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Example text
Indeed, the electromagnetic field appears if we place charged particles in a certain region of the space. This field is not homogeneous. Hence, it is possible to speak about the symmetry connected with the law of conservation of the electric charge only in a neighborhood of certain points of the space-time. x/ ! x/ ! x/ ! x/ ! x/; where k are the Pauli matrices (the constant 1=2 is introduced to express the generators via the Pauli matrices). 14) are called local gauge transformations of the first kind.
J1 , J2 , J3 /, where J1 , J2 , and J3 are the generators of the corresponding representation and J takes integer or half-integer values by analogy with the vector of the ordinary spin operator. , coincides with the dimension of the multiplet. 9) and the corresponding transformation of wave functions takes the form 1 ‰ 0 D e 2 i g J ‰: Hence, the doublet of a nucleon corresponds to J D 1=2. K C , K 0 /: In order to distinguish between these doublets, Gell-Mann and Nishijima introduced the notion of hypercharge Y (Y D 1 for nucleons and Y D 1 for „-baryons).
A ! x/ D S! x/S! 3). 1/. 13)). x/ ! x/º. x/ ! ! x/ D S! x/ ! x/ D S! x/S! @ S! x//S! 31) where is an arbitrary constant. x/; and D 1=e. 16). , coincides with the dimension of the group). 4 Groups of Local Transformations. x/º. 31) and perform the trace operation Tr. x/ ! B ! x/ C @ ! T a S! x/T b S! 35) We now substitute X D i ! 35). Further, we act by the generator T b upon the obtained equality from the right and perform the trace operation on both sides of the formula. Q ab WD if abc ! , of the dimension of the matrix S!