arXiv:2108.07219v1 [math-ph] 16 Aug 2021 oteGell-Mann the to the of generators The t eeaos h pnoeaos are matrices operators, Pauli spin the generators, Its omnyue n npyisi ae nageneraliza- of a of [3] on matrices matrices based Pauli most is the the of physics but tion in algebra, one an used of commonly generators manners different the are obtaining be- There for interaction [3]. strong gluons and the quarks tween of as symmetry chromodynamics approximate quantum in an used are matrices These swl as well as rse sflos[,5.Teeaeattlof total a are There 5]. [4, follows as pressed and ymti matrices symmetric lryi h tnadMdlo atcepyis[,2]. [1, physics particle of partic- Model physics, Standard The fundamental the in in used ularly widely are groups λ λ h eeaoso the of generators The The 1 4 S ˆ σ D = = N su 1 n −     = 2 i ler sue ecietespin- the describe used is algebra Lie (2) = su ignlmtie s-aldCra generators) Cartan (so-called matrices diagonal 1 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 λ  ( p 7 N N eea omlso h tutr osat nthe in Constants Structure the of Formulas General 0 1 1 0 = 2 ( S i ler n hi orsodn Lie corresponding their and algebra Lie ) tnsi the in stants sdfraayia n opttoa neeti Physics. in hope interest We computational generators. and the analytical of for expression the used formulas of need closed thos any involving The without expressions elements. simple non-null are constants its structure of indexes the to ˆ N n A S    ˆ     ( su epoieteaayi xrsin ftettlysymmetri totally the of expressions analytic the provide We ~ nm S n − 0 0 0 0 0 0 nm σ , 3,wihi htw sdi hswork. this in used we what is which (3), λ , λ , − 1) λ = i 2 1) = 1 2 5 / arcs[]as [3] matrices eateto hmsr,Uiest fRcetr Rochest Rochester, of University Chemistry, of Department − − nismercmatrices anti-symmetric 2 = = = 0  su ~ 2 i i 2  n k   ~ X 2 nttt fOtc,Uiest fRcetr ohse,N Rochester, Rochester, of University Optics, of Institute     su 3 i ler r proportional are algebra Lie (3) =1 − | 0 m i su 0 0 0 0 − λ , 1 i | ( 0 0 0 0 0 m N | ih − i ( k 0 − su N ih i ler.Tedrvto sbsdo eainlnigt linking relation a on based is derivation The algebra. Lie ) 8 n 0 0 ih i 0 0 i  | n i ler a eex- be can algebra Lie ) = 2 n fteGell-Mann the of and (2) k + | − | | σ , − 0 (1 + √   | i 1 n   S 3 3 ˆ λ , ih n j ucnBossion Duncan =   λ , ih m S = − ˆ 3 0 0 0 1 0 0 0 1 m j  |
6 n = = ~ 2 | 0 0 1 ,
) = λ | ,   j n ~ 2 − − N with ,   σ ih 0 0 0 0 0 0 1 1 2 j ( 0 1 0 1 0 0 0 0 0 n  2 1   N − ihthe with | system.  . 0 1 . − , 1)     (1) (2) (3) (4) (5) 1, / 2 ∗ n ege Huo Pengfei and tutr osat,weetettlyanti-symmetric totally the the constant, to structure where anti- rise give constants, and relations commutation structure These their relations. govern commutation what [15], and qudit erators the describing for a used of widely state is the 14] [5, bra nee.Tettlysmercsrcueconstant, two structure any symmetric of totally exchange The the indexes. under anti-symmetric is which ainrelation tation in[2.I unu optn,the computing, quantum Hamilto- In classical-like a [12]. between and nian mapping Hamiltonian general multi-state as proposed a also is algebra Lie ytm novn o-daaiiy[–3.The [9–13]. non-adiabaticity quantum used as involving open is of [8], systems analogy dynamics physics spin electronic-nuclear 10], the optical map [9, and to chemistry physical atomic in Georgi a In as by well of GUT [7]. of search Glashow version the possible and simplest in the [6]; as (GUT), posed model Theory quark the Unification the example, Grand For in algebra physics. particle Lie in used monly h nee S indexes The sdtrie hog h nicmuainrelation anti-commutation the through determined is a arcs epciey h xlctrlto between operator relation projection explicit diago- a The and respectively. anti-symmetric, matrices, symmetric, nal the to sponding oec te,oecndrv h elkonrsls[16] results well-known the derive orthogonal can as and one well traceless other, as are each 7, in- generators Eq. to two and the 6 that any Eq. fact of in the properties exchange the the Using under dexes. symmetric is which on nRf 5.Nt htteegnrtr r traceless are generators these that Tr[ Note [5]. Ref. in found f h rca uniist en nagbaaetegen- the are algebra an define to quantities crucial The hs ihrdimensional higher These S ijk ˆ nee n a eaayial evaluated analytically be can and indexes e i ,a ela rhnra Tr[ orthonormal as well as 0, = ] = bandt opt h auso the of values the compute to obtained − n nismercsrcuecon- structure anti-symmetric and c htteeepesoscnb widely be can expressions these that i { ~ S 2 ˆ 3 i nm d Tr , ,2, 1, saeqatmsystem. quantum -state S ˆ [  r e ok 14627 York, New er, S A , j ˆ [ } i S † wYr,14627 York, ew ˆ , = i nm S f , ˆ ijk j S su ˆ = ] ~ N | j eidxo generator a of index he m n D and 2 ] sdfie hog h commu- the through defined is , ( S δ ˆ ih N k ij i ~  n I ˆ ) ; | N su k X + n d i Algebra Lie n h eeaoscnbe can generators the and 2 =1 ijk − ( niaegnrtr corre- generators indicate ~ N 1 N f i ler r com- are algebra Lie ) k X = ijk 2 =1 − su ~ 1 2 S ˆ S 3 ˆ 5 a enpro- been has (5) d k i Tr ijk , S ˆ su j  S = ] { ( ˆ k N S ˆ , i i alge- Lie ) , ~ 2 S 2 ˆ j δ } ij su S ˆ . su k d (  ijk N (6) (6) (8) (7) . ) , 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. 14). The remaining totally anti-symmetric struc- each generator, and hence obtain the non-zero analytic ture constants are computed through the commutator ˆ ˆ expression of fSnmSn′m′ k, which are summarized in the between two diagonal generators, which is [ Dn , Dn′ ]= 2 S S first line of Eq. 14. ~ N −1 ˆ i fDnD ′ k k = 0 (see proof in Supplemental Ma- k=1 n S For a symmetric and an anti-symmetric generator, the terial, Sec II), indicating a zero value for all fDnD ′ k. P n 3

This was a known fact, as the diagonal matrices are gen- Between a symmetric and an anti-symmetric genera- erators of the Cartan subalgebra of su(N), and they com- tor, the anti-commutation relation reads mute by definition [2]. To summarize, all of the non-zero totally anti- symmetric structure constants are expressed as follows N 2−1 1 ˆ ˆ ~ ˆ f nm = f nm = f nm = , (14) Snm , An′m′ = dSnmAn′m′ k k (17) S SknAkm S SnkAkm S SkmAkn {S S } S 2 k=1 X 1 ~2 fAnmAkmAkn = , ′ ′ ′ ′ 2 = i δnm′ ( n m m n )+ δnn′ ( m m m m ) 4 | ih | − | ih | | ih | − | ih | m 1 n h ′ ′ ′ ′ fSnmAnmDm = − , fSnmAnmDn = , + δmm′ ( n n n n )+ δmn′ ( n m m n ) −r 2m 2(n 1) | ih | − | ih | | ih | − | ih | r − ~ i ′ ˆ ′ ˆ ˆ 1 = δnm An′m + δnn ( Amm′ Am′m ) fSnmAnmDk = , m < k < n. 2 S S − S s 2k(k 1) h − + δmm′ ( ˆA ′ ˆA ′ )+ δmn′ ˆA ′ , S n n − S nn S nm One of the interesting usages of these expressions is the i construction of an adjoint representation of the su(N) Lie algebra, which is a defining representation of SO(N 2 1), 2 2 − whose generators ˆi are (N 1) (N 1) matrices. The from which one can extract d nm ′ ′ . Note that based T − × − S An m k (jk)-th matrix element of ˆi is [ ˆi]jk = ifijk. Thus, ˆi on Eq. 17, there is no diagonal component ˆD, thus all are anti-symmetric, non-diagonalT T and traceless,− and weT S the dSAD = 0. The anti-commutator between symmetric provide analytic expressions to obtain them through the and diagonal generators is not necessary as we already totally anti-symmetric structure constants expressions. obtained the structure constants involving those genera- Totally Symmetric Structure Constants dijk . tors by permutation [20]. The anti-commutation relation between two symmetric generators is Computing the anti-commutators between two anti- 2 symmetric generators gives ~2 N −1 ˆ ˆ ˆ ~ ˆ Snm , S ′ ′ = δSnmS ′ ′ + dSnmS ′ ′ k k {S S n m } N n m I n m S k=1 X ~2 ′ ′ ′ ′ ′ ′ = δnm ( m n + n m )+ δnn ( m m + m m ) 2 4 | ih | | ih | | ih | | ih | ~2 N −1 ˆ ˆ ˆ ~ ˆ h ′ ′ ′ ′ Anm , An′m′ = δAnmAn′m′ + dAnmAn′m′ k k + δmm′ ( n n + n n )+ δmn′ ( n m + m n ) {S S } N I S k=1 | ih | | ih | | ih | | ih | X ~ i ~2 ′ ˆ ′ ˆ ˆ ′ ′ ′ ′ = δnm Sn′m + δnn ( Sm′m + Smm′ ) = δnn′ ( m m + m m ) δnm′ ( m n + n m ) 2 S S S 4 | ih | | ih | − | ih | | ih | h + δmm′ ( ˆS ′ + ˆS ′ )+ δmn′ ˆS ′ h ′ ′ ′ ′ n n nn nm + δmm′ ( n n + n n ) δmn′ ( n m + m n ) S ~ S S | ih | | ih | − | ih | | ih | + δnn′ δmm′ (2 m m +2 n n ) . (15) ~ i ′ ˆ ˆ ′ ˆ 2 | ih | | ih | = δnn ( Sm′m + Smm′ ) δnm Sn′m i 2 S S − S h ′ ˆ ˆ ′ ˆ We know that the last line of Eq. 15 only involves diago- + δmm ( Sn′n + Snn′ ) δmn Snm′ nal matrices. In fact, we can prove that (see Supplemen- S ~ S − S + δnn′ δmm′ (2 m m +2 n n ) , (18) tal Material, Sec III) 2 | ih | | ih | i ~2 ~2 ( m m + n n )= ˆ (16) 2 | ih | | ih | N I N ~ 2 ˆ 2 n ˆ where we recognize that the last line of Eq. 18 is identical + Dk + − Dn k(k 1)S S to the last line of Eq. 15, which can be expressed as k>n s 2n(n 1)  X − − generators in Eq. 18. We do not need to compute the n−1 p 1 ˆ m 1 ˆ anti-commutator between an asymmetric and a diagonal + Dk − Dm . 2k(k 1)S − 2m S generator as we already have the result by permutation k=m+1 s − r X  from Eq. 18 (and Eqs. 17 indicates dSAD = 0).

Thus, we can extract all the non-zero dSnmSn′m′ k, with the expressions summarized in Eq. 21. The remaining dijk values are obtained through the 4 anti-commutator between two diagonal generators Mapping Hamiltonian using the su(N) Lie Al-

N 2−1 gebra. The SU(2) representation of the Lie algebra ~2 1 ˆ ˆ ˆ ~ ˆ (spin- analogy) is often used in quantum dynamics to Dn , D ′ = δDnD ′ + dDnD ′ k k (19) 2 {S S n } N n I n S k=1 study systems with two states [9, 10, 13]. For a two X 1 n−1 level system with the Hamiltonian Hˆ = H0 ˆ + H Sˆ = ~2 I ~ · ′ ′ ′ ˆ 1 ˆ ˆ 1 ˆ = δkk ( k k + k k ) H0 + ~ (Hk Sk)= H0 + ~ [2 (V12) Sx +2 (V12) ′ ′ | ih | | ih | I k · I R · I · 2n(n 1)2n (n 1) k=1 ˆ ˆ − − h Sy +(V11 V22) Sz ], it can be shown (through the Heisen- ′ ′ X′ P− · +pδkn′ (1 n )( k n + n k ) berg equations of motion (EOMs)) that − | ih ′ | | ′ ih | + δnk′ (1 n)( n k + k n ) 3 − | ih | | ih | d 1 ′ ′ ′ i = εijkHj k, (22) + δnn′ (1 n)(1 n )( n n + n n ) dtS ~ S − − | ih | | ih | j,k ~ ~ i X ′ ˆ ′ ˆ = 2δkn Dn′ + 2δnk Dn where i = Tr[ˆρ ˆi] is the expectation value of ˆi,ρ ˆ be- 2n(n 1) S 2n′(n′ 1) S S S S − − ing the density operator, and εijk the Levi-Civita ten- ~2 n−1 p p 2 sor which is the two-dimension totally anti-symmetric + δnn′ ( k k + (1 n) n n ). n(n 1) | ih | − | ih | structure constant fijk. This is equivalent to the preces- − k=1 1 X sion of spin of a spin- 2 system around a magnetic field One can see that only diagonal matrices are involved in M = H, which is a well-known result (eg, Page 424 of the last line of Eq. 19 and there is no off-diagonal element. Ref. [21]). Those EOMs exactly obey the time-dependent We can prove that (see Supplemental Material, Sec IV) 2 Schr¨odinger equation (TDSE),c ˙i = i Vij cj . More − j=1 ~2 n−1 ~2 specifically, with an arbitrary state defined as Ψ = 2 ˆ P | ∗i ( k k + (1 n) n n )= (20) c1 1 +c2 2 , by using the transformation Sx = Re c1c2 , n(n 1) | ih | − | ih | N I | i | i 1 { } k=1 S = Im c∗c and S = ( c 2 c 2), one can show − X y 1 2 z 2 1 2 N that Eq.{ 22 is} equivalent to TDSE.| | − | | ~ 2 ˆ 2 ˆ + Dk + (2 n) Dn , For a system with N states, one can use the su(N) k(k 1)S − n(n 1)S k>n s s X − − Lie algebra for the spin analogy [8, 9, 12], describing
su which helps to determine all dDnDn′ k. the precession of the N-states system [22]. This (N) We summarize all the non-zero totally symmetric analogy is based on a reformulation of the Hamiltonian structure constants as follows Hˆ = H0 ˆ + Vˆe(Rˆ) with the generators of su(N) as fol- I 1 lows [8, 12] dSnmSknSkm = dSnmAknAkm = dSnmAmkAnk = , (21) 2 N 2−1 1 1 Hˆ = 0 ˆ + k ˆk, (23) dSnmAnkAkm = , H I ~ H S −2 k=1 X m 1 1 ˆ ˆ dSnmSnmDm = dAnmAnmDm = − , where k = ~ Tr[H k]. Similarly, for the density ma- − 2m H S 2 r 1 ˆ 1 N −1 ˆ ˆ trix [8]ρ ˆ = N + 2 k=1 k k where k = Tr[ˆρ k]. 1 suI S S S ˆ S dSnmSnmDk = dAnmAnmDk = , m < k < n, Plugging the (N) generator expression of H andρ ˆ into 2k(k 1) P s − the quantum Liouville equation i~∂ρ/∂tˆ = [H,ˆ ρˆ], one ar- 2 n rives at the following equation [23] which can be viewed dSnmSnmDn = dAnmAnmDn = − , 2n(n 1) as the generalization of the spin precession [8] − N 2−1 p 2 d 1 d nm nm = d nm nm = , n < k, S S Dk A A Dk i = fijkHj k. (24) sk(k 1) dtS ~ S − j,k=1 X 2 dDnDkDk = , k < n, where fijk is the totally anti-symmetric structure con- sn(n 1) stant of su(N). For an arbitrary state defined as Ψ = − N | i ck k , using the transformation 2 k=1 | i dDnDnDn = (2 n) . ∗ ∗ − sn(n 1) P SSnm = Re cmcn , SAnm = Im cmcn , (25) − { } { } n−1 In a recent work on deriving the quantum Liouvillian of 1 2 n 1 2 su SDn = ck − cn , (26) coupled electronic-nuclear DOFs based upon the (N) | | − 2n | | k=1 2n(n 1) r representation (see Appendix F in Ref. [12]), these dijk X − are explicitly present in the equation of motion. Hav- and the analyticp expressions of the fijk, one can show ing the above analytic expressions will facilitate future that Eq. 24 is equivalent to the TDSE . Hence Eq. 24 theoretical developments. now has a closed formula. 5

∗ Conclusion. In this letter, we provide the analytic spin analogy and application to F +H2 → F+H2, expressions of the totally symmetric and totally anti- J. Chem. Phys. 71, 2156 (1979). symmetric structure constants for the su(N) Lie algebra. [10] J. E. Runeson and J. O. Richardson, Spin-mapping We hope that these expressions can be widely used for approach for nonadiabatic molecular dynamics, J. Chem. Phys. 151, 044119 (2019). analytical and computational interest in Physics, as they [11] H. Kuratsuji and T. Suzuki, Path integral in the represen- su are valid for any dimension N of (N) Lie algebra with- tation of SU(2) coherent state and classical dynamics in a out the need to explicitly compute the commutation and generalized phase space, J. Math. Phys. 21, 472 (1980). anti-commutation relations or use any generator. The [12] J. E. Runeson and J. O. Richardson, General- structure constants bear important information on the ized spin mapping for quantum-classical dynamics, algebra they belong to, and the possibility to obtain those J. Chem. Phys. 152, 084110 (2020). constants with simple relations can bring insight into the [13] D. Bossion, S. N. Chowdhury, and P. Huo, Non-adiabatic su ring polymer molecular dynamics with spin mapping high dimensional (N) Lie algebra, which might be chal- variables, J. Chem. Phys. 154, 184106 (2021). lenging otherwise. [14] J. Patera and H. Zassenhaus, The pauli matrices in n dimensions and finest gradings of simple lie algebras of type an1, J. Math. Phys. 665-673, 40001 (1988). ACKNOWLEDGMENTS [15] Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Frontiers in This work was supported by the National Science Physics 8, 479 (2020). Foundation CAREER Award under Grant No. CHE- [16] H. E. Haber, Useful relations among the generators in the defining and adjoint representations of SU(N), 1845747. SciPost Phys. Lect. Notes , 21 (2021). [17] On page 106 in Ref. [24] (Chapter 6), the author sug- gested that “In order to determine the structure con- stants of su(N), no closed formulas are known, they have to be calculated by means [Eq. 8 in the current letter] of ∗ [email protected] performing matrix multiplications.”. † [email protected] [18] More specifically, the first N 2 − 1 generators of the [1] F. Halzen and A. Martin, Quarks and Leptons: An In- su(N + 1) Lie algebra are directly adapted from those troductory Course in Modern Particle Physics (Wiley, of su(N) with adding a (N + 1)-th row and column of ze- 1984). ros. Then one alternatively introduces the symmetric and [2] H. Georgi, Lie Algebras In Particle Physics: from Isospin anti-symmetric matrices containing the elements from To Unified Theories (CRC Press, 2000). m = 1 to m = N with n = N + 1. Lastly, one adds [3] M. Gell-Mann, Symmetries of baryons and mesons, Sˆ the diagonal matrix of DN+1 based on the expression in Phys. Rev. 125, 1067 (1962). Eq. 5. This procedure is apparent from su(2) to su(3), [4] G. Kimura, The bloch vector for n-level systems, Phys. and the example from su(3) to su(4) can be found in Lett. A (2003) 314, 339 (2003). Chapter 5 of Ref. [24]. [5] R. A. Bertlmann and P. Krammer, Bloch vectors for qu- [19] See supplemental material [url] for details of the deriva- dits, J. Phys. A: Math. Theor. 41, 235303 (2008). tions of eq. 12, eq. 16, and eq. 20. [6] J. J. J. Kokkedee, The quark model, Frontiers in physics [20] Note that dSDD′ will be same as dDD′S, and dSDS′ will be (W. A. Benjamin, New York, 1969). same as dSS′D, and dSDA will be same as dSAD. [7] H. Georgi and S. L. Glashow, Unity of All Elementary- [21] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Me- Particle Forces, Phys. Rev. Lett. 32, 438 (1974). chanics, Volume 1 (Wiley, 1997). [8] F. T. Hioe and J. H. Eberly, N-Level Coherence Vector [22] Note that in particular,. and Higher Conservation Laws in Quantum Optics and [23] This means that. Quantum Mechanics, Phys. Rev. Lett. 47, 838 (1981). [24] W. Pfeifer, The Lie Algebras SU(N): An Introduction [9] H. Meyer and W. H. Miller, Classical models (Springer, 2003). for electronic degrees of freedom: Derivation via