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General Formulas of the Structure Constants in the $\Mathfrak {Su}(N

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General Formulas of the Structure Constants in the $\Mathfrak {Su}(N General Formulas of the Structure Constants in the su(N) Lie Algebra Duncan Bossion1, ∗ and Pengfei Huo1,2, † 1Department of Chemistry, University of Rochester, Rochester, New York, 14627 2Institute of Optics, University of Rochester, Rochester, New York, 14627 We provide the analytic expressions of the totally symmetric and anti-symmetric structure con- stants in the su(N) Lie algebra. The derivation is based on a relation linking the index of a generator to the indexes of its non-null elements. The closed formulas obtained to compute the values of the structure constants are simple expressions involving those indexes and can be analytically evaluated without any need of the expression of the generators. We hope that these expressions can be widely used for analytical and computational interest in Physics. The su(N) Lie algebra and their corresponding Lie The indexes Snm, Anm and Dn indicate generators corre- groups are widely used in fundamental physics, partic- sponding to the symmetric, anti-symmetric, and diago- ularly in the Standard Model of particle physics [1, 2]. nal matrices, respectively. The explicit relation between su 1 The (2) Lie algebra is used describe the spin- 2 system. a projection operator m n and the generators can be ˆ ~ found in Ref. [5]. Note| thatih these| generators are traceless Its generators, the spin operators, are j = 2 σj with the S ˆ ˆ ˆ ~2 Pauli matrices Tr[ i] = 0, as well as orthonormal Tr[ i j ]= 2 δij . TheseS higher dimensional su(N) LieS algebraS are com- 0 1 0 i 1 0 σ = , σ = , σ = . (1) monly used in particle physics. For example, the su(6) 1 1 0 2 i −0 3 0 1 − Lie algebra in the quark model [6]; in the search of a The generators of the su(3) Lie algebra are proportional Grand Unification Theory (GUT), su(5) has been pro- ˆ ~ posed as the simplest possible version of GUT by Georgi to the Gell-Mann λ matrices [3] as j = 2 λj , with S and Glashow [7]. In atomic and optical physics [8], as 0 1 0 0 i 0 1 0 0 well as in physical chemistry [9, 10], spin analogy is used − λ1 = 1 0 0 , λ2 = i 0 0 , λ3 = 0 1 0 to map the electronic-nuclear dynamics of open quantum − 0 0 0 0 0 0 0 0 0 systems involving non-adiabaticity [9–13]. The su(N) 0 0 1 0 0 i 0 0 0 Lie algebra is also proposed as general mapping between − λ = 0 0 0 , λ = 0 0 0 , λ = 0 0 1 a multi-state Hamiltonian and a classical-like Hamilto- 4 5 6 su 1 0 0 i 0 0 0 1 0 nian [12]. In quantum computing, the (N) Lie alge- − bra [5, 14] is widely used for describing the qudit [15], 00 0 10 0 1 the state of a d-state quantum system. λ = 0 0 i , λ = 01 0 . (2) 7 8 √ The crucial quantities to define an algebra are the gen- 0 i −0 3 0 0 2 − erators and what govern their commutation and anti- These matrices are used in quantum chromodynamics as commutation relations. These relations give rise to the an approximate symmetry of the strong interaction be- structure constants, where the totally anti-symmetric tween quarks and gluons [3]. There are different manners structure constant, fijk, is defined through the commu- for obtaining the generators of an algebra, but the most tation relation commonly used one in physics is based on a generaliza- N 2−1 tion of the Pauli matrices of su(2) and of the Gell-Mann [ ˆi, ˆj ]= i~ fijk ˆk, (6) arXiv:2108.07219v1 [math-ph] 16 Aug 2021 S S S matrices [3] of su(3), which is what we used in this work. k=1 su X The generators of the (N) Lie algebra can be ex- which is anti-symmetric under the exchange of any two pressed as follows [4, 5]. There are a total of N(N 1)/2 − indexes. The totally symmetric structure constant, dijk, symmetric matrices is determined through the anti-commutation relation ~ 2 ˆ 2 N −1 Snm = m n + n m , (3) ~ S 2 | ih | | ih | ˆi, ˆj = δij ˆ + ~ dijk ˆk, (7) {S S } N I S as well as N(N 1)/2 anti-symmetric matrices k=1 − X ~ which is symmetric under the exchange of any two in- ˆ Anm = i m n n m , (4) dexes. Using the properties in Eq. 6 and Eq. 7, as well as S − 2 | ih | − | ih | the fact that the generators are traceless and orthogonal and N 1 diagonal matrices (so-called Cartan generators) to each other, one can derive the well-known results [16] − ~ n−1 2 ˆ ˆ ˆ 2 ˆ ˆ ˆ ˆ fijk = i 3 Tr [ i, j ] k ; dijk = 3 Tr i, j k . Dn = k k + (1 n) n n , (5) − ~ S S S ~ {S S }S S | ih | − | ih | 2n(n 1) k=1 (8) − X p 2 Using the above relation as well as general expressions commutation relation is of the generators (Eq. 3-5) one can compute the numer- 2 ical values of all the structure constants of the su(N) N −1 ˆ ˆ ~ ˆ [ Snm , A ′ ′ ]= i fSnmA ′ ′ k k (11) algebra through Eq. 8, as has been commonly done in S S n m n m S k=1 the literature. However, this requires laboriously efforts X 2 ~2 of different combinations of the N 1 generators of ′ ′ ′ ′ = i δnn′ ( m m + m m ) δnm′ ( m n + n m ) su(N), which remain numerous even− when considering 4 | ih | | ih | − | ih | | ih | h ′ ′ ′ ′ the symmetry properties of the structure constants when + δmn′ ( n m + m n ) δmm′ ( n n + n n ) N is large. Despite the extensive usage and crucial role | ih | | ih | − | ih | | ih | ~ i of these structure constants, to the best of our knowl- ′ ˆ ˆ ′ ˆ = i δnn ( Sm′m + Smm′ ) δnm Sn′m edge [17], we are not aware of any analytic expression 2 S S − S h ′ ˆ ′ ˆ ˆ (closed formulas) of f and d . + δmn Snm′ δmm ( Sn′n + Snn′ ) ijk ijk S − S S In this letter, we derive the analytic expressions of the ~ + δmm′ δnn′ (2 m m 2 n n ) . structure constants in the su(N) Lie algebra. The key 2 | ih |− | ih | results are summarized in Eq. 14 for fijk and Eq. 21 for i The first two lines of the last equality in Eq. 11 give dijk. We first determine the relation of the indexes of us directly several structure constants. The last line of the generators (see Eq. 3-5) Snm, Anm, and Dnm with the label n and m. We note that one can use recursive Eq. 11 contains diagonal elements, hence we know it will relations to obtain the generators of su(N + 1) from the be a combination of diagonal generators. In fact, we can generators of su(N) [18]. This helps to determine the prove (see Supplemental Material [19], Sec I) that indexes of the generators as ~2 i ( m m n n ) (12) 2 2 | ih | − | ih | Snm =n + 2(m n) 1, (9a) n−1 − − n ˆ m 1 2 ~ ˆ Dk ˆ Anm =n + 2(m n), (9b) = i Dn + S − Dm . − 2(n 1)S 2k(k 1) − 2m S 2 r − k>m r Dn =n 1. (9c) X − − p This helps to determine the rest of the structure constant Note that there is no overlap among the indexes as long fSnmAn′m′ k with the expressions documented in Eq. 14. as the conditions 1 m < n N holds. Hence, The commutation relations between symmetric and diag- there is a one to one≤ correspondence≤ between the value onal generators are not required as we already obtained i Snm, Anm, Dn with indexes n,m , which helps us all the non-zero structure constants involving diagonal to∈{ identify the type} of generator{ it labels.} This is the and symmetric generators (as we know that we cannot first key step to determine closed formulas of structure obtain a diagonal matrix through the commutator of a constants. symmetric and a diagonal generator). The Totally Anti-symmetric Structure Con- Between two anti-symmetric generators, the commu- stants fijk. The commutation relation between two tation relation is symmetric generators is N 2−1 ˆ ˆ ~ ˆ [ Anm , An′m′ ]= i fAnmAn′m′ k k (13) N 2−1 S S S k=1 ˆ ˆ ~ ˆ X [ Snm , Sn′m′ ]= i fSnmSn′m′ k k (10) ~2 S S S ′ ′ ′ ′ k=1 = δ ′ ( n m m n )+ δ ′ ( m m m m ) X nm nn ~2 4 | ih | − | ih | | ih | − | ih | ′ ′ ′ ′ ′ ′ h ′ ′ ′ ′ = δnm ( m n n m )+ δnn ( m m m m ) + δmm′ ( n n n n )+ δmn′ ( m n n m ) 4 | ih | − | ih | | ih | − | ih | | ih | − | ih | | ih | − | ih | h ′ ′ ′ ′ ~ i + δmm′ ( n n n n )+ δmn′ ( n m m n ) = i δnm′ ˆA ′ + δnn′ ( ˆA ′ ˆA ′ ) | ih | − | ih | | ih | − | ih | 2 − S n m S m m − S mm ~ i ′ ˆ ′ ˆ ˆ h = i δnm An′m + δnn ( Am′m Amm′ ) + δmm′ ( ˆA ′ ˆA ′ )+ δmn′ ˆA ′ , 2 S S − S S n n − S nn S nm h i + δmm′ ( ˆA ′ ˆA ′ ) δmn′ ˆA ′ . S n n − S nn − S nm which helps to determine the structure constants in- i volving all anti-symmetric generators (second line of With constraints from δij , we can identify the index of Eq.
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