3-Heisenberg-Robertson-Schrodinger Uncertainty Principle
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912155708686336 |
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| author | Krishna, K. Mahesh |
| author_facet | Krishna, K. Mahesh |
| contents | Let $\mathcal{X}$ be a 3-product space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$, $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ and $C: \mathcal{D}(C)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded 3-self-adjoint operators. Then for all \begin{align*}
x \in \mathcal{D}(ABC)\cap\mathcal{D}(ACB) \cap \mathcal{D}(BAC)\cap\mathcal{D}(BCA) \cap \mathcal{D}(CAB)\cap\mathcal{D}(CBA) \end{align*} with $\langle x, x, x \rangle =1$, we show that \begin{align*} (1)\quad \quad Δ_x(3, A) Δ_x(3, B) Δ_x(3, C)\geq |\langle (ABC-a BC-b AC-c AB)x, x, x\rangle +2abc|, \end{align*} where \begin{align*}
Δ_x(3, A):= \|Ax-\langle Ax, x, x \rangle x \|, \quad a:= \langle Ax, x, x \rangle, \quad b := \langle Bx, x, x \rangle, \quad c := \langle Cx, x, x \rangle. \end{align*} We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical Heisenberg-Robertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10396 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 3-Heisenberg-Robertson-Schrodinger Uncertainty Principle Krishna, K. Mahesh Functional Analysis Information Theory Mathematical Physics 46C50, 46B99 Let $\mathcal{X}$ be a 3-product space. Let $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$, $B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X}$ and $C: \mathcal{D}(C)\subseteq \mathcal{X}\to \mathcal{X}$ be possibly unbounded 3-self-adjoint operators. Then for all \begin{align*} x \in \mathcal{D}(ABC)\cap\mathcal{D}(ACB) \cap \mathcal{D}(BAC)\cap\mathcal{D}(BCA) \cap \mathcal{D}(CAB)\cap\mathcal{D}(CBA) \end{align*} with $\langle x, x, x \rangle =1$, we show that \begin{align*} (1)\quad \quad Δ_x(3, A) Δ_x(3, B) Δ_x(3, C)\geq |\langle (ABC-a BC-b AC-c AB)x, x, x\rangle +2abc|, \end{align*} where \begin{align*} Δ_x(3, A):= \|Ax-\langle Ax, x, x \rangle x \|, \quad a:= \langle Ax, x, x \rangle, \quad b := \langle Bx, x, x \rangle, \quad c := \langle Cx, x, x \rangle. \end{align*} We call Inequality (1) as 3-Heisenberg-Robertson-Schrodinger uncertainty principle. Classical Heisenberg-Robertson-Schrodinger uncertainty principle (by Schrodinger in 1930) considers two operators whereas Inequality (1) considers three operators. |
| title | 3-Heisenberg-Robertson-Schrodinger Uncertainty Principle |
| topic | Functional Analysis Information Theory Mathematical Physics 46C50, 46B99 |
| url | https://arxiv.org/abs/2412.10396 |