Causal Diamonds

Definition

Given two points $p$ and $q$ in Minkowski spacetime (or more generally a globally hyperbolic spacetime) with $q$ in the causal future of $p$, the causal diamond (also called double cone or diamond region) is

$\mathcal{D}(p,q)=J^{+}(p)\cap J^{-}(q),$

where $J^{+}(p)$ is the causal future of $p$ (the set of all points reachable from $p$ by future-directed causal curves) and $J^{-}(q)$ is the causal past of $q$.

Equivalently, for a spatial ball $B_r$ of radius $r$ centered at the origin in $t=0$ slice, the causal diamond $\mathcal{D}(B_r)$ is the domain of dependence of $B_r$: the set of all points $x$ such that every inextendible causal curve through $x$ intersects $B_r$. In coordinates:

$\mathcal{D}(B_r)=\{(t,\mathbf{x}):|\mathbf{x}|+|t|<r\}.$

This is a compact region, in contrast to the non-compact Rindler wedge.

The local von Neumann algebra $\mathcal{M}(\mathcal{D})$ associated to a causal diamond is a type III₁ factor in any reasonable QFT (assuming the Haag-Kastler axioms), and the vacuum is cyclic and separating for it. In conformal field theories, the modular Hamiltonian for the diamond algebra is known explicitly:

$K=2\pi {\int }_{B_r}\frac{r^2-|\mathbf{x}{|}^2}{2r}{T}_{tt}(\mathbf{x})d^{d-1}x,$

a result due to Casini, Huerta, and Myers.

Role in Research

Causal diamonds extend the Rindler-wedge Bell inequality program to compact, physically realistic regions. The paper Eur. Phys. J. C 85, 2025 establishes Bell-CHSH violation for the free scalar field vacuum in a bipartite diamond setup, complementing the original Rindler results.

The compactness of diamonds is physically important: it allows in-principle realizations with finite-size apparatuses. In the Rindler setting, the causal regions are infinite half-spaces — inaccessible to any finite experiment. Diamonds are the natural setting for discussions of laboratory quantum information experiments informed by relativistic QFT.

The explicit modular Hamiltonian formula in CFT (the Casini-Huerta-Myers result) makes causal diamonds particularly amenable to computing the Araki-Uhlmann relative entropy via

$S(\omega \Vert \phi )=-⟨\Omega ,\log {\Delta }_{\Phi ,\Omega }\Omega ⟩,$

since $\log \Delta$ is related to the modular Hamiltonian $K=-\log \Delta$.

Relations

• rindler-wedges — the non-compact counterpart. A wedge is a limit of diamonds, but a boost limit that pushes one tip to null infinity: taking $r\to \infty$ with the centre fixed gives all of Minkowski instead. The same algebraic methods (Tomita-Takesaki, Weyl operators) apply to both.
• tomita-takesaki-modular-theory — the modular operator for the diamond algebra is explicitly known in CFT; this makes relative entropy computations tractable
• bell-chsh-inequality — Bell-CHSH violations in diamond regions extend the Rindler program to compact setups (Eur. Phys. J. C 85, 2025)
• type-iii-von-neumann-algebras — $\mathcal{M}(\mathcal{D})$ is type III₁ for any local region in AQFT, including diamonds
• araki-uhlmann-relative-entropy — explicit modular Hamiltonians in CFT allow direct computation of relative entropy for diamond algebras
• weyl-operators — Weyl operators localized in the diamond (test functions supported in $\mathcal{D}$) generate the local algebra $\mathcal{M}(\mathcal{D})$

Papers

See bell-inequalities-qft and relative-entropy-qft for full paper lists.

Notes

• The term “causal diamond” is standard in the AQFT and quantum gravity literature. “Double cone” is the older AQFT term; “diamond region” is used in quantum information contexts.
• The modular Hamiltonian is only known explicitly for specific geometries: Rindler wedges (all QFTs), causal diamonds (CFTs via Casini-Huerta-Myers), and spherical entangling surfaces (CFTs). For general regions and non-conformal theories, it is non-local and not explicitly computable.
• Two spacelike-separated diamonds (diamonds whose closures do not intersect) provide a natural bipartite setup where Alice controls one diamond and Bob the other; microcausality ensures $[A,B]=0$ automatically.
• The modular Hamiltonian above assigns a local inverse temperature $\beta (\mathbf{x})=\pi (r^2-|\mathbf{x}{|}^2)/r$, so an observer at the centre of the diamond sees $T=1/(\pi r)$, falling with the size of the diamond, while the temperature diverges toward the edge. This is the diamond analogue of the Unruh temperature $1/(2\pi x^1)$ at proper distance $x^1$ from a Rindler horizon.