This article presents the description of the internal spaces of fermion and boson fields in d-dimensional spaces, with the odd and even “basis vectors” which are the superposition of odd and even products of operators <math><msup><mrow><mi>γ</mi></mrow><mrow><mi>a</mi></mrow></msup></math>. While the Clifford odd “basis vectors” manifest properties of fermion fields, appearing in families, the Clifford even “basis vectors” demonstrate properties of the corresponding gauge fields. In <math><mi>d</mi><mo>≥</mo><mo>(</mo><mn>13</mn><mo>+</mo><mn>1</mn><mo>)</mo></math> the corresponding creation operators manifest in <math><mi>d</mi><mo>=</mo><mo>(</mo><mn>3</mn><mo>+</mo><mn>1</mn><mo>)</mo></math> the properties of all the observed quarks and leptons, with the families included, and of their gauge boson fields, with the scalar fields included, making several predictions. The properties of the creation and annihilation operators for fermion and boson fields are illustrated on the case <math><mi>d</mi><mo>=</mo><mo>(</mo><mn>5</mn><mo>+</mo><mn>1</mn><mo>)</mo></math>, when <math><mi>S</mi><mi>O</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>1</mn><mo>)</mo></math> demonstrates the symmetry of <math><mi>S</mi><mi>U</mi><mo>(</mo><mn>3</mn><mo>)</mo><mo>×</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math>.

Slovenia

N.S. Mankoč Borštnik

How Clifford algebra helps understand second quantized quarks and leptons and corresponding vector and scalar boson fields,

In theories of the Kaluza-Klein kind there are spins or total angular moments in higher dimensions which manifest as charges in the observable d = (3 + 1). The charge conjugation requirement, if following the prescription in (3 + 1), would transform any particle state out of the Dirac sea into the hole in the Dirac sea, which manifests as an anti-particle having all the spin degrees of freedom in d , except S 03 , the same as the corresponding particle state. This is in contradiction with what we observe for the anti-particle. In this paper we redefine the discrete symmetries so that we stay within the subgroups of the starting group of symmetries, while we require that the angular moments in higher dimensions manifest as charges in d = (3 + 1). We pay attention on spaces with even d .