| Summary: | In this paper, we study possible mathematical connections of the Clifford algebra with the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mi>u</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>-Lie algebra, or in more physical terms the links between space-time symmetry (Lorentz invariance) and internal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula> gauge-symmetry for a massive spin one-half fermion described by the Dirac equation. The related matrix algebra is worked out in particular for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula> symmetry and outlined as well for the color gauge group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula>. Possible perspectives of this approach to unification of symmetries are briefly discussed. The calculations make extensive use of tensor multiplication of the matrices involved, whereby our focus is on revisiting the Coleman–Mandula theorem. This permits us to construct unified symmetries between Lorentz invariance and gauge symmetry in a direct product sense.
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