Group theoretic quantization of punctured plane
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917050004275200 |
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| author | Somvanshi, Manvendra Stargen, D. Jaffino |
| author_facet | Somvanshi, Manvendra Stargen, D. Jaffino |
| contents | We quantize punctured plane, $X=\mathbb{R}^2-\{0\}$, employing Isham's group theoretic quantization procedure. After sketching out a brief review of group theoretic quantization procedure, we apply the quantization scheme to the phase space, $M=X \times \R^2$, corresponding to the punctured plane, $X$. Particularly, we find the canonical Lie group, $\mathscr{G}$, corresponding to the phase space, $M=X \times \R^2$, to be $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$. We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, and the smooth functions, $f\in C^{\infty}(M)$, in the phase space, $M=X \times \R^2$. Making use of this homomorphism and unitary representation of the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, we deduce a quantization map that maps a subspace of classical observables, $f\in C^{\infty}(M)$, to self-adjoint operators on the Hilbert space, $\mathscr{H}$, which is the space of all square integrable functions on $X=\mathbb{R}^2-\{0\}$ with respect to the measure $\dd μ= \dd ϕ\ddρ/(2πρ)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_25794 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Group theoretic quantization of punctured plane Somvanshi, Manvendra Stargen, D. Jaffino Mathematical Physics High Energy Physics - Theory Quantum Physics We quantize punctured plane, $X=\mathbb{R}^2-\{0\}$, employing Isham's group theoretic quantization procedure. After sketching out a brief review of group theoretic quantization procedure, we apply the quantization scheme to the phase space, $M=X \times \R^2$, corresponding to the punctured plane, $X$. Particularly, we find the canonical Lie group, $\mathscr{G}$, corresponding to the phase space, $M=X \times \R^2$, to be $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$. We establish an algebra homomorphism between the Lie algebra corresponding to the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, and the smooth functions, $f\in C^{\infty}(M)$, in the phase space, $M=X \times \R^2$. Making use of this homomorphism and unitary representation of the canonical group, $\mathscr{G} = \R^2 \rtimes (SO(2)\times \R^+)$, we deduce a quantization map that maps a subspace of classical observables, $f\in C^{\infty}(M)$, to self-adjoint operators on the Hilbert space, $\mathscr{H}$, which is the space of all square integrable functions on $X=\mathbb{R}^2-\{0\}$ with respect to the measure $\dd μ= \dd ϕ\ddρ/(2πρ)$. |
| title | Group theoretic quantization of punctured plane |
| topic | Mathematical Physics High Energy Physics - Theory Quantum Physics |
| url | https://arxiv.org/abs/2510.25794 |