# Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking

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## Abstract

**:**

## 1. Introduction

## 2. Correction to Wigner–Eckart Relations by SSB

#### Spontaneous Symmetry Breaking with Symmetric Hamiltonian

## 3. Generalized Wigner–Eckart Relations and Ward Identities

**Proposition**

**1.**

**Proof.**

**Remarks:**

**1.**If the order parameter A is CPT- and rotation-invariant, $\langle A|-\mathbf{k},{k}_{0},\overline{a}\rangle =\langle A|\mathbf{k},{k}_{0},a\rangle $, Equation (20) reduces to

**2.**For the special case of relativistic systems, ${F}_{a}({\mathbf{k}}^{2},{k}_{0})={F}_{a}({k}_{0}^{2}-{\mathbf{k}}^{2})={F}_{a}\left(0\right)$, by the requirement of Lorentz invariance (see [7]).

**3.**Invariance of the ground state under rotations implies that there exists a function ${G}_{a}({\mathbf{k}}^{2},{k}_{0})$ such that

#### 3.1. Symmetry Breaking Ward Identities for Non-Symmetric Hamiltonians

**Proposition**

**2.**

**Proof.**

#### 3.2. An Example

## Author Contributions

## Funding

## Conflicts of Interest

## References

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## Share and Cite

**MDPI and ACS Style**

Heissenberg, C.; Strocchi, F.
Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking. *Symmetry* **2020**, *12*, 1120.
https://doi.org/10.3390/sym12071120

**AMA Style**

Heissenberg C, Strocchi F.
Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking. *Symmetry*. 2020; 12(7):1120.
https://doi.org/10.3390/sym12071120

**Chicago/Turabian Style**

Heissenberg, Carlo, and Franco Strocchi.
2020. "Corrections to Wigner–Eckart Relations by Spontaneous Symmetry Breaking" *Symmetry* 12, no. 7: 1120.
https://doi.org/10.3390/sym12071120