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
“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“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.
