dorsal/arxiv
View SchemaQuantum Theory from Symmetries in a General Statistical Parameter Space
| Authors | Inge S. Helland |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/9908076 |
| URL | https://arxiv.org/abs/quant-ph/9908076 |
Abstract
The aim of this paper is to show a connection between an extended theory of statistical experiments on the one hand and the foundation of quantum theory on the other hand. The main aspects of this extension are: One assumes a hyperparameter space $\Phi$ common to several potential experiments, and a basic symmetry group G associated with this space. The parameter \theta_{a} of a single experiment, looked upon as a parametric function $\theta_{a}(\cdot)$ on $\Phi$, is said to be permissible if G induces in a natural way a new group on the image space of the function. If this is not the case, it is arranged for by changing from G to a subgroup $G_{a}$. The Haar measure of this subgroup (confined to the spectrum; see below) is the prefered prior when the parameter is unknown. It is assumed that the hyperparameter itself can never be estimated, only a set of parametric functions. Model reduction is made by restricting the space of complex `wave' functions, also regarded as functions on $\Phi$, to an irreducible invariant subspace $\mathcal{M}$ under G. The spectrum of a parametric function is defined from natural group-theoretical and statistical considerations. We establish that a unique operator can be associated with every parametric functions $\theta_{a}(\cdot)$, and essentially all of the ordinary quantum theory formalism can be retrieved from these and a few related assumptions. In particular the concept of spectrum is consistent. The scope of the theory is illustrated on the one hand by the example of a spin 1/2 particle and a related EPR discussion, on the other hand by a simple macroscopic example.
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"abstract": "The aim of this paper is to show a connection between an extended theory of\nstatistical experiments on the one hand and the foundation of quantum theory on\nthe other hand. The main aspects of this extension are: One assumes a\nhyperparameter space $\\Phi$ common to several potential experiments, and a\nbasic symmetry group G associated with this space. The parameter \\theta_{a} of\na single experiment, looked upon as a parametric function $\\theta_{a}(\\cdot)$\non $\\Phi$, is said to be permissible if G induces in a natural way a new group\non the image space of the function. If this is not the case, it is arranged for\nby changing from G to a subgroup $G_{a}$. The Haar measure of this subgroup\n(confined to the spectrum; see below) is the prefered prior when the parameter\nis unknown. It is assumed that the hyperparameter itself can never be\nestimated, only a set of parametric functions. Model reduction is made by\nrestricting the space of complex `wave\u0027 functions, also regarded as functions\non $\\Phi$, to an irreducible invariant subspace $\\mathcal{M}$ under G. The\nspectrum of a parametric function is defined from natural group-theoretical and\nstatistical considerations. We establish that a unique operator can be\nassociated with every parametric functions $\\theta_{a}(\\cdot)$, and essentially\nall of the ordinary quantum theory formalism can be retrieved from these and a\nfew related assumptions. In particular the concept of spectrum is consistent.\nThe scope of the theory is illustrated on the one hand by the example of a spin\n1/2 particle and a related EPR discussion, on the other hand by a simple\nmacroscopic example.",
"arxiv_id": "quant-ph/9908076",
"authors": [
"Inge S. Helland"
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"title": "Quantum Theory from Symmetries in a General Statistical Parameter Space",
"url": "https://arxiv.org/abs/quant-ph/9908076"
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