Solving the matrix exponential function for the groups SU(3), SU(4) and Sp(2)

The well known analytical formula for $SU(2)$ matrices $U = \exp(i \vec \tau \!\cdot\! \vec \varphi\,) = \cos|\vec \varphi\,| + i\vec \tau \!\cdot\! \hat\varphi \, \sin|\vec \varphi\,|$\\ is extended to the $SU(3)$ group with eight real parameters. The resulting analytical formula involves the sum over three real roots of a cubic equation, corresponding to the so-called irreducible case, where one has to employ the trisection of an angle. When going to the special unitary group $SU(4)$ with 15 real prameters, the analytical formula involves the sum over four real roots of a quartic equation. The associated cubic resolvent equation with three positive roots belongs again to the irreducible case. Furthermore, by imposing the pertinent condition on $SU(4)$ matrices one can also treat the symplectic group $Sp(2)$ with ten real parameters. Since there the roots occur as two pairs of opposite sign, this simplifies the analytical formula for $Sp(2)$ matrices considerably. An outlook to the situation with analytical formulas for $SU(5)$, $SU(6)$ and $Sp(3)$ is also given.


Introduction and summary
In chiral effective field theories for low-energy quantum chromodynamics [1] one typically works with special unitary matrices U as the field variable, since this allows for a convenient implementation of (chiral and other) symmetry transformations. The excitations on top of the spontaneously broken groundstate, represented by the unit matrix 1, are pseudoscalar Goldstone bosons (pions, kaons and the η(548)-meson). In the two-flavor case the special unitary 2 × 2 matrices forming the compact Lie group SU(2) are often given in the exponential form U = exp(i τ · ϕ ) = cos | ϕ |1 + i τ ·φ sin | ϕ | , with τ = (τ 1 , τ 2 , τ 3 ) the Pauli matrices and | ϕ | the magnitude of a three-component real vector ϕ = (ϕ 1 , ϕ 2 , ϕ 3 ) related e.g. to the pion-fields. Since the manifold of SU(2) is identified with a three-sphere S 3 , one can provide alternative algebraic or rational parametrizations which are advantageous in calculations or for certain applications. In the three-flavor case, where the matrices U belong to the eight-dimensional compact Lie group SU (3), no such alternatives to parametrize the manifold are known and one has to stay with the exponential form U = exp(i λ· v ) in terms of eight Gell-Mann matrices λ = (λ 1 , . . . , λ 8 ) and eight real parameters v = (v 1 , . . . , v 8 ). The aim of the present work is solve the corresponding matrix exponential function. A key ingredient to limit the number of matrix powers to few independent ones is the Cayley-Hamilton relation, which states that any matrix Σ gets nullified when inserted into its characteristic polynomial P (z) = det(z1 − Σ). The coefficients [2] of the latter are given by the traces of increasing matrix powers of Σ and ultimately the determinant of Σ. When carrying out this procedure for SU(3), one encounters the problem of determining the roots of a cubic polynomial in the so-called irreducible case. It corresponds to the situation when all three roots are real and the Cardano formula exhibits under the cube-root a squareroot with a negative radicand. Then the problem gets effectively solved through a trigonometric ansatz and the trisection of an angle. When continuing the solution of the matrix exponential function to SU(4), the four real roots of a quartic polynomial are determined with the help of a cubic resolvent polynomial that also belongs to the irreducible case. As an interesting byproduct of this analysis one obtains the shape of the allowed region for certain invariants η and ζ. Moreover, be imposing the condition related to a quaternionic structure one can treat as a subgroup of SU(4) the symplectic group Sp(2) with ten real parameters. The situation with more elaborate quasi-analytical formulas for SU(5), SU(6) and Sp(3) matrices in the exponential parametrization is discussed in perspective.

Special unitary group SU(3)
One starts with the usual exponential representation of an SU(3) matrix in terms of the eight Gell-Mann matrices 1 λ = (λ 1 , . . . , λ 8 ) normalized to tr(λ a λ b ) = 2δ ab and an eightcomponent real parameter vector v = (v 1 , . . . , v 8 ). The aim is to give an analytical expression for U that involves the 3×3 unit matrix 1 and the rescaled matrix Σ = λ·v (wherev = v/| v |), each multiplied with coefficient functions that depend on the magnitude | v | = v 2 1 + · · · + v 2 8 and another invariant. Considering the traces tr Σ = 0 and tr Σ 2 = 2 one finds as a pertinent invariant the (real-valued) determinant η = det Σ. According to the Cayley-Hamilton relation the traceless hermitian matrix Σ (a Lie algebra element) gets nullified when inserted into its characteristic polynomial, which lead to the cubic relation Consequently, any power of Σ can be written as a linear combination of 1, Σ, and Σ 2 . Starting at order n with Σ n = α n 1 + β n Σ + γ n Σ 2 and multiplying with Σ one obtains via the relation in eq.(4) the expansion coefficients at order n + 1. The resulting linear recursion relation reads in vector notation: and the initial values α 0 = 1, β 0 = 0, γ 0 = 0. Through diagonalization of M 3 the exponential series exp(i| v |M 3 ) = ∞ n=0 (i| v |M 3 ) n /n! can be solved 2 and by dotting with (1, 0, 0) t from the right and (1, Σ, Σ 2 ) from the left, one ends up with the following analytical formula for an SU(3) matrix where the η-dependent quantities z 1 , z 2 , z 3 subject to the zero-sum constraint z 1 + z 2 + z 3 = 0 are the three real roots of the cubic equation Note that at the same time z 1 , z 2 , z 3 are the eigenvalues of the traceless hermitian matrix Σ = λ·v. One also observes that the trace of U is given by a much simpler formula: tr U = 3 j=1 exp(iz j | v |). Another interesting feature is that the denominator in eq.(6) is the derivative P ′ 3 (z) = 3z 2 − 1 evaluated at the root z j . Before turning to the solution of the cubic equation, one should analyze the generic behavior of the polynomial P 3 (z) = z 3 − z − η as it is sketched in Fig. 1. When having three zero-crossings, the local maximum at z = −1/ √ 3 must be positive, 2 √ 3/9 − η ≥ 0, and the local minimum at z = 1/ √ 3 must be negative, −2 √ 3/9 − η ≤ 0. Both conditions together fix the range of the determinant η to the interval In the irreducible case at present, the cubic equation 3 cos ψ which leads (via the relation cos 3ψ = 4 cos 3 ψ − 3 cos ψ) to the auxiliary equation that provides an angle ψ in the range 0 ≤ ψ ≤ π/3. It is an important solvability criterion that the argument in eq.(9) has indeed magnitude less equal to 1 as a consequence of the determinantal range derived before. The three real roots entering the analytical formula in eq.(6) are given by where in this (arbitrary) ordering the sign of z 3 is yet undetermined. The validity of the quasi-analytical formula has been checked numerically in many cases. A disadvantage for applications is its rather inexplicit dependence on η which cannot be expressed in terms of real-valued algebraic functions, but requires a detour via the real roots of a cubic equation in the irreducible case. For comparison in the case of SU(2) the iteration matrix M 2 is the Pauli matrix τ 1 and exp(i|

Special unitary group SU(4)
One starts again with the exponential representation of an SU(4) matrix 1 and the rescaled matrix Σ = λ ·v that get multiplied by functions depending on the magnitude | v | = v 2 1 + · · · + v 2 15 and further invariants. Besides the constant traces tr Σ = 0 and tr Σ 2 = 2 one finds now as the two pertinent (real-valued) invariants the determinant and the trace of the cube where the factor 1/3 is included for convenience. In present case the Cayley-Hamilton relation sets up an equation for the fourth power of Σ of the form which allows to write any higher power of Σ as a linear combination of 1, Σ, Σ 2 , and Σ 3 . Starting at order n with Σ n = α n 1 + β n Σ + γ n Σ 2 + δ n Σ 3 and multiplying with Σ one obtains via the relation in eq.(13) the expansion coefficients at order n + 1. The resulting linear recursion relation reads in vector notation:  where the (η, ζ)-dependent quantities z 1 , z 2 , z 3 , z 4 subject to the zero-sum constraint z 1 +z 2 +z 3 +z 4 = 0 are now the four real roots of the quartic equation At the same time z 1 , z 2 , z 3 , z 4 are the eigenvalues of the traceless hermitian 4×4 matrix Σ = λ·v and one gets again a simpler formula for the trace: tr U = 4 j=1 exp(iz j | v |). The denominator in eq.(15) stems from the derivative P ′ 4 (z) = 4z 3 − 2z − ζ. Borrowing results from advanced algebra [3], the determination of the four roots z j of the quartic equation P 4 (z) = 0 proceeds via three auxiliary quantities θ 1 , θ 2 , θ 3 in the following way where it has to be noted that there is only a twofold sign ambiguity in taking square roots, since the sign of the product is fixed by the condition √ θ 1 √ θ 2 √ θ 3 = ζ. The four choices of signs + +, + −, − +, − − correspond merely to a relabeling of the four roots. The θ-values derive from the roots through the inverse relations and these are all positive, since each is a square in view of the zero-sum z 1 + z 2 + z 3 + z 4 = 0. As a matter of fact [3] the three θ-values are the roots of the cubic resolvent equation [3] The generic behavior of R 3 (θ) which is shown in Fig. 2 implies several restrictions on the invariants η and ζ. In the presence of three zero-crossing on the positive θ-axis and R 3 (θ < 0) < −ζ 2 , the local maximum and local minimum must lie in between at positions θ max,min = 1 3 (2 ∓ √ 1 + 12η) ≥ 0. This implies first η ≥ −1/12 and secondly η ≤ 1/4, leading to the (narrow) range −1/12 ≤ η ≤ 1/4 for the determinant. The conditions R 3 (θ max ) ≥ 0 and R 3 (θ min ) ≤ 0 multiplied with each other lead after some manipulation to the inequality The resulting allowed range for the invariants η and ζ is the bounded region shown in Fig. 3 from which one deduces also the extremal values ζ ± = ±2 √ 6/9.

Symplectic group Sp(2)
The 15-dimensional special unitary group SU(4) contains a particular 10-dimensional subgroup, the so called (compact) symplectic group Sp(2) defined by imposing the condition (of respecting a quaternionic structure) [4] U t J U = J , on U ∈ SU(4), where t stands for transposition, and 1 and 0 denote momentarily the 2×2 unit and zero matrix. For the Lie algebra elements λ· v that depend linearly on 15 parameters v 1 , . . . , v 15 this implies the constraint which as a result eliminates five of the 15 real parameters through the linear relations The squared magnitude of the yet 15-component parameter vector v becomes a sum of ten squares after introducing the new linear combinationsṽ 3 = ( √ 3v 3 − v 8 )/2 andṽ 8 = (v 3 + √ 3v 8 )/2. Moreover, one finds that the trace of the cube vanishes, tr( λ · v ) 3 = 0, just as a consequence of the five linear relations in eq.(30) or the underlying constraint in eq.(29). Thus one is dealing for the subgroup Sp(2) of SU(4) with the special case ζ = 0 and the quartic polynomial P 4 (z) = z 4 − z 2 + η becomes biquadratic, and is effectively equivalent toP 2 (x) = x 2 − x + η. The four real roots are then given by where the determinant η must be confined to the range 0 ≤ η ≤ 1/4. Since for Sp(2) the roots of P 4 (z) occur as pairs of opposite sign, the sum in eq.(15) can be simplified to while the trace of such symplectic matrices is always real-valued: tr In perspective one can consider the 21-dimensional symplectic group Sp(3) by imposing on SU(6) matrices the condition U t J U = J, with J constructed from 3 ×3 matrices 1 and 0. The condition ( λ · v ) t = J( λ · v )J for the Lie algebra elements eliminates 14 of the original 35 parameters through linear relations, where the more interesting ones associated to the diagonal generators read 5 As a consequence of the imposed condition the traces of odd powers of λ·v vanish, and therefore one is dealing for the subgroup Sp(3) of SU(6) with the special case ζ = 0, χ = 0. The six real roots come as pairs with opposite sign ± √ x j , where x j are the three positive roots of the cubic polynomial The further analysis based on a behavior ofP 3 (x) similar to that shown in Fig. 2 leads to the following inequalities The resulting allowed range of the invariants ξ and η is shown in Fig. 4 and one recognizes as the minimal value η min = −1/27. Making the substitution x = 1 3 (1 + 2 √ 1 + 3ξ cos ψ) one obtains for the three positive roots the expressions x 1 = 1 3 (1+2 1 + 3ξ cos ψ) , x 2 = 1 3 1−2 1 + 3ξ sin ψ+ π 6 , x 3 = 1 3 1+2 1 + 3ξ sin ψ− π 6 , In the end the semi-analytical formula for Sp(3) matrices reads where the trace tr U = 2 3 j=1 cos( √ x j | v |) is again real-valued.
In passing one reminds that in low dimensions the spin groups Spin(n), defined as the two-sheeted simply-connected coverings of the special orthogonal goups SO(n), obey the following isomorphisms together with Sp(1) = SU (2). For all these compact Lie groups the analytical evaluation of the matrix exponential function has been studied in this work. Actually, the obtained formula can be evaluated most straightforwardly for the symplectic group Sp (2), whereas in the other cases one has to make a (somewhat) cumbersome detour via the three real roots of a cubic polynomial equation.