The Symmetry Laws of Nature

It was believed for a long time that the fundamental laws of nature are invariant under space inversion, and hence the conservation of space inversion symmetry (P) is a universally accepted principle. The nonconservation of this symmetry was discovered experimentally by Wu and co-workers in the (3 decay of 60Co in... 

The presentation here is short, and limited to those aspects of symmetry and group theory that are directly useful in interpreting molecular structure and spectroscopy. Nevertheless I hope that the reader will begin to sense some of the beauty of the subject. Symmetry is at the heart of our understanding of the physical laws of nature. If a reader is happy with what appears in this book, I must count this a success. But if the book motivates a reader to move deeper into the subject, I shall be gratified. 

In the first chapter, we defined the nature of a solid in terms of its building blocks plus its structure and symmetry. In the second chapter, we defined how structures of solids are determined. In this chapter, we will examine how the solid actually occurs in Nature. Consider that a solid is made up of atoms or ions that are held together by covalent/ionic forces. It is axiomatic that atoms cannot be piled together and forced to form a periodic structure without mistakes being made. The 2nd Law of Thermodynamics demands this. Such mistakes seriously affect the overall properties of the solid. Thus, defeets in the lattice are probably the most important aspect of the solid state since it is impossible to avoid defects at the atomistic level. Two factors are involved ... 

We have shown that by stacking atoms or propagation units together, a solid with specific symmetry results. If we have done this properly, a perfect solid should result with no holes or defects in it. Yet, the 2nd law of thermod5mamics demands that a certain number of point defects (vacancies) appear in the lattice. It is impossible to obtain a solid without some sort of defects. A perfect solid would violate this law. The 2nd law states that zero entropy is only possible at absolute zero temperature. Since most solids exist at temperatures far from absolute zero, those that we encounter are defect-solids. It is natural to ask what the nature of these defects might be. 

Symmetry is the natural language of the fundamental laws of physics. Group theory is its grammar. 

The principle of charge conjugation symmetry states that if each particle in a given system is replaced by its corresponding antiparticle, then it would not be possible to tell the difference, For example, if in a hydrogen atom the proton is replaced by an antiproton and the electron is replaced by a positron, then this antimatter atom will behave exactly like an ordinary atom—if observed by persons also made of antimatter. - In an antimatter universe, the laws of nature could not be distinguished from the laws of an ordinary mailer universe. 

However, the symmetry of the situation can be restored if we interchange the words right and "left in the description of the experiment at the same time that we exchange each particle with its antiparticle. In the above experiment, this is equivalent to replacing the word clockwise with counterclockwise. When this is done, the positrons arc emitted in the downward direction, just as the electrons m the original experiment. The laws of nature are thus found to be invariant to the simultaneous application of charge conjugation and mirror inversion. 

Differential equations or a set of differential equations describe a system and its evolution. Group symmetry principles summarize both invariances and the laws of nature independent of a system s specific dynamics. It is necessary that the symmetry transformations be continuous or specified by a set of parameters which can be varied continuously. The symmetry of continuous transformations leads to conservation laws. 

Fundamental phenomena and laws of nature are related to symmetry and, accordingly, symmetry is one of science s basic concepts. Perhaps it is so important in human creations because it is omnipresent in the natural world. Symmetry is beautiful although alone it may not be enough for beauty, and absolute perfection may even be irritating. Function, utility, and aesthetic appeal are the reasons for symmetry in technology and the arts. 

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop. 

The so-called laws of Nature are scientific generalizations of regularities observed in the behaviour of a system under specified conditions. Behaviour in this sense implies, almost invariably, the way in which a system of interest develops as a function of time. More basic still, more than law, call it axiom, is the all but universally accepted premise that the outcome of any scientific experiment is independent of its location and orientation in three-dimensional space, provided the experimental conditions can be replicated. A moment s reflection shows that this stipulation defines a symmetry which is equivalent to the conviction that space is both homogeneous and isotropic. The surprising conclusion is that this reproducibility, which must be assumed to enable meaningful experimentation, dictates the nature of possible observations and hence the laws that can be inferred from these observations. The conclusion is father to the thought that each law of Nature is based on an underlying symmetry. 

Like any other great idea, the symmetry principle should be used with circumspection lest the need of enquiry beyond the search for symmetry is obscured. The hazard lurks therein that nowhere in the world has mathematically precise symmetry ever been encountered. The fundamental symmetries underpinning the laws of Nature, i.e. parity (P), charge conjugation (C), and time inversion (T), are hence no more than local approximations and, although the minor exceptions may be just about undetectable, they cannot be ignored2. 

The whole concept of symmetry and law becomes more palatable against the backdrop of approximate symmetries. Inviolate laws that militate against the scientific spirit, are then prevented by broken symmetries and developments in science amount to relaxing the primitive laws, so as to describe more general situations around the special cases dictated by exact symmetries, i.e. by maximizing the parameter e. Any law that reflects a symmetry must then be considered as a useful starting point rather than a final result and conclusions based on perceived symmetries of space and time must be revisited to identify the effects of broken symmetry on the laws of Nature. 

Not only the laws of Nature but also all major scientific theories are statements of observed symmetries. The theories of special and general relativity, commonly presented as deep philosophical constructs can, for instance, be formulated as representations of assumed symmetries of space-time. Special relativity is the recognition that three-dimensional invariances are inadequate to describe the electromagnetic field, that only becomes consistent with the laws of mechanics in terms of four-dimensional space-time. The minimum requirement is euclidean space-time as represented by the symmetry group known as Lorentz transformation. 

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