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| Formato: | Preprint |
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2024
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| Acceso en línea: | https://arxiv.org/abs/2411.03952 |
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| _version_ | 1866909379122429952 |
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| author | Jeudy, Charlie Rouleux, Michel |
| author_facet | Jeudy, Charlie Rouleux, Michel |
| contents | The irreps $(SU(2),{\cal H},U)$ of SU(2) of dimension $(2S+1)^N$, i.e. operators acting on the space ${\cal H}={\cal H}_N={\bf C}^{(2S+1)^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case $S=1/2$, Dirac \cite{Dir1} discovered that there is another rep given by $({\cal S}(N),{\cal H},V)$ where ${\cal S}(N)$ is the permutation group, Thus, the standard ``linear'' Hamiltonian, or Heisenberg interaction Hamiltonian $H_0=\sum_{1\leq i\leq N}\vec S_i\cdot\vec S_j$, where $\vec σ_i=2\vec S_i$ is the vector of Pauli matrices, can be interpreted as the sum of the ``Exchange Operators'' $P_{ij}$ between particles $i$ and $j$. Schrödinger \cite{Sch} generalized to higher spin numbers $S$ the Exchange Operator $P_{ij}=P_S(\vec S_i\cdot \vec S_j)$ as a polynomial of degree $2S$ in $\vec S_i\cdot \vec S_j$. This we call the $P$-representation. There is another rep induced by the one particle permutation of states operators $\widetilde Q_α$, which we call the $Q$-rep. Our main purpose is to write some physical Hamiltonians for a few particles in the $P$- or $Q$-rep and compute their spectrum. The simplest case where there are as many particles as available states for the spin operator along the $z$-axis, i.e. $N=2S+1=3$, see Weyl \cite{Wey} or Hamermesh \cite{Ham}. Finally, we consider the relationship between permutations and rotation invariance when $S=1/2$ and $S=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03952 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized exchange operators for a system of spin-1 particles Jeudy, Charlie Rouleux, Michel Mathematical Physics The irreps $(SU(2),{\cal H},U)$ of SU(2) of dimension $(2S+1)^N$, i.e. operators acting on the space ${\cal H}={\cal H}_N={\bf C}^{(2S+1)^N}$ of $N$ identical particles with spin $S$, are described by Clebsch-Gordan decomposition into inequivalent irreps. In the special case $S=1/2$, Dirac \cite{Dir1} discovered that there is another rep given by $({\cal S}(N),{\cal H},V)$ where ${\cal S}(N)$ is the permutation group, Thus, the standard ``linear'' Hamiltonian, or Heisenberg interaction Hamiltonian $H_0=\sum_{1\leq i\leq N}\vec S_i\cdot\vec S_j$, where $\vec σ_i=2\vec S_i$ is the vector of Pauli matrices, can be interpreted as the sum of the ``Exchange Operators'' $P_{ij}$ between particles $i$ and $j$. Schrödinger \cite{Sch} generalized to higher spin numbers $S$ the Exchange Operator $P_{ij}=P_S(\vec S_i\cdot \vec S_j)$ as a polynomial of degree $2S$ in $\vec S_i\cdot \vec S_j$. This we call the $P$-representation. There is another rep induced by the one particle permutation of states operators $\widetilde Q_α$, which we call the $Q$-rep. Our main purpose is to write some physical Hamiltonians for a few particles in the $P$- or $Q$-rep and compute their spectrum. The simplest case where there are as many particles as available states for the spin operator along the $z$-axis, i.e. $N=2S+1=3$, see Weyl \cite{Wey} or Hamermesh \cite{Ham}. Finally, we consider the relationship between permutations and rotation invariance when $S=1/2$ and $S=1$. |
| title | Generalized exchange operators for a system of spin-1 particles |
| topic | Mathematical Physics |
| url | https://arxiv.org/abs/2411.03952 |