Research Article Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

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1 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2007, Article ID 20672, 0 pages doi:0.55/2007/20672 Research Article Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices Rakotonirina Christian Received December 2006; Accepted 3 February 2007 Recommended by Howard E. Bell We have expressed the tensor commutation matrix n n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3 2and2 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices. Copyright 2007 Rakotonirina Christian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.. Introduction When we had worked on Raoelina Andriambololona idea on the use of tensor product in Dirac equation [, 2],we had met the unitary matrix U 2 2 = This matrix is frequently found in quantum information theory [3 5] where one writes, by using the Pauli matrices [3 5], U 2 2 = 2 I 2 I σ i σ i.2 with I 2 the 2 2 unit matrix. We call this matrix a tensor commutation matrix 2 2. The tensor commutation matrix 3 3 is expressed by using the Gell-Mann matrices under i=

2 2 International Journal of Mathematics and Mathematical Sciences the following form [6]: U 3 3 = 3 I 3 I λ i λ i..3 We have to talk a bit about different types of matrices because in the generalization of the above formulas, we will consider a commutation matrix as a matrix of fourthorder tensor and in expressing the commutation matrices U 3 2, U 2 3, at the last section, a commutation matrix will be considered as matrix of second-order tensor. m n C denotes the set of m n matrices whose elements are complex numbers. 2. Tensor product of matrices 2.. Matrices. If the elements of a matrix are considered as the components of a secondorder tensor, we adopt the habitual notation for a matrix, without parentheses inside, whereas if the elements of the matrix are, for instance, considered as the components of sixth-order tensor, three times covariant and three times contravariant, then we represent the matrix of the following way, for example: M = M = M ii2i3 j j 2 j 3 i= , i i 2 i 3 =,2,2,22,2,22,22,222,3,32,32,322 row indices, j j 2 j 3 =,2,2,22,2,22,22,222 column indices. 2. Thefirstindicesi and j are the indices of the outside parenthesis which we call the first-order parenthesis; the second indices i 2 and j 2 are the indices of the next parentheses which we call the second-order parentheses; the third indices i 3 and j 3 are the indices of the most interior parentheses, of this example, which we call third-order parentheses. So, for instance, M2 32 = 5.

3 Rakotonirina Christian 3 If we delete the third-order parenthesis, then the elements of the matrix M are considered as the components of a fourth-order tensor, twice contravariant and twice covariant. A matrix is a diagonal matrix if deleting the interior parentheses, we have a habitual diagonal matrix. A matrix is a symmetric resp., antisymmetric matrix if deleting the interior parentheses, we have a habitual symmetric resp., antisymmetric matrix. We identify one matrix to another matrix if after deleting the interior parentheses, they are the same matrices Tensor product of matrices Definition 2.. Consider A = A i j m n C, B = B i j p r C. The matrix defined by A B... A jb... A nb... A B = A i B... A i jb... A i nb... A m B... A m j B... A m n B 2.2 is called the tensor product of the matrix A by the matrix B, A B mp nr C, A B = C ii2 j j 2 = A i 2.3 j B i2 j 2, cf., e.g., [3] where, i i 2 are row indices and j j 2 are column indices. 3. Generalized Gell-Mann matrices Let us fix n N, n 2 for all continuations. The generalized Gell-Mann matrices or n n-gell-mann matrices are the traceless Hermitian n n matrices Λ, Λ 2,...,Λ n 2 which satisfy the relation TrΛ i Λ j = 2δ ij,foralli, j {,2,...,n 2 },whereδ ij = δ ij = δ i j is the Kronecker symbol [7]. However, for the demonstration of Theorem 4.3, denote, for i<j n, thecn 2 = n!/2!n 2! n n-gell-mann matrices which are symmetric with all elements 0 except the ith row jth column and the jth row ith column which are equal to, by Λ ij ;the Cn 2 = n!/2!n 2! n n-gell-mann matrices which are antisymmetric with all elements are 0 except the ith row jth column which is equal to i and the jth row ith column which

4 4 International Journal of Mathematics and Mathematical Sciences is equal to i, byλ [ij] and by Λ d, d n, the following n n n-gell-mann matrices are diagonal: Λ 0. =...., Λ 2 = 2. 3,...,Λ n =.. 0 C 2 n n 3. For n = 2, we have the Pauli matrices. 4. Tensor commutation matrices Definition 4.. For p, q N, p 2, q 2, call the tensor commutation matrix p q the permutation matrix U p q pq pq C formed by 0 and, verifying the property U p q a b = b a 4. for all a p C, b q C. Considering U p q as a matrix of a second-order tensor, one can construct it by using the following rule [6]. Rule 4.2. Let us start in putting at first row and first column, after that let us pass into second column in going down at the rate of p rows and put at this place, then pass into third column in going down at the rate of p rows and put, and so on until there are only for us p rows for going down then we have obtained number of : q. Then pass into the next column which is the q th column, put at the second row of this column and repeat the process until we have only p 2 rows for going down then we have obtained number of : 2q. After that pass into the next column which is the 2q th column, put at the third row of this column and repeat the process until we have only p 3rows for going down then we have obtained number of : 3q. Continuing in this way, we will have that the element at p qth row and p qth column is. The other elements are 0.

5 Rakotonirina Christian 5 Theorem 4.3. One has U n n = n I n I n 2 n 2 i= Λ i Λ i. 4.2 Proof. One has I n I n = δ ii2 j j 2 = δ i j δ i2 j 2, U n n = 4.3 δ i j 2 δ i2 j, where, i i 2 are row indices and j j 2 are column indices [3]. Consider at first the C 2 n symmetric n n Gell-Mann matrices which can be written as Then Λ ij = Λ ijl k l n, k n = δ il δ j k l n, k n δ jl δk i = δ il δ j k δ jl δ i k l n, k n. l n, k n 4.4 Λ Λ ij Λ ij = ij Λ ij l l 2 = δ il δ j k δ jl δk i δ il 2 δ j δ jl2 δk i 2, 4.5 where l l 2 are row indices and k are column indices. That is, Λ ij Λ ij l l 2 = δ il δ j k δ il2 δ j δ il δ j k δ jl2 δ i δ jl δ i k δ il2 δ j δ jl δ i k δ jl2 δ i. 4.6 The Cn 2 antisymmetric n n Gell-Mann matrices can be written as Λ [ij] = Λ [ij]l k l n, k n = iδ il δ j k iδ jl δk i l n, k n. 4.7 Then Λ Λ [ij] Λ [ij] = [ij] Λ [ij] l l 2, Λ [ij] Λ [ij] l l 2 = δ il δ j k δ il2 δ j δ il δ j k δ jl2 δ i δ jl δ i k δ il2 δ j δ jl δ i k δ jl2 δ i, i<j n Λ ij Λ ij l l 2 i<j n = 2 i<j n Λ [ij] Λ [ij] l l 2 δ il δ j k δ jl2 δk i 2 δ jl δk i δ il2 δ j = 2 δ il δ j k δ jl2 δk i 2 i /=j 4.8 is the l l 2 th row, k th column of the matrix Λ ij Λ ij Λ [ij] Λ [ij]. 4.9 i<j n i<j n

6 6 International Journal of Mathematics and Mathematical Sciences Now, consider the diagonal n n Gell-Mann matrices. Let d N, d n, Λ d = Cd 2 d δk l p= δ p k dδl k δd k 4.0 and the l l 2 th row, k th of the matrix Λ d Λ d is Λ d Λ d l l 2 = d Cd 2 δ l k q= p= Cd 2 δ l k dδk d d δ q k δ p d δ p p= Cd 2 δ l k dδ d d δ p k p= Cd 2 δ l k d 2 δk d δk d 2, 4. Λ d Λ d is a diagonal matrix, so all that we have to do is to calculate the elements on the diagonal where l = k and l 2 =.Then, n Λ d Λ d l l 2 d= n = C 2 d= d n C 2 d= d d δ q d k q= dδ d k d p= p= δ p n δ p n C 2 d= d C 2 d= d dδ d d 2 δ d k δ d d δ p k p= 4.2 is the l l 2 th row, k th column of the diagonal matrix n d= Λ d Λ d with l = k and l 2 =. Let us distinguish two cases. Case. k /= or /=. Case.. k /=. If k <, n Λ d Λ d l l 2 d= Similarly, if k >, Case.2. k = /= : n Λ d Λ d l l 2 d= = = n d= C 2 d n n d= d= C 2 d C 2 = 2 [ n d= d ] k2 = 2 d n. 4.3 Λ d Λ d l l 2 = 2 n. 4.4 k2 2 C = n k2 2 C = n. 4.5

7 Rakotonirina Christian 7 Case 2. k = = : n Λ d Λ d l l 2 d= n = C 2 d= d = 2 2 n. 4.6 We can condense these cases in one formula as n d= Λ d Λ d l l 2 = 2 n δl k 2 n δ il δk i δ il2 δk i 2, 4.7 i= which yields the diagonal of the diagonal matrix n d= Λ d Λ d. For all the n n Gell-Mann matrices, we have i<j n Λ ij Λ ij l l 2 i<j n = 2 n δl k 2 = 2 n δl k 2 Λ [ij] Λ [ij] l l 2 n d= Λ d Λ d l l 2 n δ il δk i δ il2 δk i 2 2 δ il δ j k δ jl2 δk i 2 i= n j= i= n δ il δ j k δ jl2 δk i 2 i /=j 4.8 = 2 n δl k 2δ l k for all l,l 2,k, {,2,...,n}. Hence, by using 4.3, n 2 i= Λ i Λ i = 2 n I n I n 2U n n 4.9 and the theorem is proved. 5. Expression of U 3 2 and U 2 3 In this section, we derive formulas for U 3 2 and U 2 3, naturally in terms of the Pauli matrices σ = 0, σ 0 2 = 0 i 0, σ i 0 3 = 0 5.

8 8 International Journal of Mathematics and Mathematical Sciences and the Gell-Mann matrices i λ = 0 0, λ 2 = i 0 0, λ 3 = 0 0, i λ 4 = 0 0 0, λ 5 = 0 0 0, λ 6 = 0 0, 0 0 i λ 7 = 0 0 i, λ 8 = i For r N,defineE r ij as the elementary r r matrix whose elements are zeros except the ith row and jth column which is equal to. We construct U 3 2 by using Rule 4.2,and then we have Take Let with α 0, α 3, α 8 C,then U 3 2 = E 6 E 6 23 E 6 35 E 6 42 E 6 54 E E 6 = E 3 E E 3 = α 0 I 3 α 3 λ 3 α 8 λ Let α 0 = 3, α 3 = 2, α 8 = E 3 = 3 I λ 3 6 λ , 5.6 E 2 = β 0 I 2 β 3 σ with β 0, β 3 C,then β 0 = 2, β 3 = 2, E 2 = 2 I 2 2 σ So we have E 6 = 3 I λ 3 6 λ 8 2 I 2 2 σ

9 Rakotonirina Christian 9 In a similar way, we have E 6 23 = 2 λ i 2 λ 2 2 σ i 2 σ 2, E 6 35 = 2 λ 6 i 2 λ 7 2 I 2 2 σ 3, E 6 42 = 2 λ i 2 λ 2 2 I 2 2 σ 3, E 6 54 = 2 λ 6 i 2 λ 7 2 σ i 2 σ 2, E = 3 I 3 3 λ 8 2 I 2 2 σ Hence U 3 2 = 3 I λ 3 6 λ 8 2 I 2 2 σ 3 2 λ i 2 λ 2 2 σ i 2 σ 2 2 λ 6 i 2 λ 7 2 I 2 2 σ 3 2 λ i 2 λ 2 2 I 2 2 σ λ 6 i 2 λ 7 2 σ i 3 2 σ 2 3 I 3 3 λ 8 2 I 2 2 σ 3. In an analogous way, U 2 3 = 2 I 2 2 σ 3 3 I λ 3 6 λ 8 2 σ i 2 σ 2 2 λ i 2 λ 2 2 I 2 2 σ 3 2 λ 6 i 2 λ 7 2 I 2 2 σ 3 2 λ i 2 λ σ i 2 σ 2 2 λ 6 i 2 λ 7 2 I σ 3 3 I 3 3 λ 8. One can develop these formulas in employing the distributivity of the tensor product. Acknowledgments The author thanks the referee of an earlier manuscript for suggesting the topic. The author would like to thank Victor Razafinjato, Director of Civil Engineering Department of Institut Supérieur de Technologie d AntananarivoIST-T and Ratsimbarison Mahasedra for encouragement and for critical reading of the manuscript.

10 0 International Journal of Mathematics and Mathematical Sciences References [] C. Rakotonirina, ThèsedeDoctoratdeTroisième Cycle de Physique Théorique, Université d Antananarivo, Antananarivo, Madagascar, 2003, unpublished. [2] R. P. Wang, Varieties of Dirac equation and flavors of leptons and quarks, abs/hep-ph/ [3] K. Fujii, Introduction to coherent states and quantum information theory, prepared for 0th Numazu Meeting on Integrable System, Noncommutative Geometry and Quantum theory, Numazu, Shizuoka, Japan, May 2002, [4] L. D. Faddev, Algebraic aspects of the Bethe ansatz, International Journal of Modern Physics A, vol. 0, no. 3, pp , 995. [5] F. Verstraete, A Study of Entanglement in Quantum Information Theory, ThèsedeDoctorat, Katholieke Universiteit, Leuven, Belgium, [6] C. Rakotonirina, Tensor permutation matrices in finite dimensions, math.gm/ [7] S. Narison, Spectral Sum Rules, vol. 26 of World Scientific Lecture Notes in Physics, World Scientific, Singapore, 989. Rakotonirina Christian: Département du Génie Civil, Institut Supérieur de Technologie d Antananarivo IST-T, BP 822, Madagascar address: rakotopierre@refer.mg