<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0035-001X</journal-id>
<journal-title><![CDATA[Revista mexicana de física]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. mex. fis.]]></abbrev-journal-title>
<issn>0035-001X</issn>
<publisher>
<publisher-name><![CDATA[Sociedad Mexicana de Física]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0035-001X2007001000021</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[On the WKB approximation of noncommutative quantum cosmology]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Mena]]></surname>
<given-names><![CDATA[E]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Obregón]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Sabido]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<xref ref-type="aff" rid="A01"/>
</contrib>
</contrib-group>
<aff id="A01">
<institution><![CDATA[,Universidad de Guanajuato Instituto de Física ]]></institution>
<addr-line><![CDATA[León Gto]]></addr-line>
<country>México</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>08</month>
<year>2007</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>08</month>
<year>2007</year>
</pub-date>
<volume>53</volume>
<fpage>118</fpage>
<lpage>120</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_arttext&amp;pid=S0035-001X2007001000021&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_abstract&amp;pid=S0035-001X2007001000021&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.mx/scielo.php?script=sci_pdf&amp;pid=S0035-001X2007001000021&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[In this work we propose a formalism to introduce the effects of noncommutativity to classical cosmological models. The method is based on noncommutative quantum cosmology and by means of a WKB type approximation the noncommutative classical solutions can be obtained.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[En este trabajo se propone un formalismo para introducir los efectos de la no conmutatividad a modelos cosmológicos clásicos. El método esta basado en la cosmología cuántica no conmutativa en la cual por medio de la aproximación WKB se obtienen las soluciones clásicas no conmutativas.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[non-commutativity]]></kwd>
<kwd lng="en"><![CDATA[quantum cosmology]]></kwd>
<kwd lng="es"><![CDATA[No conmutatividad]]></kwd>
<kwd lng="es"><![CDATA[cosmología cuantica]]></kwd>
</kwd-group>
</article-meta>
</front><body><![CDATA[ <p align="center"><font face="verdana" size="4"><b>On the WKB approximation of noncommutative quantum cosmology</b></font></p>     <p align="center"><font face="verdana" size="2">&nbsp;</font></p>     <p align="center"><font face="verdana" size="2"><b>E. Mena, O. Obreg&oacute;n, and M. Sabido</b></font></p>     <p align="center"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><i>Instituto de F&iacute;sica de la Universidad de Guanajuato, </i><i>P.O. Box E&#150;143, 37150 Le&oacute;n Gto., M&eacute;xico, e&#150;mail: <a href="mailto:emena@fisica.ugto.mx" target="_blank">emena@fisica.ugto.mx</a>, <a href="mailto:octavio@fisica.ugto.mx" target="_blank">octavio@fisica.ugto.mx</a>, <a href="mailto:msabido@fisica.ugto.mx" target="_blank">msabido@fisica.ugto.mx</a></i></font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2">Recibido el 1 de mayo de 2006    <br>   Aceptado el 1 de noviembre de 2006</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>Abstract</b></font></p>     ]]></body>
<body><![CDATA[<p align="justify"><font face="verdana" size="2">In this work we propose a formalism to introduce the effects of noncommutativity to classical cosmological models. The method is based on noncommutative quantum cosmology and by means of a WKB type approximation the noncommutative classical solutions can be obtained.</font></p>     <p align="justify"><font face="verdana" size="2"><b>Keywords: </b>non&#150;commutativity; quantum cosmology.</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>Resumen</b></font></p>     <p align="justify"><font face="verdana" size="2">En este trabajo se propone un formalismo para introducir los efectos de la no conmutatividad a modelos cosmol&oacute;gicos cl&aacute;sicos. El m&eacute;todo esta basado en la cosmolog&iacute;a cu&aacute;ntica no conmutativa en la cual por medio de la aproximaci&oacute;n WKB se obtienen las soluciones cl&aacute;sicas no conmutativas.</font></p>     <p align="justify"><font face="verdana" size="2"><b>Descriptores: </b>No conmutatividad; cosmolog&iacute;a cuantica. </font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2">PACS: 02.40.Gh; 04.60.Kz; 98.80.Qc</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><a href="/pdf/rmf/v53s4/v53s4a21.pdf">DESCARGAR ART&Iacute;CULO EN FORMATO PDF</a></font></p>     ]]></body>
<body><![CDATA[<p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>Acknowledgments</b></font></p>     <p align="justify"><font face="verdana" size="2">We will like to thank M.P. Ryan for enlightening discussions on quantum cosmology. This work was partially supported by CONACyT grants 47641 and 51306, also by PROMEP grants UGTO&#150;CA&#150;3 and PROMEP&#150;PTC&#150;085.</font></p>     <p align="justify"><font face="verdana" size="2">&nbsp;</font></p>     <p align="justify"><font face="verdana" size="2"><b>References</b></font></p>     <!-- ref --><p align="justify"><font face="verdana" size="2">1. H. Snyder, <i>Phys. Rev </i><b>71 </b>(1947) 38.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341413&pid=S0035-001X200700100002100001&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">2. M.P. Ryan,  in:   <i>Hamiltonian Cosmology </i>(Springer, Berlin, 1972).</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341414&pid=S0035-001X200700100002100002&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">3. M. MacCallum, in: <i>General Relativity: An Einstein Centenary Survey, </i>edited by S. Hawking and W. Israel (Cambridge University Press, Cambridge, England, 1979).</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341415&pid=S0035-001X200700100002100003&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">4. D. Bohm, <i>Quantum Theory </i>(Prentice&#150;Hall, Englewood Clifss, NJ, 1955);    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341416&pid=S0035-001X200700100002100004&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> Ph M. Morse and H. Feshbach, <i>Methods of Theoretical Physics </i>(McGraw&#150;Hill, New York, 1953) Vol. 2;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341417&pid=S0035-001X200700100002100005&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> L.I. Schiff, <i>Quantum Mechanics </i>(McGraw&#150;Hill, New York, 1955);    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341418&pid=S0035-001X200700100002100006&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> A. Messiah, <i>Quantum Mechanics </i>(North&#150;Holland, Amsterdan, 1961) Vol. 1.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341419&pid=S0035-001X200700100002100007&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">5. D.N. Page, "Lectures on Quantum Cosmology", in <i>Gravitation: A Banff Summer Institute, </i>edited by R. Mann and P. Wesson (World Scientific, Singapore, 1991).</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341420&pid=S0035-001X200700100002100008&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">6. B.S. DeWitt, <i>Phys. Rev. </i><b>160 </b>(1967) 1113.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341421&pid=S0035-001X200700100002100009&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">7. J.M. Romero and J.A. Santiago, <i>Mod. Phys. Lett. A </i><b>20 </b>(2005) 78.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341422&pid=S0035-001X200700100002100010&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">8. R. Brandenberger and P.M. Ho,    <i>Phys. Rev. D </i><b>66 </b>(2002) 023517;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341423&pid=S0035-001X200700100002100011&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> Q.G. Huang and M. Li,   <i>Nucl. Phys. B </i><b>713 </b>(2005) 219;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341424&pid=S0035-001X200700100002100012&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> Q.G. Huang and M. Li, <i>JHEP </i><b>0306 </b>(2003) 014;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341425&pid=S0035-001X200700100002100013&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> H.s. Kim, G.S. Lee, H.W. Lee, and Y.S. Myung, <i>Phys. Rev. D </i><b>70 </b>(2004) 043521;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341426&pid=S0035-001X200700100002100014&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> H.s. Kim, G.S. Lee, and Y.S. Myung, <i>Mod. Phys. Lett. A </i><b>20 </b>(2005) 271;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341427&pid=S0035-001X200700100002100015&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> D.J. Liu and X.Z. Li, Phys. Rev. D <b>70 </b>(2004) 123504.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341428&pid=S0035-001X200700100002100016&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">9. H. Garc&iacute;a&#150;Compe&aacute;n, O. Obreg&oacute;n, and C. Ram&iacute;rez, <i>Phys. Rev. Lett. </i><b>88</b> (2002) 161301.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341429&pid=S0035-001X200700100002100017&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">10. I Gamboa, M. Loewe, and J.C. Rojas, Phys. <i>Rev. D </i><b>64 </b>067901;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341430&pid=S0035-001X200700100002100018&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> M. Chaichian, M.M. Sheikh&#150;Jabbari, and A. Tureanu., <i>Phys. Rev. Lett. </i><b>86</b> 2716.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341431&pid=S0035-001X200700100002100019&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">11. H. Garc&iacute;a&#150;Compe&aacute;n, O. Obreg&oacute;n, C. Ram&iacute;rez, and M. Sabido, <i>Phys.   Rev.   D   </i><b>68</b>   (2003)   044015;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341432&pid=S0035-001X200700100002100020&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref -->    H.   Garc&iacute;a&#150;Compe&aacute;n, O. Obreg&oacute;n, C. Ram&iacute;rez, and M. Sabido,   <i>Phys. Rev. D </i><b>68</b> (2003) 045010;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341433&pid=S0035-001X200700100002100021&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref -->  M. Banados, O. Chandia, N. Grandi, F.A. Schaposnik, and G.A. Silva, <i>Phys. Rev. D </i><b>64 </b>(2001) 084012;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341434&pid=S0035-001X200700100002100022&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> H. Nishino and S. Rajpoot, <i>Phys. Lett. B </i><b>532 </b>(2002) 334;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341435&pid=S0035-001X200700100002100023&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> V.P. Nair, "Gravitational Fields on a Noncommutative Space", hep&#150;th/0112114;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341436&pid=S0035-001X200700100002100024&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> S. Cacciatori, D. Klemm, L. Martucci, and D. Zanon, <i>Phys. Lett. B </i><b>536 </b>(2002) 101;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341437&pid=S0035-001X200700100002100025&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> S. Cacciatori, A.H. Chamseddine, D. Klemm, L. Martucci, W.A. Sabra, and D. Zanon, "Noncommutative Gravity in Two Dimensions", hep&#150;th/0203038;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341438&pid=S0035-001X200700100002100026&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref -->  Y. Abe and V.P. Nair,"Noncommutative Gravity:     Fuzzy   Sphere   and   Others",   hep&#150;th/0212270;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341439&pid=S0035-001X200700100002100027&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref -->    M.A. Cardella and Daniela Zanon, "Noncommutative Deformation of Four&#150;dimensional Einstein Gravity", hep&#150;th/0212071;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341440&pid=S0035-001X200700100002100028&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> A.H. Chamseddine, "Invariant Actions for Noncommutative Gravity", hep&#150;th/0202137;    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341441&pid=S0035-001X200700100002100029&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --> J.W. Moffat, <i>Phys. Lett. B </i><b>491 </b>(2000) 345; <i>    &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341442&pid=S0035-001X200700100002100030&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref -->Phys. Lett. B </i><b>493 </b>(2000) 142.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341443&pid=S0035-001X200700100002100031&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">12. R. Kantowski and R. Sachs, <i>J. Math. Phys. </i><b>7</b> (1966) 443.</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341444&pid=S0035-001X200700100002100032&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --><!-- ref --><p align="justify"><font face="verdana" size="2">13. G.D. Barbosa and N. Pinto&#150;Neto, <i>Phys. Rev. D </i><b>70 </b>(2004) 103512, (arXiv:hep&#150;th/0407111).</font>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;[&#160;<a href="javascript:void(0);" onclick="javascript: window.open('/scielo.php?script=sci_nlinks&ref=8341445&pid=S0035-001X200700100002100033&lng=','','width=640,height=500,resizable=yes,scrollbars=1,menubar=yes,');">Links</a>&#160;]<!-- end-ref --> ]]></body><back>
<ref-list>
<ref id="B1">
<label>1</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Snyder]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>1947</year>
<numero>71</numero>
<issue>71</issue>
<page-range>38</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>2</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ryan]]></surname>
<given-names><![CDATA[M.P]]></given-names>
</name>
</person-group>
<source><![CDATA[Hamiltonian Cosmology]]></source>
<year>1972</year>
<publisher-loc><![CDATA[Berlin ]]></publisher-loc>
<publisher-name><![CDATA[Springer]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B3">
<label>3</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[MacCallum]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Hawking]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Israel]]></surname>
<given-names><![CDATA[W]]></given-names>
</name>
</person-group>
<source><![CDATA[General Relativity: An Einstein Centenary Survey]]></source>
<year>1979</year>
<publisher-loc><![CDATA[Cambridge ]]></publisher-loc>
<publisher-name><![CDATA[Cambridge University Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B4">
<label>4</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Bohm]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<source><![CDATA[Quantum Theory]]></source>
<year>1955</year>
<publisher-loc><![CDATA[Englewood Clifss^eNJ, NJ,]]></publisher-loc>
<publisher-name><![CDATA[Prentice-Hall]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B5">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Morse]]></surname>
<given-names><![CDATA[Ph M]]></given-names>
</name>
<name>
<surname><![CDATA[Feshbach]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
</person-group>
<source><![CDATA[Methods of Theoretical Physics]]></source>
<year>1953</year>
<volume>2</volume>
<publisher-loc><![CDATA[New York ]]></publisher-loc>
<publisher-name><![CDATA[McGraw-Hill]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B6">
<nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Schiff]]></surname>
<given-names><![CDATA[L.I]]></given-names>
</name>
</person-group>
<source><![CDATA[Quantum Mechanics]]></source>
<year>1955</year>
<publisher-loc><![CDATA[New York ]]></publisher-loc>
<publisher-name><![CDATA[McGraw-Hill]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B7">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Messiah]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Quantum Mechanics]]></source>
<year>1961</year>
<volume>1</volume>
<publisher-loc><![CDATA[North-Holland ]]></publisher-loc>
</nlm-citation>
</ref>
<ref id="B8">
<label>5</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Page]]></surname>
<given-names><![CDATA[D.N]]></given-names>
</name>
</person-group>
<article-title xml:lang="en"><![CDATA["Lectures on Quantum Cosmology"]]></article-title>
<person-group person-group-type="editor">
<name>
<surname><![CDATA[Mann]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
<name>
<surname><![CDATA[Wesson]]></surname>
<given-names><![CDATA[P]]></given-names>
</name>
</person-group>
<source><![CDATA[Gravitation: A Banff Summer Institute]]></source>
<year>1991</year>
<publisher-loc><![CDATA[Singapore ]]></publisher-loc>
<publisher-name><![CDATA[World Scientific]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B9">
<label>6</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[DeWitt]]></surname>
<given-names><![CDATA[B.S]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>1967</year>
<numero>160</numero>
<issue>160</issue>
<page-range>1113</page-range></nlm-citation>
</ref>
<ref id="B10">
<label>7</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Romero]]></surname>
<given-names><![CDATA[J.M]]></given-names>
</name>
<name>
<surname><![CDATA[Santiago]]></surname>
<given-names><![CDATA[J.A]]></given-names>
</name>
</person-group>
<source><![CDATA[Mod. Phys. Lett]]></source>
<year>2005</year>
<numero>A 20</numero>
<issue>A 20</issue>
<page-range>78</page-range></nlm-citation>
</ref>
<ref id="B11">
<label>8</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Brandenberger]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
<name>
<surname><![CDATA[Ho]]></surname>
<given-names><![CDATA[P.M]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2002</year>
<numero>D 66</numero>
<issue>D 66</issue>
<page-range>023517</page-range></nlm-citation>
</ref>
<ref id="B12">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Huang]]></surname>
<given-names><![CDATA[Q.G]]></given-names>
</name>
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[Nucl. Phys]]></source>
<year>2005</year>
<numero>B 713</numero>
<issue>B 713</issue>
<page-range>219</page-range></nlm-citation>
</ref>
<ref id="B13">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Huang]]></surname>
<given-names><![CDATA[Q.G]]></given-names>
</name>
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[JHEP]]></source>
<year>2003</year>
<numero>0306</numero>
<issue>0306</issue>
<page-range>014</page-range></nlm-citation>
</ref>
<ref id="B14">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kim]]></surname>
<given-names><![CDATA[H.s]]></given-names>
</name>
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[G.S]]></given-names>
</name>
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[H.W]]></given-names>
</name>
<name>
<surname><![CDATA[Myung]]></surname>
<given-names><![CDATA[Y.S]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year></year>
<numero>D 70</numero>
<issue>D 70</issue>
<page-range>043521</page-range></nlm-citation>
</ref>
<ref id="B15">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kim]]></surname>
<given-names><![CDATA[H.s]]></given-names>
</name>
<name>
<surname><![CDATA[Lee]]></surname>
<given-names><![CDATA[G.S]]></given-names>
</name>
<name>
<surname><![CDATA[Myung]]></surname>
<given-names><![CDATA[Y.S]]></given-names>
</name>
</person-group>
<source><![CDATA[Mod. Phys. Lett]]></source>
<year>2005</year>
<numero>A 20</numero>
<issue>A 20</issue>
<page-range>271</page-range></nlm-citation>
</ref>
<ref id="B16">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Liu]]></surname>
<given-names><![CDATA[D.J]]></given-names>
</name>
<name>
<surname><![CDATA[Li]]></surname>
<given-names><![CDATA[X.Z]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2004</year>
<numero>D 70</numero>
<issue>D 70</issue>
<page-range>123504</page-range></nlm-citation>
</ref>
<ref id="B17">
<label>9</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[García-Compeán]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Obregón]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Ramírez]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev. Lett]]></source>
<year>2002</year>
<numero>88</numero>
<issue>88</issue>
<page-range>161301</page-range></nlm-citation>
</ref>
<ref id="B18">
<label>10</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gamboa]]></surname>
<given-names><![CDATA[I]]></given-names>
</name>
<name>
<surname><![CDATA[Loewe]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Rojas]]></surname>
<given-names><![CDATA[J.C]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year></year>
<numero>D 64</numero>
<issue>D 64</issue>
<page-range>067901</page-range></nlm-citation>
</ref>
<ref id="B19">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Chaichian]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Sheikh-Jabbari]]></surname>
<given-names><![CDATA[M.M]]></given-names>
</name>
<name>
<surname><![CDATA[Tureanu]]></surname>
<given-names><![CDATA[A]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev. Lett]]></source>
<year></year>
<numero>86</numero>
<issue>86</issue>
<page-range>2716</page-range></nlm-citation>
</ref>
<ref id="B20">
<label>11</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[García-Compeán]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Obregón]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Ramírez]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Sabido]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2003</year>
<numero>D 68</numero>
<issue>D 68</issue>
<page-range>044015</page-range></nlm-citation>
</ref>
<ref id="B21">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[García-Compeán]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Obregón]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Ramírez]]></surname>
<given-names><![CDATA[C]]></given-names>
</name>
<name>
<surname><![CDATA[Sabido]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2003</year>
<numero>D 68</numero>
<issue>D 68</issue>
<page-range>045010</page-range></nlm-citation>
</ref>
<ref id="B22">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Banados]]></surname>
<given-names><![CDATA[M]]></given-names>
</name>
<name>
<surname><![CDATA[Chandia]]></surname>
<given-names><![CDATA[O]]></given-names>
</name>
<name>
<surname><![CDATA[Grandi]]></surname>
<given-names><![CDATA[N]]></given-names>
</name>
<name>
<surname><![CDATA[Schaposnik]]></surname>
<given-names><![CDATA[F.A]]></given-names>
</name>
<name>
<surname><![CDATA[Silva]]></surname>
<given-names><![CDATA[G.A]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2001</year>
<numero>D 64</numero>
<issue>D 64</issue>
<page-range>084012</page-range></nlm-citation>
</ref>
<ref id="B23">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Nishino]]></surname>
<given-names><![CDATA[H]]></given-names>
</name>
<name>
<surname><![CDATA[Rajpoot]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Lett]]></source>
<year>2002</year>
<numero>B 532</numero>
<issue>B 532</issue>
<page-range>334</page-range></nlm-citation>
</ref>
<ref id="B24">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Nair]]></surname>
<given-names><![CDATA[V.P]]></given-names>
</name>
</person-group>
<source><![CDATA["Gravitational Fields on a Noncommutative Space"]]></source>
<year></year>
</nlm-citation>
</ref>
<ref id="B25">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cacciatori]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Klemm]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Martucci]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Zanon]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Lett]]></source>
<year>2002</year>
<numero>B 536</numero>
<issue>B 536</issue>
<page-range>101</page-range></nlm-citation>
</ref>
<ref id="B26">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cacciatori]]></surname>
<given-names><![CDATA[S]]></given-names>
</name>
<name>
<surname><![CDATA[Chamseddine]]></surname>
<given-names><![CDATA[A.H]]></given-names>
</name>
<name>
<surname><![CDATA[Klemm]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
<name>
<surname><![CDATA[Martucci]]></surname>
<given-names><![CDATA[L]]></given-names>
</name>
<name>
<surname><![CDATA[Sabra]]></surname>
<given-names><![CDATA[W.A]]></given-names>
</name>
<name>
<surname><![CDATA[Zanon]]></surname>
<given-names><![CDATA[D]]></given-names>
</name>
</person-group>
<source><![CDATA["Noncommutative Gravity in Two Dimensions"]]></source>
<year></year>
</nlm-citation>
</ref>
<ref id="B27">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Abe]]></surname>
<given-names><![CDATA[Y]]></given-names>
</name>
<name>
<surname><![CDATA[Nair]]></surname>
<given-names><![CDATA[V.P]]></given-names>
</name>
</person-group>
<source><![CDATA[Noncommutative Gravity: Fuzzy Sphere and Others]]></source>
<year></year>
</nlm-citation>
</ref>
<ref id="B28">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Cardella]]></surname>
<given-names><![CDATA[M.A]]></given-names>
</name>
<name>
<surname><![CDATA[Zanon]]></surname>
<given-names><![CDATA[Daniela]]></given-names>
</name>
</person-group>
<source><![CDATA["Noncommutative Deformation of Four-dimensional Einstein Gravity"]]></source>
<year></year>
</nlm-citation>
</ref>
<ref id="B29">
<nlm-citation citation-type="">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Chamseddine]]></surname>
<given-names><![CDATA[A.H]]></given-names>
</name>
</person-group>
<source><![CDATA["Invariant Actions for Noncommutative Gravity"]]></source>
<year></year>
</nlm-citation>
</ref>
<ref id="B30">
<nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Moffat]]></surname>
<given-names><![CDATA[J.W]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Lett]]></source>
<year>2000</year>
<numero>B 491</numero>
<issue>B 491</issue>
<page-range>345</page-range></nlm-citation>
</ref>
<ref id="B31">
<nlm-citation citation-type="journal">
<source><![CDATA[Phys. Lett]]></source>
<year>2000</year>
<numero>B 493</numero>
<issue>B 493</issue>
<page-range>142</page-range></nlm-citation>
</ref>
<ref id="B32">
<label>12</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kantowski]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
<name>
<surname><![CDATA[Sachs]]></surname>
<given-names><![CDATA[R]]></given-names>
</name>
</person-group>
<source><![CDATA[J. Math. Phys]]></source>
<year>1966</year>
<numero>7</numero>
<issue>7</issue>
<page-range>443</page-range></nlm-citation>
</ref>
<ref id="B33">
<label>13</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Barbosa]]></surname>
<given-names><![CDATA[G.D]]></given-names>
</name>
<name>
<surname><![CDATA[Pinto-Neto]]></surname>
<given-names><![CDATA[N]]></given-names>
</name>
</person-group>
<source><![CDATA[Phys. Rev]]></source>
<year>2004</year>
<numero>D 70</numero>
<issue>D 70</issue>
<page-range>103512</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
