The disappearance of causality at small scale in almost-commutative manifolds
This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by the so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consid...
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description | This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by the so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form
C
(
M
)
⊗
A
f
, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in
C
(
M
)
⊗
A
f
, with A
f
noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale. The usual Lorentzian causality would then only arise as an approximation of this noncommutative causality. |
doi_str_mv | 10.1063/1.4985929 |
format | Article |
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C
(
M
)
⊗
A
f
, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in
C
(
M
)
⊗
A
f
, with A
f
noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale. The usual Lorentzian causality would then only arise as an approximation of this noncommutative causality.</description><identifier>ISSN: 0022-2488</identifier><identifier>EISSN: 1089-7658</identifier><identifier>DOI: 10.1063/1.4985929</identifier><identifier>CODEN: JMAPAQ</identifier><language>eng</language><publisher>New York: American Institute of Physics</publisher><subject>Algebra ; Approximation ; Causality ; Cones ; General Relativity and Quantum Cosmology ; Mathematical analysis ; Mathematical Physics ; Physics ; Topological manifolds</subject><ispartof>Journal of mathematical physics, 2017-09, Vol.58 (9), p.1</ispartof><rights>Author(s)</rights><rights>Copyright American Institute of Physics Sep 2017</rights><rights>2017 Author(s). Published by AIP Publishing.</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c354t-b4f624f0435c2c67d6224585ccb4f04375ee1b31d71ac2df663a59d5eedcd8c43</citedby><cites>FETCH-LOGICAL-c354t-b4f624f0435c2c67d6224585ccb4f04375ee1b31d71ac2df663a59d5eedcd8c43</cites><orcidid>0000-0003-4038-2640 ; 0000-0003-2643-6980</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubs.aip.org/jmp/article-lookup/doi/10.1063/1.4985929$$EHTML$$P50$$Gscitation$$H</linktohtml><link.rule.ids>230,314,780,784,794,885,4512,27924,27925,76384</link.rule.ids><backlink>$$Uhttps://hal.science/hal-01645170$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Bizi, Nadir</creatorcontrib><creatorcontrib>Besnard, Fabien</creatorcontrib><title>The disappearance of causality at small scale in almost-commutative manifolds</title><title>Journal of mathematical physics</title><description>This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by the so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form
C
(
M
)
⊗
A
f
, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in
C
(
M
)
⊗
A
f
, with A
f
noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale. The usual Lorentzian causality would then only arise as an approximation of this noncommutative causality.</description><subject>Algebra</subject><subject>Approximation</subject><subject>Causality</subject><subject>Cones</subject><subject>General Relativity and Quantum Cosmology</subject><subject>Mathematical analysis</subject><subject>Mathematical Physics</subject><subject>Physics</subject><subject>Topological manifolds</subject><issn>0022-2488</issn><issn>1089-7658</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><recordid>eNp9kFFLwzAUhYMoOKcP_oOATwqduWmSpo9jqBMmvszncJemLKNtZtMO9u_t3NC3PV04fJxz-Qi5BzYBptJnmIhcy5znF2QETOdJpqS-JCPGOE-40Pqa3MS4YQxACzEiH8u1o4WPuN06bLGxjoaSWuwjVr7bU-xorLGqaLRYOeobilUdYpfYUNd9h53fOVpj48tQFfGWXJVYRXd3umPy9fqynM2Txefb-2y6SGwqRZesRKm4KJlIpeVWZYXiXEgtrV39ppl0DlYpFBmg5UWpVIoyL4a0sIW2Ih2Tx2PvGiuzbX2N7d4E9GY-XZhDxkAJCRnbwcA-HNltG757FzuzCX3bDO8ZDqAGaTnn5yjIswESqdL_u7YNMbau_BsHZg7-DZiT_4F9OrLR-oOn0JyBfwCpSoMT</recordid><startdate>20170901</startdate><enddate>20170901</enddate><creator>Bizi, Nadir</creator><creator>Besnard, Fabien</creator><general>American Institute of Physics</general><general>American Institute of Physics (AIP)</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7U5</scope><scope>8FD</scope><scope>H8D</scope><scope>JQ2</scope><scope>L7M</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0003-4038-2640</orcidid><orcidid>https://orcid.org/0000-0003-2643-6980</orcidid></search><sort><creationdate>20170901</creationdate><title>The disappearance of causality at small scale in almost-commutative manifolds</title><author>Bizi, Nadir ; Besnard, Fabien</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c354t-b4f624f0435c2c67d6224585ccb4f04375ee1b31d71ac2df663a59d5eedcd8c43</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Algebra</topic><topic>Approximation</topic><topic>Causality</topic><topic>Cones</topic><topic>General Relativity and Quantum Cosmology</topic><topic>Mathematical analysis</topic><topic>Mathematical Physics</topic><topic>Physics</topic><topic>Topological manifolds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Bizi, Nadir</creatorcontrib><creatorcontrib>Besnard, Fabien</creatorcontrib><collection>CrossRef</collection><collection>Solid State and Superconductivity Abstracts</collection><collection>Technology Research Database</collection><collection>Aerospace Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Journal of mathematical physics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Bizi, Nadir</au><au>Besnard, Fabien</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The disappearance of causality at small scale in almost-commutative manifolds</atitle><jtitle>Journal of mathematical physics</jtitle><date>2017-09-01</date><risdate>2017</risdate><volume>58</volume><issue>9</issue><spage>1</spage><pages>1-</pages><issn>0022-2488</issn><eissn>1089-7658</eissn><coden>JMAPAQ</coden><abstract>This paper continues the investigations of noncommutative ordered spaces put forward by one of the authors. These metaphoric spaces are defined dually by the so-called isocones which generalize to the noncommutative setting the convex cones of order-preserving functions. In this paper we will consider the case of isocones inside almost-commutative algebras of the form
C
(
M
)
⊗
A
f
, with M a compact metrizable space. We will give a family of isocones in such an algebra with the property that every possible isocone is contained in exactly one member of the family. We conjecture that this family is in fact a complete classification, a hypothesis related with the noncommutative Stone-Weierstrass conjecture. We also obtain that every isocone in
C
(
M
)
⊗
A
f
, with A
f
noncommutative, induces an order relation on M with the property that every point in M lies in a neighbourhood of incomparable points. Thus, if the causal order relation on spacetime is induced by an isocone in an almost-commutative (but not commutative) algebra, then causality must disappear at small scale. The usual Lorentzian causality would then only arise as an approximation of this noncommutative causality.</abstract><cop>New York</cop><pub>American Institute of Physics</pub><doi>10.1063/1.4985929</doi><tpages>20</tpages><orcidid>https://orcid.org/0000-0003-4038-2640</orcidid><orcidid>https://orcid.org/0000-0003-2643-6980</orcidid></addata></record> |
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subjects | Algebra Approximation Causality Cones General Relativity and Quantum Cosmology Mathematical analysis Mathematical Physics Physics Topological manifolds |
title | The disappearance of causality at small scale in almost-commutative manifolds |
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