dorsal/arxiv
View SchemaInsufficient reason and entropy in quantum theory
| Authors | Ariel Caticha |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9810074 |
| URL | https://arxiv.org/abs/quant-ph/9810074 |
| DOI | 10.1023/A:1003692916756 |
| Journal | Found.Phys.30:227-251,2000 |
Abstract
The objective of the consistent-amplitude approach to quantum theory has been to justify the mathematical formalism on the basis of three main assumptions: the first defines the subject matter, the second introduces amplitudes as the tools for quantitative reasoning, and the third is an interpretative rule that provides the link to the prediction of experimental outcomes. In this work we introduce a natural and compelling fourth assumption: if there is no reason to prefer one region of the configuration space over another then they should be `weighted' equally. This is the last ingredient necessary to introduce a unique inner product in the linear space of wave functions. Thus, a form of the principle of insufficient reason is implicit in the Hilbert inner product. Armed with the inner product we obtain two results. First, we elaborate on an earlier proof of the Born probability rule. The implicit appeal to insufficient reason shows that quantum probabilities are not more objective than classical probabilities. Previously we had argued that the consistent manipulation of amplitudes leads to a linear time evolution; our second result is that time evolution must also be unitary. The argument is straightforward and hinges on the conservation of entropy. The only subtlety consists of defining the correct entropy; it is the array entropy, not von Neumann's. After unitary evolution has been established we proceed to introduce the useful notion of observables and we explore how von Neumann's entropy can be linked to Shannon's information theory. Finally, we discuss how various connections among the postulates of quantum theory are made explicit within this approach.
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"abstract": "The objective of the consistent-amplitude approach to quantum theory has been\nto justify the mathematical formalism on the basis of three main assumptions:\nthe first defines the subject matter, the second introduces amplitudes as the\ntools for quantitative reasoning, and the third is an interpretative rule that\nprovides the link to the prediction of experimental outcomes. In this work we\nintroduce a natural and compelling fourth assumption: if there is no reason to\nprefer one region of the configuration space over another then they should be\n`weighted\u0027 equally. This is the last ingredient necessary to introduce a unique\ninner product in the linear space of wave functions. Thus, a form of the\nprinciple of insufficient reason is implicit in the Hilbert inner product.\nArmed with the inner product we obtain two results. First, we elaborate on an\nearlier proof of the Born probability rule. The implicit appeal to insufficient\nreason shows that quantum probabilities are not more objective than classical\nprobabilities. Previously we had argued that the consistent manipulation of\namplitudes leads to a linear time evolution; our second result is that time\nevolution must also be unitary. The argument is straightforward and hinges on\nthe conservation of entropy. The only subtlety consists of defining the correct\nentropy; it is the array entropy, not von Neumann\u0027s. After unitary evolution\nhas been established we proceed to introduce the useful notion of observables\nand we explore how von Neumann\u0027s entropy can be linked to Shannon\u0027s information\ntheory. Finally, we discuss how various connections among the postulates of\nquantum theory are made explicit within this approach.",
"arxiv_id": "quant-ph/9810074",
"authors": [
"Ariel Caticha"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech",
"gr-qc",
"math-ph",
"math.MP"
],
"doi": "10.1023/A:1003692916756",
"journal_ref": "Found.Phys.30:227-251,2000",
"title": "Insufficient reason and entropy in quantum theory",
"url": "https://arxiv.org/abs/quant-ph/9810074"
},
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