Octonion Internal Space Algebra for the Standard Model
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...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-05-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/9/5/222 |
| _version_ | 1797598170265419776 |
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| author | Ivan Todorov |
| author_facet | Ivan Todorov |
| author_sort | Ivan Todorov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T03:15:35Z |
| format | Article |
| id | doaj.art-95502810f42a4dd39eb06771d821abaa |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-11T03:15:35Z |
| publishDate | 2023-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-95502810f42a4dd39eb06771d821abaa2023-11-18T03:34:52ZengMDPI AGUniverse2218-19972023-05-019522210.3390/universe9050222Octonion Internal Space Algebra for the Standard ModelIvan Todorov0Institut des Hautes Études Scientifiques, 35 Route de Chartres, 91440 Bures-sur-Yvette, FranceThis 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.https://www.mdpi.com/2218-1997/9/5/222Clifford algebracomposition algebratrialityJordan algebracomplex structuresuperselection rules |
| spellingShingle | Ivan Todorov Octonion Internal Space Algebra for the Standard Model Universe Clifford algebra composition algebra triality Jordan algebra complex structure superselection rules |
| title | Octonion Internal Space Algebra for the Standard Model |
| title_full | Octonion Internal Space Algebra for the Standard Model |
| title_fullStr | Octonion Internal Space Algebra for the Standard Model |
| title_full_unstemmed | Octonion Internal Space Algebra for the Standard Model |
| title_short | Octonion Internal Space Algebra for the Standard Model |
| title_sort | octonion internal space algebra for the standard model |
| topic | Clifford algebra composition algebra triality Jordan algebra complex structure superselection rules |
| url | https://www.mdpi.com/2218-1997/9/5/222 |
| work_keys_str_mv | AT ivantodorov octonioninternalspacealgebraforthestandardmodel |