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ON CUTOFFS IN QUANTUM MECHANICS

Jacques Mandelbrojt, Szolem Mandelbrojt

Published Jan 27, 1964• 0 citations
OpenAlex

Abstract

The procedure used to obtain numerical results is analyzed. An inequality between the half-interval and normalization constant associated with a particular cutoff is established. (D.C.W.)

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A bstract We explore constraints that arise from associating an entanglement wedge (EW) to subregions of a cutoff boundary at a finite distance in AdS/CFT, using a subcritical end-of-the-world (ETW) brane acting as a cutoff. In particular, we consider the case of two intervals in the holographic dual to a BCFT, with one interval A located at the asymptotic boundary and the second interval B located on the ETW brane. We discuss in detail subtleties that arise near the RT end-points when defining the EW for this configuration, particularly in the connected phase. Entanglement wedge nesting (EWN) requires that $$ {\mathcal{W}}_E(A) $$ W E A ∪ $$ {\mathcal{W}}_E(B) $$ W E B ⊆ $$ {\mathcal{W}}_E\left(A\cup B\right) $$ W E A ∪ B . We demonstrate that already in the simplest example of an AdS 3 bulk geometry, EWN can be violated even if A and B are spacelike separated through the bulk and instead we must require the stronger condition that $$ {\mathcal{W}}_E(A) $$ W E A be spacelike separated from $$ {\mathcal{W}}_E(B) $$ W E B , which highlights the non-local nature of the cutoff theory. Our prescription to associate EWs to subregions on the ETW brane is different from the restricted maximin procedure [1] but will agree within the subset of parameter space where EWN is respected. Additionally, we study EWN in a two sided BTZ black hole geometry with an ETW brane in one of the exteriors. In the BTZ black hole example we find that our condition for EWN disallows configurations where the RT surface goes from the brane to the black hole singularity.

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Significance of a certain restriction in the derivation of Bell's inequality from quantum mechanics

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Physics Letters A•Aug 1, 1983•1 citation
It has been known since at least 1969, that there exist quantum states of a free particle on the line with purely positive momentum that exhibit probability transfer towards the negative half-line. This phenomenon, the quantum backflow effect, is the subject of this thesis. The largest amount of backflow any state can exhibit over a single time interval, the Bracken–Melloy constant, was initially calculated to be cBM ≈ 0.04. In the first half of this thesis, we consider the question of how much quantum backflow states can exhibit over disjoint time intervals. This is done by formulating the question in terms of the spectra of bounded operators. When considering more than one disjoint time interval, we discuss a new phenomenon where a state exhibits more probability transfer in the same direction as its momentum than any classical state. We call this effect quantum overflow. Given a number M of disjoint time intervals, we give bounds on the maximum amount of backflow and overflow a state can exhibit over any M many intervals. The limiting cases in which two disjoint intervals merge into one is particularly studied. Finally, we show plots of the time t = 0 momentum space wavefunction of states exhibiting multiple quantum backflow and quantum overflow. The second half of the thesis contains a detailed numerical investigation into the single backflow problem. Based on previous conjectures, we construct the maximum backflow state from a given set of normalised states closely resembling the sinc function. The Bracken—Melloy constant is then bounded from below by the largest solution to a generalised eigenvalue problem. Using a multi-step numerical procedure, we solve a high precision large dense generalised eigenvalue problem. From this we obtain new numerical approximation to the backflow state and the best known rigorous lower bound of the Bracken—Melloy constant.
A bstract We explore constraints that arise from associating an entanglement wedge (EW) to subregions of a cutoff boundary at a finite distance in AdS/CFT, using a subcritical end-of-the-world (ETW) brane acting as a cutoff. In particular, we consider the case of two intervals in the holographic dual to a BCFT, with one interval A located at the asymptotic boundary and the second interval B located on the ETW brane. We discuss in detail subtleties that arise near the RT end-points when defining the EW for this configuration, particularly in the connected phase. Entanglement wedge nesting (EWN) requires that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A ∪ WE(B) {\mathcal{W}}_E(B) WE​(B) W E B ⊆ WE(A∪B) {\mathcal{W}}_E\left(A\cup B\right) WE​(A∪B) W E A ∪ B . We demonstrate that already in the simplest example of an AdS 3 bulk geometry, EWN can be violated even if A and B are spacelike separated through the bulk and instead we must require the stronger condition that WE(A) {\mathcal{W}}_E(A) WE​(A) W E A be spacelike separated from WE(B) {\mathcal{W}}_E(B) WE​(B) W E B , which highlights the non-local nature of the cutoff theory. Our prescription to associate EWs to subregions on the ETW brane is different from the restricted maximin procedure [1] but will agree within the subset of parameter space where EWN is respected. Additionally, we study EWN in a two sided BTZ black hole geometry with an ETW brane in one of the exteriors. In the BTZ black hole example we find that our condition for EWN disallows configurations where the RT surface goes from the brane to the black hole singularity.