Supersymmetric Klein-Gordon and Dirac oscillators
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912288507691008 |
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| author | Popov, Alexander D. |
| author_facet | Popov, Alexander D. |
| contents | We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$ is a homogeneous Kähler-Einstein manifold $Z_6$=AdS$_7$/U(1)=U(3,1)/U(3)$\times$U(1). It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space $Z_6$ of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space $Z_6$. Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of $Z_6$. After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on $Z_6$ with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Supersymmetric Klein-Gordon and Dirac oscillators Popov, Alexander D. High Energy Physics - Theory Mathematical Physics Quantum Physics We have recently shown that the space of initial data (covariant phase space) of the relativistic oscillator in Minkowski space $\mathbb{R}^{3,1}$ is a homogeneous Kähler-Einstein manifold $Z_6$=AdS$_7$/U(1)=U(3,1)/U(3)$\times$U(1). It was also shown that the energy eigenstates of the quantum relativistic oscillator form a direct sum of two weighted Bergman spaces of holomorphic (particles) and antiholomorphic (antiparticles) square-integrable functions on the covariant phase space $Z_6$ of the classical oscillator. Here we show that the covariant phase space of the supersymmetric version of the relativistic oscillator (oscillating spinning particle) is the odd tangent bundle of the space $Z_6$. Quantizing this model yields a Dirac oscillator equation on the phase space whose solution space is a direct sum of two spinor spaces parametrized by holomorphic and antiholomorphic functions on the odd tangent bundle of $Z_6$. After expanding the general solution in Grassmann variables, we obtain components of the spinor field that are holomorphic and antiholomorphic functions from Bergman spaces on $Z_6$ with different weight functions. Thus, the supersymmetric model under consideration is exactly solvable, Lorentz covariant and unitary. |
| title | Supersymmetric Klein-Gordon and Dirac oscillators |
| topic | High Energy Physics - Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2411.19578 |