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 γ ^j (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 (η _jk (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

(i) when m = 2n, (n ≥ 3), ${\mathfrak{H}}{\left( m \right)}$ is the group generated by the set of matrices

$$ \begin{array}{*{20}c} {{\left\{ {\left. {\text{T}} \right|{\text{T = }}\frac{{\text{1}}} {{{\sqrt \xi }}}{\left( {\begin{array}{*{20}c} {{{\rm I} + K}} & {0} \\ {0} & {{{\rm I} - K}} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {U} & {0} \\ {0} & {U} \\ \end{array} } \right)},} \right.}} \\ {{\left. {U \in {\mathfrak{H}}{\left( {m - 1} \right)},\;K = {\sum\limits_{j = 0}^{m - 2} {a^{j} \gamma _{j} {\left( {m - 1} \right)}} },\;\xi = 1 - {\sum\limits_{k,j = 0}^{m - 2} {\eta _{{kj}} a^{k} a^{j} } } > 0} \right\};}} \\ \end{array} $$

(ii) when m = 2n + 1 (n ≥ 3), ${\mathfrak{H}}{\left( m \right)}$ is generated by the set of matrices

$$ \begin{array}{*{20}c} {{\left\{ {\left. {\text{T}} \right|{\text{T = }}\frac{{\text{1}}} {{{\sqrt \xi }}}{\left( {\begin{array}{*{20}c} {{\rm I}} & {K} \\ {{ - K^{ - } }} & {{\rm I}} \\ \end{array} } \right)}U,U \in {\mathfrak{H}}{\left( {m - 1} \right)},\;\xi = 1 - {\sum\limits_{k,j = 0}^{m - 2} {\eta _{{kj}} a^{k} a^{j} } } > 0,} \right.}} \\ {{\left. {K = {\text{i}}{\left[ {{\sum\limits_{j = 0}^{m - 3} {a^{j} \gamma _{j} {\left( {m - 2} \right)}} } + a^{{m - 2}} {\rm I}_{n} } \right]},\;K^{ - } = {\text{i}}{\left[ {{\sum\limits_{j = 0}^{m - 3} {a^{j} \gamma _{j} {\left( {m - 2} \right)}} } - a^{{m - 2}} {\rm I}_{n} } \right]}} \right\}.}} \\ \end{array} $$