The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing ...The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.展开更多
The most complicated problems to solve in science are multidisciplinary problems, which include the problem of explaining of physical reality and physical essenceofimaginary numbers discovered more than five hundred y...The most complicated problems to solve in science are multidisciplinary problems, which include the problem of explaining of physical reality and physical essenceofimaginary numbers discovered more than five hundred years ago. However, the authors of the special relativity theory created more than a hundred years ago, were trying to unprovenly claim that they have solved this problem and that the imaginary numbers are not physically real with a principle of non-exceeding the speed of light. And physical experiments MINOS at the Tevatron collider and OPERA at the Large Hadron Collider were even seem to confirm such an assertion, since they could not refute the principle of non-exceeding the speed of light. However, since mathematics is the universal language of all exact sciences, the principle of the physical reality of imaginary numbers was still proven, but proven otherwise—with the theoretical and experimental studies of the vibrational processes in linear electric circuits that can be repeated and verified in any radioelectronic and electrical laboratory. And that is why they are definitely reliable and conclusive. Using the very general scientific principle of the physical reality of imaginary numbers allowed to correct the relativistic formula of special relativity theory and on this basis to cre-ate a theory of hidden Multiverse containing more than two mutually invisible parallel universes. The nature of their invisibility is explained. It is proven that the existence of these unseen parallel universes explains the phenomenon of dark matter and dark energy. Using data obtained by the spacecrafts WMAP and Planck led to the discovery of quaternion structure of the hidden Multiverse and show that our hidden Multiverse is not the only one in nature.展开更多
A class of meta-invariant operators over Cayley-Dickson algebra is studied. Their spectral theory is investigated. More- over, theorems about spectra of generalized unitary operators and their semigroups are demonstra...A class of meta-invariant operators over Cayley-Dickson algebra is studied. Their spectral theory is investigated. More- over, theorems about spectra of generalized unitary operators and their semigroups are demonstrated.展开更多
“All is Number”—the universe follows a few profound mathematical rules,and pure thought can grasp reality.This paper explores the first principles of fundamental physics,focusing on the principle of relativity,the ...“All is Number”—the universe follows a few profound mathematical rules,and pure thought can grasp reality.This paper explores the first principles of fundamental physics,focusing on the principle of relativity,the principle of least action,and the principle of regularity.By illustrating the principle of relativity with an example of coordinate transformation,the paper clarifies the nature of spacetime:spacetime exists objectively,whereas coordinate systems are merely mathematical constructs.It also discusses the uniqueness of natural coordinate systems and their roles in both quantum and classical mechanics.Clifford geometric algebra is introduced as a mathematical framework for physical theories,and the principle of least action is analyzed,emphasizing the Lagrangian as an intrinsic characteristic of physical systems,which can be expressed as a linear combination of the system’s energy terms.Through examples from complicated systems,the paper demonstrates the structural characteristics,applications,and limitations of this principle.Furthermore,it examines the fundamental distinction between the finite and the infinite,noting that infinity is merely an analytical variable rather than a number.If a physical equation yields a solution with infinite energy density,the theory must be revised to maintain consistency.These first principles not only constitute the foundation of physics but also unveil profound symmetries and universal patterns in mathematical structures.展开更多
文摘The Clifford algebra is a unification and generalization of real number, complex number, quaternion, and vector algebra, which accurately and faithfully characterizes the intrinsic properties of space-time, providing a unified, standard, elegant, and open language and tools for numerous complicated mathematical and physical theories. So it is worth popularizing in the teaching of undergraduate physics and mathematics. Clifford algebras can be directly generalized to 2<sup>n</sup>-ary associative algebras. In this generalization, the matrix representation of the orthonormal basis of space-time plays an important role. The matrix representation carries more information than the abstract definition, such as determinant and the definition of inverse elements. Without this matrix representation, the discussion of hypercomplex numbers will be difficult. The zero norm set of hypercomplex numbers is a closed set of special geometric meanings, like the light-cone in the realistic space-time, which has no substantial effect on the algebraic calculus. The physical equations expressed in Clifford algebra have a simple formalism, symmetrical structure, standard derivation, complete content. Therefore, we can hope that this magical algebra can complete a new large synthesis of modern science.
文摘The most complicated problems to solve in science are multidisciplinary problems, which include the problem of explaining of physical reality and physical essenceofimaginary numbers discovered more than five hundred years ago. However, the authors of the special relativity theory created more than a hundred years ago, were trying to unprovenly claim that they have solved this problem and that the imaginary numbers are not physically real with a principle of non-exceeding the speed of light. And physical experiments MINOS at the Tevatron collider and OPERA at the Large Hadron Collider were even seem to confirm such an assertion, since they could not refute the principle of non-exceeding the speed of light. However, since mathematics is the universal language of all exact sciences, the principle of the physical reality of imaginary numbers was still proven, but proven otherwise—with the theoretical and experimental studies of the vibrational processes in linear electric circuits that can be repeated and verified in any radioelectronic and electrical laboratory. And that is why they are definitely reliable and conclusive. Using the very general scientific principle of the physical reality of imaginary numbers allowed to correct the relativistic formula of special relativity theory and on this basis to cre-ate a theory of hidden Multiverse containing more than two mutually invisible parallel universes. The nature of their invisibility is explained. It is proven that the existence of these unseen parallel universes explains the phenomenon of dark matter and dark energy. Using data obtained by the spacecrafts WMAP and Planck led to the discovery of quaternion structure of the hidden Multiverse and show that our hidden Multiverse is not the only one in nature.
文摘A class of meta-invariant operators over Cayley-Dickson algebra is studied. Their spectral theory is investigated. More- over, theorems about spectra of generalized unitary operators and their semigroups are demonstrated.
文摘“All is Number”—the universe follows a few profound mathematical rules,and pure thought can grasp reality.This paper explores the first principles of fundamental physics,focusing on the principle of relativity,the principle of least action,and the principle of regularity.By illustrating the principle of relativity with an example of coordinate transformation,the paper clarifies the nature of spacetime:spacetime exists objectively,whereas coordinate systems are merely mathematical constructs.It also discusses the uniqueness of natural coordinate systems and their roles in both quantum and classical mechanics.Clifford geometric algebra is introduced as a mathematical framework for physical theories,and the principle of least action is analyzed,emphasizing the Lagrangian as an intrinsic characteristic of physical systems,which can be expressed as a linear combination of the system’s energy terms.Through examples from complicated systems,the paper demonstrates the structural characteristics,applications,and limitations of this principle.Furthermore,it examines the fundamental distinction between the finite and the infinite,noting that infinity is merely an analytical variable rather than a number.If a physical equation yields a solution with infinite energy density,the theory must be revised to maintain consistency.These first principles not only constitute the foundation of physics but also unveil profound symmetries and universal patterns in mathematical structures.
