# Spontaneous Breakdown of the Time Reversal Symmetry

## Abstract

**:**

## 1. Introduction

## 2. Classical Effective Theories

## 3. A Single Degree of Freedom

#### 3.1. Classical Chronon-Dynamics

#### 3.2. Harmonic Systems

#### 3.3. Generalized ϵ-Prescription

- The coupling at the final time within a chronon, Equation (16), is transformed into an infinitesimal, time translation invariant coupling.
- The initial conditions are represented by the homogeneous solution of the equation of motion which belong to the null-space of the equation of motion operator. They influence the action of the generalized ϵ-prescription in $\mathcal{O}(\u03f5)$ which is enough to generate a finite effect via the null-space singularity of Equation (A17).
- Since the initial conditions are handled by the imaginary terms of the action the time flows in such a the direction which makes $\Im S\left[\widehat{x}\right]$ increasing during the motion, cf. Equation (15).

#### 3.4. Mechanical Time Arrow

## 4. Open Systems in the Thermodynamical Limit

#### 4.1. Effective Chronon Theory

#### 4.2. Energy Balance

#### 4.3. Normal Modes I.: Finite System

#### 4.4. Normal Modes II.: Infinite System

- $\mathrm{\Delta}\tilde{\omega}>0$: The observations, carried out in time $T\gg 1/\mathrm{\Delta}\tilde{\omega}$, can resolve all normal modes and the effective theory for ${x}_{T}$ is conservative. We can reproduce such observations with an effective theory where the integration over the spectral variable is performed first, followed by the summation. The linear equation of motion operator is given by Equation (68) and the motion is reversible at frequencies which do not belong to the normal mode spectrum.
- $\mathrm{\Delta}\tilde{\omega}=0$: The normal frequency spectrum has an accumulation point and the long but finite time measurements leave infinitely many normal modes unresolved. The infinitely many unresolved normal modes act as an uncontrollable absorber of the system energy and the effective dynamics for ${x}_{T}$ with finite T contains dissipative forces. The summation must be carried out first in the first equation in Equation (70) in this case, leaving the integration for the second step.

#### 4.5. Toy Model

- Discrete environment spectrum without condensation point: The breakdown of the time reversal symmetry is not universal. The experiments, performed in a sufficiently long time reveal the time reversal invariant system dynamics, realized at frequencies which do not belong to the environment spectrum.
- Discrete environment spectrum with condensation point: The continuous spectral function becomes a good approximation for finite time observations at the condensation point and indicate the irresistible loss of energy to those modes.
- The continuous environment spectrum: We find soft irreversibility at any frequency. The analysis of the discrete spectrum reveals that the loss of energy is due to the environment, whose modes are degenerate with the system.

## 5. Finite Life-Time and Decoherence

#### 5.1. Quantum Chronon-Dynamics

#### 5.2. Open Quantum Systems

#### 5.3. Propagator

#### 5.4. Double Role of the Environment Induced Time Arrow

- The transfer of the mechanical time arrow from the environment to the system by the spontaneous breakdown of the time reversal symmetry determines the direction of the time in which the dissipative work, performed by the system is positive and the motion is stable. The norm of the system state is non-increasing in the time direction.
- The positive norm of the system states makes $\Im {S}_{2}\ge 0$, where ${S}_{2}$ is defined by the help of the decomposition Equation (43) of the effective action and renders the interference among the possible system histories destructive rather than constructive in the same time direction.

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Non-Uniform Convergence and Symmetry Breaking

#### Appendix A.1. Long Distance Cutoff: Phase Transition

**Figure A1.**The qualitative dependence of the order parameter M on the explicit symmetry breaking, h, for $V=1$ (dotted line), $V=5$ (dashed line) $V=100$ (thick line).

**P**and

**L**, respectively.

#### Appendix A.2. Short Distance Cutoff: Dynamically Modified Relations

#### Appendix A.3. Short Time Cutoff: Propagation Along Fractals

#### Appendix A.4. Long Time Cutoff: Null-Space Singularities and Auxiliary Conditions

**Figure A2.**The function $G(\u03f5T)$ plotted against ϵ for $T=1$ (dotted line), $T=5$ (dashed line) and $T=100$ (thick line).

## Appendix B. Green Function of a Harmonic Oscillator

#### Appendix B.1. Discretized Time

#### Appendix B.2. Effective Action

#### Appendix B.3. Removal of the UV and the IR Cutoffs

#### Appendix B.4. Quantum Oscillator

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**Figure 1.**The system undergoes a time reversal transformation, the time arrow is flipped at $t={t}_{f}$, and the motion is followed back to its initial state.

**Figure 2.**The iteration of the equation of motion Equation (34) can be visualized by tree-graphs. Here the lines represent the Green functions, ${D}^{r}$, the dots stand for the vertex g and the crosses represent the source, j. The line AB of the third graph is referred to in Section 4.4.

**Figure 3.**The system coordinate, ${x}_{1}$, acts as an external source on the environment coordinate ${x}_{n}$, $n>1$, cf. Figure 1. The dashed lines represent ${D}_{n{\sigma}^{\prime}}^{+}({t}_{1},{t}_{2}){z}_{1}^{{\sigma}^{\prime}}({t}_{2})$, ${t}_{2}^{-}={t}_{2}$, ${t}_{2}^{-}=2{t}_{f}-{t}_{i}-{t}_{2}^{+}$, the response of ${z}_{n}({t}_{1})$ on ${z}_{1}({t}_{2})$.

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Polonyi, J.
Spontaneous Breakdown of the Time Reversal Symmetry. *Symmetry* **2016**, *8*, 25.
https://doi.org/10.3390/sym8040025

**AMA Style**

Polonyi J.
Spontaneous Breakdown of the Time Reversal Symmetry. *Symmetry*. 2016; 8(4):25.
https://doi.org/10.3390/sym8040025

**Chicago/Turabian Style**

Polonyi, Janos.
2016. "Spontaneous Breakdown of the Time Reversal Symmetry" *Symmetry* 8, no. 4: 25.
https://doi.org/10.3390/sym8040025