On the Gamma Matrix Representations of SO(8) and Clifford Algebras

Simulik, V. M.

doi:10.1007/s00006-018-0906-3

Cited by 6 publications

(36 citation statements)

References 59 publications

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“…Recently (see, e.g., [16]), we put the gamma matrix representations of the Clifford algebra Cℓ R (0,6) and the Lie algebra SO (8) into consideration, which are defined over the field of real numbers. The comparison with the well-known gamma matrix representations of the algebras Cℓ C (1,3) (in the physical literature, the Clifford-Dirac algebra) and SO(1,5) defined over the field of complex numbers demonstrates that the new representations contain much more useful elements and perspectives for applications.…”

Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

“…The Bose symmetries were found as well [19]. Below, we will use these new algebraic objects [16] for the problem of finding the hidden symmetries of a relativistic hydrogen atom. The necessary part of the consideration from [16] is presented in a compact version.…”

Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

“…Due to the evident fact that only six operators of (12) are linearly independent, 4 = = − 7 1 2 3 , it is the representation of the Clifford algebra Cℓ R (0,6) of the dimension equal to 2 6 = 64. The first 16 operators are given in Table 1 of [16], the next 16 ones can be found from them with the help of the multiplication by imaginary unit = √ −1. Last 32 operators follow from first 32 ones with the help of the multiplication by the operator^of com-plex conjugation.…”

Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

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Symmetries of Relativistic Hydrogen Atom

Simulik

Gordievich²

2019

Ukr. J. Phys.

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The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above-mentioned algebras commute with the operator of the Dirac equation in an external Coulomb field and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s = (1, 0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with an external Coulomb field. A relativistic hydrogen atom is modeled by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers in the case of a non-interacting spinor field. New symmetry operators are found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which are known from our recent papers as well. Hidden symmetries were found both in the canonical Foldy–Wouthuysen and covariant Dirac representations. The found symmetry operators, which are pure matrix ones in the Foldy–Wouthuysen representation, become non-local in the Dirac model.

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Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

Section: Briefly On the Gamma Matrix Representations Of The Real Clifmentioning

confidence: 99%

See 1 more Smart Citation

Symmetries of Relativistic Hydrogen Atom

Simulik

Gordievich²

2019

Ukr. J. Phys.

Self Cite

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show abstract

“…Recently, see, e.g., [16], we put into consideration the gamma matrix representations of the Clifford algebra Cℓ R (0,6) and the Lie algebra SO (8), which are defined over the field of real numbers. Comparison with well known gamma matrix representations of algebras Cℓ C (1,3) (in physical literature the Clifford-Dirac algebra) and SO (1,5), defined over the field of complex numbers, demonstrates that new representations contain much more useful elements and perspectives for application.…”

Section: Briefly On the Gamma Matrix Representations Of The Realmentioning

confidence: 99%

“…Last 32 are found from first 32 with the help of multiplication by operatorĈ of complex conjugation. Thus, if to introduce the notation "stand CD" ("stand" and CD are taken from standard Clifford-Dirac) for the set of 16 matrices from the Table 1 in [16], then the set of 64 elements of Cℓ R (0,6) algebra representation will be given by (stand CD) ∪ i · (stand CD) ∪Ĉ · (stand CD) ∪ iĈ · (stand CD) .…”

Section: Briefly On the Gamma Matrix Representations Of The Realmentioning

confidence: 99%

Hidden symmetries of relativistic hydrogen atom

Simulik

Gordievich

2019

J. Phys.: Conf. Ser.

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The Dirac equation in the external Coulomb field is proved to possess the symmetry determined by the 31 operators, which form the 31-dimensional algebra. Two different fermionic realizations of the SO(1,3) algebra of the Lorentz group are found. Two different bosonic realizations of this algebra are found as well. All generators of the above mentioned algebras commute with the operator of the Dirac equation in the external Coulomb field, and, therefore, determine the algebras of invariance of such Dirac equation. Hence, the spin s=(1,0) Bose symmetry of the Dirac equation for the free spinor field, proved recently in our papers, is extended here for the Dirac equation interacting with external Coulomb field. Relativistic hydrogen atom is modeling here by such Dirac equation. We are able to prove for the relativistic hydrogen atom both the fermionic and bosonic symmetries known from our papers about the case of non-interacting spinor field. New symmetry operators were found on the basis of new gamma matrix representations of the Clifford and SO(8) algebras, which were found recently in our papers. Hidden symmetries were found both in the canonical Foldy-Wouthuysen and in the covariant Dirac representations. The symmetry operators, which are simple and graceful in the Foldy-Wouthuysen representation, become non-local in the Dirac model.

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On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation

Simulik,

Vyikon

2023

Adv. Appl. Clifford Algebras

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On the Gamma Matrix Representations of SO(8) and Clifford Algebras

Cited by 6 publications

References 59 publications

Symmetries of Relativistic Hydrogen Atom

Symmetries of Relativistic Hydrogen Atom

Hidden symmetries of relativistic hydrogen atom

On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation

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