# Broken Time Translation Symmetry as a Model for Quantum State Reduction

## Abstract

**:**

**PACS**11.30.Qc; 03.65.Ta; 04.20.Cv

## 1. Introduction

## 2. Spontaneous symmetry breaking

#### 2.1. The harmonic crystal

#### 2.2. Breaking the symmetry

## 3. Spontaneously broken unitarity

#### 3.1. The time scales of non-unitary dynamics

#### 3.2. The order parameter field

#### 3.3. Gravity’s influence on quantum mechanics

## 4. Spontaneously broken unitarity as a model for quantum state reduction

#### 4.1. The dynamics of quantum state reduction

#### 4.2. quantum measurement

#### 4.3. Born’s rule

## 5. Comparison to other models of quantum state reduction

#### 5.1. Experimental implications

## 6. Conclusions

## Acknowledgments

## References

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**Figure 1.**A nearly balanced pencil. The limits of making the pencil infinitely sharp ($b\to 0$) and perfectly balanced ($\theta \to 0$) do not commute, so that even the smallest deflection will tip over a sharp enough pencil.

**Figure 2.**The dynamics of the crystal wavefunction under the influence of a non-unitary field.

**Left**: the evolution of equation (11), starting from a perfectly homogeneous (${\mathbf{p}}_{\mathrm{tot}}=0$) initial state, and resulting in a localised state at the origin. The parameters used in this figure are $Nm=1.5\xb7{10}^{-9}$ kg and $\omega =0.1$ kHz. The inset shows the spread in position as a function of $tNm{\omega}^{2}$ for several simulations of the same process, with parameters varying between $1.0\xb7{10}^{-11}<Nm<1.0\xb7{10}^{-9}$ kg and $0.1<\omega <1.0$ kHz. It indicates that the time scale over which the reduction from an unordered, delocalised state to an ordered, localised state takes place, is proportional to $\hslash /\left(Nm{\omega}^{2}{x}_{0}^{2}\right)$, where ${x}_{0}^{2}$ is related to the spread of the initial wavepacket. The final spread in each individual case is determined by the competition between the two terms in equation (11), and is given by $\langle {x}^{2}\rangle =\hslash \sqrt{1/Nm}/\sqrt{Nm{\omega}^{2}}=\hslash /\left(Nm\omega \right)$.

**Right**: the evolution starting from an already localised state. The final state will again be a localised state at the origin, but the translation of the initial state towards the origin requires both terms in equation (11), resulting in a timescale for this process proportional to $1/\omega $. The parameters used in this figure are $Nm=1.0\xb7{10}^{-11}$ kg and $\omega =1.0$ kHz. The inset shows the average position as a function of $t\omega $ for several simulations of the same process, with parameters varying between $1.0\xb7{10}^{-11}<Nm<1.0\xb7{10}^{-10}$ kg and $0.1<\omega <1.0$ kHz. The curves in both figures have been renormalised at each time step for clarity.

**Figure 3.**The massive superposition $|\psi \rangle =\sqrt{1/2}\left(\right|{\psi}_{1}\rangle +|{\psi}_{2}\rangle )$ of a block of width L and mass M over a small distance x.

**Figure 4.**The dynamics of the crystal wavefunction under the influence of a non-unitary field, starting from an initial superposition of two localised states. The final state, localised at the origin, is reached via two consecutive processes. First the superposition is reduced to just a single localised wavefunction within a timescale proportional to $1/\left(Nm{\omega}^{2}\right)$, analogous to the process in the left of figure 2. Then the single localised state is translated towards the origin within a time scale proportional to $1/\omega $, as in the right of figure 2. The parameter values used in this figure are $Nm=2.0\xb7{10}^{-11}$ kg and $\omega =1.0$ kHz. The insets show the average position as a function of $tNm{\omega}^{2}$ (top) and $t\omega $ (bottom), for several simulations of the same process, with parameters varying between $0.5\xb7{10}^{-11}<Nm<5.0\xb7{10}^{-11}$ kg while $\omega =1.0$ kHz. The curves have been renormalised at each time step for clarity.

**Figure 5.**The distribution of times after which the crystal dynamics in the presence of a fluctuating field yields a localised wavefunction. The normalised number density $n\left(t\right)$ (in arbitrary units) indicates the number of simulations out of 50000 in which a single component first dominates the wavefunction at time t. The data is plotted as a function of t in the upper inset. The same data is shown in the main figure, but as a function of $tNm{\omega}^{2}$, indicating that the onset of the distribution is proportional to $1/\left(Nm{\omega}^{2}\right)$. The lower inset displays the same data again, but plotted against $t{\left(Nm{\omega}^{2}\right)}^{2}$, showing that the width of the distribution is proportional to $1/{\left(Nm{\omega}^{2}\right)}^{2}$. The different lines represent sets of simulations with parameter values ranging between $0.5<\omega <1.0$ kHz and $0.5\xb7{10}^{-11}<Nm<1.5\xb7{10}^{-11}$ kg.

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Van Wezel, J.
Broken Time Translation Symmetry as a Model for Quantum State Reduction. *Symmetry* **2010**, *2*, 582-608.
https://doi.org/10.3390/sym2020582

**AMA Style**

Van Wezel J.
Broken Time Translation Symmetry as a Model for Quantum State Reduction. *Symmetry*. 2010; 2(2):582-608.
https://doi.org/10.3390/sym2020582

**Chicago/Turabian Style**

Van Wezel, Jasper.
2010. "Broken Time Translation Symmetry as a Model for Quantum State Reduction" *Symmetry* 2, no. 2: 582-608.
https://doi.org/10.3390/sym2020582