- #1
CAF123
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Higher dimensional groups are parametrised by several parameters (e.g the three dimensional rotation group SO(3) is described by the three Euler angles). Consider the following ansatz: $$\rho_1 = \mathbf{1} + i \alpha^a T_a + \frac{1}{2} (i\alpha^a T_a)^2 + O(\alpha^3)$$
$$\rho_2 = \mathbf{1} + i \beta^b T_b + \frac{1}{2} (i\beta^b T_b)^2 + O(\beta^3)$$
$$\rho_3 = \mathbf{1} + i \gamma^c T_c + \frac{1}{2} (i\gamma^c T_c)^2 + O(\gamma^3),$$ where summation over indices ##a,b,c## is understood and ##a,b,c = 1 \dots \text{dim(Lie Algebra)}, ##(e.g for SO(3), a,b,c = 1,..3 the three Euler angles).
I don't really understand what these equations are saying - could someone explain? My thoughts are that the ##\rho_i## are rotation matrices in 3-space, where the rotation is about an axis ##\underline{n} = \langle \alpha^1, \alpha^2, \alpha^3 \rangle##, the ##\alpha^i## specifying a choice of Euler angles. In which case, the ##T_a## would be the generators of the SO(3) Lie algebra and the ##\rho_i## are elements of the group SO(3)?
Many thanks.
$$\rho_2 = \mathbf{1} + i \beta^b T_b + \frac{1}{2} (i\beta^b T_b)^2 + O(\beta^3)$$
$$\rho_3 = \mathbf{1} + i \gamma^c T_c + \frac{1}{2} (i\gamma^c T_c)^2 + O(\gamma^3),$$ where summation over indices ##a,b,c## is understood and ##a,b,c = 1 \dots \text{dim(Lie Algebra)}, ##(e.g for SO(3), a,b,c = 1,..3 the three Euler angles).
I don't really understand what these equations are saying - could someone explain? My thoughts are that the ##\rho_i## are rotation matrices in 3-space, where the rotation is about an axis ##\underline{n} = \langle \alpha^1, \alpha^2, \alpha^3 \rangle##, the ##\alpha^i## specifying a choice of Euler angles. In which case, the ##T_a## would be the generators of the SO(3) Lie algebra and the ##\rho_i## are elements of the group SO(3)?
Many thanks.