By Jean Hladik (auth.)

Invented by means of Dirac in growing his relativistic quantum concept of the electron, spinors are vital in quantum idea, relativity, nuclear physics, atomic and molecular physics, and condensed subject physics. primarily, they're the mathematical entities that correspond to electrons within the similar approach that normal wave services correspond to classical debris (including photons). as a result of their kinfolk to the rotation crew SO(n) and the unitary crew SU(n), the dialogue might be of curiosity to utilized mathematicians in addition to physicists.

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**Sample text**

If it can be proved that the only matrix commuting with all the matrices of a representation of a group is a matrix which is a multiple of the unit matrix, then the representation is irreducible. In fact, in the presence of a true stable subspace (the case of reducibility) it is easy to construct a matrix which commutes and which is not a multiple of the unit matrix. The Irreducibility of a Representation r(j) First of all let us consider diagonal matrices M(U) of a representation of 8U(2). We then have b = 0, lal = 1, and we can write a = eiCi / 2 with arbitrary a.

34) so for diagonal matrices Amm/(e im'Ci - eimCi ) = O. 35) and the matrix A must be dia- gonal. For the moment let us consider an arbitrary matrix M(U).

J - m)! (j + m')! ]1/2 (j + m - A)! (j - m' - A)! (m' - m + A)! xai+m-A(a*)j-ml-AbA( _b*)m -m+A. 27) l The summation over A is carried out over all the values which actually correspond to a term in the binomial formula, that is, A integral, j + m - A ~ 0, j' - m' - A ~ 0, A ~ 0, m' - m + A ~ O. These values correspond to the factorials of the denominator which are not infinite. For example, for m' = j the denominator becomes infinite except for A = O. , the factorial of a negative integer, is infinite as a result of the relation between the gamma functions and the factorials, n!