A New $q$-Heisenberg Algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918046116872192 |
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| author | Quiceno, Julio Cesar Jaramillo |
| author_facet | Quiceno, Julio Cesar Jaramillo |
| contents | This work introduces a novel $q$-$\hbar$ deformation of the Heisenberg algebra, designed to unify and extend several existing $q$-deformed formulations. Starting from the canonical Heisenberg algebra defined by the commutation relation $[\hat{x}, \hat{p}] = i\hbar$ on a Hilbert space \cite{Zettili2009}, we survey a variety of $q$-deformed structures previously proposed by Wess \cite{Wess2000}, Schmüdgen \cite{Schmudgen1999}, Wess--Schwenk \cite{Wess-Schwenk1992}, Gaddis \cite{Jasson-Gaddis2016}, and others. These frameworks involve position, momentum, and auxiliary operators that satisfy nontrivial commutation rules and algebraic relations incorporating deformation parameters. Our new $q$-$\hbar$ Heisenberg algebra $\mathcal{H}_q$ is generated by elements $\hat{x}_α$, $\hat{y}_λ$, and $\hat{p}_β$ with $α, λ, β\in \{1,2,3\}$, and is defined through generalized commutation relations parameterized by real constants $n, m, l$ and three dynamical functions $Ψ(q)$, $Φ(q)$, and $Π(q)$ depending on the deformation parameter $q$ and the generators. By selecting appropriate values for these parameters and functions, our framework recovers several well-known algebras as special cases, including the classical Heisenberg algebra for $q = 1$ and $Ψ= 1$, $Φ= Π= 0$, and various $q$-deformed algebras for $q \neq 1$. The algebraic consistency of these generalizations is demonstrated through a series of explicit examples, and the resulting structures are shown to align with quantum planes \cite{Yuri-Manin2010} and enveloping algebras associated with Lie algebra homomorphisms \cite{Reyes2014a}. This construction offers a flexible and unified formalism for studying quantum deformations, with potential applications in quantum mechanics, noncommutative geometry, and quantum group theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_04248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A New $q$-Heisenberg Algebra Quiceno, Julio Cesar Jaramillo Mathematical Physics Quantum Physics This work introduces a novel $q$-$\hbar$ deformation of the Heisenberg algebra, designed to unify and extend several existing $q$-deformed formulations. Starting from the canonical Heisenberg algebra defined by the commutation relation $[\hat{x}, \hat{p}] = i\hbar$ on a Hilbert space \cite{Zettili2009}, we survey a variety of $q$-deformed structures previously proposed by Wess \cite{Wess2000}, Schmüdgen \cite{Schmudgen1999}, Wess--Schwenk \cite{Wess-Schwenk1992}, Gaddis \cite{Jasson-Gaddis2016}, and others. These frameworks involve position, momentum, and auxiliary operators that satisfy nontrivial commutation rules and algebraic relations incorporating deformation parameters. Our new $q$-$\hbar$ Heisenberg algebra $\mathcal{H}_q$ is generated by elements $\hat{x}_α$, $\hat{y}_λ$, and $\hat{p}_β$ with $α, λ, β\in \{1,2,3\}$, and is defined through generalized commutation relations parameterized by real constants $n, m, l$ and three dynamical functions $Ψ(q)$, $Φ(q)$, and $Π(q)$ depending on the deformation parameter $q$ and the generators. By selecting appropriate values for these parameters and functions, our framework recovers several well-known algebras as special cases, including the classical Heisenberg algebra for $q = 1$ and $Ψ= 1$, $Φ= Π= 0$, and various $q$-deformed algebras for $q \neq 1$. The algebraic consistency of these generalizations is demonstrated through a series of explicit examples, and the resulting structures are shown to align with quantum planes \cite{Yuri-Manin2010} and enveloping algebras associated with Lie algebra homomorphisms \cite{Reyes2014a}. This construction offers a flexible and unified formalism for studying quantum deformations, with potential applications in quantum mechanics, noncommutative geometry, and quantum group theory. |
| title | A New $q$-Heisenberg Algebra |
| topic | Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2506.04248 |