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由 张轩中 著作
斯方程。拉普拉斯算子的四维形式是达朗贝耳算子。 .... 第二个定理异常强大，他几乎用肉眼看出角. 动量守恒定理，说的是行星矢径在单位时间扫过的面积相同。第三个摘 要
对称和守恒作为一个古老而又常新的概念，经历了从分立走向综合的漫长发展历程．对称性对物理学的发展起到了重要作用．本文通过对牛顿定律与对称性的关系，相对论力学系统中对称性与新守恒律的理论研究，可控理力学系统中的对称性与守恒律的研究，空间反演不变性与宇称守恒，时空对称性与能量守恒定律，空间平移不变性与动量守恒定律，空间转动不变性与角动量守恒等的研究来讨论对称性与守恒律的关系，使人们进一步了解对称中的守恒和守恒中的对称，使它成为学习物理学的工具，使人们从这一点出发发现物理学中的某些理论的欠缺，为物理学的发展做出贡献。从物理学的发展来看，正是“对称一不对称一新的对称”的不断循环往复，才使物理学理论从较低的对称层次向较高的对称层次发展，从较小范围的统一向较大范围的统一发展，使人类对自然界的认识不断深化。
关键词 对称；守恒；牛顿运动定律；空间不变性
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Abstract
Symmetry and conservation as a regular old and new concepts, evolved from discrete to integrated long development process. Symmetry of the physics has played an important role. Based on the Newton's law and symmetry, the symmetry of relativistic mechanical system with the new theory of conservation laws, controllable mechanical system management and the conservation laws of symmetry, space inversion invariant and parity conservation, space-time symmetry and energy conservation, space translation invariance and momentum conservation law, space rotation invariance and conservation of angular momentum and other studies to discuss the relationship between symmetries and conservation laws, Raise awareness of conservation and conservation of symmetry in the symmetry, it become a tool for learning physics, so that people from that point some of the theories of physics discovered the lack of, in order to contribute to the development of physics. From the perspective of the development of physics is "symmetrical asymmetry a new symmetry of a" continuing the cycle, it makes theoretical physics from lower to higher levels of symmetry level of development, from a small range of uniform larger range unity to the development of mankind on the natural world, with deepening understanding.
Keywords: symmetry; conservation; Newton's laws of motion; space invariance.
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目 录
摘要 „„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„2
ABSTRACT „„„„„„„„„„„„ „„„„„„ „„„„„„„ „„„3
1 引言 „„„„„„„„„„„„„„„„„„„„„„„„„„„„5 2 物理学中的对称性 „„„„„„„„„„„„„„„„„„„„„„6 2．1 物理对称性的分类 „„„„„„„„„„„„„„„„„„„„„„6 2．1．1 直观对称 „„„„„„„„„„„„„„„„„„„„„„„„„„6 2．1．2 抽象对称 „„„„„„„„„„„„„„„„„„„„„„„„„„7 2．1．3 数学对称 „„„„„„„„„„„„„„„„„„„„„„„„„„7 2．1．4 对称破缺 „„„„„„„„„„„„„„„„„„„„„„„„„„7 2．2 对称操作与对称性 „„„„„„„„„„„„„„„„„„„„„„8 3 牛顿运动定律与对称性 „„„„„„„„„„„„„„„„„„„„8 4 因果关系和对称性原理 „„„„„„„„„„„„„„„„„„„„9 4．1 因果性的等价原理 „„„„„„„„„„„„„„„„„„„„„„9 4．2 因果关系的对称性 „„„„„„„„„„„„„„„„„„„„„„10 4．3 应用举例 „„„„„„„„„„„„„„„„„„„„„„„„„„10 5 对称性与守恒律之间的依存关系 „„„„„„„„„„„„„„„„10 5．1 机械能对空间坐标平移的对称性与动量守恒 „„„„„„„„„„„12 5．2 机械能对空间坐标系转动的对称性与角动量守恒 „„„„„„„„„13 5．3 机械能对时间平移的对称性与机械能守恒 „„„„„„„„„„„„13 6 在对称守恒指引下创新的典型范例 „„„„„„„„„„„„„„„14 7 对称性的深入认识与对称性破缺 „„„„„„„„„„„„„„„„16 8 量子力学中的对称性与守恒律 „„„„„„„„„„„„„„„„„17 8. 1 粒子流密度与粒子数守恒定律 „„„„„„„„„„„„„„„„„17 8．2 量子力学中对称的原因产生对称的结果的例子 „„„„„„„„„„18 8．3 微观粒子的全同性原理 „„„„„„„„„„„„„„„„„„„„21 8．4 量子力学中的守恒律 „„„„„„„„„„„„„„„„„„„„„21 9 结语 „„„„„„„„„„„„„„„„„„„„„„„„„„„„25 参考文献 „„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„„26
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1、引言
作为物理学的最原始、最基本的概念，对称和守恒各自有着深刻的思想渊源．人类对于对称和守恒的认识也是从表面深入到内部，而对称和守恒也经历了从分立走向综合的漫长发展历程．特别是在现代物理学中，对称性和守恒律对科学家来说始终具有非凡的吸引力，是一个非常有趣和深刻的话题．在探索千变万化、纷繁复杂的自然现象的普遍规律的过程中，守恒量与守恒定律是物理学家们长期倾心关注的议题。现代物理学研究表明，自然界中的守恒定律与相应的对称性是密切相关的。因此，认识现代物理学对称性的深刻内涵，明确对称性与守恒律之间的密切联系，对于探究自然规律、揭示宇宙奥秘是十分重要的。近代科学表明．自然界的所有重要的规律均与某种对称性有关，甚至所有自然界中的相互作用，都具有某种特殊的对称性——所谓“规范对称性”。实际上，对称性的研究日益深入，已越来越广泛的应用到物理学的各个分支：量子论、高能物理、相对论、原子分子物理、晶体物理、原子核物理。以及化学(分子轨道理论、配位场理论等)、生物(DNA的构型对称性等)和工程技术。
何谓对称性?按照英国《韦氏国际辞典)中的定义：对称性乃是分界线或中央平面两侧各部分在大小、形状和相对位置的对应性。这里讲的是人们观察客观事物形体上的最直观特征而形成的认识，也就是所谓的几何对称性。关于对称性和守恒定律的研究一直是物理学中的一个重要领域。对称性与守恒定律的本质和它们之间的关系一直是人们研究的重要内容。在经典力学中，从牛顿方程出发，在一定条件下可以导出力学量的守恒定律。粗看起来，守恒定律似乎是运动方程的结果．但从本质上来看，守恒定律运动方程更为基本，因为它表述了自然界的一些普遍法则，支配着自然界的所有过程，制约着不同领域的运动方程．物理学关于对称性探索的一个重要进展是诺特定理的建立，定理指出，如果运动定律在某一变换下具有不变性．必相应地存在一条守恒定律．
简言之，物理定律的一种对称性，对应地存在一条守恒定律．经典物理范围内的对称性和守恒定律相联系的诺特定理后来经过推广，在量子力学范围内也成立．在量子力学和粒子物理学中，又引入了一些新的内部自由度，认识了一些新的抽象空间的对称性以及与之相应的守恒定律，这就给解决复杂的微观问题带来好处，尤其现在根据量子体系对称性用群论的方法处理问题，更显优越。在物理学中，尤其是在理论物理学中，我们所说的对称性指的是体系的拉格朗日量或者哈密顿量在某种变换下的不变性。这些变换一般可分为连续变换、分立变换和对于内禀参量的变换。每一种变换下的不变性，都对应一种守恒律，意味着存在某种不可观测量。例如，时间平移不变性，对应能量守恒，意味着时间的原点不可观测；空间平移不变性，对应
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