| Summary: | This paper surveys recent progress in our search for an appropriate internal space algebra for the standard model (SM) of particle physics. After a brief review of the existing approaches, we start with the Clifford algebras involving operators of left multiplication by octonions. A central role is played by a distinguished complex structure that implements the splitting of the octonions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">O</mi><mo>=</mo><mi mathvariant="double-struck">C</mi><mo>⊕</mo><msup><mrow><mi mathvariant="double-struck">C</mi></mrow><mn>3</mn></msup></mrow></semantics></math></inline-formula>, which reflect the lepton-quark symmetry. Such a complex structure on the 32-dimensional space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mo>ℓ</mo><mn>10</mn></msub></mrow></semantics></math></inline-formula> Majorana spinors is generated by the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mo>ℓ</mo><mn>6</mn></msub><mrow><mo>(</mo><mo>⊂</mo><mi>C</mi><msub><mo>ℓ</mo><mn>10</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> volume form, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>6</mn></msub><mo>=</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>⋯</mo><msub><mi>γ</mi><mn>6</mn></msub></mrow></semantics></math></inline-formula>, and is left invariant by the Pati–Salam subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>10</mn><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mrow><mi>PS</mi></mrow></msub><mo>=</mo><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow><mo>×</mo><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow><mo>/</mo><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. While the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>p</mi><mi>i</mi><mi>n</mi><mo>(</mo><mn>10</mn><mo>)</mo></mrow></semantics></math></inline-formula> invariant volume form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ω</mi><mn>10</mn></msub><mo>=</mo><msub><mi>γ</mi><mn>1</mn></msub><mo>…</mo><msub><mi>γ</mi><mn>10</mn></msub></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mo>ℓ</mo><mn>10</mn></msub></mrow></semantics></math></inline-formula> is known to split <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">S</mi></semantics></math></inline-formula> on a complex basis into left and right chiral (semi)spinors, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">P</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>i</mi><msub><mi>ω</mi><mn>6</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is interpreted as the projector on the 16-dimensional <i>particle subspace</i> (which annihilates the antiparticles).The standard model gauge group appears as the subgroup of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mrow><mi>PS</mi></mrow></msub></semantics></math></inline-formula> that preserves the sterile neutrino (which is identified with the Fock vacuum). The <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-graded internal space algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is then included in the projected tensor product <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">A</mi><mo>⊂</mo><mi mathvariant="script">P</mi><mi>C</mi><msub><mo>ℓ</mo><mn>10</mn></msub><mi mathvariant="script">P</mi><mo>=</mo><mi>C</mi><msub><mo>ℓ</mo><mn>4</mn></msub><mo>⊗</mo><mi>C</mi><msubsup><mo>ℓ</mo><mn>6</mn><mn>0</mn></msubsup></mrow></semantics></math></inline-formula>. The Higgs field appears as the scalar term of a superconnection, an element of the odd part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msubsup><mo>ℓ</mo><mn>4</mn><mn>1</mn></msubsup></mrow></semantics></math></inline-formula> of the first factor. The fact that the projection of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mo>ℓ</mo><mn>10</mn></msub></mrow></semantics></math></inline-formula> only involves the even part <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msubsup><mo>ℓ</mo><mn>6</mn><mn>0</mn></msubsup></mrow></semantics></math></inline-formula> of the second factor guarantees that the color symmetry remains unbroken. As an application, we express the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><msub><mi>m</mi><mi>H</mi></msub><msub><mi>m</mi><mi>W</mi></msub></mfrac></semantics></math></inline-formula> of the Higgs to the <i>W</i> boson masses in terms of the cosine of the <i>theoretical</i> Weinberg angle.
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