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Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations
Eugene Lytvynov 
Communications in Mathematical Physics, Volume: 351, Issue: 2, Pages: 653 - 687
Swansea University Author:
Eugene Lytvynov
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DOI (Published version): 10.1007/s00220-016-2786-5
Abstract
Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distribut...
| Published in: | Communications in Mathematical Physics |
|---|---|
| ISSN: | 0010-3616 1432-0916 |
| Published: |
2017
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| Online Access: |
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| URI: | https://cronfa.swan.ac.uk/Record/cronfa29639 |
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2016-08-24T12:53:47Z |
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2020-07-08T18:45:55Z |
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<?xml version="1.0"?><rfc1807><datestamp>2020-07-08T16:24:06.7686738</datestamp><bib-version>v2</bib-version><id>29639</id><entry>2016-08-24</entry><title>Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations</title><swanseaauthors><author><sid>e5b4fef159d90a480b1961cef89a17b7</sid><ORCID>0000-0001-9685-7727</ORCID><firstname>Eugene</firstname><surname>Lytvynov</surname><name>Eugene Lytvynov</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2016-08-24</date><deptcode>MACS</deptcode><abstract>Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=\kappa^2\mathbf 1$ ($\kappa>0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$.</abstract><type>Journal Article</type><journal>Communications in Mathematical Physics</journal><volume>351</volume><journalNumber>2</journalNumber><paginationStart>653</paginationStart><paginationEnd>687</paginationEnd><publisher/><issnPrint>0010-3616</issnPrint><issnElectronic>1432-0916</issnElectronic><keywords/><publishedDay>31</publishedDay><publishedMonth>12</publishedMonth><publishedYear>2017</publishedYear><publishedDate>2017-12-31</publishedDate><doi>10.1007/s00220-016-2786-5</doi><url/><notes/><college>COLLEGE NANME</college><department>Mathematics and Computer Science School</department><CollegeCode>COLLEGE CODE</CollegeCode><DepartmentCode>MACS</DepartmentCode><institution>Swansea University</institution><apcterm/><lastEdited>2020-07-08T16:24:06.7686738</lastEdited><Created>2016-08-24T11:18:55.8799080</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>Eugene</firstname><surname>Lytvynov</surname><orcid>0000-0001-9685-7727</orcid><order>1</order></author></authors><documents><document><filename>0029639-24082016112000.pdf</filename><originalFilename>quasi-free_anyons.pdf</originalFilename><uploaded>2016-08-24T11:20:00.3870000</uploaded><type>Output</type><contentLength>415035</contentLength><contentType>application/pdf</contentType><version>Accepted Manuscript</version><cronfaStatus>true</cronfaStatus><embargoDate>2017-10-18T00:00:00.0000000</embargoDate><copyrightCorrect>true</copyrightCorrect></document></documents><OutputDurs/></rfc1807> |
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2020-07-08T16:24:06.7686738 v2 29639 2016-08-24 Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations e5b4fef159d90a480b1961cef89a17b7 0000-0001-9685-7727 Eugene Lytvynov Eugene Lytvynov true false 2016-08-24 MACS Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=\kappa^2\mathbf 1$ ($\kappa>0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$. Journal Article Communications in Mathematical Physics 351 2 653 687 0010-3616 1432-0916 31 12 2017 2017-12-31 10.1007/s00220-016-2786-5 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University 2020-07-08T16:24:06.7686738 2016-08-24T11:18:55.8799080 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Eugene Lytvynov 0000-0001-9685-7727 1 0029639-24082016112000.pdf quasi-free_anyons.pdf 2016-08-24T11:20:00.3870000 Output 415035 application/pdf Accepted Manuscript true 2017-10-18T00:00:00.0000000 true |
| title |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
| spellingShingle |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations Eugene Lytvynov |
| title_short |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
| title_full |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
| title_fullStr |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
| title_full_unstemmed |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
| title_sort |
Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations |
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e5b4fef159d90a480b1961cef89a17b7 |
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e5b4fef159d90a480b1961cef89a17b7_***_Eugene Lytvynov |
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Eugene Lytvynov |
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Eugene Lytvynov |
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Communications in Mathematical Physics |
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10.1007/s00220-016-2786-5 |
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Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=\kappa^2\mathbf 1$ ($\kappa>0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$. |
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2017-12-31T19:05:08Z |
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1821433458718670848 |
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11.047609 |