Dirac reduction algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911081722544128 |
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| author | Dorang, Matthew Hartwig, Jonas T. Williams II, Dwight Anderson |
| author_facet | Dorang, Matthew Hartwig, Jonas T. Williams II, Dwight Anderson |
| contents | There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ to the Weyl-Clifford superalgebra $W(2n|n)$ with $2n$ even Weyl algebra generators and $n$ odd Clifford algebra generators. Under this homomorphism, the positive odd root vector $x\in\mathfrak{osp}(1|2)$ is sent to the Dirac operator $γ^μ\partial_μ\in W(2n|n)$ and generates a left ideal $I$. The corresponding reduction (super)algebra, denoted $Z_n$, is the normalizer of $I$ in $W(2n|n)$ modulo $I$. By construction, $Z_n$ acts on the space of all Clifford algebra-valued polynomial solutions to the (massless) Dirac equation. In this paper, we find a complete presentation of (a localization of) this so-termed Dirac reduction algebra. Furthermore, we use the Dirac reduction algebra to generate all polynomial solutions to the Dirac equation in $n$-dimensional flat spacetime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_21730 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dirac reduction algebra Dorang, Matthew Hartwig, Jonas T. Williams II, Dwight Anderson Representation Theory Mathematical Physics 17B10, 17B60, 17B81, 17B37, 34L40 There is a homomorphism of associative superalgebras from the enveloping algebra of the orthosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ to the Weyl-Clifford superalgebra $W(2n|n)$ with $2n$ even Weyl algebra generators and $n$ odd Clifford algebra generators. Under this homomorphism, the positive odd root vector $x\in\mathfrak{osp}(1|2)$ is sent to the Dirac operator $γ^μ\partial_μ\in W(2n|n)$ and generates a left ideal $I$. The corresponding reduction (super)algebra, denoted $Z_n$, is the normalizer of $I$ in $W(2n|n)$ modulo $I$. By construction, $Z_n$ acts on the space of all Clifford algebra-valued polynomial solutions to the (massless) Dirac equation. In this paper, we find a complete presentation of (a localization of) this so-termed Dirac reduction algebra. Furthermore, we use the Dirac reduction algebra to generate all polynomial solutions to the Dirac equation in $n$-dimensional flat spacetime. |
| title | Dirac reduction algebra |
| topic | Representation Theory Mathematical Physics 17B10, 17B60, 17B81, 17B37, 34L40 |
| url | https://arxiv.org/abs/2507.21730 |