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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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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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2017-12-31T03:57:43Z |
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1851092174068776960 |
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11.089386 |