A symplectic subgroup of a pseudounitary group as a subset of Clifford algebra

A Theory of Quantized Fields Based on Orthogonal and Symplectic Clifford Algebras

The transition from a classical to quantum theory is investigated within the context of orthogonal and symplectic Clifford algebras, first for particles, and then for fields. It is shown that the generators of Clifford algebras have the role of quantum mechanical operators that satisfy the Heisenberg equations of motion. For quadratic Hamiltonians, the latter equations are obtained from the classical equations of motion, rewritten in terms of the phase space coordinates and the corresponding basis vectors. Then, assuming that such equations hold for arbitrary path, i.e., that coordinates and m…

Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism

In this paper we start from a basic notion of process, which we structure into two groupoids, one orthogonal and one symplectic. By introducing additional structure, we convert these groupoids into orthogonal and symplectic Clifford algebras respectively. We show how the orthogonal Clifford algebra, which include the Schröodinger, Pauli and Dirac formalisms, describe the classical light-cone structure of space-time, as well as providing a basis for the description of quantum phenomena. By constructing an orthogonal Clifford bundle with a Dirac connection, we make contact with quantum mechanics…

关于 Clifford 代数中包含 spin 群的一些 Lie 群

On some Lie groups containing spin group in Clifford algebra

本文研究了复化 Clifford 代数中的 Lie 群，利用共轭与矩阵运算的关系证明了其与经典矩阵群在任意维度和符号差下的同构，并给出了包含 Spin 群的经典矩阵 Lie 群。

有限交换群关联的克莱因群分解

Splitting of Clifford groups associated to finite abelian groups

本文研究有限交换群关联的克莱因群与辛群的扩展，证明当群阶不被4整除时扩展可分解为半直积，扩展了Korbelář和Tolar的循环结果。

复合有限量子系统的Clifford群结构

Structure of Clifford groups of composite finite quantum systems

研究配置空间为Z_{n1}⊕...⊕Z_{nk}的多部分量子系统的Clifford群，证明当总维数N为奇数时Clifford群是自然半直积，对于偶数N，证明当N不被4整除时Clifford群和射影Clifford群也是自然半直积。

Representations of orthogonal and symplectic Yangians

Exteded Yangian algebras of orthogonal and symplectic types are defined by the Yang-Baxter RLL relation involving the fundamental R-matrix with $so(n)$ or $sp(2m)$ symmetry. We study representations of highest weight characterized by weight function ratios. We consider the algebra relations for the linear and the quadratic evaluations and the resulting conditions imposed on the representation weights. We present expressions of L-operators constructed on underlying Clifford and Heisenberg algebras and characterize their representations.

Clifford 张量幂的对偶理论

Duality theory for Clifford tensor powers

本文为涵盖 qubit 系统的 Clifford 群张量幂表示提供了统一的对偶理论框架，推广了 eta 对应，并提出了高效实现黑盒 Clifford 幺正演化复共轭的协议。

Grade free product formulae from Grassmann Hopf gebras

In the traditional approaches to Clifford algebras, the Clifford product is evaluated by recursive application of the product of a one-vector (span of the generators) on homogeneous i.e. sums of decomposable (Grassmann), multivectors and later extended by bilinearity. The Hestenesian 'dot' product, extending the one-vector scalar product, is even worse having exceptions for scalars and the need for applying grade operators at various times. Moreover, the multivector grade is not a generic Clifford algebra concept. The situation becomes even worse in geometric applications if a meet, join or co…

Mathematics for theoretical physics

This book intends to give the main definitions and theorems in mathematics which could be useful for workers in theoretical physics. It gives an extensive and precise coverage of the subjects which are addressed, in a consistent and intelligible manner.The first part addresses the Foundations (mathematical logic, set theory, categories), the second Algebra (algebraic strucutes, groups, vector spaces tensors, matrices, Clifford algebra). The third Analysis (general topology, measure theory, Banach Spaces, Spectral theory). The fourth Differential Geometry (derivatives, manifolds, tensorial bund…

关于Clifford群的特征标表

On character table of Clifford groups

本文构造了Clifford群在n=1,2,3时的特征标表，并证明当n≥3时该群仅有平凡线性特征标、Pauli群是其唯一非平凡正规真子群，且矩阵表示忠实。

Meeting Descartes and Klein Somewhere in a Noncommutative Space

We combine the coordinate method and Erlangen program in the framework of noncommutative geometry through an investigation of symmetries of noncommutative coordinate algebras. As the model we use the coherent states construction and the wavelet transform in functional spaces. New examples are a three dimensional spectrum of a non-normal matrix and a quantisation procedure from symplectomorphisms. Keywords: Heisenberg group, special linear group, symplectic group, Hardy space, Segal-Bargmann space, Clifford algebra, Cauchy-Riemann-Dirac operator, Möbius transformations, functional calculus, Wey…