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Matthias Schmidt - One of the best experts on this subject based on the ideXlab platform.
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the hilbert space costratification for the orbit type strata of su 2 lattice gauge theory
arXiv: Mathematical Physics, 2018Co-Authors: Erik Fuchs, Peter D. Jarvis, Gerd Rudolph, Matthias SchmidtAbstract:We construct the Hilbert space costratification of $G=\mathrm{SU}(2)$-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work where we have implemented the classical gauge orbit strata on quantum level within a suitable holomorphic picture. In this picture, each element $\tau$ of the classical stratification corresponds to the zero locus of a finite subset $\{p_i\}$ of the algebra $\mathcal R$ of $G$-invariant representative functions on the complexification of $G^N$. Viewing the invariants as Multiplication operators $\hat p_i$ on the Hilbert space $\mathcal H$, the union of their images defines a subspace of $\mathcal H$ whose orthogonal complement $\mathcal H_\tau$ is the element of the costratification corresponding to $\tau$. To construct $\mathcal H_\tau$, one has to determine the images of the $\hat p_i$ explicitly. To accomplish that goal, we construct an orthonormal basis in $\mathcal H$ and determine the Multiplication Law for the basis elements, that is, we determine the structure constants of $\mathcal R$ in this basis. This part of our analysis applies to any compact Lie group $G$. For $G = \mathrm{SU}(2)$, the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators $\hat p_i$ as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner's $3nj$ symbols. The latter are further expressed in terms of $9j$ symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.
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The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory
Journal of Mathematical Physics, 2018Co-Authors: Erik Fuchs, Peter D. Jarvis, Gerd Rudolph, Matthias SchmidtAbstract:We construct the Hilbert space costratification of G = SU(2)-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work [F. Furstenberg, G. Rudolph, and M. Schmidt, J. Geom. Phys. 119, 66–81 (2017)], where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture. In this picture, each element τ of the classical stratification corresponds to the zero locus of a finite subset {pi} of the algebra R of G-invariant representative functions on GCN. Viewing the invariants as Multiplication operators p^i on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement Hτ is the element of the costratification corresponding to τ. To construct Hτ, one has to determine the images of the p^i explicitly. To accomplish this goal, we construct an orthonormal basis in H and determine the Multiplication Law for the basis elements; that is, we determine the structure constants of R in this basis. This part of our analysis applies to any compact Lie group G. For G = SU(2), the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators p^i as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner’s 3nj symbols. The latter are further expressed in terms of 9j symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.We construct the Hilbert space costratification of G = SU(2)-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work [F. Furstenberg, G. Rudolph, and M. Schmidt, J. Geom. Phys. 119, 66–81 (2017)], where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture. In this picture, each element τ of the classical stratification corresponds to the zero locus of a finite subset {pi} of the algebra R of G-invariant representative functions on GCN. Viewing the invariants as Multiplication operators p^i on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement Hτ is the element of the costratification corresponding to τ. To construct Hτ, one has to determine the images of the p^i explicitly. To accomplish this goal, we construct an orthonormal basis in H and determine the Multiplication Law for the basis elements; that is, we determine the structure constants of R in thi...
S Vaidya - One of the best experts on this subject based on the ideXlab platform.
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spin and statistics on the groenewold moyal plane pauli forbidden levels and transitions
International Journal of Modern Physics A, 2006Co-Authors: A P Balachandran, G Mangano, A Pinzul, S VaidyaAbstract:The Groenewold–Moyal plane is the algebra ${\mathcal A}_\theta({\mathbb R}^{d+1})$ of functions on ℝd+1 with the *-product as the Multiplication Law, and the commutator $[\hat{x}_\mu,\hat{x}_\nu] =i\theta_{\mu \nu}\, (\mu,\nu=0,1,\ldots,d)$ between the coordinate functions. Chaichian et al.1 and Aschieri et al.2 have proved that the Poincare group acts as automorphisms on ${\mathcal A}_\theta({\mathbb R}^{d+1})$ if the coproduct is deformed. (See also the prior work of Majid,3 Oeckl4 and Grosse et al.5) In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Ref. 2. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.
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spin and statistics on the groenewold moyal plane pauli forbidden levels and transitions
arXiv: High Energy Physics - Theory, 2005Co-Authors: A P Balachandran, G Mangano, A Pinzul, S VaidyaAbstract:The Groenewold-Moyal plane is the algebra A_\theta(R^(d+1)) of functions on R^(d+1) with the star-product as the Multiplication Law, and the commutator [x_\mu,x_\nu] =i \theta_{\mu \nu} between the coordinate functions. Chaichian et al. and Aschieri et al. have proved that the Poincare group acts as automorphisms on A_\theta(R^(d+1))$ if the coproduct is deformed. (See also the prior work of Majid, Oeckl and Grosse et al). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Aschieri et al. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.
Erik Fuchs - One of the best experts on this subject based on the ideXlab platform.
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the hilbert space costratification for the orbit type strata of su 2 lattice gauge theory
arXiv: Mathematical Physics, 2018Co-Authors: Erik Fuchs, Peter D. Jarvis, Gerd Rudolph, Matthias SchmidtAbstract:We construct the Hilbert space costratification of $G=\mathrm{SU}(2)$-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work where we have implemented the classical gauge orbit strata on quantum level within a suitable holomorphic picture. In this picture, each element $\tau$ of the classical stratification corresponds to the zero locus of a finite subset $\{p_i\}$ of the algebra $\mathcal R$ of $G$-invariant representative functions on the complexification of $G^N$. Viewing the invariants as Multiplication operators $\hat p_i$ on the Hilbert space $\mathcal H$, the union of their images defines a subspace of $\mathcal H$ whose orthogonal complement $\mathcal H_\tau$ is the element of the costratification corresponding to $\tau$. To construct $\mathcal H_\tau$, one has to determine the images of the $\hat p_i$ explicitly. To accomplish that goal, we construct an orthonormal basis in $\mathcal H$ and determine the Multiplication Law for the basis elements, that is, we determine the structure constants of $\mathcal R$ in this basis. This part of our analysis applies to any compact Lie group $G$. For $G = \mathrm{SU}(2)$, the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators $\hat p_i$ as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner's $3nj$ symbols. The latter are further expressed in terms of $9j$ symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.
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The Hilbert space costratification for the orbit type strata of SU(2)-lattice gauge theory
Journal of Mathematical Physics, 2018Co-Authors: Erik Fuchs, Peter D. Jarvis, Gerd Rudolph, Matthias SchmidtAbstract:We construct the Hilbert space costratification of G = SU(2)-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work [F. Furstenberg, G. Rudolph, and M. Schmidt, J. Geom. Phys. 119, 66–81 (2017)], where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture. In this picture, each element τ of the classical stratification corresponds to the zero locus of a finite subset {pi} of the algebra R of G-invariant representative functions on GCN. Viewing the invariants as Multiplication operators p^i on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement Hτ is the element of the costratification corresponding to τ. To construct Hτ, one has to determine the images of the p^i explicitly. To accomplish this goal, we construct an orthonormal basis in H and determine the Multiplication Law for the basis elements; that is, we determine the structure constants of R in this basis. This part of our analysis applies to any compact Lie group G. For G = SU(2), the above procedure boils down to a problem in combinatorics of angular momentum theory. Using this theory, we obtain the union of the images of the operators p^i as a subspace generated by vectors whose coefficients with respect to our basis are given in terms of Wigner’s 3nj symbols. The latter are further expressed in terms of 9j symbols. Using these techniques, we are also able to reduce the eigenvalue problem for the Hamiltonian of this theory to a problem in linear algebra.We construct the Hilbert space costratification of G = SU(2)-quantum gauge theory on a finite spatial lattice in the Hamiltonian approach. We build on previous work [F. Furstenberg, G. Rudolph, and M. Schmidt, J. Geom. Phys. 119, 66–81 (2017)], where we have implemented the classical gauge orbit strata on the quantum level within a suitable holomorphic picture. In this picture, each element τ of the classical stratification corresponds to the zero locus of a finite subset {pi} of the algebra R of G-invariant representative functions on GCN. Viewing the invariants as Multiplication operators p^i on the Hilbert space H, the union of their images defines a subspace of H whose orthogonal complement Hτ is the element of the costratification corresponding to τ. To construct Hτ, one has to determine the images of the p^i explicitly. To accomplish this goal, we construct an orthonormal basis in H and determine the Multiplication Law for the basis elements; that is, we determine the structure constants of R in thi...
A P Balachandran - One of the best experts on this subject based on the ideXlab platform.
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spin and statistics on the groenewold moyal plane pauli forbidden levels and transitions
International Journal of Modern Physics A, 2006Co-Authors: A P Balachandran, G Mangano, A Pinzul, S VaidyaAbstract:The Groenewold–Moyal plane is the algebra ${\mathcal A}_\theta({\mathbb R}^{d+1})$ of functions on ℝd+1 with the *-product as the Multiplication Law, and the commutator $[\hat{x}_\mu,\hat{x}_\nu] =i\theta_{\mu \nu}\, (\mu,\nu=0,1,\ldots,d)$ between the coordinate functions. Chaichian et al.1 and Aschieri et al.2 have proved that the Poincare group acts as automorphisms on ${\mathcal A}_\theta({\mathbb R}^{d+1})$ if the coproduct is deformed. (See also the prior work of Majid,3 Oeckl4 and Grosse et al.5) In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Ref. 2. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.
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spin and statistics on the groenewold moyal plane pauli forbidden levels and transitions
arXiv: High Energy Physics - Theory, 2005Co-Authors: A P Balachandran, G Mangano, A Pinzul, S VaidyaAbstract:The Groenewold-Moyal plane is the algebra A_\theta(R^(d+1)) of functions on R^(d+1) with the star-product as the Multiplication Law, and the commutator [x_\mu,x_\nu] =i \theta_{\mu \nu} between the coordinate functions. Chaichian et al. and Aschieri et al. have proved that the Poincare group acts as automorphisms on A_\theta(R^(d+1))$ if the coproduct is deformed. (See also the prior work of Majid, Oeckl and Grosse et al). In fact, the diffeomorphism group with a deformed coproduct also does so according to the results of Aschieri et al. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.
Florentin Smarandache - One of the best experts on this subject based on the ideXlab platform.
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refined literal indeterminacy and the Multiplication Law of sub indeterminacies
viXra, 2015Co-Authors: Florentin SmarandacheAbstract:In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, and elementary neutrosophic algebraic structures. After- wards, their generalizations to refined neutrosophic set, re- spectively refined neutrosophic numerical and literal com- ponents, then refined neutrosophic numbers and refined neutrosophic algebraic structures. The aim of this paper is to construct examples of splitting the literal indeterminacy
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Refined Literal Indeterminacy and the Multiplication Law of Sub-Indeterminacies
Neutrosophic Sets and Systems, 2015Co-Authors: Florentin SmarandacheAbstract:Abstract. In this paper, we make a short history about: the neutrosophic set, neutrosophic numerical components and neutrosophic literal components, neutrosophic numbers, neutrosophic intervals, neutrosophic hypercomplex numbers of dimension n, and elementary neutrosophic algebraic structures. Afterwards, their generalizations to refined neutrosophic set, respectively refined neutrosophic numerical and literal components, then refined neutrosophic numbers and refined neutrosophic algebraic structures. The aim of this paper is to construct examples of splitting the literal indeterminacy ሺࡵሻ into literal sub-indeterminacies ሺࡵ ,ࡵ,...,࢘ࡵሻ, and to define a Multiplication Law of these literal sub-indeterminacies in order to be able to build refined ࡵ െ neutrosophic algebraic structures. Also, examples of splitting the numerical indeterminacy ሺሻ into numerical sub-indeterminacies, and examples of splitting neutrosophic numerical components into neutrosophic numerical sub-components are given.
