Ball Regions and Conformal Modular Hamiltonians

For the vacuum of a conformal field theory, the modular flow of a spatial ball is exactly geometric and its modular Hamiltonian is a local stress-tensor integral. The result follows by conformally mapping the ball’s causal diamond to a Rindler wedge. Conformal covariance and the vacuum are essential: the same local formula is not exact for a massive theory, a generic region, or a generic excited state.

Required background. Conformal geometry and maps supplies the diamond–wedge transformation, the conformal stress tensor supplies the charge, and the Bisognano–Wichmann theorem supplies the wedge modular flow.

Helpful background. The cylinder map and CFT interval, sphere, and cylinder examples relate this flow to thermal descriptions.

The conformal Killing flow of a ball

Let $B=\{\mathbf x:r<R\}$ be the ball at $t=0$, centered at the origin. Its domain of dependence is the causal diamond $\lvert t\rvert+r<R$. The conformal Killing vector that preserves the diamond is

$$\zeta_B= \frac{\pi}{R} \left[(R^2-t^2-r^2)\,\partial_t -2t x^i\partial_i\right].$$

It is future-directed inside the diamond and vanishes on the diamond’s null boundary. At $t=0$,

$$\zeta_B^0=\frac{\pi}{R}(R^2-r^2).$$

The vacuum modular Hamiltonian is therefore

$$K_B= 2\pi\int_{r<R}d^{d-1}x\; \frac{R^2-r^2}{2R}\,T_{00}(0,\mathbf x)+c.$$

The normalization agrees with the wedge formula near any smooth point of the entangling surface: writing the inward proper distance as $\ell=R-r$, the weight obeys $(R^2-r^2)/(2R)=\ell+O(\ell^2/R)$. The additive $c$ fixes $\operatorname{Tr}e^{-K_B}=1$ only in a regulated type-I description and does not affect the flow.

The structural map places Ball Regions and Conformal Modular Hamiltonians between intrinsic modular data and the additional hypotheses that permit a geometric interpretation.

Tomita polar decomposition intrinsically produces $J$ and $\Delta$. Wedge boosts and CFT ball flow are special consequences of locality, covariance, the spectrum condition, the vacuum, and—only for the ball—the conformal map. The diagram is schematic and not to scale.

From the wedge to the diamond

A special conformal transformation maps the right Rindler wedge to the causal diamond of $B$. In a CFT, this map is unitarily implemented on the vacuum representation, up to the standard Weyl transformation of local fields. Conjugating the Bisognano–Wichmann modular group by that unitary gives the ball modular group, as derived in Casini, Huerta, and Myers 2011, §2.

For a scalar primary $\mathcal O$ of scaling dimension $\Delta_{\mathcal O}$, the active flow has the form

$$\sigma_s(\mathcal O(x)) =\Omega_s(x)^{\Delta_{\mathcal O}}\, \mathcal O(x_s),$$

where $x_s$ follows the integral curves of $\zeta_B$ with the modular convention fixed above, and $\Omega_s$ is the conformal scale factor. Spin indices receive the corresponding local Lorentz transformation. The explicit coordinate fractions depend on the chosen modular-time sign; the invariant content is that the flow preserves the diamond and fixes its tips and entangling sphere.

This construction explains locality. A generic logarithm of a reduced state need not be an integral of $T_{00}$. Here the logarithm is the conserved charge associated with a conformal Killing field because the algebra–state pair is unitarily related to the wedge pair.

Hyperbolic thermal frame

The causal diamond is also conformal to the hyperbolic cylinder $\mathbb R_\tau\times H^{d-1}$ with curvature radius $R$. Under this map, the reduced vacuum state becomes a thermal state for translations in $\tau$ at

$$T_{\rm hyp}=\frac{1}{2\pi R}.$$

This thermal description is especially useful for Rényi entropies and spherical entanglement. It is not a statement that the original Minkowski ball has a uniform local temperature. The redshifted inverse temperature with respect to Minkowski time varies as the inverse of the weight:

$$\beta_{\rm loc}(r)=2\pi\,\frac{R^2-r^2}{2R}.$$

It tends to zero at the entangling surface, reproducing the universal Rindler behavior, and reaches $\pi R$ at the center in this coordinate normalization.

Two-dimensional interval check

For a $1+1$-dimensional CFT interval $(-R,R)$, the formula reduces to

$$K_{(-R,R)} =2\pi\int_{-R}^{R}dx\; \frac{R^2-x^2}{2R}\,T_{00}(x)+c.$$

For a general interval $(u,v)$ on the line, translation and rescaling give

$$K_{(u,v)} =2\pi\int_u^v dx\; \frac{(x-u)(v-x)}{v-u}\,T_{00}(x)+c.$$

The weight is positive inside, vanishes linearly at both endpoints, and has length dimension one. These three checks catch most sign and normalization mistakes.

Limits of the result

The exact formula uses the CFT vacuum and a round ball. A relevant deformation introduces a scale and destroys the unitary conformal equivalence to the wedge. An excited state changes the modular operator even if the algebra is unchanged. A deformed entangling surface changes the causal domain and generates response terms, including stress-tensor integrals on null boundaries and contact contributions. Those corrections are addressed through shape perturbation, not by substituting a new radius into the ball kernel.

The formula also does not determine the spectrum of $K_B$ as a discrete list. In continuum QFT the local algebra is type III; the hyperbolic thermal picture organizes correlators but does not manufacture a trace-class reduced density matrix.

Common pitfalls

Using conformal symmetry without the conformal vacuum. A CFT Hamiltonian does not make every state conformally related to the wedge vacuum. The state is part of the modular data.

Calling $\beta_{\rm loc}(r)$ a global temperature. It is a position-dependent conversion between modular flow and Minkowski time. The uniform temperature belongs to the hyperbolic frame.

Forgetting the complement. The displayed integral is one-sided. The full standard-form modular generator acts with the complementary contribution of opposite orientation.

Before assigning geometric meaning to this modular statement, use the validity map to check standardness, spectral domains, normalization, geometric hypotheses, and approximation control independently.

A modular-flow claim is only as strong as its algebra–state standardness, spectral domain, normalization, geometric hypotheses, and approximation control. Dashed branches mark conditional steps; the terminal warnings identify common out-of-domain uses. The diagram is schematic and not to scale.

References

• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 2011, no. 5 (2011): 036. DOI; arXiv.

Further reading

• Hislop, Peter D., and Roberto Longo. “Modular Structure of the Local Algebras Associated with the Free Massless Scalar Field Theory.” Communications in Mathematical Physics 84 (1982): 71–85. Project Euclid.