# Causality Is an Effect, II

## Abstract

**:**

## 1. Introduction

## 2. Perturbation

## 3. Review of Previous Work

## 4. Quantum Version

Even for identical particles this leads to cancellation. In one dimension the wave function for a pair of Gaussians is (not normalized)

(The variables are ${x}_{1}$ and ${x}_{2}$; all the others are constants.) The diagonal elements of the density matrix (calculated from Equation (2)) already show signs of cancellation, as follows:

As is evident, if ${x}_{\alpha}$ is significantly different from ${x}_{\beta}$ or ${k}_{\alpha}$ from ${k}_{\beta}$, then there is already cancellation or rapid oscillation. With more particles the effect is stronger. This, by the way, is the reason that isolated systems can be analyzed without paying attention to symmetrization with respect to all identical particles in the universe.

## 5. Classical Proof

## 6. The Set ${\mathbf{\u03f5}}_{\mathbf{0}}\cap {\mathbf{\varphi}}^{-\mathbf{T}}{\mathbf{\u03f5}}_{\mathbf{T}}$ Is Not Empty under Quantum Evolution

## 7. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Justification of Schottky

## Appendix B. More on History

## Appendix C. The “Cat Map”

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**Figure 1.**The entropy as a function of time for the “cat map” is plotted. In all three figures the unperturbed entropy is shown. In the second figure the perturbation (with circles around values) takes place at time $t=3$, while in the third figure the perturbation is at time $t=13$. In the latter two cases the change in macroscopic behavior is subsequent in the sense of increasing entropy. (The graphs are inverted from the usual definition of entropy.)

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Schulman, L.S. Causality Is an Effect, II. *Entropy* **2021**, *23*, 682.
https://doi.org/10.3390/e23060682

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Schulman LS. Causality Is an Effect, II. *Entropy*. 2021; 23(6):682.
https://doi.org/10.3390/e23060682

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Schulman, Lawrence S. 2021. "Causality Is an Effect, II" *Entropy* 23, no. 6: 682.
https://doi.org/10.3390/e23060682