Symmetry and the Laws of Nature

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. 

2Despite known deviations from each of the individual symmetries, it is widely believed by theoreticians that the combined PCT operation is an exact symmetry. 

To keep this proposition in focus, equivalence relationships, in the sense of symmetries in the physical world, may be defined in terms of a metric for the state space of a system. The metric [14] is a real non-negative function d(, ) with the following properties for all states u, v, w  

Another term for approximate symmetry is broken symmetry. The symmetry breaking factor is whatever factor is responsible for the deviation of e from zero. As an example, any crystal has broken translational symmetry. The exact symmetry limit is an infinite crystal, obviously unattainable in practice. 

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