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Technical note · Algebraic QFT / Modular theory

The algebra generated by $K_0$ and $G_f$ on a Rindler horizon

Status of this document. This is a technical note that reassembles known results within a deliberately restricted scope. It is not a claim of new theory or of new research results. The only generators considered are the common Rindler boost $K_0$ and the transversally smeared null translations $G_f$. Transverse‑dependent dilations $D_a$, a full affine current algebra, and special conformal transformations are not constructed.
Known theorem Null-plane QFT input Standing assumption / definition

1Overview

Consider the future horizon of the right Rindler wedge in the Minkowski vacuum, and the family of von Neumann algebras attached to null cuts of that horizon. Two kinds of generator appear here:

The relations obtained are $$[K_0,G_f]=-iG_f,\qquad [G_f,G_g]=0,$$ and the resulting real Lie algebra is the semidirect product $$\mathfrak{g}=\mathbb{R}K_0\ltimes\bigl(\mathscr{F}/\ker G\bigr),$$ in which $\mathbb{R}K_0$ acts on the abelian ideal by a uniform scalar multiple, with no $f$-dependent weighting.

The derivation is split into two stages, and the note keeps them apart throughout.

  1. What a single half-sided modular inclusion gives. Here we restrict entirely to the reference cut $A=0$, where half-sidedness follows from Bisognano–Wichmann alone. The Borchers–Wiesbrock theorem then yields a positive self-adjoint generator and its boost covariance.
  2. What requires null-plane QFT input. The whole family of cuts $\{A\}$, the identity $K_A=K_0-G_A$, the independence of the generator from the reference cut, and the mutual commutativity of the $G_f$ are introduced in §4 and §5 as results of Casini–Teste–Torroba (and Wall), not as consequences of the abstract inclusion.

In particular, the generator supplied by a single inclusion is a priori $G_{A;f}$, carrying a label for the reference cut; the statement $G_{A;f}=G_f$ is an input, not a theorem of the modular-inclusion framework.

What this note does not claim is listed in §9.

Definitions & conventions

2Setting and conventions

2.1 Geometry

Minkowski spacetime $\mathbb{R}^{1,d-1}$ with signature $(-,+,\dots,+)$, coordinates $(x^0,x^1,y)$ with transverse coordinate $y=(x^2,\dots,x^{d-1})\in\mathbb{R}^{d-2}$, and null coordinates $$x^\pm=x^0\pm x^1 .$$

2.2 Test-function space

We fix an explicit space rather than writing $C^\infty$ loosely. Let $$\mathscr{F}_0:=C_c^\infty(\mathbb{R}^{d-2};\mathbb{R}),$$ and let $\mathbf{1}$ denote the constant function $y\mapsto 1$. Since $\mathbf{1}\notin\mathscr{F}_0$, the constant mode is adjoined separately:

$$\mathscr{F}:=\mathscr{F}_0\oplus\mathbb{R}\mathbf{1}$$

as a real vector space. Elements are written $f=f_0+c\mathbf{1}$ with $f_0\in\mathscr{F}_0$ and $c\in\mathbb{R}$. Set $$\mathscr{F}_+:=\{f\in\mathscr{F}\ :\ f(y)\ge0\ \text{for all }y\},$$ so that $\mathscr{F}=\mathscr{F}_+-\mathscr{F}_+$.

The constant mode is treated separately because, as explained in §4.4, the generator attached to $\mathbf{1}$ is the null translation generator $P_+$, whose existence is available from Poincaré covariance of the vacuum theory, whereas the compactly supported modes are the ones supplied by the inclusions of §3. No claim is made about a larger space of bounded or slowly decaying $f$; extending $\mathscr{F}$ beyond $\mathscr{F}_0\oplus\mathbb{R}\mathbf{1}$ is a separate problem and is not addressed.

2.3 Null cuts and local algebras

A null cut is a map $A:\mathbb{R}^{d-2}\to\mathbb{R}$. For a cut $A$ put $$\mathcal{H}_A:=\{(x^+,y)\in\mathcal{H}\ :\ x^+>A(y)\},$$ and let $\mathcal{A}_A$ denote the von Neumann algebra attached to the causal completion of $\mathcal{H}_A$. In particular $\mathcal{A}_0=\mathcal{A}(W_R)$.

Assumption (V). The Minkowski vacuum $\Omega$ is cyclic and separating for every $\mathcal{A}_A$ occurring below. Tomita–Takesaki theory then supplies a modular operator $\Delta_A$ and a modular conjugation $J_A$.

2.4 Normalization of modular generators

$$\Delta_A^{it}=e^{-2\pi itK_A},\qquad K_A=-\frac{1}{2\pi}\log\Delta_A .$$ With this convention the Bisognano–Wichmann theorem reads: $e^{iuK_0}$ implements the boost of rapidity $u$. Each $K_A$ is self-adjoint with $K_A\Omega=0$ and $J_AK_AJ_A=-K_A$. These are full modular Hamiltonians (of the form $K_{\rm out}-K_{\rm in}$), not one-sided quantities.

Known theorems only — restricted to $A=0$

3What a single half-sided modular inclusion gives — restricted to $A=0$

This section uses only the reference cut $A=0$. Nothing here refers to a general cut family; general cuts enter in §4.

3.1 The inclusion and its half-sidedness

Let $f\in\mathscr{F}_+$ and take the cut $A=f$. Isotony gives $$\mathcal{A}_f\subset\mathcal{A}_0 .$$ By Bisognano–Wichmann, $\Delta_0^{it}$ implements the boost, acting on $\mathcal{H}$ as $x^+\mapsto e^{-2\pi t}x^+$. Hence $\mathcal{H}_f$ is mapped to $\{x^+>e^{-2\pi t}f(y)\}$, and for $t\le 0$ we have $e^{-2\pi t}f\ge f$, so $$\Delta_0^{it}\,\mathcal{A}_f\,\Delta_0^{-it}\subset\mathcal{A}_f\qquad(t\le0).$$ Thus $\mathcal{A}_f\subset\mathcal{A}_0$ is a half-sided modular inclusion (HSMI) with common cyclic and separating vector $\Omega$. Only Bisognano–Wichmann was used; no information about $\Delta_A^{it}$ for $A\neq0$ is needed.

We do not verify cyclicity of $\Omega$ for the relative commutant $\mathcal{A}_f'\cap\mathcal{A}_0$, and therefore do not use the term “standard HSMI”; we write simply HSMI.

3.2 The Borchers–Wiesbrock theorem

For an HSMI $\mathcal{N}\subset\mathcal{M}$ with common cyclic separating $\Omega$, Borchers (1992) and Wiesbrock (1993), with the technical completion of Araki–Zsidó (2005), give:

There is a unique strongly continuous one-parameter unitary group $s\mapsto U(s)=e^{isG}$ with $$G=G^*\ge0,\qquad G\Omega=0,$$ $$\Delta_\mathcal{M}^{it}U(s)\Delta_\mathcal{M}^{-it}=U(e^{-2\pi t}s),\qquad J_\mathcal{M}U(s)J_\mathcal{M}=U(-s),$$ $$U(1)\,\mathcal{M}\,U(1)^*=\mathcal{N}.$$

Applied to $\mathcal{A}_f\subset\mathcal{A}_0$ with $f\in\mathscr{F}_+$, this produces a generator denoted $$G_{0;f}.$$

The reference-cut label is retained deliberately. The theorem asserts uniqueness for a given inclusion. It says nothing about whether generators arising from different reference cuts coincide, and in this section no other reference cut is available.

What follows from §3 alone, for $f\in\mathscr{F}_+$:

This exhausts what the abstract framework yields here. Each inclusion contributes one generator; together with $K_0$ this is a two-dimensional algebra. Relations among generators belonging to different $f$, and any statement about $K_A$ for $A\neq0$, lie outside §3.

Null-plane QFT input

4Null-plane QFT input

Everything in this section is an additional physical input. None of it follows from §3.

4.1 The cut family and its modular Hamiltonians

Wall (2011) and Casini–Teste–Torroba (2017) give, for the vacuum and for algebras attached to cuts of a null plane, that $\Delta_A^{it}$ acts on $\mathcal{H}$ generator-by-generator as $$x^+\longmapsto A(y)+e^{-2\pi t}\bigl(x^+-A(y)\bigr),$$ and, in the normalization of §2.4, $$K_A=K_0-G_A. \tag{4.1}$$

Formally, $$K_A\ \text{“}=\text{”}\ \int d^{d-2}y\int dx^+\,\bigl(x^+-A(y)\bigr)\,T_{++}(x^+,y), \tag{4.2}$$ $$G_f\ \text{“}=\text{”}\ \int d^{d-2}y\,f(y)\int dx^+\,T_{++}(x^+,y). \tag{4.3}$$

(4.2)–(4.3) are formal identifications, not definitions. What is defined is the self-adjoint operator supplied by §3.2, or — for the cut family — by the input (4.1); the integral expressions record the correspondence in models where $T_{++}$ is meaningful.

Caution. The object $$\mathcal{P}(y):=\int dx^+\,T_{++}(x^+,y)$$ is not to be treated as an ordinary densely defined self-adjoint operator for each fixed $y$. $T_{++}$ is an operator-valued distribution, and smearing it along the whole null generator while localizing sharply in $y$ does not in general define an operator. The meaningful objects are the transversally smeared $G_f$ with $f\in\mathscr{F}$. The symbol $\mathcal{P}(y)$ is not used again.

4.2 Independence of the reference cut

Applying (4.1) to the pair $A$ and $A+f$, together with the sum rule read off from the Wiesbrock relation, $$G_{A;f}=K_A-K_{A+f},$$ gives $$G_{A;f}=(K_0-G_A)-(K_0-G_{A+f})=G_{A+f}-G_A,$$ whose right-hand side, by the linearity of (4.3), equals $G_f$ and contains no reference to $A$:

$$G_{A;f}=G_f\tag{4.4}$$

This is an input-derived statement. Its support is either the explicit form (4.1)–(4.3) of Casini–Teste–Torroba, or the coherence of the generators across the whole cut family in the known null-plane Lie algebra. It does not follow from the Borchers–Wiesbrock theorem, which knows only about one inclusion at a time. From here on we write $G_f$ and drop the label.

4.3 Formal linear extension to general $f$

Section 3 supplies generators only for $f\in\mathscr{F}_+$. For general $f=f_+-f_-\in\mathscr{F}$ we set $$G_f:=G_{f_+}-G_{f_-}$$ and treat this as a formal linear extension on a common core, in the following precise sense.

Assumption (L). There is a dense subspace $\mathcal{D}$ with $\Omega\in\mathcal{D}$, invariant under the relevant operators, on which every $G_f$ ($f\in\mathscr{F}$) is defined, and on which $$G_{af+bg}\,\psi=(aG_f+bG_g)\,\psi\qquad(a,b\in\mathbb{R},\ \psi\in\mathcal{D}).$$

No claim is made that $G_f$ is essentially self-adjoint on $\mathcal{D}$ for general $f$, and no attempt is made here to prove it. All algebraic statements below involving general $f\in\mathscr{F}$ are statements about relations holding on $\mathcal{D}$. Statements at the level of unitary groups are made only under the separate hypothesis of §5.2.

Positivity $G_f\ge0$ is retained only for $f\in\mathscr{F}_+$; it fails for general $f\in\mathscr{F}$.

4.4 The constant mode

For $f=\mathbf{1}$ the cut is the rigidly translated horizon, and $G_\mathbf{1}$ is the generator of $x^+\mapsto x^++s$, i.e. the null translation $$G_\mathbf{1}=P_+ ,$$ which exists as a positive self-adjoint operator with $P_+\Omega=0$ directly from Poincaré covariance of the vacuum theory (and, in modular language, from Borchers' theorem applied to the translated wedge). Since $\mathbf{1}\notin C_c^\infty$, this mode is not produced by the argument of §3.1 within $\mathscr{F}_0$; it is adjoined by hand as in §2.2, using its independently available construction. Assumption (L) is understood to include $\mathbf{1}$ in $\mathscr{F}$.

Operator-theoretic caveats

5Commutativity and operator-theoretic caveats

5.1 $[G_f,G_g]=0$ does not follow from a single HSMI

Stated explicitly: a single half-sided modular inclusion relates one generator to the modular generator of one reference algebra, and nothing more. The mutual relation of $G_f$ and $G_g$ for different $f,g$ requires an independent input. We use one of:

5.2 Strong commutativity versus vanishing commutators on a core

For unbounded self-adjoint operators, the formal relation $$G_{f+g}=G_f+G_g\quad\text{on }\mathcal{D}$$ does not imply commutativity of the unitary groups $e^{isG_f}$ and $e^{itG_g}$. Vanishing of the commutator on a common dense core is compatible with non-commuting spectral projections (the phenomenon exhibited by Nelson's example). We therefore distinguish:

The Lie-algebraic statements of §6 are made at the level of the core. Group-level statements — in particular that the algebra exponentiates to a group — are made only under the additional hypothesis of strong commutativity, i.e. under input (C2). We do not prove that (C1) implies (C2).

5.3 On decompositions of the null-plane algebra

For a general interacting QFT the following are logically distinct and are not identified here:

The last is beset by type-theoretic difficulties (continuous tensor products of type III$_1$ factors) and is not asserted. This note asserts none of the three; it uses only the weaker and explicitly stated input (C1) or (C2).

Theorem + input, combined

6Main relations and the resulting Lie algebra

6.1 Inputs and assumptions

(V)
$\Omega$ cyclic and separating for each $\mathcal{A}_A$
standard assumption
(BW)
Bisognano–Wichmann: $\Delta_0^{it}$ is the boost
known theorem
(BWi)
Borchers–Wiesbrock HSMI theorem
known theorem (with Araki–Zsidó)
(M)
$K_A=K_0-G_A$; formal identification (4.3); $G_{A;f}=G_f$
null-plane QFT input (§4)
(L)
formal linear extension of $f\mapsto G_f$ on a core $\mathcal{D}$
technical assumption (§4.3)
(C)
(C1) or (C2): commutativity of the $G_f$
null-plane QFT input (§5.1)
(P)
$G_\mathbf{1}=P_+$ adjoined
Poincaré covariance (§4.4)

6.2 Derivations

(i) $[K_0,G_f]=-iG_f$.
For $f\in\mathscr{F}_0\cap\mathscr{F}_+$ this is (3.1), obtained in §3 from (V), (BW), (BWi) alone. For $f=\mathbf{1}$ it is the standard boost–translation relation, available from (P). Extension to all $f\in\mathscr{F}$ is by (L), on the core:

$$[K_0,G_f]=-iG_f\tag{6.1}$$

The coefficient on the right is the single constant $-i$, independent of $f$: $K_0$ acts uniformly on the family $\{G_f\}$ and assigns no $f$-dependent weight. This is used in §8.

(ii) $[G_f,G_g]=0$.
By input (C), on the core $\mathcal{D}$:

$$[G_f,G_g]=0\tag{6.2}$$

Under (C2) this holds in the strong sense as well; see §5.2.

(iii) $K_A=K_0-G_A$.
Input (M), for cuts $A\in\mathscr{F}$:

$$K_A=K_0-G_A\tag{6.3}$$

(iv) $[K_A,K_B]=iG_{B-A}$.
For $A,B\in\mathscr{F}$ we have $B-A\in\mathscr{F}$, and from (6.1)–(6.3) with (L), $$[K_A,K_B]=[K_0-G_A,\,K_0-G_B]=-[K_0,G_B]+[K_0,G_A]=iG_B-iG_A,$$ so

$$[K_A,K_B]=iG_{B-A}\tag{6.4}$$

Also, for every $A\in\mathscr{F}$, $$[K_A,G_f]=[K_0,G_f]-[G_A,G_f]=-iG_f, \tag{6.5}$$ independent of $A$.

6.3 The Lie algebra, with the kernel quotiented out

Injectivity of $f\mapsto G_f$ is not assumed. Set $$\ker G:=\{f\in\mathscr{F}\ :\ G_f\psi=0\ \text{for all }\psi\in\mathcal{D}\},$$ a real linear subspace of $\mathscr{F}$ by (L). Because $\operatorname{ad}(K_0)$ acts on $\{G_f\}$ as a uniform scalar (6.1), every linear subspace is preserved by the action; in particular $\ker G$ is, so the quotient carries the induced action with no further hypothesis.

Introduce anti-self-adjoint generators $$\kappa:=iK_0,\qquad \gamma_{[f]}:=iG_f\quad\bigl([f]\in\mathscr{F}/\ker G\bigr),$$ well defined on the quotient. Then, on the core $\mathcal{D}$, $$[\kappa,\gamma_{[f]}]=\gamma_{[f]},\qquad [\gamma_{[f]},\gamma_{[g]}]=0,\qquad \gamma_{[af+bg]}=a\gamma_{[f]}+b\gamma_{[g]} .$$

The resulting real Lie algebra is

$$\mathfrak{g}=\mathbb{R}K_0\ltimes\bigl(\mathscr{F}/\ker G\bigr),\qquad \mathscr{F}=C_c^\infty(\mathbb{R}^{d-2};\mathbb{R})\oplus\mathbb{R}\mathbf{1}$$

with

Each $K_A$ with $A\in\mathscr{F}$ lies in $\mathfrak{g}$ as $K_0-G_A$, and (6.4) closes inside $\mathfrak{g}$.

Group level. Only under the strong-commutativity hypothesis (C2) does this exponentiate to a group $\bigl(\mathscr{F}/\ker G\bigr)\rtimes\mathbb{R}$, with $\mathbb{R}$ acting by the overall rescaling $f\mapsto e^{u}f$. Without (C2), the content of the boxed statement is the set of core-level relations above.

Geometric reading

7Geometric reading

On $\mathcal{H}$ the correspondence with vector fields is $$K_0\ \longleftrightarrow\ x^+\partial_+,\qquad G_f\ \longleftrightarrow\ f(y)\,\partial_+ ,$$ and the Lie brackets $$[x^+\partial_+,\,f\partial_+]=-f\partial_+,\qquad [f\partial_+,\,g\partial_+]=0$$ match (6.1)–(6.2) under $[\cdot,\cdot]_{\rm op}=i[\cdot,\cdot]_{\rm Lie}$; with $K_A\leftrightarrow(x^+-A(y))\partial_+$ the same holds for (6.4).

This correspondence is stated with the following restrictions.

Consequence of the algebra

8A structural limitation

Every element of $\mathfrak{g}$ has the form $$X=\alpha K_0+G_f,\qquad \alpha\in\mathbb{R},\ [f]\in\mathscr{F}/\ker G,$$ and by (6.1)–(6.2), for every $g$, $$\operatorname{ad}(X)\,G_g=\alpha[K_0,G_g]+[G_f,G_g]=-i\alpha\,G_g .$$

Thus every element of $\mathfrak{g}$ acts on the family $\{G_g\}$ by the uniform scalar $-i\alpha$. Consequently there is no element of $\mathfrak{g}$ implementing $$G_g\ \longmapsto\ G_{ag}$$ for non-constant $a$: no transverse-dependent dilation generator lies inside the algebra constructed here.

The scope of this conclusion.

9Explicitly not claimed

The note likewise does not establish, and does not attempt to establish: standardness of the inclusions (§3.1), essential self-adjointness of $G_f$ for general $f$ (§4.3), the implication (C1) $\Rightarrow$ (C2) (§5.2), injectivity of $f\mapsto G_f$ (§6.3), or any extension of $\mathscr{F}$ beyond $C_c^\infty\oplus\mathbb{R}\mathbf{1}$ (§2.2). These remain stated hypotheses.

10Conclusion

Combining the positivity and boost covariance supplied by a single half-sided modular inclusion at the reference cut $A=0$ with the known null-plane modular Hamiltonian structure ($K_A=K_0-G_A$, reference-cut independence of the generator, and mutual commutativity), one obtains a semidirect-product algebra $$\mathfrak{g}=\mathbb{R}K_0\ltimes\bigl(\mathscr{F}/\ker G\bigr),\qquad \mathscr{F}=C_c^\infty(\mathbb{R}^{d-2};\mathbb{R})\oplus\mathbb{R}\mathbf{1},$$ generated by the common boost $K_0$ and the commuting transversally smeared null translations $G_f$. The relations hold on a common core; the corresponding group statement holds under the additional hypothesis of strong commutativity.

No stronger claim is made.

—References