dorsal/arxiv
View SchemaGeneral Formulation of Quantum Analysis
| Authors | Masuo Suzuki |
|---|---|
| Categories | |
| ArXiv ID | physics/9803009 |
| URL | https://arxiv.org/abs/physics/9803009 |
| DOI | 10.1142/S0129055X9900009X |
Abstract
A general formulation of noncommutative or quantum derivatives for operators in a Banach space is given on the basis of the Leibniz rule, irrespective of their explicit representations such as the G\^ateaux derivative or commutators. This yields a unified formulation of quantum analysis, namely the invariance of quantum derivatives, which are expressed by multiple integrals of ordinary higher derivatives with hyperoperator variables. Multivariate quantum analysis is also formulated in the present unified scheme by introducing a partial inner derivation and a rearrangement formula. Operator Taylor expansion formulas are also given by introducing the two hyperoperators $ \delta_{A \to B} \equiv -\delta_A^{-1} \delta_B$ and $d_{A \to B} \equiv \delta_{(-\delta_A^{-1}B) ; A}$ with the inner derivation $\delta_A : Q \mapsto [A,Q] \equiv AQ-QA$. Physically the present noncommutative derivatives express quantum fluctuations and responses.
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"abstract": "A general formulation of noncommutative or quantum derivatives for operators\nin a Banach space is given on the basis of the Leibniz rule, irrespective of\ntheir explicit representations such as the G\\^ateaux derivative or commutators.\nThis yields a unified formulation of quantum analysis, namely the invariance of\nquantum derivatives, which are expressed by multiple integrals of ordinary\nhigher derivatives with hyperoperator variables. Multivariate quantum analysis\nis also formulated in the present unified scheme by introducing a partial inner\nderivation and a rearrangement formula. Operator Taylor expansion formulas are\nalso given by introducing the two hyperoperators $ \\delta_{A \\to B} \\equiv\n-\\delta_A^{-1} \\delta_B$ and $d_{A \\to B} \\equiv \\delta_{(-\\delta_A^{-1}B) ;\nA}$ with the inner derivation $\\delta_A : Q \\mapsto [A,Q] \\equiv AQ-QA$.\nPhysically the present noncommutative derivatives express quantum fluctuations\nand responses.",
"arxiv_id": "physics/9803009",
"authors": [
"Masuo Suzuki"
],
"categories": [
"math-ph",
"math.MP"
],
"doi": "10.1142/S0129055X9900009X",
"title": "General Formulation of Quantum Analysis",
"url": "https://arxiv.org/abs/physics/9803009"
},
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