四元数在量子力学中的应用

把双四元数推广到了二级双四元数，并设计了一种态函数的四元数表示法，从而用四元数表述了相对论量子力学，使四元数物理学形成了系统，用四元数表示的算符和状态，对于导出算符间的对易关系和状态的洛伦兹变换性质是方便的。     

三维转动的四元数表述

用四元数表达三维的旋转与使用矩阵相比具有两个优点：第一．几何意义明确;第二,计算简单．因此,四元数在数学、物理学和计算机图形学中具有很高的应用价值．本文详细叙述了四元数的这一功能,讨论了四元数的定义、运算、性质、几何意义和它的3种表达形式,给出了使用四元数处理点的各种几何变换的一般结论．     

彩色图像四元数矩不变量的研究

为了研究彩色图像的矩不变量特性,采用四元数进行彩色图像处理,以充分利用彩色图像的整体信息,实现彩色图像RGB并行处理。本文把传统灰度图像的矩不变量理论推广应用到四元数层面上来,定义了彩色图像的四元数矩并构造了该矩函数的仿射不变量。实验结果表明：所提出的彩色图像的四元数矩不变量的稳定性要优于L.V.Gool等人提出的彩色矩仿射不变量,其σ/u值提高了2个数量级。所提出的四元数仿射矩不变量可以作为模式识别中彩色目标的特征描述子来实现彩色图像目标的识别与跟踪。     

彩色图像四元数矩不变量的研究

为了研究彩色图像的矩不变量特性,采用四元数进行彩色图像处理,以充分利用彩色图像的整体信息,实现彩色图像RGB并行处理。本文把传统灰度图像的矩不变量理论推广应用到四元数层面上来,定义了彩色图像的四元数矩并构造了该矩函数的仿射不变量。实验结果表明:所提出的彩色图像的四元数矩不变量的稳定性要优于L.V.Gool等人提出的彩色矩仿射不变量,其σ/u值提高了2个数量级。所提出的四元数仿射矩不变量可以作为模式识别中彩色目标的特征描述子来实现彩色图像目标的识别与跟踪。     

狄拉克γ矩阵与四元数空间

四元数及其空间的性质早已被研究。将时空间选择为（双）四元数空间，同样可导出狭义相对论，并可简洁地叙述经典力学中的刚体运动理论。重要的是，用四元数空间可描写量子力学中的波函数及其相位值，还可描写量子电动力学中的γ矩阵及狄拉克矩阵。     

再谈γ矩阵与四元数空间

在“狄拉克γ矩阵与四元数空间”一文的基础上，四元数及其空间的性质可以作进一步的研究。在四元数空间中，从度规方程出发，也可导出量子电动力学中的γ矩阵；重要的是，用四元数空间不仅能给出平直空间中的γ矩阵，还可给出弯曲空间中的γ矩阵。     

基于光的反射与折射定律的表述方法推导及应用

光的反射和光的折射是几何光学应用中最基本的方法。在现代精密光学元件加工过程中,其不同的表述方法可以为光线追迹、棱镜误差分析以及棱镜装调提供不同的解决思路。介绍了光的反射定律和折射定律的传统形式表述方法,并推导了光的反射定律和折射定律的矢量、矩阵及四元数3种表述方法。通过Matlab辅助下的模拟计算,得到施密特棱镜检验光路中2个不同区域入射光线的反射在水平方向上对称分布于前表面反射像两侧,与实际应用相符,实现了矢量、矩阵和四元数表述方法在施密特棱镜检验光路中的应用。3种表述方法可以为光线追迹、棱镜误差分析以及棱镜装调提供科学、实用的解决方法。     

基于离散四元数傅里叶变换的双随机相位加密技术

应用光学图像加密的思想,将离散四元数傅里叶变换(DQFT)与双随机相位加密技术相结合,提出了一种应用于彩色图像的双随机相位加密新技术. 基于DQFT的双随机相位加密技术可将彩色图像作为一个整体进行加密,从而在保持了系统的保密性能的同时,有效地降低了复杂性. 阐述了加密和解密的原理,并通过实验对其鲁棒性进行了验证. 关键词： 四元数 离散四元数傅里叶变换(DQFT) 双随机相位加密     

四元数在建立波动方程中的应用

用四元数对经典的能量，动量关系式进行因式分解，建立起了现今人们已知的所有不同形式的波动方程，这样既表明了不同微观粒子不同的波动方程有着同一个来源，也为建立未知粒子的未知的波动方程提供了可能性。     

四旋翼飞行器姿态解算与滤波

四旋翼飞行器的运动控制关键在于对飞行过程中的实时姿态角控制。目前实时姿态角信息还不能直接测量出来。为了能利用已有的传感器数据解算出更准确的姿态角,通过物理实验详细分析了四旋翼飞行器姿态角的解算和滤波算法。首先,通过联立欧拉方向余弦矩阵与四元数矩阵,得到用四元数表达的姿态角表达式。然后,结合加速度计和磁强计实时测量的数据,分别采用互补滤波和卡尔曼滤波两种方法来补偿四元数结果,分别分析如何选取最佳参数,并对比分析了两种滤波方式的优缺点。在一定精度要求范围内,这两种滤波方式都能获得更加准确的姿态角,但是互补滤波相对卡尔曼滤波有一定的解算时延。因此在精度要求一般的系统中,这两种滤波方式都可以用来求解姿态角,卡尔曼滤波方法则更适于对实时性要求更高的系统。     

Thermodynamics and Preferred Frame

The Lorentz covariant statistical physics and thermodynamics is formulated within the preferred frame approach. The transformation laws for geometrical and mechanical quantities such as volume and pressure as well as the Lorentz-invariant measure on the phase space are found using Lorentz transformations in absolute synchronization. Next, the probability density and partition function are investigated using the preferred frame approach, and the transformation laws for internal energy, entropy, temperature and other thermodynamical potentials are established. The Lorentz covariance of basic thermodynamical relations, including Clapeyron's equation and Maxwell's relations is shown. Finally, the relation of presented approach to the previous approaches to relativistic thermodynamics is briefly discussed.     

Conformal Symmetry and Quantum Relativity

The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincare symmetry of field theory can be extended to the larger conformal symmetry. We use these symmetries to define quantum observables associated with positions in space-time, in the spirit of Einstein theory of relativity. This conception of localization may be applied to massive as well as massless fields. Localization observables are defined as to obey Lorentz covariant commutation relations and in particular include a time observable conjugated to energy. While position components do not commute in the presence of a nonvanishing spin, they still satisfy quantum relations which generalize the differential laws of classical relativity. We also give of these observables a representation in terms of canonical spatial positions, canonical spin components, and a proper time operator conjugated to mass. These results plead for a new representation not only of space-time localization but also of motion.     

Are There Dynamical Laws?

The nature of a physical law is examined, and it is suggested that there may not be any fundamental dynamical laws. This explains the intrinsic indeterminism of quantum theory. The probabilities for transition from a given initial state to a final state then depends on the quantum geometry that is determined by symmetries, which may exist as relations between states in the absence of dynamical laws. This enables the experimentally well-confirmed quantum probabilities to be derived from the geometry of Hilbert space and gives rise to effective probabilistic laws. An arrow of time which is consistent with the one given by the second law of thermodynamics, regarded as an effective law, is obtained. Symmetries are used as the basis for a new proposed paradigm of physics. This naturally gives rise to the gravitational and gauge fields from the symmetry group of the standard model and a general procedure for obtaining interactions from any symmetry group.     

Relativistic quantum logic

On the basis of the well-known quantum logic and quantum probability a formal language of relativistic quantum physics is developed. This language incorporates quantum logical as well as relativistic restrictions. It is shown that relativity imposes serious restrictions on the validity regions of propositions in space-time. By an additional postulate this relativistic quantum logic can be made consistent. The results of this paper are derived exclusively within the formal quantum language; they are, however, in accordance with well-known facts of relativistic quantum physics in Hilbert space.     

Occam’s Razor as a Formal Basis for a Physical Theory

We introduce the principle of Occam's Razor in a form that can be used as a basis for economical formulations of physics. This allows us to explain the general structure of the Lagrangian for a composite physical system, as well as some other artificial postulates behind the variational formulations of physical laws. As an example, we derive Hamilton's principle of stationary action together with the Lagrangians for the cases of Newtonian mechanics, relativistic mechanics and a relativistic particle in an external gravitational field.     

Why gravity experiments are so exciting

Being the most fundamental interaction gravity not only describes a particular interaction between matter, but also covers issues like the notion of space and time, the role of the observer and the relativistic measurement process. Gravity is geometry and, as a consequence, allows for the existence of black holes, non-trivial topologies, a cosmological big bang, time-travel, warp drive, and other phenomena not known from non-relativistic physics. Here we present the experimental basis of General Relativity, in particular its foundations encoded in the Einstein Equivalence Principle and its predictions in the weak and strong gravity regime. We discuss various routes to search for effects possibly signaling effects of the looked for quantum theory of gravity. We lay emphasis on assumptions to be tested which are only rarely discussed in the literature like tests of Newton’s axioms, tests of conservation laws, etc. We propose an experiment testing the order of time derivatives in the equation of motion.     

麦克斯韦-玻耳兹曼统计分布律的相对论修正

为了避免光速极限带来的复杂性,首先给出了理想气体分子动量的分布律公式．在此基础上,进一步讨论了麦克斯韦一玻耳兹曼统计分布律的相对论修正．解析和数值计算均表明,这一修正几乎没有可观测的物理效应．这说明,大学物理教学中介绍的麦克斯韦一玻耳兹曼速度分布律公式对实际应用已经足够精确．当然,我们决不可据此近似公式断言气体分子有一定的概率可以超光速运动．     

Second quantized quaternion quantum theory

The basic structure of a second quantized relativistic quantum theory is outlined. The vector space is over the ring of complex quaternions instead of the usual field of complex numbers. This is motivated by the simple quaternion structure of the Dirac equation.