Some statistical aspects of the spinor field Fermi-Bose duality

Abstract

The structure of 29-dimensional extended real Clifford-Dirac algebra, which has been introduced in our paper Phys. Lett. A, 2011, 375, 2479, is considered in brief. Using this algebra, the property of Fermi-Bose duality of the Dirac equation with nonzero mass is proved. It means that Dirac equation can describe not only the fermionic but also the bosonic states. The proof of our assertion based on the examples of bosonic symmetries, solutions and conservation laws is given. Some statistical aspects of the spinor field Fermi-Bose duality are discussed.

On the Relativistic Canonical Quantum Mechanics and Field Theory of Arbitrary Spin

“…The relationship (45) between the Clifford-Dirac algebra Cl R (0,6) and algebra SO (8) is similar to the relationship between the standard Clifford-Dirac algebra Cl C (1,3) and algebra SO (3,3) found in [99,100]; for the relationship between Cl C (1,3) and SO(1,5) see in [85][86][87][88][89][90][91]. Note that subalgebra SO(6)⊂SO (8) is the algebra of invariance of the Dirac equation in the FW representation [6,7].…”

“…It has been explained in [1] (and in [85][86][87][88][89][90][91] in details) that the Clifford-Dirac algebra should be introduced into consideration in the FW representation [6] of the spinor field. The reasons are as follows.…”

“…In [85][86][87][88][89][90][91] such additional algebras have been introduced for the purposes of finding links between the fermionic and bosonic states of the spinor field. New algebra can be formed by the generators of Cl C (1,3) together with the generators of the Pauli-Gürsey-Ibragimov algebra [96][97][98].…”

“…Therefore, in application to Clifford algebras operators (51) generate the 64-dimensional Cl R (0,6) algebra as well. The maximal pure matrix algebra of invariance of the FW equation (34) (1,5), are principal in description of the Bose states in the framework of the Dirac theory [85][86][87][88][89][90][91]. The algebra SO(8) includes two independent spin s=1/2 SU(2) subalgebras (…”

“…Therefore, the relativistic hydrogen atom has wide additional symmetries given by the 31 nontrivial generators from SO(6) ⊕ iγ 0 · SO(6) ⊕ iγ 0 . Moreover, the relativistic hydrogen atom has additional spin 1 symmetries, which for the free Dirac equation are known from [85][86][87][88][89][90][91] and which are the consequences of the algebra SO(6) ⊕ iγ 0 · SO(6) ⊕ iγ 0 . Indeed, more wide symmetries than SO(4) spin 1/2 symmetry of V. Fock [101] or further symmetries found in [102][103][104][105][106] (see the brief analysis in [107]) can be found on the basis of this 31-dimensional algebra.…”