Symmetry Rules pp 261-281 | Cite as

# Symmetry in Processes

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## Abstract

Taking the active view of transformations, the notion of symmetry of the laws of nature, or symmetry of evolution, is formalized. A symmetry transformation of the laws of nature is a transformation that commutes with the evolution transformation. The fundamental point underlying the formalism is that symmetry of evolution is an indifference of nature, whereby the laws of nature ignore some aspect of states of physical systems. The meaning of a time reversed process and time reversal symmetry of the laws of nature is considered as well. It is shown that symmetry of the laws of nature determines an equivalence relation in a state space of a system, the equivalence of states that are indistinguishable by the laws of nature. Applying the equivalence principle, one obtains the equivalence principle for processes in quasi-isolated systems: Equivalent states, as initial states, must evolve into equivalent states, as final states, while inequivalent states may evolve into equivalent states. That leads to the symmetry principle for processes in quasi-isolated systems: The “initial” symmetry group (that of the cause) is a subgroup of the “final” symmetry group (that of the effect). And that in turn leads to the general symmetry evolution principle: For a quasi-isolated physical system the degree of symmetry cannot decrease as the system evolves, but either remains constant or increases. The latter principle is too general to be of much use. So rather than consider the symmetry group of all of state space, one looks at the symmetry group of only a single state, the group of permutations of the equivalence subspace (equivalence class in state space) to which it belongs. With the assumption of nonconvergent evolution, that leads to the special symmetry evolution principle: The degree of symmetry of the state of a quasi-isolated system cannot decrease during evolution, but either remains constant or increases. Or equivalently: As a quasi-isolated system evolves, the populations of the equivalence subspaces of the sequence of states through which it passes cannot decrease, but either remain constant or increase. For systems, such as a gas, that possess nonconvergent evolution of microstates and convergent evolution of macrostates to macrostates of stable equilibrium, the special symmetry evolution principle gives the theorem that the degree of symmetry of a macrostate of stable equilibrium must be relatively high.

## Keywords

State Space Symmetry Group Equivalence Principle Convergent Evolution Symmetry Transformation## Preview

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