Rindler Positivity in Quantum Field Theory

• Rindler positivity is a property in relativistic quantum field theories that ensures modular-reflected two-point functions yield non-negative vacuum expectations.
• It imposes infinite hierarchies of inequalities on Rényi mutual information, enforcing a decreasing and convex behavior with increasing separation.
• In 1+1-dimensional CFT, Rindler positivity leads to bootstrap-type constraints on twist-operator correlators, refining OPE coefficients to be positive.

Rindler positivity is a fundamental property of relativistic quantum field theories (QFTs), encapsulating the positivity of modular‐reflected two‐point functions between spacelike regions and their CPT-like reflected counterparts in Rindler wedges. This principle, when combined with vacuum clustering, implies an infinite set of sharp inequalities for Rényi mutual information (RMI) between a region and its mirror image. In 1+1–dimensional conformal field theory (CFT), these inequalities strengthen into concrete monotonicity and operator‐product‐expansion (OPE) constraints, including a novel bootstrap‐type positivity condition for twist-operator blocks (Blanco et al., 2019).

1. Definition of Rindler Positivity

Rindler positivity originates from Tomita–Takesaki theory, which asserts that for any local operator ${\cal O}(A)$ supported in a spacelike region $A$ in the right Rindler wedge and its CPT-like reflected image ${\cal O}(\bar A)$ in the left wedge, the vacuum expectation value satisfies: $$\bigl\langle 0\big|\;{\cal O}(\bar A)\;{\cal O}(A)\;\big|0\bigr\rangle \;\ge\; 0.$$ This reflects the positivity of the modular‐reflected two‐point function.

2. Rényi Mutual Information and Its Quantitative Formulation

For disjoint spatial regions $A_i$ (right wedge) and $\bar{A}_j$ (left wedge reflection), the $n$th Rényi entropy is: $S_n(V) = \frac{1}{1 - n}\,\log\Tr\bigl(\rho_V^n\bigr),$ yielding the Rényi mutual information: $I_n(A_i, \bar{A}_j) = S_n(A_i) + S_n(\bar{A}_j) - S_n(A_i \cup \bar{A}_j) = \frac{1}{n-1}\log\frac{\Tr\,\rho_{A_i \cup \bar{A}_j}^n}{\Tr\,\rho_{A_i}^n\,\Tr\,\rho_{\bar{A}_j}^n}.$ This measures correlations in the $n$th Rényi index between regions and their mirror reflections.

3. Positive-Definite Kernel Induced by Rindler Positivity

For integer $n > 1$, expanding a superposition of twist-operator insertions and utilizing reflection symmetry produces the $N \times N$ matrix: $$M_{ij} = \exp\left[(n-1)\,I_n(A_i, \bar{A}_j)\right] = F_n(A_i, \bar{A}_j),$$ which must be positive semidefinite. For translation-invariant families $A(u)$, this reduces to $F_n(\eta)$, where $\eta$ represents the distance parameter, and positivity of all minors dictates that $F_n(\eta)$ is a positive-definite function on $(0, \infty)$ in the sense of Schoenberg.

4. Cluster Property and Complete Monotonicity

Vacuum clustering ensures $I_n(\eta)\to 0$ as $\eta\to\infty$, forcing $F_n(\eta)$ to be bounded at infinity. The Widder–Bernstein theorem then stipulates that a continuous positive-definite function that is bounded at infinity must be completely monotonic (CM): $$(-1)^k\frac{d^k}{d\eta^k}F_n(\eta) \ge 0, \quad \forall k = 0,1,2,\dots$$ Since $F_n = e^{(n-1)I_n}$, this yields an infinite tower of local inequalities; notably,

$I_n'(\eta) \le 0,\quad I_n''(\eta) \ge 0,$

implying that $I_n(\eta)$ is a decreasing convex function of separation.

5. 1+1–Dimensional CFT, Cross-Ratios, and OPE Constraints

In 1+1–dimensional CFT, RMI between intervals of length $L$ separated by $\eta$ depends solely on the cross-ratio $x$: $$x = \frac{(v_1-u_1)(v_2-u_2)}{(u_2-u_1)(v_2-v_1)} = \left[1+\frac{\eta}{L}\right]^{-2},\quad \rho = -\log x = 2\log(1+\eta/L).$$ Complete monotonicity now applies to $F_n(x)$ in $\rho$: $$(-1)^k\frac{d^k}{d\rho^k}\exp\left[(n-1)I_n(x(\rho))\right] \ge 0,\quad k=0,1,2,\dots$$ and implies monotonicity with respect to $x$

$\frac{d}{dx}I_n(x) \ge 0,$

which is equivalent to strong subadditivity of Rényi entropy here. Through replica-twist-field construction,

$$F_n(x) = \exp\left[(n-1)I_n(x)\right] = \langle\sigma_1(0)\sigma_{-1}(x)\sigma_1(1)\sigma_{-1}(\infty)\rangle = \sum_{m}C_{m}x^{d_{m}},$$

where $d_{m}\ge 0$ are untwisted operator dimensions and $C_{m}$ are quadratic combinations of OPE coefficients. The CM condition mandates $C_{m}\ge 0$, giving an OPE positivity constraint.

6. Summary of Key Inequalities and Physical Significance

• Rindler positivity for one-parameter families:

$$F_n(\eta) = \exp[(n-1)I_n(\eta)] \implies F_n(\eta) \text{ is }$$

positive-definite on $(0,\infty)$.

• Clustering $\implies$ complete monotonicity:

$$(-1)^k F_n^{(k)}(\eta) \ge 0, \qquad I_n'(\eta)\le 0,\quad I_n''(\eta)\ge 0$$

$I_n(\eta)$ is decreasing and convex.

• CFT cross-ratio formulation:

$F_n(x) = \sum_{m}C_{m} x^{d_m},\quad C_{m}\ge 0$

OPE positivity constraint.

• Physical meaning: These inequalities, though Rényi mutual information for general $n\neq1$ does not obey strong subadditivity, enforce an infinite hierarchy of sign‐alternation constraints on dependence with separation. In 1+1–CFT, they become monotonicity and bootstrap‐type positivity requirements on twist-field correlators.

7. Applications and Implications

The established constraints yield new nontrivial bounds on the operator content of CFTs, especially via OPE positivity in four-point functions of twist operators. The resulting hierarchical inequalities for RMI as a function of separation offer quantitative probes for modular reflection properties and clustering in any relativistic QFT. In the context of CFTs, the positivity constraints on OPE coefficients derived from complete monotonicity are directly testable and provide novel input into the conformal bootstrap program (Blanco et al., 2019).