EXPRESSING A TENSOR PERMUTATION MA- TRIX p n IN TERMS OF THE GENERALIZED GELL-MANN MATRICES

Transcription

1 EXPRESSING A TENSOR PERMUTATION MA- TRIX p n IN TERMS OF THE GENERALIZED GELL-MANN MATRICES Christian RAKOTONIRINA Institut Supérieur de Technologie d Antananarivo, IST-T, BP 8122, Madagascar rakotopierre@refer.mg Rakotoson HANITRIARIVO Institut National des Sciences et Techniques Nuclaires, Madagascar-INSTN rakotosonhanitriarivo@yahoo.fr Abstract We have shown how to express a tensor permutation matrix p n as a linear combination of the tensor products of p p-gell-mann matrices. We have given the expression of a tensor permutation matrix as a linear combination of the tensor products of the Pauli matrices. Introduction The expression of the tensor commutation matrix 2 2 as a linear combination of the tensor products of the Pauli matrices U 2 2 = = 1 2 I 2 I σ i σ i with I 2 the 2 2 unit matrix, [1], [2] are frequently found in quantum theory. The tensor commutation matrix 3 3 is expressed as a linear combination of the tensor products of the 3 3-Gell-Mann matrices [3] U 3 3 = = 1 3 I 3 I λ i λ i and as generalization of these two formulae the tensor commutation matrix p p is expressed as a linear combination of the tensor products of the p p-gell-mann 1

2 matrices[4] U p p = 1 p I p I p p Λ a Λ a (1) where I p the p p-unit matrix. It is natural to think to more general formulae in the direction from tensor commutation matrix to tensor permutation matrix. However, the aim of this paper is not to construct a more general formula but to show only how to express a tensor permutation matrix p n as a linear combination of the tensor products of the p p-gell-mann matrices, with for p = 2 the expression is in terms of the Pauli matrices. In the firt four sections we will construct the tools which we will need for the examples in the last section. The main idea is to decompose the tensor permutation matrix p n as a product of some tensor transposition matrices. 1 Tensor product of matrices Theorem 1 Consider (A i ) 1 i n m a basis of M n m (C), (B j ) 1 j p r a basis of M p r (C). Then, (A i B j ) 1 i n m,1 j p r is a basis of M np mr (C). Proposition 2 Consider (A i ) 1 i n a set of elements of M p r (C), (B i ) 1 i n a set of elements of M l m (C), A M p r (C) and B M l m (C). If then, for any matrix K, A B = A K B = A i B i (1.2) A i K B i 2 Tensor permutation matrices Definition 3 For p, q N, p 2, q 2, we call tensor commutation matrix p q the permutation matrix U p q M pq pq (C) formed by 0 and 1, verifying the property U p q.(a b) = b a for all a M p 1 (C), b M q 1 (C). More generally, for k N, k > 2 and for σ permutation on {1, 2,..., k}, we call σ-tensor permutation matrix n 1 n 2... n k the permutation matrix U n1 n 2... n k (σ) M n1n 2...n k n 1n 2...n k (C) formed by 0 and 1, verifying the property U n1 n 2... n k (σ) (a 1 a 2... a k ) = a σ(1) a σ(2)... a σ(k) for all a i M ni 1 (C), (i {1, 2,..., k}). 2

3 Definition 4 For k N, k > 2 and for σ permutation on {1, 2,..., k}, we call σ-tensor transposition matrix n 1 n 2... n k a σ-tensor permutation matrix n 1 n 2... n k with σ is a transposition. Consider the σ-tensor transposition matrix n 1 n 2... n k, U n1 n 2... n k (σ) with σ the transposition (i j). U n1 n 2... n k (σ) (a 1... a i a i+1... a j a j+1... a k ) = a 1... a i 1 a j a i+1... a j 1 a i a j+1... a k for any a l M nl 1 (C). If (B li ) 1 li n jn i, ( B lj )1 l j n in j are respectively bases of M nj n i (C) and M ni n j (C), then the tensor commutation matrix U ni n j can be decomposed as a linear combination of the basis ( B li B lj )1 l i n jn i,1 l j n in j of M nin j n in j (C). We want to prove that U n1 n 2... n k (σ) is a linear combination of ( In1n2...n B i 1 l i I ni+1n i+2...n B ) j 1 l j I nj+1n j+2...n k 1 l i n jn i,1 l j n in j. For doing so, it suffices to prove the following theorem by using the proposition 2. ( ) Theorem 5 Suppose σ = = ( 1 3 ) permutation on {1, 2, 3}, (B i1 ) 1 i1 N 3N 1, (B i3 ) 1 i3 N 1N 3 bases respectively of M N3 N 1 (C) and M N1 N 3 (C). If U N1 N 3 = N 1N 3 U N1 N 2 N 3 (σ) = N 1N 3 i 1=1 i 3=1 N 1N 3 N 1N 3 i 1=1 α i1i3 B i1 B i3, α i1i3 C, then i 3=1 α i1i3 B i1 I N2 B i3. 3 Decomposition of a tensor permutation matrix Notation 6 Let σ S n, that is σ a permutation on {1, 2,..., n}, p N, p 2, we denote the tensor permutation matrix Up p... p (σ) by U p n (σ),[5]. }{{} n times We use the following lemma for demonstrating the theorem below. Lemma 7 Let σ S n, whose cycle is C σ = (i 1 i 2 i 3... i n 1 i n ) then, C σ = (i 1 i 2 i 3... i n 1 ) (i n 1 i n ) 3

4 Theorem 8 For n N, n > 1, σ S n whose cycle is with k N, k > 2. Then or C σ = (i 1 i 2 i 3... i k 1 i k ) U p n (σ) = U p n ((i 1 i 2... i k 1 )) U p n ((i k 1 i k )) U p n (σ) = U p n ((i 1 i 2 )) U p n ((i 2 i 3 ))... U p n ((i k 1 i k )) with p N, p > 2. So, a tensor permutation matrix can be expressed as a product of tensor transposition matrices. Corollary 9 For n N, n > 2, σ S n whose cycle is Then, C σ = (i 1 i 2 i 3... i n 1 n) U p n (σ) = [ U p (n 1) ((i 1 i 2... i n 2 i n 1 )) I p ] Up n ((i n 1 n)) 4 n n-gell-mann matrices The n n-gell-mann matrices are hermitian, traceless matrices Λ 1,Λ 2,...,Λ n satisfying the commutation relations (Cf. for example [6], [7]) [Λ a, Λ b ] = 2i f abc Λ c (4.1) where f abc are the structure constants which are reals and totally antisymmetric, and T r (Λ a Λ b ) = 2δ ab where δ ab the Kronecker symbol. For n = 2, the 2 2-Gell-Mann matrices are the usual Pauli matrices. They satisfy also the anticommutation relation (Cf. for example[6], [7]) {Λ a, Λ b } = 4 n n δ abi n + 2 d abc Λ c (4.2) where I n denotes the n-dimensional unit matrix and the constants d abc are reals and totally symmetric in the three indices, and by using the relations (4.1) and (4.2), we have Λ a Λ b = 2 n n δ ab + d abc Λ c + i f abc Λ c (4.3) 4

5 The structure constants satisfy the relation (Cf. for example[6]) e=1 f abe f cde = 2 n n (δ acδ bd δ ad δ bc ) + 5 Examples e=1 d ace d dbe e=1 d ade d bce (4.4) Now, we have some theorems and relations on the generalized Gell-Mann matrices which we need for expressing a tensor permutation matrix in terms of the generalized Gell-Mann matrices. In this section, we treat some examples. 5.1 U n 3(σ) σ = (1 2 3) By employing the Lemma 7 and by using the Theorem 8 However, using (1) and σ = (1 2)(2 3) U n 3 ((1 2 3)) = U n 3 ((1 2)) U n 3 ((2 3)) (5.1) U n 3 ((1 2)) = 1 n I n I n I n U n 3 ((2 3)) = 1 n I n I n I n So, by using (5.1) we have U n 3 ((1 2 3)) = 1 n 2 I n I n I n + 1 I n Λ a Λ a + 1 Λ a Λ a I n Λ a Λ a I n I n Λ a Λ a b=1 Λ a Λ a Λ b Λ b 5

6 Hence, employing the relation (4.3) U n 3 ((1 2 3)) = 1 n 2 I n I n I n + 1 I n Λ a Λ a i Λ a Λ a I n + 1 Λ a I n Λ a b=1 f abc Λ a Λ b Λ c b=1 d abc Λ a Λ b Λ c (5.2) 5.2 U 2 3(σ), σ S 3 Now we give a formula giving U 2 3(σ), naturally in terms of the Pauli matrices ( ) ( ) ( ) i 1 0 σ 1 =, σ =, σ i 0 3 = 0 1 Using the relation (Cf. for example [8]) σ l σ k = δ lk I 2 + i ε lkm σ m where ε ijk is totally antisymmetric in the three indices, which is equal 1 if (i j k) = (1 2 3), we have U 2 3(1 2 3) = 1 4 I 2 I 2 I Conclusion m=1 I 2 σ l σ l l=1 σ l σ l I 2 i 4 l=1 j=1 k=1 σ l I 2 σ l l=1 ε ijk σ i σ j σ k Based on the fact that a tensor permutation matrix is a product of tensor transposition matrices and on the Theorem 5, with the help of the expression of a tensor commutation matrix in terms of the generalized Gell-Mann matrices, we can express a tensor permutation matrix as linear combination of the tensor products of the generalized Gell-Mann matrices. We have no intention for searching a general formula. However, we have shown that any tensor permutation matrix can be expressed in terms of the generalized Gell-Mann matrices and then the expression can be simplified by using the relations between these Matrices. 6

7 Acknowledgements The author would like to thank Ratsimbarison Mahasedra for encouragement and for critical reading the manuscript. References [1] FADDEV.L.D, Int.J.Mod.Phys.A, Vol.10, No 13, May,1848 (1995). [2] FUJII.K, Introduction to Coherent States and Quantum Information Theory, arxiv: quant-ph/ , prepared for 10th Numazu Meeting on Integral System, Noncommutative Geometry and Quantum theory, Numazu, Shizuoka, Japan, 7-9 Mai [3] RAKOTONIRINA.C, arxiv:math/ [4] RAKOTONIRINA.C, International Journal of Mathematics and Mathematical Sciences, Volume 2007, Article ID 20672, 10 pages, [5] RAOELINA ANDRIAMBOLOLONA, Algèbre linéaire et Multilinéaire. Applications, collection LIRA, tome 2, (1986). [6] NARISON.S, Spectral sum Rules, World Scientific Lecture Notes in Physics - Vol.26, 501, (1989). [7] PICH.A, arxiv:hep-ph/ [8] RAOELINA ANDRIAMBOLOLONA, Ann.Univ.Madagascar, Série Sc. Nature et Math, n9, (1972). 7