{"title":"Isotropic random spin weighted functions on 𝑆² vs isotropic random fields on 𝑆³","authors":"Michele Stecconi","doi":"10.1090/tpms/1177","DOIUrl":null,"url":null,"abstract":"<p>We show that an isotropic random field on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S upper U left-parenthesis 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>S</mml:mi>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">SU(2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is not necessarily isotropic as a random field on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">S^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed\">\n <mml:semantics>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:annotation encoding=\"application/x-tex\">S^3</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"d\">\n <mml:semantics>\n <mml:mi>d</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">d</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet minus StartFraction d Over 2 EndFraction comma ellipsis comma StartFraction d Over 2 EndFraction EndSet\">\n <mml:semantics>\n <mml:mrow>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-OPEN\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">{</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mfrac>\n <mml:mi>d</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mo>,</mml:mo>\n <mml:mo>…<!-- … --></mml:mo>\n <mml:mo>,</mml:mo>\n <mml:mfrac>\n <mml:mi>d</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:mfrac>\n <mml:mstyle scriptlevel=\"0\">\n <mml:mrow class=\"MJX-TeXAtom-CLOSE\">\n <mml:mo maxsize=\"1.2em\" minsize=\"1.2em\">}</mml:mo>\n </mml:mrow>\n </mml:mstyle>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\bigl \\{-\\frac {d}{2},\\dots ,\\frac {d}{2}\\bigr \\}</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, each of which is isotropic in the sense of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S upper U left-parenthesis 2 right-parenthesis\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>S</mml:mi>\n <mml:mi>U</mml:mi>\n <mml:mo stretchy=\"false\">(</mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mo stretchy=\"false\">)</mml:mo>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">SU(2)</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.</p>\n\n<p>In addition we will give an overview of the theory of spin weighted functions and Wigner <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper D\">\n <mml:semantics>\n <mml:mi>D</mml:mi>\n <mml:annotation encoding=\"application/x-tex\">D</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal ð ModifyingAbove normal ð With bar\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi mathvariant=\"normal\">ð<!-- ð --></mml:mi>\n <mml:mover>\n <mml:mi mathvariant=\"normal\">ð<!-- ð --></mml:mi>\n <mml:mo accent=\"false\">¯<!-- ¯ --></mml:mo>\n </mml:mover>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\eth \\overline {\\eth }</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> and the horizontal Laplacian of the Hopf fibration <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper S cubed right-arrow upper S squared\">\n <mml:semantics>\n <mml:mrow>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>3</mml:mn>\n </mml:msup>\n <mml:mo stretchy=\"false\">→<!-- → --></mml:mo>\n <mml:msup>\n <mml:mi>S</mml:mi>\n <mml:mn>2</mml:mn>\n </mml:msup>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">S^3\\to S^2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>, in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]</p>","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1177","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
We show that an isotropic random field on SU(2)SU(2) is not necessarily isotropic as a random field on S3S^3, although the two spaces can be identified. The ambiguity is due to the fact that the notion of isotropy on a group and on a sphere are different, the latter being much stronger. We show that any isotropic random field on S3S^3 is necessarily a superposition of uncorrelated random harmonic homogeneous polynomials, such that the one of degree dd is necessarily a superposition of uncorrelated random spin weighted functions of every possible spin weight in the range {−d2,…,d2}\bigl \{-\frac {d}{2},\dots ,\frac {d}{2}\bigr \}, each of which is isotropic in the sense of SU(2)SU(2). Moreover, for a random field of fixed degree, each spin weight appears with the same magnitude, in a sense to be specified.
In addition we will give an overview of the theory of spin weighted functions and Wigner DD-matrices, with the purpose of gathering together many different points of view and adding ours. As a byproduct of this survey we will prove some new properties of the Wigner matrices and a formula relating the operators ðð¯\eth \overline {\eth } and the horizontal Laplacian of the Hopf fibration S3→S2S^3\to S^2, in the sense of [Bérard Bergery and Bourguignon, Illinois J. Math. 26 (1982), no. 2, 181–200.]