GGeometric invariant decomposition of SU(3)
Martin Roelfs a ∗† a KU Leuven Campus Kortrijk–Kulak, Department of Physics, Etienne Sabbelaan 53 bus 7657, 8500Kortrijk, Belgium
Abstract
A novel invariant decomposition of diagonalizable n × n matrices into n com-muting matrices is presented. This decomposition is subsequently used to splitthe fundamental representation of su (3) Lie algebra elements into at most threecommuting elements of u (3). As a result, the exponential of an su (3) Lie alge-bra element can be split into three commuting generalized Euler’s formulas, orconversely, a Lie group element can be factorized into at most three generalizedEuler’s formulas. After the factorization has been performed, the logarithm followsimmediately. The aim of this paper is to identify the quantities left invariant by a given SU(3) trans-formation, and to describe the role these quantities play as the generators of the trans-formation. Consider a traceless skew-Hermitian 3 × B ; an element of the Liealgebra su (3). We will demonstrate that such a matrix can be decomposed into at mostthree commuting matrices b i ∈ u (3): B = b + b + b . (1)The b i are said to be simple because b i = λ i , where λ i ∈ R , λ i ≤
0. Defining β i := √− λ i , it is easily verified that ˆ b i := b i /β i squares to − . Therefore each b i canbe normalized to behave like an imaginary unit.Because the SU(3) element corresponding to B is U = exp [ B ], it follows from thecommutativity of the b i that the exponential can be split into the product of threegeneralized Euler’s formulas: U = e B = e b e b e b (2)= (cid:89) i =1 (cid:104) cos β i + ˆ b i sin β i (cid:105) , (3)where e b i ∈ U(3), but the product e b e b e b ∈ SU(3). As each b i is invariant under thetransformation Ub i U † , the decomposition of eq. (1) is called the invariant decomposition of B . ∗ [email protected] † The research of Martin Roelfs is supported by KU Leuven IF project C14/16/067. a r X i v : . [ m a t h - ph ] F e b he logarithm of U i = exp [ b i ] is not unique, in the same way that the complexlogarithm is not unique [1]. Following a similar strategy to complex analysis, we firstdefine a principal logarithm , for which 0 ≤ β i ≤ π . To this end, let us definec( b i ) := 12 (cid:16) U i + U † i (cid:17) = cos β i (4)s( b i ) := 12 (cid:16) U i − U † i (cid:17) = ˆ b i sin β i . (5)As 0 ≤ β i ≤ π implies 0 ≤ sin β i ≤
1, only the cosine function has to be inverted toobtain the principal logarithm:Ln U i = (cid:91) s( b i ) arccos (cid:18)
13 tr [c( b i )] (cid:19) . (6)= ˆ b i arccos(cos β i ) . (7)Because the sign information of s( b i ) is not carried by sin β i , but rather by ˆ b i , a full2 π range for the principal logarithm is maintained. Consequently, by factoring a groupelement U into U U U , a principal logarithm for U follows directly, asLn( U ) = Ln( U ) + Ln( U ) + Ln( U ) . (8)Closed forms for the exponential function of su (3) elements have been published before,see e.g. [2, 3]. However, the invariant decomposition of eq. (1) presents an intuitiveapproach, which is also easy to invert to give a closed form logarithm for SU(3) elements.Additionally, the invariant decomposition of eq. (1) gives the invariants b i a very stronggeometric interpretation, as the invariants of the transformations they generate.This paper is organized as follows. Firstly, in section 2.1 we define the invariantdecomposition of diagonalizable n × n matrices. Secondly, in section 2.2 we define theexponential function of SU(3) elements using the invariant decomposition. Thirdly, insection 2.3 we describe how the factorization U = U U U is performed. Fourthly,in section 2.4 we describe the logarithm of U ∈ SU(3). Lastly, we apply the invariantdecomposition to the Gell-Mann matrices in section 2.5.
We start by defining the invariant decomposition for 3 × n × n case follows directly. Theorem 2.1. A × diagonalizable matrix B , can be decomposed into at most threecommuting normal matrices b i , satisfying b i = λ i . Here λ i ∈ R , λ i ≤ .Proof. As the 3 × B is diagonalizable, it can be written as B = PDP − , where D = diag( α , α , α ) is a diagonal matrix whose diagonal entries are the eigenvalues α i ∈ C . Assuming a decomposition B = (cid:80) i =1 b i into commuting b i exists, it followsthat all b i would be simultaneously diagonalizable, and thus D = P − b P + P − b P + P − b P . (9)We then make the ansatz P − b P = ( α − tr [ B ]) diag(+1 , − , −
1) (10) P − b P = ( α − tr [ B ]) diag( − , +1 , − P − b P = ( α − tr [ B ]) diag( − , − , +1) , B = (cid:80) i =1 b i , b i = ( α i − tr [ B ]) , and [ b i , b j ] = 0. Therefore thesought-after decomposition has been found. When the eigenvalues α i are degenerate,this decomposition is no longer unique.We notice that a decomposition of this type exists for any diagonalizable n × n matrix B . If n ≥ P − b P = ( α − n − tr [ B ]) diag(+1 , − , . . . , − , −
1) (11) P − b P = ( α − n − tr [ B ]) diag( − , +1 , . . . , − − P − b n P = ( α n − n − tr [ B ]) diag( − , − , . . . , − , +1) . The proof follows along the same lines as that of theorem 2.1. To investigate if all theproperties of the invariant decomposition discussed in this paper are also valid for n > × B can be split into at mostthree commuting matrices b i , we remark that when B is traceless and skew-Hermitian,and thus B ∈ su (3), the values of λ i can alternatively be calculated as the roots of = (cid:0) b − λ i (cid:1)(cid:0) b − λ i (cid:1)(cid:0) b − λ i (cid:1) (12)= ⇒ − λ i + 14 tr (cid:2) B (cid:3) λ i − (cid:18)
14 tr (cid:2) B (cid:3)(cid:19) λ i + (cid:20) det( B )8 (cid:21) . (13)When all λ i are distinct, the b i are found by solving b i = (cid:20) B + 18 λ i det ( B ) (cid:21)(cid:20) + 12 λ i (cid:18) B − tr (cid:0) B (cid:1)(cid:19)(cid:21) − . (14)Thus, when all λ i are distinct, the invariant decomposition can be performed withoutperforming diagonalization, but at the cost of an inverse. Equation (14) is the matrixrepresentation of the orthogonal decomposition of bivectors, given in Clifford algebra togeometric calculus [4]. However, the invariant decomposition of theorem 2.1 can alwaysbe performed, even when the λ i are degenerate. su (3) element Since elements of SU(3) can be written as exp [ B ], with B ∈ su (3), we would likea simple and intuitive way to compute exponentials of su (3) elements. The invariantdecomposition of theorem 2.1 provides such a method, since the exponential of B ∈ su (3)follows straightforwardly after performing the decomposition of B into { b i } . Define β i := √− λ i , where λ i = tr (cid:2) b i (cid:3) . Then an SU(3) group element U = exp [ B ] becomes U = e B = e b e b e b (15)= (cid:89) i =1 [c( b i ) + s( b i )]= (cid:89) i =1 (cid:20) cos( β i ) + b i β i sin( β i ) (cid:21) , where c( b i ) and c( b i ) were previously defined in eq. (5). We can also form a family ofelements generated by the b i : U ( θ , θ , θ ) = e θ b e θ b e θ b . (16)3ach b i is an invariant of U ( θ , θ , θ ) ∈ U(3), as due to the commutativity of the b i , U ( θ , θ , θ ) b i U ( θ , θ , θ ) † = b i . (17)Therefore, B is invariant under the entire family of transformations U ( θ , θ , θ ) U ( θ , θ , θ ) BU ( θ , θ , θ ) † = B , (18)and so is any other linear combination of the b i , i.e. A ( A , A , A ) = A b + A b + A b . (19)Given the invariant decomposition of B , three parameters θ i determine the group el-ements U ( θ , θ , θ ) which leave B invariant, and { A i b i } spans the three parameterinvariant subspace of U ( θ , θ , θ ). For U ( θ , θ , θ ) to be an element of SU(3) it needsto satisfy the constraint tr [ θ b + θ b + θ b ] = 0; and for A ( A , A , A ) to be anelement of su (3) it has to satisfy tr A = 0. Therefore there are only two degrees offreedom in these scenarios. (3) element From the exponential map of section 2.2, we know that U ∈ SU(3) can be written as U U U , where U i = exp [ b i ] and b i is simple . So given U ∈ SU(3), how do we findthe U i ? Splitting U ( B ) = e B into cosine and sine, givesc( B ) := 12 (cid:2) U + U † (cid:3) (20)s( B ) := 12 (cid:2) U − U † (cid:3) . (21)Using c( B ) and s( B ), the grades of U are defined as (cid:104) U (cid:105) := c( b ) c( b ) c( b ) = + tr [c( B )] (22) (cid:104) U (cid:105) := (cid:88) i =1 s( b i ) (cid:89) j (cid:54) = i c( b j ) = s( B ) − (cid:104) U (cid:105) (23) (cid:104) U (cid:105) := (cid:88) i =1 c( b i ) (cid:89) j (cid:54) = i s( b j ) = c( B ) − (cid:104) U (cid:105) (24) (cid:104) U (cid:105) := s( b ) s( b ) s( b ) = tr [s( B )] . (25)It is important to note that (cid:104) U (cid:105) is not traceless, though it closely resembles the tracelessprojection commonly used in lattice Quantum Chromodynamics [5, 6, 7]:s( B ) (cid:12)(cid:12) traceless = s( B ) − tr [s( B )] . (26)However, as tr B = 0, it follows that in general tr (cid:104) U (cid:105) (cid:54) = 0: it contains a contributionproportional to the diagonal u (3) generator i . But because the generator i is identicalto the matrix representation of the pseudoscalar i , it is overzealous to discard theentire trace: only the part corresponding to the pseudoscalar (cid:104) U (cid:105) has to be subtractedto obtain (cid:104) U (cid:105) .An invariant decomposition of A = (cid:104) U (cid:105) + (cid:104) U (cid:105) results in complex eigenvalues α i = γ i + iδ i , where plugging only the real part γ i in eq. (10) yields the Hermitianmatrices H i = c( b i ) (cid:89) j (cid:54) = i s( b j ) , (27)4hile the imaginary part iδ i yields the skew-Hermitian matrices S i = s( b i ) (cid:89) j (cid:54) = i c( b j ) . (28)This yields a decomposition of U into its 8 invariants:s( b )c( b ) c( b ) c( b )s( b ) s( b )c( b ) c( b ) c( b ) c( b )s( b )c( b ) s( b )c( b )s( b ) s( b ) s( b ) s( b )c( b ) c( b )s( b ) s( b ) s( b )c( b )Therefore there are a number of equivalent ways to perform the factorization. We definethe matrix normalization procedure asˆ M i := M i (cid:107) M i (cid:107) , (cid:107) M i (cid:107) := (cid:114)
13 tr (cid:104) M i M † i (cid:105) . (29)Then, the equivalent ways of calculating e.g. U are, up to the normalization of eq. (29), U ∝ (cid:104) U (cid:105) + S (30) ∝ + H S − ∝ + H S − (31) ∝ + (cid:104) U (cid:105) H − . (32)The other U i are obtained in analogous fashion. In order to maintain the distinctionbetween ± U , it is recommended to calculate only e.g. U and U , after which the lastfactor follows as U = U † U † U .When (cid:104) U (cid:105) (cid:54) = 0, it is computationally most efficient to use eq. (30), as eqs. (31)and (32) come at the cost of an inverse. However, this cost becomes unavoidable when (cid:104) U (cid:105) = 0. (3) element With the factorization U = U U U of section 2.3 in hand, the principal logarithm issimply Ln U = Ln U + Ln U + Ln U , (33)where Ln U i is given by Ln U i = (cid:91) s( b i ) arccos (cid:18)
13 tr [c( b i )] (cid:19) (34)= ˆ b i arccos (cid:18)
13 Re tr [ U i ] (cid:19) . (35)The logarithm is by no means unique. The principal logarithm Ln U i is one such loga-rithm, but so is ln U i = Ln U i + 2 πk i ˆ b i , (36)where k i ∈ Z . It follows that all possible logarithms of U ∈ SU(3) are given byln U = (cid:88) i =1 Ln U i + 2 πk i ˆ b i . (37)Each of the logarithms ln U i behaves just like the complex logarithm of complex analysis[1], and so the theory of complex analysis can be brought to bear on studying theirproperties. 5 .5 Decomposition of Gell-Mann matrices The Gell-Mann matrices are the Hermitian generators of su (3), and play an importantrole in Quantum Chromodynamics [5, 8, 9]. It is therefore important to discuss howthey decompose under theorem 2.1. Consider the Gell-Mann matrix λ : λ = = + − . (38)If we define ρ ± = (cid:16) ± (cid:17) , then ρ ± is an invertible Hermitian matrix satisfying ρ ± = . By repeating this process for the other Gell-Mann matrices we find a total of15 linearly dependent Hermitian matrices, which we arrange with indices ranging from − ρ ± a = λ a ± (cid:16) (cid:17) a = 1 , , λ a ± (cid:16) (cid:17) a = 4 , λ a ± (cid:16) (cid:17) a = 6 , (cid:16) − (cid:17) a = 0 . (39)The linear combinations of ρ ± a build up the Gell-Mann matrices: λ a = (cid:40) ρ + a + ρ − a a = 1 , , . . . , √ ρ − − √ ρ +3 + √ ρ a = 8 . (40)Using the invariant decomposition, the exponentials of individual Gell-Mann matricescan easily be calculated. For λ a with a = 1 , , . . . ,
7, we have the exponential factoriza-tion e iθ λ a = e i θ ρ + a e i θ ρ − a = (cid:0) cos (cid:0) θ (cid:1) + i ρ + a sin (cid:0) θ (cid:1)(cid:1)(cid:0) cos (cid:0) θ (cid:1) + i ρ − a sin (cid:0) θ (cid:1)(cid:1) = (1 + cos θ ) − (1 − cos θ ) ρ + a ρ − a + i λ a sin θ = (cid:0) − λ a (cid:1) + λ a cos θ + i λ a sin θ. This is identical to eq. (7) of [2], as it should. The c( iθ λ a ) part of these matricestherefore has an equilibrium position (cid:0) − λ a (cid:1) = (cid:0) − ρ + a ρ − a (cid:1) when θ = ± π , about which is being rotated from at θ = 0, to − ρ + a ρ − a when θ = ± π .The exponential of λ is most easily calculated from its diagonal form, or using theinvariant decomposition and a bit more calculus, as e iθ λ a = diag (cid:18) e i θ √ , e i θ √ , e − i θ √ (cid:19) . Combined with the identity matrix there are 16 Hermitian matrices { , ρ ± a } , whichsquare to , and 16 skew-Hermitian matrices { i , i ρ ± a } , which square to − . Thismaps onto the even subalgebra of the geometric algebra G (6), which C. Doran et al. [10]proved can be used to describe SU(3). Investigating this link further will be the topicof future research. 6 Conclusion
A novel decomposition for n × n matrices was found. When applying this decompositionto B ∈ su (3), we found B could be split into three commuting matrices: B = b + b + b ,where each b i is called simple , because its square is λ i , with λ i ∈ R .As the group element U = exp [ B ] leaves each of the b i invariant under the transfor-mation Ub i U † = b i , we named this decomposition the invariant decomposition of B .We then found that the invariants b i play an important role in both the exponentials andthe logarithms of U ∈ SU(3), as they are both the geometric invariants, and generators,of U .The invariant decomposition offers an easy and intuitive way to perform computa-tions in SU(3), bringing Abelian intuitions into this non-Abelian space. The author would like to thank Prof. David Dudal, Prof. Anthony Lasenby, and StevenDe Keninck for valuable discussions about this research. Additional gratitude goes tothe insights provided by geometric algebra, and [4, 10] in particular, which were thedriving force behind this research.
References [1] M. J. Ablowitz and A. S. Fokas.
Complex Variables: Introduction and Applications .2nd ed. Cambridge Texts in Applied Mathematics. Cambridge University Press,2003. doi : .[2] T. L. Curtright and C. K. Zachos. “Elementary results for the fundamental repre-sentation of SU(3)”. In: Rept. Math. Phys.
76 (2015), pp. 401–404. doi : .[3] T. S. Van Kortryk. “Matrix exponentials, SU(N) group elements, and real poly-nomial roots”. In: Journal of Mathematical Physics doi : .[4] D. Hestenes and G. Sobczyk. Clifford algebra to geometric calculus : a unifiedlanguage for mathematics and physics . Dordrecht; Boston; Hingham, MA, U.S.A.:D. Reidel ; Distributed in the U.S.A. and Canada by Kluwer Academic Publishers,1984.[5] T. DeGrand and C. DeTar.
Lattice Methods for Quantum Chromodynamics . 2006. doi : .[6] J. E. Mandula and M. Ogilvie. “The Gluon Is Massive: A Lattice Calculationof the Gluon Propagator in the Landau Gauge”. In: Phys. Lett. B
185 (1987),pp. 127–132. doi : .[7] L. Giusti et al. “Problems on lattice gauge fixing”. In: Int. J. Mod. Phys. A doi : .[8] M. Gell-Mann. “The Eightfold Way: A Theory of strong interaction symmetry”.In: (Mar. 1961). doi : .[9] M. E. Peskin and D. V. Schroeder. An Introduction To Quantum Field Theory .Frontiers in Physics. Avalon Publishing, 1995. isbn : 9780813345437.[10] C. Doran et al. “Lie groups as spin groups”. In:
Journal of Mathematical Physics doi :10.1063/1.530050