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A mathematical model for the Einstein-Podolsky-Rosen argument

Riccardo Adami, Luigi Barletti, Alessandro Teta

Abstract

We study a nonrelativistic system made of two quantum particles constrained to move on a line and a spin located at a fixed point of the line. Initially the two particles are in a maximally entangled state and the spin is down. The first particle interacts with the spin while the second particle is free, i.e., it does not interact neither with the first particle nor with the spin. We rigorously prove that there is a correlation between the state of the spin and the state of the second particle. More precisely, we show that, in a suitable scaling limit, if the first particle flips the spin, then the second particle possesses a definite momentum in the direction opposite to the spin.

A mathematical model for the Einstein-Podolsky-Rosen argument

Abstract

We study a nonrelativistic system made of two quantum particles constrained to move on a line and a spin located at a fixed point of the line. Initially the two particles are in a maximally entangled state and the spin is down. The first particle interacts with the spin while the second particle is free, i.e., it does not interact neither with the first particle nor with the spin. We rigorously prove that there is a correlation between the state of the spin and the state of the second particle. More precisely, we show that, in a suitable scaling limit, if the first particle flips the spin, then the second particle possesses a definite momentum in the direction opposite to the spin.
Paper Structure (6 sections, 5 theorems, 115 equations)

This paper contains 6 sections, 5 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Estimate of $\mathcal{R}(t)$
  4. Estimate of $Q_1$ and $Q_2$
  5. Estimate of ${\mathcal{J}} (t)$
  6. Estimate of $\mathcal{L}(t)$

Key Result

Theorem 1.1

Assume that $V$ and $f$ belong to the Schwartz space $\mathcal{S}(\mathbb{R})$, with $f$ real, even and $\|f\|_{L^2(\mathbb{R})} =1$, and let us fix $t> T_{coll}$. Then for $\varepsilon \to 0$ we have where and the positive constant $C$ is independent of $\varepsilon$ and depends on $t$ and on the parameters of the model.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 3.1
  • proof
  • Theorem 3.2: Estimate of $Q_1$ and $Q_2$
  • proof
  • Proposition 3.3
  • proof
  • Proposition 4.1
  • proof