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2 /: The generators satisfy the relation l D l and are linear operators (matrices) in the Minkowski space. l /˛ˇ D g ˛ı ˇ C g ˛ı ˇ: It is easy to see that they satisfy the following commutation relations: Œl ; l˛ˇ D g ˛l ˇ The linear span of the matrices l the Lorentz group. ƒ; a/. 20) with ƒ 2 LC form a 10parameter continuous Lie group. 6. 13). C/ correspond to the transformations with det ƒ D C1 and det ƒ D 1; respectively. 2). The generalization to the case of complex inhomogeneous transformations is trivial.

Prove that the homogeneous Lorentz transformation x 0 ! x/ ! 1. 23). 2. Show that the matrices S! 18). 3. x/. 4. 31) form a group. Part II Classical Theory of the Free Fields Chapter 4 Lagrangian and Hamiltonian Formalisms of the Classical Field Theory For the construction of the theory of classical ﬁelds, we use the Lagrange approach because the Lagrangian formalism proves to be the most convenient tool for the development of the relativistic theory in view of its obvious covariance. The Hamiltonian formalism (or the formalism of canonical variables) appears to be useful in ﬁnding the similarities between the classical and quantum theories.

The required symmetry was proposed by Gell-Mann and Ne’eman in 1961 [63, 137, 65]. 3/. 10) where fj kl are the structural constants antisymmetric with respect to the indices j , k, and l. They take the following values: f123 D 1; 1 f156 D f246 D f257 D f345 D f367 D ; 2 p 3 f458 D f678 D : 2 The structural constants corresponding to the other values of indices are equal to zero. 3/ triplet contains the isodoublet ‰1 , ‰2 and the isosinglet ‰3 . 3/-based theory, the generators of representation are connected not only with the components of isospin but also with the hypercharge Y .