A Representation of the Lorentz Spin Group and Its Application

Abstract

For an integer m ≥ 4, we define a set of \( 2^{{{\left[ {\frac{m} {2}} \right]}}} \times 2^{{{\left[ {\frac{m} {2}} \right]}}} \) matrices γj (m), (j = 0, 1, . . . , m − 1) which satisfy \( \gamma _{j} {\left( m \right)}\gamma _{k} {\left( m \right)} + \gamma _{k} {\left( m \right)}\gamma _{j} {\left( m \right)} = 2\eta _{{jk}} {\left( m \right)}{\rm I}_{{{\left[ {\frac{m} {2}} \right]}}} \), where (ηjk(m))0≤j,k≤m−1 is a diagonal matrix, the first diagonal element of which is 1 and the others are −1, \( {\rm I}_{{{\left[ {\frac{m} {2}} \right]}}} \) is a \( 2^{{{\left[ {\frac{m} {2}} \right]}}} \times 2^{{{\left[ {\frac{m} {2}} \right]}}} \) identity matrix with \( _{{{\left[ {\frac{m} {2}} \right]}}} \) being the integer part of \( {\frac{m} {2}} \) . For m = 4 and 5, the representation \( {\mathfrak{H}}{\left( m \right)} \) of the Lorentz Spin group is known. For m ≥ 6, we prove that