Chiral symmetry: An analytic 
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 unitary matrix

Robilotta, M. R.

doi:10.1103/physrevd.106.054027

Cited by 1 publication

(2 citation statements)

References 34 publications

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“…Both angles specify to which element on a maximal torus SO(2) × SO(2) = S 1 × S 1 in SO(4) = S 3 × RP 3 the given rotation matrix R 4 (⃗ v ) is related by conjugation [5]. Note that a conventional maximal torus in SO(4) consists of independent rotations that take place in the x 1 x 2 -plane and the x 3 x 4 -plane of four-dimensional space.…”

Section: So(4)mentioning

confidence: 99%

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Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

Kaiser

2023

Mathematics

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In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

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Section: So(4)mentioning

confidence: 99%

“…This straightforward solution should be compared with the overly long expositions about an analytic SU(3) matrix in ref. [3]. When going to the special unitary group SU(4) with 15 real parameters, the analytical formula involved the sum over four real roots of a quartic equation.…”

Section: Introductionmentioning

confidence: 99%

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

Kaiser

2023

Mathematics

View full text Add to dashboard Cite

show abstract

Chiral symmetry: An analytic SU(3) unitary matrix

Cited by 1 publication

References 34 publications

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

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