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Обобщенная алгебра Клиффорда – универсальный инструмент описания и объединения фундаментальных полей
This article presents an overview of the construction of Clifford algebra in pseudo-Riemannian space R1,3 for describing and unifying fundamental fields: gravitational, electromagnetic, and Dirac fields. The quadratic differential form ∇A = ∇•A + ∇∧A was defined as a measure of the local inhomogeneity of a vector field in R1,3. Instead of the classical scalar and vector products in ∇A, the Clifford product was used: the inner ∇•A and outer ∇∧A ones. In addition, instead of the orthonormal basis in the Minkowski space (E1,3) {t,x,y.z}, the so-called canonical basis, composed of the Dirac matrices {γ0, γ1,γ2,γ3}, was used, which significantly expanded the meaning and capabilities of the classical basis. The associativity property of the product ∇∇A = (∇∇A) = ∇(∇A) made it possible to obtain the following: a) the Einstein equation from (∇∇A); and b) the Maxwell equation from ∇(∇A). Simultaneously, two independent Maxwell systems were combined into a single equation too. Transforming ∇A into a sum of three independent biquaternions ∇A = Σαℬα (α = 1,2,3) allowed us to obtain bispinors (antibispinors), rotations on the spatial and temporal planes in R1,3, and representation functions and generators of the Lorentz group in R1,3. The gradient of the bispinors allowed us to derive three pairs of Dirac-type equations for the three generations of fermions and bosons. Transformations of vectors using generalized biquaternions (x′ = ℬα•x•ℬ̃̃α), or more precisely, Lorentz transformations in curvilinear coordinates, led to the universal form of the Doppler law. The approximation function of the proposed nonlinear Hubble law was calculated from experimental data on the dependence of the redshift on the distance to the observed stars. This nonlinearity of the Hubble law explains the mechanism for the accelerated increase in redshift for the steady-state model without the Big Bang.
В статье представлен обзор конструкции алгебры Клиффорда в псевдоримановом пространстве R1,3 для описания и объединения фундаментальных полей: гравитационного, электромагнитного и поля Дирака. Квадратичная дифференциальная форма ∇A = ∇•A + ∇∧A была определена как мера локальной неоднородности векторного поля в R1,3. В ∇A вместо классических скалярных и векторных произведений было применено произведение Клиффорда: внутреннее ∇•A и внешнее ∇∧A. Также вместо ортонормированного базиса в пространстве Минковского (E1,3) {t,x,y.z} был использован, так называемый, канонический базис, составленный из матриц Дирака {γ0, γ1,γ2,γ3}, что значительно расширяет смысл и возможности классического базиса. Свойство ассоциативности произведения ∇∇A = (∇∇A) = ∇(∇A) позволило получить: а) уравнение Эйнштейна из (∇∇A); б) уравнение Максвелла из ∇(∇A). При этом две независимые системы Максвелла также объединились в единое уравнение. Преобразование ∇A как сумма трёх независимых бикватернионов ∇A = Σαℬα (α = 1,2,3) позволило получить биспиноры (антибиспиноры), повороты на пространственных и временных плоскостях в R1,3, а также функции представления и генераторы группы Лоренца в R1,3. Градиент биспиноров позволил вывести три пары уравнений типа Дирака для трёх поколений фермионов и бозонов. Преобразования векторов с помощью обобщенных бикватернионов (x′ = ℬα•x•ℬ̃̃α), а точнее, преобразования Лоренца в криволинейных координатах привели к универсальному виду закона Доплера. Аппроксимирующая функция предполагаемого нелинейного закона Хаббла была вычислена из экспериментальных данных по измерению зависимости красного смещения от расстояния до наблюдаемых звезд. Эта нелинейность закона Хаббла объясняет механизм ускоренного увеличения красного смещения для модели Стационарной Вселенной без модели «Большого Взрыва»
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