi _j)] =0$.

For example if we specify that $\phi _i = \varOmega$, cyclic and separating, and $\phi _j = \psi$ we find that:

$$\begin{aligned} J_\varOmega J_{\psi |\varOmega } = \varTheta '_{\varOmega |\psi }\,, \qquad (\varTheta '_{\varOmega |\psi })^\dagger (\varTheta '_{\varOmega |\psi }) = \pi '(\psi )\,, \qquad (\varTheta '_{\varOmega |\psi }) (\varTheta '_{\varOmega |\psi })^\dagger = J_{\varOmega } \pi (\psi ) J_\varOmega \end{aligned}$$

(337)

where the support of this operator is: $\mathrm{supp}(\varTheta _{\varOmega |\psi }') = \pi '(\psi ) {\mathcal {H}}$ and $\mathrm{supp}(\varTheta _{\varOmega |\psi }')^\dagger =J_\varOmega \pi (\psi ) J_{\varOmega } {\mathcal {H}}$. We also have the useful relations:

$$\begin{aligned} J_{\psi |\varOmega } = J_\varOmega \varTheta '_{\varOmega |\psi } \qquad J_{\varOmega |\psi } = ( \varTheta '_{\varOmega |\psi })^\dagger J_\varOmega \qquad J_\psi = (\varTheta '_{\varOmega |\psi })^\dagger J_\varOmega \varTheta '_{\varOmega |\psi }. \end{aligned}$$

(338)

There is also a triple relation:

$$\begin{aligned} \varTheta '_{1|2} \varTheta '_{2|3} = \varTheta '_{1|3} \qquad \mathrm{if} \,\,\,\, \begin{matrix} \pi (\phi _1)\pi (\phi _2) = \pi (\phi _1) \,\, \mathrm{or} \\ \pi (\phi _2)\pi (\phi _3) = \pi (\phi _3) \end{matrix} \end{aligned}$$

(339)

where again this later condition can be achieved only under the conditions specified for the support projectors.