New representation of Dirac-Yang-Mills equations

Marchuk, N. G.

doi:10.1134/s1061920806040030

Cited by 3 publications

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“…The following proposition is proved in [7]: The group Sp(Cℓ(1, 3)) is isomorphic to the group Sp(2, R) and the Lie algebra sp(Cℓ (1,3)) is isomorphic to the Lie algebra sp(2, R), i.e.…”

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confidence: 99%

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confidence: 99%

“…In [2,3,4,5,8] we develop a new approach to field theory, which based on so called model equations of field theory. In this paper we introduce Dirac-Yang-Mills model equations for spin 1/2 fermions interacting with two gauge fields simultaneously.…”

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confidence: 99%

“…Let Cℓ k (1,3), (k = 0, 1, 2, 3, 4) be subspaces of rank k Clifford algebra elements and Cℓ Even (1,3), Cℓ Odd (1,3) be the subspaces of even and odd Clifford algebra elements respectively. By Cℓ R (1, 3) denote the real Clifford algebra.…”

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confidence: 99%

“…Denote β = e 0 ∈ Cℓ (1,3). Consider an operation of pseudo-Hermitian conjugation * : Cℓ(1, 3) → Cℓ (1,3) such that (e a ) * = e a , a = 0, 1, 2, 3 and (λU ) * = λU * , (U V ) * = V * U * , (U + V ) * = U * + V * , where U, V are arbitrary elements of Cℓ(1, 3) and λ ∈ C. Now we can define an operation of Hermitian conjugation of Clifford algebra elements by the formula [6] U † = βU * β.…”

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Mass generation mechanism for spin 1/2 fermions in Dirac--Yang--Mills model equations with a symplectic gauge symmetry

Marchuk

2010

Preprint

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In the Standard Model of electroweak interactions the fundamental fermions acquire masses by the Yukawa interaction with the (spin 0) Higgs field. In our model spin 1/2 fermions acquire masses by an interaction with (spin 1) gauge field with symplectic symmetry.In [2,3,4,5,8] we develop a new approach to field theory, which based on so called model equations of field theory. In this paper we introduce Dirac-Yang-Mills model equations for spin 1/2 fermions interacting with two gauge fields simultaneously. One field has a unitary gauge symmetry and another has a symplectic gauge symmetry. There is no fermion's mass (m) term in the model Dirac equation. But there is the term 3m 3 /16 in the right hand part of the Yang-Mills equations with symplectic gauge symmetry. Hence, the constant 3m 3 /16 can be considered as a constant (charge) describing the interaction of a fermion with the symplectic gauge field.

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“…The following proposition is proved in [7]: The group Sp(Cℓ(1, 3)) is isomorphic to the group Sp(2, R) and the Lie algebra sp(Cℓ (1,3)) is isomorphic to the Lie algebra sp(2, R), i.e.…”

mentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Mass generation mechanism for spin 1/2 fermions in Dirac--Yang--Mills model equations with a symplectic gauge symmetry

Marchuk

2010

Preprint

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Model Dirac-Maxwell equations with pseudounitary symmetry

Marchuk

2008

Theor Math Phys

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We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system.In mathematical and theoretical physics, the Dirac equation is used to describe particles with spin 1/2 [1]- [3]. The Maxwell equations describe the electromagnetic field (photons). The system of DiracMaxwell equations models the interaction of electrons with the electromagnetic field.Sommerfeld discovered ([4], pp. 219-229 in the Russian translation) that two approaches can be used to prove the covariance of the Dirac equation for electrons in the case of the Lorentzian replacements of coordinates. In the first case, it is assumed that the Dirac matrices γ µ transform as components of a vector and the wave function ψ of the electron remains unchanged. In this case, all the quantities in the Dirac equation are tensor fields (vectors, covectors, and scalars). In the second case, it is assumed that the Dirac γ-matrices remain unchanged and the wave function ψ of the electron transforms according to the spinor representation of the Lorentz group, i.e., ψ is a Dirac spinor.Accordingly, we can speak about the presence of two representations of the Dirac equation for the electron: tensor and spinor. We can conclude from Sommerfeld's arguments that he had no basis to consider one representation right and the other wrong. We note that Usachev [5] studied the problem of spins of particles described by the Dirac equation in the tensor representation. He showed that this equation describes particles with spin 1/2 just like the Dirac equation in the spinor representation. The spinor representation of the Dirac equation is generally accepted in modern theoretical and mathematical physics.Papers have been devoted to developing a new approach to the Dirac equation and systems of DiracMaxwell and Dirac-Yang-Mills equations [6]- [22]; this approach is based on using the Clifford algebra and the tensor representation of the Dirac equation.Here, we propose and consider a model system of Dirac-Maxwell equations constructed based on the matrix technique and the tensor representation of the Dirac equation for electrons. We show that in contrast to the previously discussed equations, this system of equations has a new global symmetry with respect to the pseudounitary group SU (2, 2). We also study a version of the model system of Dirac-Maxwell equations with a local (gauge) pseudounitary symmetry.

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Модельные Уравнения Дирака - Максвелла С Псевдоунитарной Симметрией

Marchuk¹

2008

ТМФ

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Вводится новая система уравнений, называемая модельной системой уравне-ний Дирака-Максвелла. Эта система уравнений воспроизводит основные свой-ства стандартной системы уравнений Дирака-Максвелла. В то же время мо-дельная система уравнений имеет ряд отличий от стандартной системы урав-нений, в частности она является тензорной и обладает новой симметрией по отношению к псевдоунитарной группе. Предложен также вариант модельной системы уравнений Дирака-Максвелла с локальной (калибровочной) псевдо-унитарной симметрией. Показано, что любое спинорное решение стандартной системы уравнений Дирака-Максвелла может быть получено из соответствую-щего тензорного решения модельной системы уравнений Дирака-Максвелла.Ключевые слова: уравнение Дирака, уравнения Максвелла, спиноры, псевдоунитар-ная симметрия, калибровочная симметрия, спиноризация.В математической и теоретической физике уравнение Дирака [1]-[3] используется для описания частиц спина 1/2. Уравнения Максвелла описывают электромагнит-ное поле (фотоны). Система уравнений Дирака-Максвелла моделирует взаимодей-ствие электрона с электромагнитным полем.Зоммерфельд обнаружил ([4], с. 219-229), что возможны два подхода к доказа-тельству ковариантности уравнения Дирака для электрона при лоренцевых заменах координат. В первом случае предполагается, что матрицы Дирака γ µ преобразуют-ся как компоненты вектора, а волновая функция электрона ψ не меняется. При этом все математические объекты, входящие в уравнение Дирака, являются тензорными полями (векторами, ковекторами, скалярами). Во втором случае предполагается, что γ-матрицы Дирака не меняются, а волновая функция электрона ψ преобразуется по спинорному представлению группы Лоренца, т.е. ψ является спинором Дирака.Соответственно можно говорить о наличии двух представлений уравнения Дира-ка для электрона -тензорном и спинорном. Из рассуждений Зоммерфельда можно сделать вывод о том, что он не видит оснований считать одно представление пра-вильным, а другое -ошибочным. Отметим, что Усачев [5] рассмотрел вопрос о спине частиц, описываемых уравнением Дирака в тензорном представлении. Он * Математический институт им. В. А. Стеклова РАН, Москва, Россия.

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New representation of Dirac-Yang-Mills equations

Cited by 3 publications

References 13 publications

Mass generation mechanism for spin 1/2 fermions in Dirac--Yang--Mills model equations with a symplectic gauge symmetry

Mass generation mechanism for spin 1/2 fermions in Dirac--Yang--Mills model equations with a symplectic gauge symmetry

Model Dirac-Maxwell equations with pseudounitary symmetry

Модельные Уравнения Дирака - Максвелла С Псевдоунитарной Симметрией

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