# Symmetry in Physics

Part of the The Frontiers Collection book series (FRONTCOLL)

## Abstract

The chapter discusses symmetry of evolution of quasi-isolated physical systems and symmetry of states of such systems. The former symmetry, also called symmetry of the laws of nature, is a manifestation of nature’s indifference to certain aspects of physical systems. When systems differ only in such aspects, nature treats the systems in essentially the same manner so that they evolve in essentially the same way. States of systems are symmetric when changes can be applied to them that affect only those of their aspects to which nature is indifferent. Reference frames are needed to endow transformations with physical significance and must themselves be asymmetric under the transformations. The notion of active and passive transformations is introduced, where the former affect physical systems (and not reference frames), while the latter act only on reference frames (and not on systems). That leads to active and passive formulations of symmetry of evolution. Both formulations involve different evolving physical systems. In the active formulation the evolutions have different descriptions with respect to the same reference frame, while in the passive formulation they have the same description with respect to different reference frames. These concepts are introduced: global reference frame, which is a single frame that is valid for all space and all time; inertial reference frame, which is a reference frame in which inertial motion appears as such; and local reference frame, which is the assignment of an individual reference frame to every space-time point, i.e., the assignment of an individual time varying reference frame to every point in space. Gauge issues are discussed. A global transformation is one that has the same effect at all locations and instants. Gauge transformations are passive transformations that in general have different effects at different locations and instants (although, as a particular case, a gauge transformation might have the same effect at all locations and instants, making it a global transformation). A gauge group is a group of gauge transformations that are obtained by taking a parametrized group of global transformations and making its parameters space-time dependent. That leads to the notion of gauge symmetry, which is symmetry of evolution under a gauge group. Gauge symmetry relates inertial and dynamic evolutions. Gauge theories are theories that possess gauge symmetry. Indeed, the most successful theories of the fundamental particles and their interactions are gauge theories. Further, the gauge symmetries essentially determine these theories. The relation between symmetries and conservations, also known as conservation laws, is discussed. Each conservation is fundamentally linked to a symmetry of evolution, or symmetry of the laws of nature. The symmetry that lies at the foundation of physics comprises the symmetry lying at the foundation of science—reproducibility, predictability, and reduction, with the help of analogy (all discussed in the previous chapter)—and the realizations of symmetry that are particular to physics—symmetry of evolution, symmetry of states, gauge symmetry, and the symmetry inherent to quantum theory.

## Keywords

Reference Frame Gauge Group Gauge Transformation Gauge Symmetry Canonical Variable
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