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Additional info for Asymptotic methods in nonlinear Wavw Theory
Where Xl = Oil Xl. Generalizing the previous steps for SO(2), we know that the set of 3 x 3, real, and orthogonal matrices 0(3) leaves this quantity invariant. The condition of orthogonality reduces the number of independent numbers down to 9 - 6 = 3 elements. 44) where r i has purely imaginary elements. 4. Representations of 50(3) and 5U(2) of freedom. Therefore 0(3) is a three-parameter Lie group, parametrized by three angles. 47) where E 123 = +1. 48) This is an example of a Lie algebra (not to be confused with the Lie group).
Instead of a one-particle, classical description of a point particle in terms of coordinates, we now have a classical formalism defined in terms of fields ¢(x). In constructing field theories, we clearly see that the study of symmetries plays a crucial role. In fact, over the past two decades, physicists have come to the realization that symmetries are perhaps the most powerful tool that we have in describing the physiCal universe. As a consequence, it will prove beneficial to begin a more systematic discussion of symmetries and group theory.
52) For higher tensor fields, we must be careful to select out only the irreducible fields. The easiest way to decompose a reducible a tensor is to take various symmetric and anti-symmetric combinations of the indices. Irreducible representations can be extracted by using the two constant tensors, ~ij and €ijk. 53) More generally, we can use the method of Young Tableaux described in the Appendix to find more irreducible representations. As in the case of 0(2), we can also find a relationship between 0(3) and a unitary group.