16 problems

Let $A$ be a smooth $*$-algebra, let $\sigma$ be a regular automorphism of $A$, and let $F$ be its fixed point algebra. Let ${\mathcal S}$ be the semigroup of modular homotopy clas…

Let $4\mathcal{A}(\mathcal{O})4$ be the algebra of a massive theory associated with a double cone $4\mathcal{O}4$, and let $4\omega_{m=0}4$ denote the conformally invariant vacuum…

Consider a massive quantum field theory and its holographic projection onto the lightcone horizon. Write the lightcone coordinates as $x_{+}=x^{0}+\left|\vec{x}\right|$ and…

Let a massive quantum field theory be localized in a double cone and let its modular objects define the corresponding modular transformations. The source considers transformations…

Modular symmetry-group size conjecture. Modular symmetry groups are generically larger than the isometry and conformal groups of the underlying space-times.

Let $\mathcal{M}$ be a local von Neumann factor, let $\sigma_s$ be its modular action, and let $\mathcal{R}(\mathcal{M},\sigma_s)$ be the algebra generated by $\mathcal{M}$ togethe…

Let $a^{(+)}(k)$ denote real-space creation or annihilation variables, and let $a^{(+)}(k,\beta)$ and $\tilde{a}^{(+)}(k,\beta)$ denote temperature-dependent primordial variables.…

Let $g_{ik}(x)$ be a spacetime metric, let $dt=dx^0$ be coordinate time, and consider an ideal clock fixed at spatial position $x^i$, $i=1,2,3$. Its proper time satisfies … Let…

Let ${\mathcal A}_a=U(a){\mathcal A}U(a)^*\subset{\mathcal A}$ be a half-sided modular inclusion, let $\rho$ and $\sigma$ be the states appearing in the strengthened monotonicity t…

Let $M$ be a von Neumann algebra with a faithful normal state $\varphi$, let $A$ be a $\mathrm{JW}$-algebra, and let $P\colon M\to M$ be a faithful normal positive unital projectio…

Second-derivative Rényi conjecture. The second null (or spacelike) derivative of the sandwiched Rényi divergence is non-negative.

Morozova–Chencov–Petz standard-representation conjecture. The following hold: (1) \H_{\hhh,\phi} is a Hilbert space; (2)…

The discussion concerns vacuum fluctuations and entanglement in bounded open regions of spacetime, together with the thermal properties of local algebras of observables. The author…

Let the massive free scalar theory be considered in the vacuum representation, localized in a double cone, and let its modular-group generator be compared with the corresponding ge…

Quantum Einstein equation conjecture. The Tomita–Takesaki theorem should play the role of a quantum Einstein equation, providing a way to associate to a suitable “state” of the phy…

Determination conjecture. There exists a unitary operator $W$ such that