# Spin-Field Correspondence

## Abstract

**:**

**2016**, 759, 424–429, a new class of field theories called Nonlinear Field Space Theory was proposed. In this approach, the standard field theories are considered as linear approximations to some more general theories characterized by nonlinear field phase spaces. The case of spherical geometry is especially interesting due to its relation with the spin physics. Here, we explore this possibility, showing that classical scalar field theory with such a field space can be viewed as a perturbation of a continuous spin system. In this picture, the spin precession and the scalar field excitations are dual descriptions of the same physics. The duality is studied in the example of the Heisenberg model. It is shown that the Heisenberg model coupled to a magnetic field leads to a non-relativistic scalar field theory, characterized by quadratic dispersion relation. Finally, on the basis of analysis of the relation between the spin phase space and the scalar field theory, we propose the Spin-Field correspondence between the known types of fields and the corresponding spin systems.

## 1. Introduction

## 2. The Heisenberg Model

## 3. The Spin-Field Correspondence

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

## References

- Kirillov, A.A. Lectures on the Orbit Method; American Mathematical Society: Providence, RI, USA, 2004. [Google Scholar]
- Gell-Mann, M.; Levy, M. The axial vector current in beta decay. Il Nuovo Cimento
**1960**, 16, 705–726. [Google Scholar] [CrossRef] - Prabhakar, A.; Stancil, D.D. Spin Waves, Theory and Applications; Springer: New York, NY, USA, 2009. [Google Scholar]
- Bilski, J.; Brahma, S.; Marcianò, A.; Mielczarek, J. Klein-Gordon field from the XXZ model. Unpublished work. 2017. [Google Scholar]
- Kosterlitz, J.M.; Thouless, D.J. Metastability and Phase Transitions in Two-Dimensional Systems. J. Phys. C
**1973**, 6, 1181–1203. [Google Scholar] [CrossRef] - Holstein, T.; Primakoff, H. Field Dependence of the Intrinsic Domain Magnetization of a Ferromagnet. Phys. Rev.
**1940**, 58, 1098–1113. [Google Scholar] - Mielczarek, J.; Trześniewski, T. The Nonlinear Field Space Theory. Phys. Lett. B
**2016**, 759, 424–429. [Google Scholar] [CrossRef] - Ashtekar, A.; Lewandowski, J. Background independent quantum gravity: A Status report. Class. Quantum Gravity
**2004**, 21, R53–R152. [Google Scholar] [CrossRef] - Mielczarek, J.; Trześniewski, T. Nonlinear Field Space Cosmology. Unpublished work. 2017. [Google Scholar]
- Amelino-Camelia, G.; Freidel, L.; Kowalski-Glikman, J.; Smolin, L. The principle of relative locality. Phys. Rev. D
**2011**, 84, 084010. [Google Scholar] [CrossRef]

**Figure 1.**Graphical representation of the precession of the vector $\overrightarrow{J}$ around the x axis (the magnetic field $\overrightarrow{B}=({B}_{x},0,0)$). For small precession angles, the arrow of the vector outlines a circle on the $(q,p)$ plane. The picture captures the idea of local approximation of the spin phase space (${S}^{2}$) by the ${\mathbb{R}}^{2}$ phase space, where the relation precession of $\overrightarrow{J}$ = oscillation of q is satisfied.

**Table 1.**The correspondence between the different multiplicities of spins and the corresponding field theories.

Field Type | Field Spin (s) | $\mathbf{Dim}\left({\Gamma}_{\mathit{x}}\right)$ | ${\mathit{N}}_{\mathit{x}}$ |
---|---|---|---|

Scalar field | 0 | 2 | 1 |

Spinor field | 1/2 | 4 | 2 |

Vector field | 1 | 6 | 3 |

Rarita–Schwinger field | 3/2 | 8 | 4 |

Tensor field | 2 | 10 | 5 |

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**MDPI and ACS Style**

Mielczarek, J. Spin-Field Correspondence. *Universe* **2017**, *3*, 29.
https://doi.org/10.3390/universe3020029

**AMA Style**

Mielczarek J. Spin-Field Correspondence. *Universe*. 2017; 3(2):29.
https://doi.org/10.3390/universe3020029

**Chicago/Turabian Style**

Mielczarek, Jakub. 2017. "Spin-Field Correspondence" *Universe* 3, no. 2: 29.
https://doi.org/10.3390/universe3020029