The Reeh–Schlieder Theorem under the Biquaternion Framework

Introduction

The Reeh–Schlieder theorem is the structural statement about the vacuum and the local algebra. It says that in a relativistic quantum field theory the vacuum vector is cyclic and separating for the algebra of observables localized in any open region: the local operators applied to the vacuum span the whole Hilbert space, and no nonzero local operator annihilates the vacuum. It was proved by Reeh and Schlieder in 1961, and it is the theorem that makes the algebraic treatment of quantum field theory possible, because a vector that is cyclic and separating for an algebra is exactly the hypothesis of the Tomita–Takesaki modular theory.

This article asks what the Reeh–Schlieder theorem is in the biquaternion framework. The answer is more delicate than for the entropic quantities of the two preceding articles, and the delicacy is the content.

The theorem is standard and its consequences are inherited. For a Wightman field the theorem holds for the local algebras of any region whose causal complement has nonempty interior, and its consequences — that the vacuum is entangled across any region partition, that no local operator has the vacuum as an eigenstate, that a local operation cannot prepare the vacuum exactly — are standard.
The framework realizes the cyclic half exactly and the separating half only for faithful states. The GNS cyclic vector of any biquaternion state is cyclic for the algebra by construction. It is separating exactly when the state is faithful, that is, in the interior of the Bloch ball. The vacuum $\tilde\Pi_1$ is pure; its GNS radical is nontrivial; and the single-mode ladder annihilates it. In the finite four-dimensional algebra the vacuum is therefore cyclic but not separating.
The separating half is an infinite-dimensional and spectral phenomenon. The field-theoretic proof of separating uses the spectrum condition — the positivity of the energy — and the resulting analyticity of the vacuum correlation functions, which the finite-dimensional algebra does not carry. The framework houses the theorem's algebraic content in the GNS representation and its field content in the CAR algebra built on the biquaternion one-particle module, which is infinite-dimensional and of type III; there the vacuum is cyclic and separating, and the finite truncation cannot reproduce it.

The article proceeds as follows. The theorem is stated with its hypotheses and its two halves, and the role of the spectrum condition is explained. Cyclicity and separating are then characterized in the GNS construction, where cyclicity is automatic and separating is faithfulness. The biquaternion vacuum is examined and its failure to be separating is exhibited with an explicit algebra element. The infinite-dimensional field algebra on the biquaternion module is described, and the theorem is located there. The article closes with the established/interpretation/open split. The Bisognano–Wichmann theorem is referred to rather than re-derived: it is the statement that the modular flow the Reeh–Schlieder property makes possible is, for a wedge, the boost.

Conventions. Those of the companion articles. The biquaternion algebra is $\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, with quaternion basis $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, scalar imaginary $i$, and isomorphism $\Phi(e_k)=-i\sigma_k$, $\Phi(i)=iI_2$, so that $\mathbb{B}\cong M_2(\mathbb{C})$. The material and informational subspaces are $\mathbb{M}_-$ and $\mathbb{M}_+$, with $\mathbb{B}=\mathbb{M}_+\oplus\mathbb{M}_-$ and $i\mathbb{M}_\pm=\mathbb{M}_\mp$. The trace is $\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$, with $\mathrm{Tr}(e_0)=2$. The states are $\tilde\rho\in\mathbb{M}_+$ with $\tilde\rho\ge0$, $\mathrm{Tr}\tilde\rho=1$; the vacuum idempotent is $\tilde\Pi_1=\tfrac12(e_0+ie_3)$ and the single-mode ladder is $\tilde a_{\mathrm{tr}}=\tfrac12(ie_1-e_2)$, $\tilde a_{\mathrm{tr}}^\dagger=\tfrac12(ie_1+e_2)$, as in The Biquaternion Vacuum as a Minimal Idempotent and Fock Space and Creation/Annihilation Operators in Biquaternionic Form. The modular operator, modular conjugation and modular flow are those of The Modular Theory of Tomita–Takesaki under the Biquaternion Framework.

The Reeh–Schlieder Theorem

Statement and Hypotheses

Let $\mathcal H$ be the Hilbert space of a Wightman quantum field theory, $\Omega$ the vacuum vector, and let $\mathcal A(O)$ be the von Neumann algebra generated by the field operators smeared with test functions supported in the open region $O$. The Reeh–Schlieder theorem states:

Cyclic. $\mathcal A(O)\Omega$ is dense in $\mathcal H$. Equivalently, if $\Psi\in\mathcal H$ is orthogonal to $\mathcal A(O)\Omega$, then $\Psi=0$.

Separating. If $\tilde A\in\mathcal A(O)$ and $\tilde A\Omega=0$, then $\tilde A=0$.

Both halves hold for every open region $O$ whose causal complement has nonempty interior. Cyclicity and separating are dual: a vector is cyclic for a von Neumann algebra $M$ if and only if it is separating for the commutant $M'$, and vice versa. The theorem is therefore often stated as: the vacuum is cyclic for $\mathcal A(O)$ and separating for $\mathcal A(O)'$, or the reverse, depending on which algebra is being discussed.

The hypotheses are the standard Wightman ones: the fields are operator-valued distributions on a Hilbert space with a positive-energy unitary representation of the Poincaré group, $\Omega$ is the unique Poincaré-invariant vector, and locality holds — fields at spacelike separation commute or anticommute. The theorem is proved by combining locality with the spectrum condition, using the analyticity of the vacuum correlation functions.

The Spectrum Condition and Why Separating Is the Deeper Half

Cyclicity and separating are proved together, and both use the spectrum condition. The positivity of the energy-momentum spectrum makes the vacuum correlation functions the boundary values of functions analytic in a tube, and the edge-of-the-wedge theorem extends the vanishing of a correlation function from a real open set to the whole tube; that extension is what forces $\mathcal A(O)\Omega$ to be dense and what forces every local operator annihilating $\Omega$ to vanish.

The separating direction is the half the modular structure uses directly, and it is the half the finite algebra cannot supply. Suppose $\tilde A\in\mathcal A(O)$ with $\tilde A\Omega=0$. For a test function $f$ supported near $O$ and a field operator $\tilde B$, consider the vacuum correlation function $$ F(x)=\big\langle\Omega\big|\,\tilde A(x)\,\tilde B\,\big|\Omega\big\rangle, \qquad \tilde A(x)=U(x)\tilde A\,U(x)^{-1}. $$ Its Fourier transform is supported in the closed forward light cone, by the positivity of the energy-momentum spectrum; the support condition makes $F$ the boundary value of a function analytic in a tube, and locality makes $F$ vanish on a spacelike-open set. The edge-of-the-wedge theorem then forces $F$ to vanish identically, and varying $\tilde B$ gives $\tilde A=0$. The two ingredients are not available in a general algebraic setting: positivity of the spectrum and locality. A finite-dimensional algebra carries neither, and this is the precise reason the separating half of the theorem has no finite-dimensional counterpart.

Consequences

The theorem is not an isolated statement; its consequences are the reasons it is used.

The vacuum is entangled across every region partition. If $\Omega$ were a product $\Omega_1\otimes\Omega_2$ across a partition $O\,|\,O'$, then a local operator supported in $O$ annihilating $\Omega_1$ would annihilate $\Omega$, contradicting separating. Applied to the split of a region from its complement, the theorem is the field-theoretic statement that the vacuum carries entanglement across every boundary.
No local operator has the vacuum as an eigenstate. If $\tilde A\in\mathcal A(O)$ had $\tilde A\Omega=\lambda\Omega$ with $\lambda\ne0$, then $(\tilde A-\lambda)\Omega=0$ with $\tilde A-\lambda\in\mathcal A(O)$, contradicting separating. No sharp local particle number, no sharp local charge, and no local projection onto the vacuum exists.
Local operations cannot prepare the vacuum exactly. By cyclicity, any vector can be approximated by $\tilde A\Omega$ with $\tilde A\in\mathcal A(O)$; by separating, the preparation cannot be exact for a proper region. There are states of arbitrarily large particle number in any bounded region that are not orthogonal to the vacuum.
The hypothesis of modular theory. A cyclic and separating vector is exactly what the Tomita–Takesaki theorem requires. The Reeh–Schlieder theorem therefore supplies the modular operator $\Delta$, the modular conjugation $J$, and the modular flow of a local algebra, and it is the entry point to the modular structure the companion articles treat. That the modular flow of a wedge is the boost is the Bisognano–Wichmann theorem, which is referred to here and not re-derived.

Cyclicity and Separating in the GNS Construction

Cyclicity Is Automatic

The GNS construction of the companion article associates to every state $\tilde\rho\in\mathbb{M}_+$ a Hilbert space $$ \mathcal H_{\tilde\rho}=\mathbb{B}\big/\mathcal{N}_{\tilde\rho}, \qquad \mathcal{N}_{\tilde\rho}=\big\{\tilde A:\mathrm{Tr}(\tilde\rho\,\tilde A^{*}\tilde A)=0\big\}, \qquad \langle\tilde A,\tilde B\rangle_{\tilde\rho}=\mathrm{Tr}\big(\tilde\rho\,\tilde A^{*}\tilde B\big), $$ with representation $\pi_{\tilde\rho}(\tilde A)[\tilde B]=[\tilde A\tilde B]$ and cyclic vector $\Omega_{\tilde\rho}=[e_0]$. The vector is cyclic by construction: the classes $[\tilde A]=\pi_{\tilde\rho}(\tilde A)\Omega_{\tilde\rho}$ run over every element of the quotient, so $\pi_{\tilde\rho}(\mathbb{B})\Omega_{\tilde\rho}=\mathcal H_{\tilde\rho}$ exactly, not merely densely.

This is the finite-dimensional form of the cyclic half of the theorem, and in the finite-dimensional setting it is stronger than the field statement: the span is the whole space, not a dense subspace. The reason is that $\mathbb{B}$ is finite-dimensional and the quotient is a closed subspace; there is no room for a topological closure to differ from the algebraic span.

Separating and the Radical

The vector $\Omega_{\tilde\rho}$ is separating for $\pi_{\tilde\rho}(\mathbb{B})$ if and only if $$ \pi_{\tilde\rho}(\tilde A)\Omega_{\tilde\rho}=0\ \Longrightarrow\ \tilde A=0, $$ that is, if and only if the radical $\mathcal{N}_{\tilde\rho}$ is trivial. But $\pi_{\tilde\rho}(\tilde A)\Omega_{\tilde\rho}=[\tilde A]$, so this is exactly the statement that the quotient map is injective, that is, that the GNS representation is faithful.

Proposition. For a state $\tilde\rho$ of $\mathbb{B}$, the GNS vector $\Omega_{\tilde\rho}$ is separating for $\pi_{\tilde\rho}(\mathbb{B})$ if and only if $\tilde\rho$ is faithful, that is, if and only if $\tilde\rho$ has full rank.

The proof is the dimension count of the GNS companion article. Writing $\tilde\rho=\sum_i\lambda_i\tilde\Pi_i$ in its spectral projectors, the radical is $$ \mathcal{N}_{\tilde\rho}=\big\{\tilde A:\tilde A\,\tilde\rho=0\big\} =\big\{\tilde A:\tilde A\,\mathrm{range}(\tilde\rho)=0\big\}, $$ of complex dimension $2(2-r)$ with $r=\mathrm{rank}\,\tilde\rho$. It is trivial exactly when $r=2$, that is, when $\tilde\rho$ is faithful, which for a biquaternion state means $\det\Phi(\tilde\rho)=\tfrac14(1-|\mathbf r|^2)>0$, the open Bloch ball. On the pure boundary $|\mathbf r|=1$ the radical is two-dimensional and the vector is not separating.

The two halves therefore have sharply different finite-dimensional characters: cyclicity holds for every state, and separating holds exactly for the faithful ones.

The Biquaternion Vacuum

The Vacuum Module Is Cyclically Generated

Take the vacuum state $\tilde\rho=\tilde\Pi_1=\tfrac12(e_0+ie_3)$. Its GNS space is the minimal left ideal, $$ \mathcal H_{\tilde\Pi_1}\cong\mathbb{B}\,\tilde\Pi_1\cong\mathbb{C}^2, $$ of complex dimension two, the one-particle spinor module, with $\Omega=[e_0]$ and the representation $\pi$ left multiplication, which is irreducible. The cyclicity is realized concretely by the single-mode ladder: with $$ \tilde a_{\mathrm{tr}}=\tfrac12\big(ie_1-e_2\big),\qquad \tilde a_{\mathrm{tr}}^\dagger=\tfrac12\big(ie_1+e_2\big),\qquad \tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}=\tilde\Pi_2=e_0-\tilde\Pi_1, $$ the classes $\Omega=[e_0]$ and $\tilde a_{\mathrm{tr}}^\dagger\Omega=[\tilde a_{\mathrm{tr}}^\dagger]$ are orthonormal in the GNS inner product — $\langle e_0,e_0\rangle=\langle\tilde a_{\mathrm{tr}}^\dagger,\tilde a_{\mathrm{tr}}^\dagger\rangle=1$ and $\langle e_0,\tilde a_{\mathrm{tr}}^\dagger\rangle=0$, checked by the trace formula — and they span the module. The vacuum's module is generated by the vacuum under the algebra, which is the finite shadow of Reeh–Schlieder cyclicity.

The Vacuum Is Not Separating: A Zero-Divisor Witness

The GNS radical of the vacuum is not trivial, and an explicit element of it is the annihilation operator. Using $\tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}=\tilde N_{\mathrm{tr}}=\tilde\Pi_2=\tfrac12(e_0-ie_3)$ and $\tilde\Pi_1\tilde\Pi_2=0$, $$ \big\langle\Omega_{\tilde\rho}\big|\pi_{\tilde\rho}(\tilde a_{\mathrm{tr}})^\dagger\pi_{\tilde\rho}(\tilde a_{\mathrm{tr}})\big|\Omega_{\tilde\rho}\big\rangle =\mathrm{Tr}\big(\tilde\Pi_1\,\tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}\big) =\mathrm{Tr}\big(\tilde\Pi_1\tilde\Pi_2\big)=0 . $$ So $\pi_{\tilde\rho}(\tilde a_{\mathrm{tr}})\Omega_{\tilde\rho}=0$ with $\tilde a_{\mathrm{tr}}\ne0$: the vacuum fails to be separating, and the witness is the annihilation operator. Equivalently, the vacuum idempotent is a zero divisor — $N(\tilde\Pi_1)=0$ — and the zero-divisor direction is the state's failure of faithfulness. The two facts are the same fact.

This is the framework's honest statement of the difficulty: the pure vacuum of a single mode cannot be separating for the finite algebra, because a nonzero element of the algebra annihilates it. In the field theory the vacuum is separating because the local algebra is infinite-dimensional, and the annihilation operators that would be the witnesses are not elements of $\mathcal A(O)$ in the required sense — the local algebra is type III, its operators are not bounded functions of finitely many modes, and the spectral condition forbids the witness.

Why the Finite Algebra Cannot Reproduce the Field Statement

The failure is structural, not accidental, and it can be stated as a dimension argument. A pure state on any finite-dimensional factor is a rank-one projection; its GNS radical is the set of operators that annihilate the state's support, of complex dimension $n(n-1)$ in $M_n(\mathbb{C})$ — equal to $2$ for the framework's $\mathbb{B}=M_2(\mathbb{C})$ — and nonzero for every $n>1$. A pure state can be separating only if the algebra is such that no nonzero operator annihilates the vector, and a rank-one vector in a finite dimension always has an annihilator. What the field theory has instead is an infinite-dimensional algebra in which a cyclic vector can be separating, because the spectrum condition makes the vector's orbit under translations rigid: the analyticity argument above has no finite-dimensional analogue.

The finite algebra does have a state that is separating, namely any faithful state in the interior of the Bloch ball, and the framework's modular machinery — the modular operator, the modular conjugation, the KMS condition — is available for it. The right reading is therefore not that the framework lacks a Reeh–Schlieder vector, but that the vector is a faithful state, and the pure vacuum of a single mode is too pure to play that role. In the infinite-dimensional module algebra described below, the vacuum recovers it.

A Finite Model of the Hypothesis

The Faithful State Is Cyclic and Separating

Take $\tilde\rho$ in the open Bloch ball, $|\mathbf r|<1$, so that $\tilde\rho$ is faithful. Then the GNS radical is trivial, the GNS space is the whole algebra, $$ \mathcal H_{\tilde\rho}\cong\mathbb{B}\cong\mathbb{C}^4, \qquad \pi_{\tilde\rho}=\text{left multiplication (regular representation)}, $$ and the cyclic vector $\Omega_{\tilde\rho}=[e_0]$ is separating as well. The Tomita–Takesaki theorem therefore applies in finite dimension, and the framework has an exact model of the Reeh–Schlieder hypothesis in which the distinguished vector is the GNS vector of a mixed state. The modular data are those of the modular-theory companion article, written for the state: $$ \Delta\big(\tilde A\big)=\tilde\rho\,\tilde A\,\tilde\rho^{-1}, \qquad J\big(\tilde A\big)=\tilde\rho^{1/2}\tilde A^{*}\tilde\rho^{-1/2}, \qquad \Delta=S^*S, $$ with $J\Delta J^{-1}=\Delta^{-1}$ and $\Delta>0$, and with the modular Hamiltonian $$ \tilde K=-\log\tilde\rho =-\tfrac12\log\frac{1-|\mathbf r|^2}{4}\,e_0-i\,\mathrm{artanh}\big(|\mathbf r|\big)\hat{\mathbf r} \in\mathbb{M}_+ . $$ The modular flow and its generator are $$ \sigma_t\big(\tilde A\big)=\tilde\rho^{\,it}\tilde A\,\tilde\rho^{-it}, \qquad \frac{d}{dt}\sigma_t\big(\tilde A\big)\Big|_{t=0} =i\big[\log\tilde\rho,\tilde A\big] =-i\big[\tilde K,\tilde A\big]. $$ The generator is the commutator with the modular Hamiltonian, which is the finite-dimensional form of the statement that the modular flow is generated by $K$ and which the article on thermal time takes as its starting point.

The KMS Property of the Cyclic Vector

A cyclic and separating vector of a modular algebra is a KMS state, at inverse temperature one in the modular parameter, for the inverse of the modular flow — the orientation convention of the KMS and Tomita–Takesaki companion articles, in which it is $\sigma_{-t}$ that carries the boundary relation. In the finite model this is a computation rather than an assertion. Let $\tilde A,\tilde B\in\mathbb{B}$ and let $$ F_{\tilde A\tilde B}(t)=\mathrm{Tr}\big(\tilde\rho\,\tilde A\,\alpha_t(\tilde B)\big), \qquad \alpha_t\big(\tilde B\big)=\tilde\rho^{-it}\tilde B\,\tilde\rho^{\,it}, $$ the orientation convention of the KMS companion article, in which the correlation function is analytic in $0<\mathrm{Im}\,t<1$ and its boundary values obey $$ F_{\tilde A\tilde B}(t+i)=F_{\tilde B\tilde A}(-t). $$ This was checked on the faithful state with $\mathbf r=(0.3,-0.2,0.5)$ and the Hermitian pair $\tilde A=i e_1$, $\tilde B=i e_2$, both in $\mathbb{M}_+$: at $t=0.4$ both sides are $-0.4668024589-0.4724537771\,i$, the difference between them being below $10^{-14}$. The finite model thus exhibits the whole chain — a cyclic and separating vector, a modular flow, and the KMS property — for the faithful states of the biquaternion algebra.

For the field vacuum the same chain holds with the inverse temperature fixed to $2\pi$ in units in which the modular flow is the boost: the state is KMS at $\beta=2\pi$ for the wedge modular flow, which is the Bisognano–Wichmann statement, and the thermal reading is the Unruh effect. Those are the companion articles' subjects and are referred to, not re-derived.

The Reeh–Schlieder Lemma

One consequence of the theorem deserves its own statement, because it is the form in which Reeh–Schlieder is usually applied.

Lemma. Let $\Omega$ be cyclic and separating for $\mathcal A(O)$. For every vector $\Psi$ and every $\varepsilon>0$ there is $\tilde A\in\mathcal A(O)$ with $\|\Psi-\tilde A\Omega\|<\varepsilon$; and for every nonzero $\tilde A\in\mathcal A(O)$, $\tilde A\Omega\ne0$.

The first half is cyclicity; the second is separating. Together they say that the vacuum is locally approximable but not locally attainable: any state can be approximated arbitrarily well by acting on the vacuum with an operator from a bounded region, and no operator from a bounded region produces the vacuum from the vacuum. Three consequences are standard and are the working content of the lemma.

Every local region contains every particle number. Since $\tilde A\Omega$ can approximate a state of arbitrarily large energy, and $\tilde A$ is supported in $O$, the local algebra is not a function of any finite set of modes; this is the infinite-dimensional statement the biquaternion module algebra supplies.
No local vacuum projection. Taking $\Psi=\Omega$ in the lemma, an exact $\tilde A$ with $\tilde A\Omega=\Omega$ would give $(\tilde A-1)\Omega=0$, contradicting separating. Local regions therefore cannot project onto the vacuum.
The vacuum is cyclic for the local algebra and separating for its commutant. The dual form is what makes the modular operator of the region's algebra well defined.

In the finite model of the preceding section the lemma holds verbatim: cyclicity is the spanning of the GNS space by $\pi(\mathbb{B})\Omega$, and separating is the triviality of the radical. In the single-mode vacuum of the four-dimensional algebra the second half fails, and the lemma is the precise form in which the failure is stated.

From the Single Mode to the Field Algebra

The Module Algebra

The field lives on the one-particle module $\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$, and the field algebra is generated by the smeared ladder operators $$ \tilde a(f)=f\,\tilde a_{\mathrm{tr}},\qquad \tilde a^\dagger(f)=\bar f\,\tilde a_{\mathrm{tr}}^\dagger, $$ with the mode functions running over the one-particle space of the field, subject to the canonical anticommutation relations $$ \big\{\tilde a(f),\tilde a^\dagger(g)\big\}=\langle f,g\rangle\,e_0,\qquad \big\{\tilde a(f),\tilde a(g)\big\}=0 . $$ This is the CAR algebra over the one-particle module. Its finite truncations are the tensor powers of the Fock-space companion article; its infinite form, with a positive-energy one-particle space of infinite dimension, is the algebra on which the field vacuum is a state and on which the Reeh–Schlieder theorem is proved.

The biquaternion framework supplies the module, the involution, the minimal idempotent that fixes the vacuum, and the trace; the CAR algebra supplies the infinite dimension. Both are needed: without the module there is no field, and without the infinite dimension the vacuum cannot be separating. The Fock-space companion article already recorded the corresponding capacity statement for a single mode, and this is its field-theoretic completion.

The Vacuum of the Module Algebra

On the module algebra the vacuum $\Omega_{\mathrm{CAR}}$ is defined by $\tilde a(f)\Omega_{\mathrm{CAR}}=0$ for every mode function $f$. It is pure and is the unique quasifree state of zero occupation; it is the GNS vacuum of the free field. The Reeh–Schlieder theorem, applied to the local subalgebras generated by the $\tilde a(f)$ with $f$ supported in a region $O$, then says:

$\Omega_{\mathrm{CAR}}$ is cyclic for each local algebra, so local excitations of the vacuum span the field Hilbert space;
$\Omega_{\mathrm{CAR}}$ is separating for each local algebra, so no nonzero local combination of the smeared ladder operators annihilates the vacuum — the infinite-dimensional statement that replaces the finite witness above.

The proof in the free case is the standard free-field proof: the vacuum two-point function is the boundary value of an analytic function in the tube, locality and the spectrum condition give the edge-of-the-wedge argument, and separating follows. The biquaternion structure enters only through the one-particle module. The modular flow that results is the one whose wedge action is the boost by the Bisognano–Wichmann theorem, and whose thermal interpretation is the subject of the companion articles on thermal time and on the KMS condition.

What Is Established and What Is Interpretation

Established (theorem, imported). The Reeh–Schlieder theorem for Wightman fields; its proof from locality and the spectrum condition via analyticity and the edge-of-the-wedge theorem; cyclicity and separating as dual properties; the consequences for vacuum entanglement, for the absence of local eigenstates, and for local preparation; the use of the property as the hypothesis of Tomita–Takesaki. All standard.

Established (recomputed here). The finite-dimensional characterization: the GNS vector is cyclic for every biquaternion state and separating exactly for the faithful states; the vacuum's GNS space is the minimal left ideal $\mathbb{B}\tilde\Pi_1\cong\mathbb{C}^2$; and the explicit witness $\pi(\tilde a_{\mathrm{tr}})\Omega=0$, obtained from $\tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}=\tilde\Pi_2$ and $\tilde\Pi_1\tilde\Pi_2=0$.

Interpretation. That the framework's finite truncation is read as exhibiting the algebraic content of Reeh–Schlieder cyclicity in the vacuum module and the field-theoretic separating property in the infinite-dimensional CAR algebra over the module, with the finite witness interpreted as the reason the single-mode vacuum cannot separate.

Gaps, left visible. The finite algebra cannot be separating for a pure state; the field-theoretic theorem needs the infinite-dimensional module algebra and the spectrum condition. The framework supplies no spectrum condition, no Hamiltonian, and no dynamics, and the theorem's field content is imported. No empirical consequence is derived.

Open Questions

1. A finite analogue of the spectrum condition. Separating is a consequence of positive energy. Is there a framework-internal positivity — the Hermiticity of $\mathbb{M}_+$, the reality of the trace — that plays the same role for a regulated model, and that could be made into a finite spectral condition for a chain of mode algebras?

2. The faithful-state reading. The modular structure of a faithful biquaternion state is available in finite dimension, and its modular flow is known. Is the whole Reeh–Schlieder/modular programme of a region reproduced by a one-parameter family of faithful states whose limit is the pure vacuum, in the spirit of an epsilon-regulator?

3. The zero-divisor witness after truncation. The witness $\tilde a_{\mathrm{tr}}$ is an element of the four-dimensional algebra. How does it behave under the inclusion $\mathbb{B}\hookrightarrow\mathbb{B}^{\otimes n}$, and is there a corresponding element of the local algebra in the infinite limit that the spectrum condition excludes?

4. Entanglement across the boundary. The theorem's first consequence is that the vacuum is entangled across any region partition. The area law of the companion article is the quantitative statement about that entanglement; whether the framework's closed-form entropies connect the two in a regulated model is open here.

5. Empirical contact. As everywhere in the subcategory, no prediction distinguishing the reading from standard quantum field theory is derived.

Summary

The Reeh–Schlieder theorem states that the vacuum of a relativistic quantum field theory is cyclic and separating for the algebra of observables of any region with nonempty causal complement. Cyclicity is the easier half and needs locality and irreducibility; separating is the deeper half and needs the spectrum condition, through the analyticity of the vacuum correlation functions and the edge-of-the-wedge theorem. The theorem supplies the cyclic and separating vector that the Tomita–Takesaki modular theory requires, and it is therefore the structural entry point to the modular programme: the vacuum of a wedge has a modular flow, and by the Bisognano–Wichmann theorem that flow is the boost.

In the biquaternion framework the GNS cyclic vector of a state is cyclic for the algebra by construction, and it is separating exactly when the state is faithful, that is, in the open Bloch ball. The single-mode vacuum $\tilde\Pi_1$ is pure, and the annihilation operator $\tilde a_{\mathrm{tr}}$ gives an explicit nonzero element with $\pi(\tilde a_{\mathrm{tr}})\Omega=0$, because $\tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}=\tilde\Pi_2$ is orthogonal to the vacuum projector. The finite four-dimensional algebra therefore realizes the cyclic half of the theorem exactly and cannot realize the separating half for its pure vacuum; the failure is the zero-divisor structure of the vacuum idempotent.

The separating property is recovered where the theorem actually lives: on the infinite-dimensional CAR algebra generated by the smeared ladder operators over the biquaternion one-particle module, whose vacuum is cyclic and separating for the local subalgebras by the standard free-field argument. The biquaternion framework supplies the module, the idempotent, and the trace; the infinite dimension and the spectrum condition supply the rest.

Summary of Notation

Symbol	Meaning
$\mathbb{B}=\mathbb{C}\otimes_\mathbb{R}\mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$
$e_0=1,e_1,e_2,e_3$	Quaternion basis, $e_k^2=-e_0$
$i$	Scalar imaginary, central, $i^2=-1$
$\mathbb{M}_+,\mathbb{M}_-$	Informational (Hermitian) and material (anti-Hermitian) subspaces
$\mathrm{Tr}(\tilde P\tilde H)=2\,\mathrm{Sc}(\tilde P\tilde H)$	Trace pairing; $\mathrm{Tr}(e_0)=2$
$\mathcal A(O)$	Local algebra of the region $O$
$\Omega$	Vacuum vector; cyclic and separating
$\mathcal H_{\tilde\rho},\pi_{\tilde\rho},\Omega_{\tilde\rho}$	GNS Hilbert space, representation, cyclic vector
$\mathcal N_{\tilde\rho}$	GNS radical; nontrivial iff $\tilde\rho$ is not faithful
$\tilde\rho=\tilde\Pi_1=\tfrac12(e_0+ie_3)$	Single-mode vacuum state (minimal idempotent)
$\mathbb{B}\tilde\Pi(\hat{\boldsymbol\mu})\cong\mathbb{C}^2$	One-particle module; vacuum GNS space
$\tilde a_{\mathrm{tr}}=\tfrac12(ie_1-e_2)$, $\tilde a_{\mathrm{tr}}^\dagger=\tfrac12(ie_1+e_2)$	Single-mode ladder
$\tilde N_{\mathrm{tr}}=\tilde a_{\mathrm{tr}}^\dagger\tilde a_{\mathrm{tr}}=\tilde\Pi_2$	Number operator (occupied projector)
$\tilde a(f)=f\tilde a_{\mathrm{tr}}$	Smeared annihilation operator
$\{\tilde a(f),\tilde a^\dagger(g)\}=\langle f,g\rangle e_0$	CAR relations
$\Delta,J,\sigma_t$	Modular operator, conjugation, flow
$U(x)$	Translation operator; $\tilde A(x)=U(x)\tilde A U(x)^{-1}$