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Kronecker Commutation Matrices and Particle Physics Kronecker Commutation Matrices and Particle Physics Christian Rakotonirina∗ Civil engineering Department, Institut Sup´erieurde Technologie d'Antananarivo, IST-T Laboratoire de la Dynamique de l'Atmosph`ere,du Climat et des Oc´eans,DYACO, University of Antananarivo Madagascar December 5, 2016 Abstract In this paper, formulas giving a Kronecker commutation matrices (KCMs) in terms of some matrices of particles physics and formulas giving electric charge operators (ECOs) for fundamental fermions in terms of KCMs have been reviewed. Physical meaning have been given to the eigenvalues and eigenvectors of a KCM. Introduction The Kronecker or tensor commutation matrices (KCMs) are matrices which commute Kronecker or tensor product of matrices. So, we can think of using them where Kronecker product is used. Kronecker product is used in many branches of physics and mathematics: in quantum information theory, op- tics, matrix equations and algebraic Bethe ansatz. One can remark that a wave function of two identical fermions is eigenfunc- tion of a KCM associated to the eigenvalue -1 and a wave function of two identical bosons is eigenfunction of a KCM associated to the eigenvalue -1. So, it is natural to think to what are the meaning we can give to these eigenvalues, their multiplicities and the eigenvectors associated, in particle physics. The KCMs have already relations with some matrices of the particle physics and we can construct an electric charge operator (OCE) for fundamental fermions by using a KCM. After the first section which will speak about kronecker product and the mathematical definition of a KCM, in the second ∗[email protected] 1 section we will review the expression of the KCMs in terms of some matrices of particle physics, after that in the third section we will review the ECOs built with the KCMs. Finally, in the fourth section we will give the eigenval- ues of KCMs with their multiplicities and the eigenvectors associated, and some examples giving their meaning in particle physics. We will take as system of units the natural units ~ = c = 1 and as unit of charge the charge of an electron e. 1 Kronecker Commutation Matrices i m×n The Kronecker product of a matrix A = (Aj) 2 C by other matrix B : 0 1 1 1 1 0 1 1 1 1 A1 :::Aj :::An A1B :::Aj B :::AnB B . C B . C B . C B . C B C B C A⊗B = B Ai :::Ai :::Ai C⊗B = B Ai B :::Ai B :::Ai BC B 1 j n C B 1 j n C B . C B . C @ . A @ . A m m m m m m A1 :::Aj :::An A1 B :::Aj B :::An B is not commutative. That is A ⊗ B 6= B ⊗ A K the Kronecker commutation matrix (KCM) K (a ⊗ b) = b ⊗ a with a, b are unicolumn matrices. The KCM K2⊗2 commutes two row and unicolumn matrices a1 b1 b1 a1 K2⊗2 ⊗ = ⊗ a2 b2 b2 a2 0 1 0 1 0 1 0 1 a1 b1 b1 a1 K3⊗3 @a2A ⊗ @b2A = @b2A ⊗ @a2A a3 b3 b3 a3 and so on. 01 0 0 0 0 0 0 0 01 B0 0 0 1 0 0 0 0 0C B C 0 1 B0 0 0 0 0 0 1 0 0C 1 0 0 0 B C B0 1 0 0 0 0 0 0 0C B0 0 1 0C B C K2⊗2 = B C ;K3⊗3 = B0 0 0 0 1 0 0 0 0C @0 1 0 0A B C B0 0 0 0 0 0 0 1 0C 0 0 0 1 B C B0 0 1 0 0 0 0 0 0C B C @0 0 0 0 0 1 0 0 0A 0 0 0 0 0 0 0 0 1 2 2 KCMs and Matrices of Particle Physics The KCMs K2⊗2 and K3⊗3 can be expressed respectively in terms of the Pauli matrices (See, for example, [1]) and the Gell-Mann matrices [2] by the following ways 3 1 1 X K = I ⊗ I + σ ⊗ σ 2⊗2 2 2 2 2 i i i=1 1 0 where I = the 2 × 2 unit matrix and 2 0 1 0 1 0 −i 1 0 σ = , σ = , σ = 1 1 0 2 i 0 3 0 −1 the Pauli matrices 8 1 1 X K = I ⊗ I + λ ⊗ λ 3⊗3 3 3 3 2 i i i=1 where 0 0 1 0 1 0 0 −i 0 1 0 1 0 0 1 λ1 = @ 1 0 0 A , λ2 = @ i 0 0 A , λ3 = @ 0 −1 0 A, 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 −i 1 0 0 0 0 1 0 0 0 0 1 λ4 = @ 0 0 0 A, λ5 = @ 0 0 0 A, λ6 = @ 0 0 1 A, λ7 = @ 0 0 −i A 1 0 0 i 0 0 0 1 0 0 i 0 0 1 0 0 1 λ = p1 0 1 0 8 3 @ A 0 0 −2 are the Gell-Mann matrices or 3 × 3 Gell-Mann matrices. A variant of this last formula with other definition of the Gell-Mann matrices [3] have recently application in optics. We say variant because this paper has expressed the KCM K3⊗3 in terms of other Gell-Mann matrices, with its definition of the Gell-Mann matrices. As a generalization [4], n2−1 1 1 X K = I ⊗ I + Λ ⊗ Λ n⊗n n n n 2 i i i=1 where Λi's are the n × n Gell-Mann matrices. 3 KCMs and Charges of Fundamental Fermions In this section we are going to construct ECOs for fundamental fermions in using KCMs [5]. An ECO we are going to construct here can have charges 3 of leptons and quarks, together as eigenvalues. Let us recall at first that fundamental fermions have the quantum numbers J3, the isospin and Y , the hypercharge. The electric charge Q of a fermion is given by the Gell-Mann-Nishijima formula Y Q = J + (1) 3 2 For the fermions of the standard model (SM) these quantum numbers are given in the following table [6] QJ3 Y Neutral Leptons νeL, νµL, ντL 0 1=2 −1 Charged Leptons eL, µL, τL −1 −1=2 −1 eR, µR, τR −1 0 −2 r b g r b g r b g Quarks u, c, t uL, uL, uL, cL, cL, cL, tL, tL, tL 2=3 1=2 1=3 r b g r b g r b g uR, uR, uR, cR, cR, cR, tR, tR, tR 2=3 0 4=3 r b g r b g r b g Quarks d, s, b dL, dL, dL, sL, sL, sL, bL, bL, bL −1=3 −1=2 1=3 r b g r b g r b g dR, dR, dR, sR, sR, sR, bR, bR, bR −1=3 0 −2=3 A matrix relation of the Gell-Mann-Nishijima for eight leptons and quarks r b g r b of the SM of the same generation, for example eL, νeL, uL, uL, uL, dL, dL, g et dL has been proposed in [7]. 3 ! 1 1 X Q = σ0 ⊗ σ0 ⊗ σ3 + σi ⊗ σi ⊗ σ0 (2) 2 | {z } 6 Y i=1 | {z } J3 The same ECO can be obtained from a relation between the following ECO 0 0 2=3 0 of leptons Q = and ECO of quarks Q = L 0 −1 Q 0 −1=3 2 Q − Q = I Q L 3 2 This leads to the following ECO (2), but in terms of the KCM K2⊗2. 1 Q = σ ⊗ σ ⊗ Q + (K + σ ⊗ σ ) ⊗ σ 0 0 L 3 2⊗2 0 0 0 or 1 Q = σ ⊗ σ ⊗ Q + (K − σ ⊗ σ ) ⊗ σ 0 0 Q 3 2⊗2 0 0 0 We can remark in these formulas that the eigenvalue +1 of the KCM K2⊗2 is associated to the charges of leptons whereas the eigenvalue -1 is associated to the charges of quarks. 4 For including more than eight fundamental fermions, let us construct an ECO in terms of the KCM K3⊗3. For doing so, let us take the following 00 0 0 1 ECOs respectively of leptons and quarks, QL = @0 −1 0 A whose diag- 0 0 −1 onal is formed by the electric charge of a neutrino and the electric charges 02=3 0 0 1 of two charged leptons and QQ = @ 0 −1=3 0 A whose diagonal is 0 0 −1=3 formed by the electric charge of a quark u (a quark c (charm) or a quark t (top)) and the electric charges of quark d, quark s (strange) or quark b (bottom). 2 Q − Q = I (3) Q L 3 3 Then we have an ECO for some fermions of the SM 1 Q = I ⊗ I ⊗ Q + (K + I ⊗ I ) ⊗ I 3 3 L 3 3⊗3 3 3 3 or 1 Q = I ⊗ I ⊗ Q + (K − I ⊗ I ) ⊗ I 3 3 Q 3 3⊗3 3 3 3 4 Eigenvalues of KCMs in Particle physics The following relations give the eigenvalues of the KCMs with their multi- plicities. K2⊗2 ≡ diag(−1; +1; +1; +1) | {z } 3 times K3⊗3 ≡ diag(−1; −1; −1; +1; +1; +1; +1; +1; +1) | {z } | {z } 3 times 6 times K4⊗4 ≡ diag(−1; −1; −1; −1; −1; −1; +1; +1; +1; +1; +1; +1; +1; +1; +1; +1) | {z } | {z } 6 times 10 times and so on.
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