Analyticity and the Unruh effect: a study of local modular flow

TL;DR

The paper provides general, non-case-specific constraints on when geometric modular flow can arise in relativistic QFTs, showing that such flow must be generated by a conformal Killing vector and, in well-behaved analytic states, must be future-directed. It unifies the Unruh effect with broader modular-flow phenomena via Tomita–Takesaki theory, the disk method, and analyticity arguments, and demonstrates how a conformal (but not necessarily isometric) modular flow can only occur within conformal field theories. It also discusses constructive approaches, notably path-integral methods, to realize states with local modular flow in analytic spacetimes, and outlines key open problems, including scaling near region edges, removing analyticity assumptions, and extending constructions beyond free theories. Together, these results advance understanding of energy, entropy, and geometry in QFT and quantum gravity, with implications for entanglement structure and holographic reconstruction. A central theme is that local modular flow tightly constrains the allowed geometric and dynamical structures in quantum field theories.

Abstract

The Unruh effect can be formulated as the statement that the Minkowski vacuum in a Rindler wedge has a boost as its modular flow. In recent years, other examples of states with geometrically local modular flow have played important roles in understanding energy and entropy in quantum field theory and quantum gravity. Here I initiate a general study of the settings in which geometric modular flow can arise, showing (i) that any geometric modular flow must be a conformal symmetry of the background spacetime, and (ii) that in a well behaved class of "weakly analytic" states, geometric modular flow must be future-directed. I further argue that if a geometric transformation is conformal but not isometric, then it can only be realized as modular flow in a conformal field theory. Finally, I discuss a few settings in which converse results can be shown -- i.e., settings in which a state can be constructed whose modular flow reproduces a given vector field.

Figures (14)

• Figure 1: The Bisognano-Wichmann formulation of the Unruh effect. The modular flow of the Minkowski vacuum $|\Omega\rangle,$ restricted to a Rindler wedge, is the Lorentz boost generated by $\xi^a = 2 \pi \left[x \left( \frac{\partial}{\partial t} \right)^a + t \left(\frac{\partial}{\partial x} \right)^a\right]$.
• Figure 2: A Cauchy slice $\Sigma$ together with the domain of dependence $A$ for a partial Cauchy slice, and the complementary domain $A'.$
• Figure 3: A two-dimensional disk embedded in a complex space $\mathbb{C}^n.$ If the embedding is a holomorphic function of the complex disk $|\zeta| \leq 1,$ and if $f : \mathbb{C}^n \to \mathbb{C}$ is holomorphic, then the value of $f$ at the tip of the disk can be written as an integral over values of $f$ at the disk's edge. In some settings, this idea can be used to extend a function defined in a vicinity of the disk's edge all the way into the disk's interior.
• Figure 4: Left: A pair of spacelike separated points $x$ and $y$ that are mapped, via the diffeomorphism $\psi_{t},$ to a pair of causally separated points. By moving $t$ continuously toward zero, we can find a parameter $t_0$ such that $\psi_{t_0}(x)$ and $\psi_{t_0}(y)$ are null separated. Right: Neighborhoods $\Omega_1$ and $\Omega_2$ of $x$ and $y$ that are small enough to be entirely spacelike separated. Their images under $\psi_{t_0}$ are null separated, so one can always construct a pair of non-commuting smeared fields in $\psi_{t_0}(\Omega_1)$ and $\psi_{t_0}(\Omega_2).$
• Figure 5: A neighborhood $D_1$ of the point $x$ in which $\xi$ is not future-directed, together with a neighborhood $D_2$ that is spacelike separated from $D_1 + \epsilon \xi$ and null separated from $D_1 - \epsilon \xi.$ The region $D_1$ is spacelike separated from $D_2 + \epsilon \left( \frac{\partial}{\partial x^0} \right),$ and null separated from $D_2 - \epsilon \left( \frac{\partial}{\partial x^0} \right).$

Theorems & Definitions (5)

• Theorem 3.1
• proof
• Definition 3.2: Weakly analytic state
• Definition 3.3: Analytic state
• Theorem 3.4