# The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem

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## Abstract

**:**

## 1. Introduction

## 2. Spontaneous Symmetry Breaking: The Linear $\mathit{\sigma}$ Model

#### 2.1. The Nambu-Goldstone Theorem: Standard Counting Rule

#### 2.2. Charge Conservation

## 3. The Quantum Yang-Baxter Conditions

#### 3.1. The Relations between the Order Parameter and the Conserved Charges

#### Exchange of the Intermediate Particles $n\to {n}^{\prime}$

## 4. The Spontaneous Symmetry Breaking Condition and the Yang-Baxter Relations

#### 4.1. Pairs of Nambu-Goldstone Bosons Representing the Same Degree of Freedom

#### 4.2. Pairs of Nambu-Goldstone Bosons Independent

## 5. The Counting of Nambu-Goldstone Bosons Based on the Yang-Baxter Relations

## 6. Sum of Vacuums and Theorems Connected with the Spontaneous Symmetry Breaking Phenomena

#### 6.1. The Justification behind the Sum over the Vacuums

#### Further Justifications of the Sum over the Degenerate Vacuum

#### 6.2. The Theorems Contained inside the Yang-Baxter Relations

**Theorem**

**1.**

**Corollary**

**1.**

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The potential for the spontaneous breaking of symmetry of the $O\left(N\right)$ symmetry, for the case of $N=2$. Taken from [33].

**Figure 2.**The Quantum Yang Baxter relations defined by Equation (32).

**Figure 3.**The effect of exchanging $n\to {n}^{\prime}$. The upper relation corresponds to the QYBE defined in Equation (32). The lower relation corresponds to Equation (36). The figures marked with the same color are the twist map (mirror image) of each other. The arrows represent the flow of time for each phase.

**Figure 4.**The effect of exchanging n by ${n}^{\prime}$. The figures connected by blue lines are the mirror image of each other.

**Figure 5.**The degenerate vacuum corresponding to the Nambu potential. The black line crossing the circumference suggests that, for each positive value of the order parameter on a given vacuum (expectation value), there is a corresponding negative value with the same magnitude but defined in a different vacuum. Then, summing over the expectation values of the of the order parameters, obtained along the degenerate vacuum gives a trivial (vanishing) result. The arrows illustrate the direction of the sum. The right-hand side of the figure is the potential term defined in Equation (48) and the left-hand side is the same potential observed over the ${\varphi}_{1}-{\varphi}_{2}$ plane. The right-hand side is partially taken from [38] and briefly modified.

**Figure 6.**The triangular representation of the vacuum degeneracy. At the finite volume approximation, the triangles are large (left hand-side). At the infinite volume limit, the triangles become narrow and the three lines (in the triangle) become almost parallel (right-hand side of the figure). The equality between triangles in the figure represents the QYBE and each triangle corresponds to one term in the expansion (41) after summing over the degenerate vacuum.

**Figure 7.**The effect of external perturbations. In the infinite volume approximation, any small external perturbation selects only one vacuum arbitrarily. The left-hand side represents the situation before the perturbation appears. In such a case, the QYBE are valid and we can sum over the degenerate vacuum. Once the perturbation selects one particular vacuum, the degenerate vacuum disappears (red dotted line) and we have the usual interaction of two particles (two Nambu-Goldstone bosons). The Nambu-Goldstone bosons, however, only move through the degenerate vacuum. The slopes of the lines are exaggerated in the figure, but, in the large volume limit, the lines are almost horizontal.

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Arraut, I.
The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem. *Symmetry* **2019**, *11*, 803.
https://doi.org/10.3390/sym11060803

**AMA Style**

Arraut I.
The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem. *Symmetry*. 2019; 11(6):803.
https://doi.org/10.3390/sym11060803

**Chicago/Turabian Style**

Arraut, Ivan.
2019. "The Quantum Yang-Baxter Conditions: The Fundamental Relations behind the Nambu-Goldstone Theorem" *Symmetry* 11, no. 6: 803.
https://doi.org/10.3390/sym11060803