RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866913702704316416 |
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| author | Dey, Rukmini |
| author_facet | Dey, Rukmini |
| contents | In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension $n$ into ${\mathbb C}P^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of ${\mathbb C}P^n$. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space ${\mathbb C}P^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02838 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold Dey, Rukmini Mathematical Physics High Energy Physics - Theory Differential Geometry Functional Analysis Quantum Physics In this article we define Berezin-type and Odzijewicz-type quantizations on compact smooth manifolds. The method is we embed the smooth manifold of real dimension $n$ into ${\mathbb C}P^n$ and induce the quantizations from there. The standard way by which reproducing kernel Hilbert spaces are defined on submanifolds gives a way to define the pullback coherent states. In Berezin-type quantization the Hilbert space of quantization is the pullback (by the embedding) of the Hilbert space of geometric quantization of ${\mathbb C}P^n$. In the Odzijewicz-type quantization one has to consider a tensor product of the geometric quantization line bundle with holomorphic $n$-forms. In the Berezin case, the operators that are quantized are those induced from the ambient space ${\mathbb C}P^n$. The Berezin-type quantization exhibited here is a generalization of an earlier work of the author and Ghosh. In both Berezin and Odzijewicz-type quantizations we first exhibit this quantization explicitly on ${\mathbb C}P^n$ and we induce the quantization on the smooth compact embedded manifold from ${\mathbb C}P^n$. |
| title | RKHS, Berezin and Odzijewicz-type quantizations on arbitrary compact smooth manifold |
| topic | Mathematical Physics High Energy Physics - Theory Differential Geometry Functional Analysis Quantum Physics |
| url | https://arxiv.org/abs/2405.02838 |