[1903.07496] The notion of observable and the moment problem for *-algebras and their GNS representations

<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"> <head> <title>[1903.07496] The notion of observable and the moment problem for *-algebras and their GNS representations</title> <meta name="viewport" content="width=device-width, initial-scale=1"> <link rel="apple-touch-icon" sizes="180x180" href="/static/browse/0.3.4/images/icons/apple-touch-icon.png"> <link rel="icon" type="image/png" sizes="32x32" href="/static/browse/0.3.4/images/icons/favicon-32x32.png"> <link rel="icon" type="image/png" sizes="16x16" href="/static/browse/0.3.4/images/icons/favicon-16x16.png"> <link rel="manifest" href="/static/browse/0.3.4/images/icons/site.webmanifest"> <link rel="mask-icon" href="/static/browse/0.3.4/images/icons/safari-pinned-tab.svg" color="#5bbad5"> <meta name="msapplication-TileColor" content="#da532c"> <meta name="theme-color" content="#ffffff"> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/arXiv.css?v=20240822" /> <link rel="stylesheet" type="text/css" media="print" href="/static/browse/0.3.4/css/arXiv-print.css?v=20200611" /> <link rel="stylesheet" type="text/css" media="screen" href="/static/browse/0.3.4/css/browse_search.css" /> <script language="javascript" src="/static/browse/0.3.4/js/accordion.js" /></script> <link rel="canonical" href="https://arxiv.org/abs/1903.07496"/> <meta name="description" content="Abstract page for arXiv paper 1903.07496: The notion of observable and the moment problem for *-algebras and their GNS representations"><meta property="og:type" content="website" /> <meta property="og:site_name" content="arXiv.org" /> <meta property="og:title" content="The notion of observable and the moment problem for *-algebras and their GNS representations" /> <meta property="og:url" content="https://arxiv.org/abs/1903.07496v4" /> <meta property="og:image" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:secure_url" content="/static/browse/0.3.4/images/arxiv-logo-fb.png" /> <meta property="og:image:width" content="1200" /> <meta property="og:image:height" content="700" /> <meta property="og:image:alt" content="arXiv logo"/> <meta property="og:description" content="We address some usually overlooked issues concerning the use of $*$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $*$-algebra describing a quantum system and $蠅\colon\mathfrak{A}\to\mathbb{C}$ a state, we focus in particular on the interpretation of $蠅(a)$ as expectation value for an algebraic observable $a=a^*\in\mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{蠅(a^n)\}_{n\in\mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $蟺_蠅(a)$ in the GNS representation of $蠅$ and thus it has important consequences for the interpretation of $a$ as an observable. We provide physical examples (also from QFT) where the moment problem for $\{蠅(a^n)\}_{n\in\mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{蠅_b(a^n)\}_{n\in\mathbb{N}}$, being $b\in\mathfrak{A}$ and $蠅_b(\cdot):=蠅(b^*\cdot b)$. Letting $渭_{蠅_b}^{(a)}$ be a solution of the moment problem for the sequence $\{蠅_b(a^n)\}_{n\in\mathbb{N}}$, we introduce a consistency relation on the family $\{渭_{蠅_{b}}^{(a)}\}_{b\in\mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{渭_{蠅_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $蟺_蠅(a)$. In particular there exists a unique consistent family of $\{渭_{蠅_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ if and only if $蟺_蠅(a)$ is maximally symmetric. This result suggests that a better physical understanding of the notion of observable for general $*$-algebras should be based on POVMs rather than projection-valued measure (PVM)."/> <meta name="twitter:site" content="@arxiv"/> <meta name="twitter:card" content="summary"/> <meta name="twitter:title" content="The notion of observable and the moment problem for *-algebras and..."/> <meta name="twitter:description" content="We address some usually overlooked issues concerning the use of $*$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $*$-algebra describing a quantum system and..."/> <meta name="twitter:image" content="https://static.arxiv.org/icons/twitter/arxiv-logo-twitter-square.png"/> <meta name="twitter:image:alt" content="arXiv logo"/> <link rel="stylesheet" media="screen" type="text/css" href="/static/browse/0.3.4/css/tooltip.css"/><link rel="stylesheet" media="screen" type="text/css" href="https://static.arxiv.org/js/bibex-dev/bibex.css?20200709"/> <script src="/static/browse/0.3.4/js/mathjaxToggle.min.js" type="text/javascript"></script> <script src="//code.jquery.com/jquery-latest.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/js-cookie@2/src/js.cookie.min.js" type="text/javascript"></script> <script src="//cdn.jsdelivr.net/npm/dompurify@2.3.5/dist/purify.min.js"></script> <script src="/static/browse/0.3.4/js/toggle-labs.js?20241022" type="text/javascript"></script> <script src="/static/browse/0.3.4/js/cite.js" type="text/javascript"></script><meta name="citation_title" content="The notion of observable and the moment problem for *-algebras and their GNS representations" /><meta name="citation_author" content="Drago, Nicol貌" /><meta name="citation_author" content="Moretti, Valter" /><meta name="citation_doi" content="10.1007/s11005-020-01277-x" /><meta name="citation_date" content="2019/03/18" /><meta name="citation_online_date" content="2020/02/17" /><meta name="citation_pdf_url" content="http://arxiv.org/pdf/1903.07496" /><meta name="citation_arxiv_id" content="1903.07496" /><meta name="citation_abstract" content="We address some usually overlooked issues concerning the use of $*$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $*$-algebra describing a quantum system and $\omega\colon\mathfrak{A}\to\mathbb{C}$ a state, we focus in particular on the interpretation of $\omega(a)$ as expectation value for an algebraic observable $a=a^*\in\mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{\omega(a^n)\}_{n\in\mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $\pi_\omega(a)$ in the GNS representation of $\omega$ and thus it has important consequences for the interpretation of $a$ as an observable. We provide physical examples (also from QFT) where the moment problem for $\{\omega(a^n)\}_{n\in\mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, being $b\in\mathfrak{A}$ and $\omega_b(\cdot):=\omega(b^*\cdot b)$. Letting $\mu_{\omega_b}^{(a)}$ be a solution of the moment problem for the sequence $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, we introduce a consistency relation on the family $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $\pi_\omega(a)$. In particular there exists a unique consistent family of $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ if and only if $\pi_\omega(a)$ is maximally symmetric. This result suggests that a better physical understanding of the notion of observable for general $*$-algebras should be based on POVMs rather than projection-valued measure (PVM)." /> </head> <body class="with-cu-identity"> <div class="flex-wrap-footer"> <header> <a href="#content" class="is-sr-only">Skip to main content</a>  <div class="columns is-vcentered is-hidden-mobile" id="cu-identity"> <div class="column" id="cu-logo"> <a href="https://www.cornell.edu/"><img src="/static/browse/0.3.4/images/icons/cu/cornell-reduced-white-SMALL.svg" alt="Cornell University" /></a> </div><div class="column" id="support-ack"> <span id="support-ack-url">We gratefully acknowledge support from the Simons Foundation, <a href="https://info.arxiv.org/about/ourmembers.html">member institutions</a>, and all contributors.</span> <a href="https://info.arxiv.org/about/donate.html" class="btn-header-donate">Donate</a> </div> </div> <div id="header" class="is-hidden-mobile"> <a aria-hidden="true" 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representations</h1> <div class="authors"><span class="descriptor">Authors:</span><a href="https://arxiv.org/search/math-ph?searchtype=author&query=Drago,+N" rel="nofollow">Nicol貌 Drago</a>, <a href="https://arxiv.org/search/math-ph?searchtype=author&query=Moretti,+V" rel="nofollow">Valter Moretti</a></div> <div id="download-button-info" hidden>View a PDF of the paper titled The notion of observable and the moment problem for *-algebras and their GNS representations, by Nicol\`o Drago and Valter Moretti</div> <a class="mobile-submission-download" href="/pdf/1903.07496">View PDF</a> <blockquote class="abstract mathjax"> <span class="descriptor">Abstract:</span>We address some usually overlooked issues concerning the use of $*$-algebras in quantum theory and their physical interpretation. If $\mathfrak{A}$ is a $*$-algebra describing a quantum system and $\omega\colon\mathfrak{A}\to\mathbb{C}$ a state, we focus in particular on the interpretation of $\omega(a)$ as expectation value for an algebraic observable $a=a^*\in\mathfrak{A}$, studying the problem of finding a probability measure reproducing the moments $\{\omega(a^n)\}_{n\in\mathbb{N}}$. This problem enjoys a close relation with the self-adjointeness of the (in general only symmetric) operator $\pi_\omega(a)$ in the GNS representation of $\omega$ and thus it has important consequences for the interpretation of $a$ as an observable. We provide physical examples (also from QFT) where the moment problem for $\{\omega(a^n)\}_{n\in\mathbb{N}}$ does not admit a unique solution. To reduce this ambiguity, we consider the moment problem for the sequences $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, being $b\in\mathfrak{A}$ and $\omega_b(\cdot):=\omega(b^*\cdot b)$. Letting $\mu_{\omega_b}^{(a)}$ be a solution of the moment problem for the sequence $\{\omega_b(a^n)\}_{n\in\mathbb{N}}$, we introduce a consistency relation on the family $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$. We prove a 1-1 correspondence between consistent families $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ and positive operator-valued measures (POVM) associated with the symmetric operator $\pi_\omega(a)$. In particular there exists a unique consistent family of $\{\mu_{\omega_{b}}^{(a)}\}_{b\in\mathfrak{A}}$ if and only if $\pi_\omega(a)$ is maximally symmetric. This result suggests that a better physical understanding of the notion of observable for general $*$-algebras should be based on POVMs rather than projection-valued measure (PVM). </blockquote>  <div class="metatable"> <table summary="Additional metadata"> <tr> <td class="tablecell label">Comments:</td> <td class="tablecell comments mathjax">44 pages, no figures, accepted for publication in Letters in Mathematical Physics</td> </tr> <tr> <td class="tablecell label">Subjects:</td> <td class="tablecell subjects"> <span class="primary-subject">Mathematical Physics (math-ph)</span>; High Energy Physics - Theory (hep-th); Operator Algebras (math.OA)</td> </tr><tr> <td class="tablecell label">Cite as:</td> <td class="tablecell arxivid"><span class="arxivid"><a href="https://arxiv.org/abs/1903.07496">arXiv:1903.07496</a> [math-ph]</span></td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxividv">(or <span class="arxivid"> <a href="https://arxiv.org/abs/1903.07496v4">arXiv:1903.07496v4</a> [math-ph]</span> for this version) </td> </tr> <tr> <td class="tablecell label"> </td> <td class="tablecell arxivdoi"> <a href="https://doi.org/10.48550/arXiv.1903.07496" id="arxiv-doi-link">https://doi.org/10.48550/arXiv.1903.07496</a><div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> arXiv-issued DOI via DataCite</div> </div> </td> </tr> <tr> <td class="tablecell label">Journal reference:</td> <td class="tablecell jref">Lett Math Phys (2020)</td> </tr> <tr> <td class="tablecell label"> <abbr title="Digital Object Identifier">Related DOI</abbr>: </td> <td class="tablecell doi"><a href="https://doi.org/10.1007/s11005-020-01277-x" data-doi="10.1007/s11005-020-01277-x" class="link-https link-external" rel="external noopener nofollow">https://doi.org/10.1007/s11005-020-01277-x</a>  <div class="button-and-tooltip"> <button class="more-info" aria-describedby="more-info-desc-1"> <svg height="15" role="presentation" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256 8C119.043 8 8 119.083 8 256c0 136.997 111.043 248 248 248s248-111.003 248-248C504 119.083 392.957 8 256 8zm0 110c23.196 0 42 18.804 42 42s-18.804 42-42 42-42-18.804-42-42 18.804-42 42-42zm56 254c0 6.627-5.373 12-12 12h-88c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h12v-64h-12c-6.627 0-12-5.373-12-12v-24c0-6.627 5.373-12 12-12h64c6.627 0 12 5.373 12 12v100h12c6.627 0 12 5.373 12 12v24z" class=""></path></svg> <span class="visually-hidden">Focus to learn more</span> </button>  <div role="tooltip" id="more-info-desc-1"> <span class="left-corner"></span> DOI(s) linking to related resources </div> </div> </td> </tr> </table> </div> </div> </div> <div class="submission-history"> <h2>Submission history</h2> From: Nicol贸 Drago [<a href="/show-email/45f2c626/1903.07496" rel="nofollow">view email</a>] <br/> <strong><a href="/abs/1903.07496v1" rel="nofollow">[v1]</a></strong> Mon, 18 Mar 2019 15:12:08 UTC (36 KB)<br/> <strong><a href="/abs/1903.07496v2" rel="nofollow">[v2]</a></strong> Tue, 24 Sep 2019 10:11:19 UTC (44 KB)<br/> <strong><a href="/abs/1903.07496v3" rel="nofollow">[v3]</a></strong> Sat, 28 Dec 2019 09:26:56 UTC (47 KB)<br/> <strong>[v4]</strong> Mon, 17 Feb 2020 12:15:54 UTC (51 KB)<br/> </div> </div>  <div class="extra-services"> <div class="full-text"> <a name="other"></a> <span class="descriptor">Full-text links:</span> <h2>Access Paper:</h2> <ul> <div id="download-button-info" hidden> View a PDF of the paper titled The notion of observable and the moment problem for *-algebras and their GNS representations, by Nicol\`o Drago and Valter Moretti</div><li><a href="/pdf/1903.07496" aria-describedby="download-button-info" accesskey="f" class="abs-button download-pdf">View PDF</a></li><li><a href="/src/1903.07496" class="abs-button download-eprint">TeX Source</a></li><li><a href="/format/1903.07496" class="abs-button download-format">Other Formats</a></li></ul> <div class="abs-license"><a href="http://arxiv.org/licenses/nonexclusive-distrib/1.0/" title="Rights to this article">view license</a></div> </div>  <div class="browse"> Current browse context: <div class="current">math-ph</div> <div class="prevnext"> <span class="arrow"> <a 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[1903.07496] The notion of observable and the moment problem for *-algebras and their GNS representations