A Representation of the Lorentz Spin Group and Its Application期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

A Representation of the Lorentz Spin Group and Its Application

Authors:

Affiliation:

(1) Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P. R. China;(2) School of Mathematical Science, Capital Normal University, Beijing, 100037, P. R. China

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 $gamma _{j} {left( m right)}gamma _{k} {left( m right)} + gamma _{k} {left( m right)}gamma _{j} {left( m right)} = 2eta _{{jk}} {left( m right)}{rm I}_{{{left[ {frac{m} {2}} right]}}}$ , 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

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

(ii) when m = 2n + 1 (n ≥ 3), $$
{mathfrak{H}}{left( m right)}
$$

is generated by the set of matrices

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

Partially supported by Chinese NNSF Projects (10231050/A010109, 10375038, 904030180) and NKBRPC Project (2004CB318000)

Keywords:

Lorentz spin group representation Yang-Mills equation

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