Lie Algebra and Representation of SU(4)

Transcription

1 EJTP, No Electronic Journal of Theoretical Physics Lie Algebra and Representation of SU() Mahmoud A. A. Sbaih, Moeen KH. Srour, M. S. Hamada and H. M. Fayad Department of Physics, Al Aqsa University, P. O. Box 5, Gaza, Palestine Received 5 December 8, Accepted 6 September 9, Published 5 January Abstract: In this paper we give a new representation for the Lie unimodular group SU(). Both the nonzero structure constant f kμν and the symmetric invariant tensor d kμν one calculated. We also bring out a new representation of SU() in terms of Pauli matrices constructed. Finally, the weight of the first fundamental representation of SU() is also obtained. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Group Theory; Lie Algebra; SU() PACS ():..Sv;..-v;..Hj;.65.Fd. Introduction The group of all unitary matrices of order n is known as U(n), whereas the group of all unitary matrices of order n with determinant + is denoted by SU(n) [, ]. It is clear that the SU(n) group is a subgroup of U(n), continuous, connected, lie group, and possesses n real independent parameters. The unitary unimodular group SU() in two complex dimensions is the simplest nontrivial example of a non abelian group which describes the two level quantum system, at the same time SU() is homomorphic on SO(), which have three generators σ x =, σ y = i, σ z = () i which are a set of three independent traceless hermitian matrices of order. mahmoud sbaih@yahoo.com moeensrour@hotmail.com

2 Electronic Journal of Theoretical Physics, No The next group after SU() is SU() which possesses eight dimensional unimodular matrices in three complex dimension. The description of three level systems [,, 5] in general quantum mechanics also involves SU(). It has been discussed a generalization of SU() [6],their commutation and anticommutation relations, structure constant f kμν and the symmetric invariant tensor d kμν. The next group after SU() is SU(). In SU() we have 5 generators (n ) of SU() as matrices. The group SU() which can be broken to anther subgroup such that SU() SU() SU(), often exploits the canonical subgroup chain SU() SU() SU(), also it is useful to study the members of the baryon M of SU(). In the quark model there are at least four quarks, the four quarks are members of the first fundamental quartet of SU(). This paper is organized as follows. Section II deals with the definition of the group SU() and the generators λ-matrices in the defining representation. For their commutation relations, we constructed the structure constant f kμν and for their anticommutator, we constructed the symmetric invariant tensor d kμν. The nonzero structure constant f kμν and symmetric invariant d kμν are listed. Section III deals with another kind of generators which are obtained from the λ μ (μ = 5), and from these generators we will find the weights of the first fundamental representation of SU().. Generators of SU() and λ-matrices The group SU() is represented as SU() = { A= comlex matrix A A=, det(a)= } () The hermitian matrix generators of SU(), analogous to the pauli matrices of SU() and the Gell Mann matrices of SU() which are:

3 Electronic Journal of Theoretical Physics, No λ =, λ = i i, λ = λ =, λ 5 = i, λ 6 = i i λ 7 =, i λ 8 =, λ 9 = () i λ =, i λ =, λ = i i λ =, λ =, i i λ 5 = 6 The λ μ matrices are orthogonal and satisfy Tr (λ μ ) =; μ = 5 ()

4 Electronic Journal of Theoretical Physics, No These matrices λ μ obey both the characteristic commutation and the Jacobi identity relations [[λ ν,λ μ ],λ k ]+[[λ μ,λ k ],λ ν ]+[[λ k,λ ν ],λ μ ]= (5) [λ k,λ μ ]=i μ f kμν λ ν ; k, μ, ν = 5 (6) Tr (λ k [λ μ,λ ν ]) = if kμν (7) where [ ] stands for the commutation relation. For SU() the number of commutation relations worked out is 5 and f kμν is denoted by the structure constant. The independent non-vanishing structure constants f kμν which are completely antisymmetric under the permutation of any two indices satisfying k<μ<νare given in the table (). For each case of anti-commutation relation { a, b}, where{a, b}= ab+ba, the calculation is more complicated. The matrices λ μ obey characteristic anti-commutation relation {λ k,λ μ } = d kμν λ ν +(A kμ ) (8) ν Tr(λ k {λ μ,λ ν })=d kμν (9) Tr(λ k λ μ λ ν )=if kμν +d kμν where A kμ is defined by matrix which satisfy if k μ; k, μ = 5 A kμ = (A) if k = μ; k, μ = 5 () This matrix A kμ is characterized by all the matrix elements is zero except the diagonal as listed in table (). In this table we have given the independent non-vanishing components of the completely symmetric d kμν under the permutation of any two indices. It is clear from tables () and () that if k, μ and ν are less than or equal to eight, the f kμν and d kμν are identical to the SU() structure constants f kμν and d kμν. It should be noted here that the equations (5,6,7,9 and ) also properties for the three dimensional λ matrices SU(). However for all group SU(n), n, nothing is new apart except that the anti- commutators as it is clear in equation (8).. Weight of SU() In the language of quantum mechanics there are always three spin operators in three dimensional space xyz, we shall call these operators H,H and H. The vector ψ j m is an eigenstate of H H ψ j m = mψ j m ()

5 Electronic Journal of Theoretical Physics, No The eigenvalue m is called a weight. In this representation, one can express the Lie algebra of SU() in term of the Pauli spin matrices. We dened these generators as follows H i = σ z () E i = σ x + iσ y () E i = σ x iσ y (5) using the above equations one can obtain the generators in the standard form in term of λ k (k = 5). These are given by

6 Electronic Journal of Theoretical Physics, No H =, H = 6, H = E =, E =, E = E =, E 5 =, E 6 = (6) E =, E =, E = E =, E 5 =, E 6 = where E i = E i, i = 6 (7)

7 Electronic Journal of Theoretical Physics, No If we operate on the four dimensional basis vector ( ) ( ) ( ) ( ) U = ; U = ; U = ; U = (8) with the diagonal vector operator H =(H,H,H ), we obtain ( ) HU =, 6, U (9) ( ) HU =, 6, U ( HU =, ), 6 U () HU = (,, ) U () Thus the four weight of the first representation of SU() are given by ( ) m() =, 6, ( ) m() =, 6, ( m() =, ), 6 ( ) m() =,, From equations () and () the weight of the second fundamental representation are just the negative of the weight of the first. () () (5) (6) Conclusion In this paper, matrices of the Lie algebra of SU() have been represented, from these matrices the calculation of the commutation and anti-commutation relations has been carried out with the given independent non-vanishing components of f kμν and d kμν,as it is clear from tables () and (), it is shown that if k, μ and ν are less than or equal to eight, the invariant tensors f kμν and d kμν are identical to SU() structure constants. As it is obvious from our analysis n, nothing is new a part in the commutation and anti-commutation relations except that as mention in equation 8, which is represented by

8 6 Electronic Journal of Theoretical Physics, No a new matrix A kμ, this matrix is diagonal matrix which is symmetric under permutation of any two indices of d kμν. In the parallel direction, we have constructed another representation of SU() in term of Palui spin matrices, from which we calculated the weight. For any representation of j of the group, there are j + different weights which belong to a different representation and if we know one weight we can obtain another weight of the representation. Thus, it is expected that our method can be systematically extended to higher dimensional group as SU(6).

9 Electronic Journal of Theoretical Physics, No References [] D. B. Lightenberg. Unitary symmetry and elementary particle. New York San Francisco London, 978. [] Griffiths. Introduction of elementary particle. John Wiley and Sons, New York, 98. [] J. Elgin. Semiclassical formalism for the treatment of three-level systems. Phys. Lett A, 8:, 98. [] P. Aravind. Matrix solution of pseudospin equations for three-level systems. J. Opt. Soc Am. B, :5, 986. [5] F. T. Hioe. Dynamic symmetries in quantum electronics. Phys. Rev. A, 8: , 98. [6] K. S. Mallesh and N. Mukunda. The algebra and geometry of su() matrices. PRAMANA, 9:7 8, 997.

10 8 Electronic Journal of Theoretical Physics, No Table Non-zero structure constant f kμν of SU() k μ ν d kμν continue on next page

11 Electronic Journal of Theoretical Physics, No Table - continued from previous page k μ ν d kμν

12 Electronic Journal of Theoretical Physics, No Table Non-zero independent element of the tensor d kμν and the matrix A kμ k μ ν d kμν A kμ continue on next page

13 Electronic Journal of Theoretical Physics, No Table D continued from previous page k μ ν d kμν A kμ continue on next page

14 Electronic Journal of Theoretical Physics, No Table D continued from previous page k μ ν d kμν A kμ continue on next page

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16 Electronic Journal of Theoretical Physics, No Table D continued from previous page k μ ν d kμν A kμ continue on next page

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18 6 Electronic Journal of Theoretical Physics, No Table D continued from previous page k μ ν d kμν A kμ