dorsal/arxiv
View SchemaThe Representation of Numbers in Quantum Mechanics
| Authors | Paul Benioff |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0103078 |
| URL | https://arxiv.org/abs/quant-ph/0103078 |
| Journal | Algorithmica, Vol.34, pp. 529-559, 2002. |
Abstract
Earlier work on modular arithmetic of k-ary representations of length L of the natural numbers in quantum mechanics is extended here to k-ary representations of all natural numbers, and to integers and rational numbers. Since the length L is indeterminate, representations of states and operators using creation and annihilation operators for bosons and fermions are defined. Emphasis is on definitions and properties of operators corresponding to the basic operations whose properties are given by the axioms for each type of number. The importance of the requirement of efficient implementability for physical models of the axioms is emphasized. Based on this, successor operations for each value of j corresponding to addition of k^{j-1} if j>0 and k^{j} if j<0 are defined. It follows from the efficient implementability of these successors, which is the case for all computers, that implementation of the addition and multiplication operators, which are defined in terms of polynomially many iterations of the successors, should be efficient. This is not the case for definitions based on the successor for j=1 only. This is the only successor defined in the usual axioms of arithmetic.
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"abstract": "Earlier work on modular arithmetic of k-ary representations of length L of\nthe natural numbers in quantum mechanics is extended here to k-ary\nrepresentations of all natural numbers, and to integers and rational numbers.\nSince the length L is indeterminate, representations of states and operators\nusing creation and annihilation operators for bosons and fermions are defined.\nEmphasis is on definitions and properties of operators corresponding to the\nbasic operations whose properties are given by the axioms for each type of\nnumber. The importance of the requirement of efficient implementability for\nphysical models of the axioms is emphasized. Based on this, successor\noperations for each value of j corresponding to addition of k^{j-1} if j\u003e0 and\nk^{j} if j\u003c0 are defined. It follows from the efficient implementability of\nthese successors, which is the case for all computers, that implementation of\nthe addition and multiplication operators, which are defined in terms of\npolynomially many iterations of the successors, should be efficient. This is\nnot the case for definitions based on the successor for j=1 only. This is the\nonly successor defined in the usual axioms of arithmetic.",
"arxiv_id": "quant-ph/0103078",
"authors": [
"Paul Benioff"
],
"categories": [
"quant-ph"
],
"journal_ref": "Algorithmica, Vol.34, pp. 529-559, 2002.",
"title": "The Representation of Numbers in Quantum Mechanics",
"url": "https://arxiv.org/abs/quant-ph/0103078"
},
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